![12 Stochastic Linear Difference Equations
The moving average piece of representation (2.3.4) is often called an impulse
response function. An impulse response function depicts the response of current
and future values of {xt} to an imposition of a random shock wt . In represen-
tation (2.3.4), the impulse response function is given by entries of the vector
sequence {Aτ
C : τ = 0, 1, . . .}.4
Shift (2.3.4) forward in time:
xt+j =
j−1
X
s=0
As
Cwt+j−s + Aj
xt. (2.3.5)
Projecting both sides of (2.3.5) on the information set {x0, wt, wt−1, . . . , w1}
gives5
Etxt+j = Aj
xt. (2.3.6)
where Et(·) ≡ E[(·) | x0, wt, wt−1, . . . , w1] = E(·) | Jt , and xt is in Jt . Equation
(2.3.6) gives the optimal j step ahead prediction.
It is useful to obtain the covariance matrix of the j -step ahead prediction
error
xt+j − Etxt+j =
j−1
X
s=0
As
Cwt−s+j (2.3.7)
We have
E(xt+j − Etxt+j) (xt+j − Etxt+j)0
=
j−1
X
k=0
Ak
CC0
Ak0
≡ vj
(2.3.8a)
Note that vj defined in (2.3.8a) can be calculated recursively via
v1 = CC0
vj = CC0
+ Avj−1A0
, j ≥ 2.
(2.3.8b)
The matrix vj is the covariance matrix of the errors in forecasting xt+j on
the basis of time t information xt . To decompose these covariances into parts
attributable to the individual components of wt , we let iτ be an N -dimensional
4 Given matrices A and C , the impulse response function can be calculated using the
MATLAB program dimpulse.m.
5 For an elementary discussion of linear least squares projections, see Sargent (1987b,
chapter IX).](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-27-320.jpg)
![14 Stochastic Linear Difference Equations
of this polynomial, denoted δ1, δ2, . . . , δm . For each δj , we construct a matrix
Dj that has the same dimension as the number of zeroes of det (ζI − A) that
are equal to δj . The diagonal entries of Dj are δj and the entries in the single
diagonal row above the main diagonal are all either zero or one. The remaining
entries of Dj are zero. Then the matrix D is block diagonal with Dj in the
jth
diagonal block.
Transform the state vector xt as follows:
x∗
t = T−1
xt. (2.4.2)
Substituting into (2.2.1), we have that
x∗
t+1 = Dx∗
t + T−1
Cwt+1. (2.4.3)
Since D is block diagonal, we can partition x∗
t according to the diagonal blocks
of D or, equivalently, according to the distinct eigenvalues of A. In the law of
motion (2.4.3), partition j of x∗
t+1 is linked only to partition j of x∗
t . In this
sense, the dynamics of system (2.4.3) are uncoupled. To calculate multi-period
forecasts and dynamic multipliers, we must raise the matrix A to integer powers
(see (2.3.6)). It is straightforward to verify that
Aτ
= T(Dτ
)T−1
. (2.4.4)
Since D is block diagonal, Dτ
is also block diagonal, where block j is just
(Dj)τ
. The matrix (Dj)τ
is upper triangular with δτ
j on the diagonal, with all
entries of the kth
upper right diagonal given by
(δj)τ−k
τ!/[k!(τ − k)!] for 0 ≤ k ≤ τ, (2.4.5)
and zeroes elsewhere. Consequently, raising D to an integer power involves
raising the eigenvalues to integer powers. Some of the eigenvalues of A may
be complex. In this case, it is convenient to use the polar decomposition of the
eigenvalues. Write eigenvalue δj in polar form as
δj = ρj exp(iθj) = ρj[cos(θj) + i sin(θj)] (2.4.6)
where ρj =| δj |. Then
δτ
j = (ρj)τ
exp(iτθj) = (ρj)τ
[cos(τθj) + i sin(τθj)]. (2.4.7)](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-29-320.jpg)
![16 Stochastic Linear Difference Equations
2.5.2. Indeterministic seasonals
We want to use (2.2.1) to represent the model
yt = α4yt−4 + wt, (2.5.2)
where wt is a martingale difference sequence and | α4 |≤ 1. We define xt =
[yt, yt−1, yt−2, yt−3]0
, n = 4,
A =
0 0 0 α4
1 0 0 0
0 1 0 0
0 0 1 0
, C =
1
0
0
0
.
With these definitions, (2.2.1) represents (2.5.2). This model displays an “in-
deterministic” seasonal. Realizations of (2.5.2) display recurrent, but aperiodic,
seasonal fluctuations.
2.5.3. Univariate autoregressive processes
We can use (2.2.1) to represent the model
yt = α1yt−1 + α2yt−2 + α3yt−3 + α4yt−4 + wt, (2.5.3)
where wt is a martingale difference sequence. We set n = 4, xt = [yt yt−1 yt−2 yt−3]0
,
A =
α1 α2 α3 α4
1 0 0 0
0 1 0 0
0 0 1 0
, C =
1
0
0
0
.
The matrix A has the form of the companion matrix to the vector
[α1 α2 α3 α4].](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-31-320.jpg)
![Examples 17
2.5.4. Vector autoregressions
Reinterpret (2.5.3) as a vector process in which yt is a (k × 1) vector, αj a
(k × k) matrix, and wt a k × 1 martingale difference sequence. Then (2.5.3) is
termed a vector autoregression. To map this into (2.2.1), we set n = k · 4,
A =
α1 α2 α3 α4
I 0 0 0
0 I 0 0
0 0 I 0
, C =
I
0
0
0
where I is the (k × k) identity matrix.
2.5.5. Polynomial time trends
Let n = 2, x0 = [0 1]0
, and
A =
1 1
0 1
, C =
0
0
. (2.5.4)
Notice that D = A in the Jordan decomposition of A. It follows from (2.4.5)
that
At
=
1 t
0 1
. (2.5.5)
Hence xt = (t, 1)0
, so that the first component of xt is a linear time trend and
the second component is a constant.
It is also possible to use (2.2.1) to represent polynomial trends of any order.
For instance, let n = 3, C = 0, x0 = (0, 0, 1)0
, and
A =
1 1 0
0 1 1
0 0 1
. (2.5.6)
Again, A = D in the Jordan decomposition of A. It follows from (2.4.5) that
At
=
1 t t(t − 1)/2
0 1 t
0 0 1
. (2.5.7)
Then x0
t = [t(t−1)/2, t, 1], so that xt contains linear and quadratic time trends.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-32-320.jpg)
![18 Stochastic Linear Difference Equations
2.5.6. Martingales with drift
We modify the linear time trend example by making C nonzero. Suppose that
N is one and C0
= [1 0]. Since A =
1 1
0 1
and At
=
1 t
0 1
, it follows that
Aτ
C =
1
0
. (2.5.8)
Substituting into the moving-average representation (2.3.4), we obtain (2.25)
x1t =
t−1
X
τ=0
wt−τ + [1 t]x0
where x1t is the first entry of xt . The first term on the right-hand side of the
preceding equation is a cumulated sum of martingale differences, and is called
a martingale, while the second term is a translated linear function of time.
2.5.7. Covariance stationary processes
Next we consider specifications of x0 and A which imply that the first two
moments of {xt : t = 1, 2, . . .} are replicated over time. Let A satisfy
A =
A11 A12
0 1
, (2.5.9)
where A11 is an (n − 1) × (n − 1) matrix with eigenvalues that have moduli
strictly less than one and A12 is an (n − 1) × 1 column vector. In addition, let
C0
= [C0
1 0]. We partition x0
t = [x0
1t x0
2t] where x1t has n − 1 entries. It follows
from (2.2.1) that
x1t+1 = A11x1t + A12x2t + C1wt+1 (2.5.10)
x2t+1 = x2t. (2.5.11)
By construction, the second component, x2t , simply replicates itself over time.
For convenience, take x20 = 1 so that x2t = 1 for t = 1, 2, . . ..
We can use (2.5.10) to compute the first two moments of x1t . Let µt =
Ex1t . Taking unconditional expectations on both sides of (2.5.10) gives
µt+1 = A11µt + A12. (2.5.12)](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-33-320.jpg)
![20 Stochastic Linear Difference Equations
the mean of x10 is (I −A11)−1
A12 , then the covariance and mean of x1t remain
constant over time. In this case, the process is said to be covariance stationary.
If the eigenvalues of A11 are all less than unity in modulus, then Vt → V as
t → ∞, starting from any initial value V0 .
From (2.3.8) and (2.5.16), notice that if all of the eigenvalues of A11 are
less than unity in modulus, then limj→∞ vj = V . That is, the covariance matrix
of j -step ahead forecast errors converges to the unconditional covariance matrix
of x as the horizon j goes to infinity.9
The matrix V can be decomposed according to the contributions of each
entry of the process {wt}. Let ιτ be an N -dimensional column vector of zeroes
except in position τ , where there is a one. Then
I =
N
X
τ=1
ιτ ι0
τ . (2.5.17)
Define a matrix Ṽτ
Ṽτ ≡
∞
X
j=o
(A11)j
C1ιτ ι0
τ C0
1(A11)j0
(2.5.18)
We have, by analogy to (2.5.15) and (2.5.16), that Ṽτ satisfies Ṽτ = A11Ṽτ A0
11+
C1iτ i0
τ C0
1 . In light of (2.5.17), (2.5.18), and (2.5.16) we have that
V =
N
X
τ=1
Ṽτ . (2.5.19)
The matrix Ṽτ has the interpretation of being the contribution to V of the
τth
component of the process {wt : t = 1, 2, . . .}. Hence, (2.5.19) gives a
decomposition of the covariance matrix V into the portions attributable to
each of the underlying economic shocks.
Next, consider the autocovariances of {xt : t = 1, 2, . . .}. From the law of
iterated expectations, it follows that
E[(x1t+τ − µ)(x1t − µ)0
] = E{E[(x1t+τ − µ) | Jt](x1t − µ)0
}
= E[Aτ
11(x1t − µ)(x1t − µ)0
]
= Aτ
11V.
(2.5.20)
9 The doubling algorithm described in chapter 9 can be used to compute the solution of
(2.5.15) via iterations that approximate (2.5.16). The algorithm is implemented in the
MATLAB programs doublej.m and doublej2.m .](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-35-320.jpg)
![Examples 21
Notice that this expected cross-product or autocovariance does not depend on
calendar time but only on the gap τ between the time indices.10
Indepen-
dence of means, covariances, and autocovariances from calendar time defines
covariance stationary processes. For the particular class of processes we are
considering, if the covariance matrix does not depend on calendar time, then
none of the autocovariance matrices does.
2.5.8. Multivariate ARMA processes
Specification (2.2.1) assumes that xt contains all the information that is avail-
able at time t to forecast xt+1 . In many applications, vector time series are
modelled as multivariate autoregressive moving-average (ARMA) processes. Let
yt be a vector stochastic process. An ARMA process {yt : t = 1, 2, . . .} has a
representation of the form:
yt = α1yt−1 + α2yt−2 + · · · + αkyt−k
+ γ0wt + γ1wt−1 + · · · + γkwt−k.
(2.5.21)
where E[wt | yt−1, yt−2, · · · yt−k+1, wt−1, wt−2, · · · wt−k+1] = 0. The require-
ment that the same number of lags of y enter (2.5.21) as the number of lags of
w is not restrictive because some coefficients can be set to zero. Hence we can
think of k as being the greater of the two lag lengths. A representation such as
(2.5.21) can be shown to satisfy (2.2.1). To see this, we define
xt =
yt
α2yt−1 + α3yt−2 · · · + αkyt−k+1 + γ1wt + γ2wt−1 · · · + γk−1wt−k+2 + γkwt−k+1
α3yt−1 · · · + αkyt−k+2 + γ2wt · · · + γk−1wt−k+3 + γkwt−k+2
.
.
.
αkyt−1 + γk−1wt + γkwt−1
γkwt
(2.5.22)
10 Equation (2.5.20) shows that the matrix autocovariogram of x1t (i.e., Γτ ≡ E[(x1t+τ −
µ)(x1t − µ)0] taken as a function of τ ) satisfies the nonrandom difference equation Γt+1 =
A11Γt with initial condition Γ0 = V .](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-36-320.jpg)
![Examples 23
We can use (2.5.26) to derive a famous formula of John F. Muth (1960).
Assume that the system (2.5.25) has been operating forever, so that the initial
time is infinitely far in the past. Then using the lag operator L, express (2.5.25)
as
(1 − α1L)yt = (γ0 + γ1L)wt.
Solving for wt gives
wt = γ−1
0
1 − α1L
1 + γ1
γ0
L
yt,
which expresses wt as a geometric distributed lag of current and past yt ’s.
Substituting this expression for wt into (2.5.26) and rearranging gives
Etyt+j = αj−1
1
h α1 + γ1
γ0
1 + γ1
γ0
L
i
yt.
In the limiting case as α1 → 1 from below, this formula becomes
Etyt+j =
h 1 + γ1
γ0
1 + γ1
γ0
L
i
yt, (2.5.27)
which is independent of the forecast horizon j . In the limiting case of α1 = 1,
it is optimal to forecast yt for any horizon as a geometric distributed lag of past
y’s. This is Muth’s finding that a univariate process whose first difference is a
first order moving average is optimally forecast via an “adaptive expectations”
scheme (i.e., a geometric distributed lag with the weights adding up to unity).
2.5.10. Growth
In much of our analysis, we assume that the eigenvalues of A have absolute
values less than or equal to one. We have seen that such a restriction still
allows for polynomial growth. Geometric growth can also be accommodated by
suitably scaling the state vector. For instance, suppose that {x+
t : t = 1, 2, . . .}
satisfies:
x+
t+1 = A+
x+
t + Cw+
t+1 (2.5.28)
where E(w+
t+1 | Jt) = 0 and E[w+
t+1(w+
t+1)0
| Jt] = (ε)t
I . The positive number
ε can be bigger than one. The eigenvalues of A+
are assumed to have absolute
values that are less than or equal to ε
1
2 , an assumption that we make to assure](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-38-320.jpg)
![24 Stochastic Linear Difference Equations
that the matrix A to be defined below has eigenvalues with modulus bounded
above by unity. We transform variables as follows:
xt = (ε)− t
2 x+
t (2.5.29)
wt = (ε)− t
2 w+
t . (2.5.30)
The transformed process {wt : t = 1, 2, . . .} is now conditionally homoskedastic
as required because E[wt+1(wt+1)0
| Jt] = I . Furthermore, the transformed
process {xt : t = 1, 2, . . .} satisfies (2.2.1) with A = ε− 1
2 A+
. The matrix
A now satisfies the restriction that its eigenvalues are bounded in modulus by
unity. The original process {x+
t : t = 1, 2, . . .} is allowed to grow over time at a
rate of up to .5 log (ε).
2.5.11. A rational expectations model
Consider a model in which a variable pt is related to a variable mt via
pt = λEtpt+1 + γmt , 0 λ 1 (2.5.31)
where
mt = Gxt (2.5.32)
and xt is governed by (2.2.1). In (2.5.31), Et(·) denotes E(·) | Jt . This is a
rational expectations version of Cagan’s model of hyperinflation (here pt is
the log of the price level and mt the log of the money supply) or a version of Le
Roy and Porter’s and Shiller’s model of stock prices (here pt is the stock price
and mt is the dividend). Recursions on (2.5.31) establish that a solution to
(2.5.31) is pt = Etγ
P∞
j=0 λj
mt+j. Using (2.3.6) and (2.5.32) in this equation
gives pt = γG
P∞
j=0 λj
Aj
xt, or pt = γG(I − λA)−1
xt. Collecting our results,
we have that (pt, mt) satisfies
pt
mt
=
γG(I − λA)−1
G
xt
xt+1 = Axt + Cwt+1.
(2.5.33)
System (2.5.33) embodies the cross-equation restrictions associated with ratio-
nal expectations models: note that the same parameters in A, G that pin down](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-39-320.jpg)
![Computer Examples 25
the stochastic process for mt also enter the equation that determines pt as a
function of the state xt .
It is useful to show how to derive (2.5.33) using the method of undetermined
coefficients. Returning to (2.5.31), we guess that a solution for pt is of the form
pt = Hxt , where H is a matrix to be determined. Given this guess and (2.2.1),
it follows that Etpt+1 = HEtxt+1 = HAxt. Substituting this and (2.5.32) into
(2.5.31) gives Hxt = λHAxt + γGxt, which must hold for all realizations xt .
This implies that H = λHA + γG or H = γG(I − λA)−1
, which agrees with
(2.5.33).
2.6. The Spectral Density Matrix
Let the mean vector of xt from the stationary distribution of an {xt} process
be denoted µ. Define the autocovariance function of the {xt} process to be
Cx(τ) = E[xt − µ] [xt−τ − µ]0
. The spectral density matrix of the {xt} process
is defined as
Sx(ω) =
∞
X
τ=−∞
Cx(τ)e−iωτ
. (2.6.1)
Consider an {xt} process governed by (2.2.1), in which xt is partitioned as in
equations (2.5.10),(2.5.11), so that x2t is the constant term. Then the spectral
density can be represented as
Sx(ω) = (I − A11e−iω
)−1
C1C0
1 (I − A0
11e+iω
)−1
. (2.6.2)
From Sx(ω),11
the autocovariances can be recovered via the inversion formula
Cx(τ) =
1
2π
Z π
−π
Sx(ω)e+iωτ
dω. (2.6.3)
These formulas enable us to compute the spectral and cross-spectral statis-
tics for any of the large variety of models that are special cases of (2.2.1).
11 The MATLAB program spectral.m can be used to compute a spectral density matrix.
The program requires that the position of the constant term, denoted nnc, in xt be specified.
The program then forms the appropriate matrices A11 and C1 in equations (2.5.10),(2.5.11),
and applies formula (2.6.2).](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-40-320.jpg)
![26 Stochastic Linear Difference Equations
2.7. Computer Examples
We now use some MATLAB programs to generate examples that fit into the
framework of this chapter.
2.7.1. Deterministic seasonal
We can use the program dlsim.m to simulate the model of the deterministic
seasonal described above. In using dlsim.m, we specify four matrices A, C, G, D
whose dimensions must be comparable. In particular, we require that A be
n × n, that C be n × k, that G be ` × n, and that D be ` × k. For the case
of an indeterministic seasonal, we want to create the following matrices:
A =
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
, C =
0
0
0
0
G =I, D =
0
0
0
0
To accomplish this, we use the following MATLAB code
a= [0 0 0 1]
A= compn (a)
(This sets A equal to the companion matrix of a.)
C = zeros(4, 1)
(This sets C equal to a 4 × 1 matrix of zeros.)
G = eye (4)
(This sets G equal to the 4 × 4 identify matrix.)
D = zeros(4, 1)](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-41-320.jpg)
![Computer Examples 27
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12 14 16 18 20
*
* * *
*
* * *
*
* * *
*
* * *
*
* * *
Fig. 2.7.1.a. Deterministic Seasonal.
-2
-1
0
1
2
3
4
0 5 10 15 20 25 30 35 40
Fig. 2.7.1.b. Indeterministic Seasonal
with unit root.
We want to simulate the system
xt+1 = Axt + Cwt+1
yt = Gxt + Dwt+1
with an “input” of wt+1 ≡ 0. We form an input vector w of length 20 by the
statement:
w = zeros (20, 1)
We set the initial condition by
x0 = [1 0 0 0]0
To generate the simulation, we set
y = dlsim(A, C, G, D, w, x0).
This generates the 20 × 4 matrix y, the ith
column of which is the time path
taken by the ith
state variable (remember that G = I and D = 0). We plot
the time path of the first component of the state vector in Fig. 2.7.1.a and
Fig. 2.7.1.b.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-42-320.jpg)
![28 Stochastic Linear Difference Equations
2.7.2. Indeterministic seasonal, unit root
We implement a model of an indeterministic seasonal by altering the pre-
ceding example by replacing w with a sequence of i.i.d. normal random variates.
We specify that Ewt+1 = 0, Ew2
t+1 = 1. We accomplish this by the MATLAB
phrase
w = randn(150, 1)
We have generated a white noise of length 150. We create the simulation by
setting
C = [1 0 0 0]0
y = dlsim (A, C, G, D, w, x0).
We report the first component of y in figure 2. Note the tendency of the system
to display explosive oscillations. We invite the reader to calculate the variance
of x1t as a function of t.
2.7.3. Indeterministic seasonal, no unit root
We now set
a= [0 0 0 .7]
A = compn(a)
With all other matrices defined as in the preceding example, we form
y = dlsim (A, C, G, D, w, x0)
We plot the component x1t in figure 2.7.2. Notice that the explosive oscillations
that were present in Fig. 2.7.1.b are no longer present.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-43-320.jpg)
![Computer Examples 29
-3
-2
-1
0
1
2
3
0 5 10 15 20 25 30 35 40
Figure 2.7.2: Indeterministic seasonal with no unit root.
2.7.4. First order autoregression
We want to simulate the first order autoregression
x1t+1 = .9x1t + wt+1,
where wt+1 is a normally distributed white noise with unit variance. We accom-
plish this by modifying the MATLAB code of the previous example as follows:
x0= [0 0 0 0]0
a= [.9 0 0 0]
A= compn(a)
y= dlsim(A, C, G, D, w, x0)
Fig. 2.7.3.a graphs the first component of y, which is the process {x1t}.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-44-320.jpg)
![30 Stochastic Linear Difference Equations
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 20 40 60 80 100 120 140 160
Fig. 2.7.3.a. Simulation of first-order
autoregression.
-6
-4
-2
0
2
4
6
0 20 40 60 80 100 120 140 160
Fig. 2.7.3.b Simulation of second-order
autoregression.
2.7.5. Second order autoregression
We simulate the system
x1t+1 = 1.2x1t − .3x1t−1 + wt+1
by modifying the code of the preceding example as follows:
a= [1.2 − .3 0 0]
A= compn (a)
y= dlsim (A, C, G, D, w, x0)
Fig. 2.7.3.b displays the output of {x1t}.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-45-320.jpg)
![Computer Examples 31
2.7.6. Growth with homoskedastic noise
We want to simulate the model
x1t+1 = 1.025x1t + wt+1
where {wt+1} continues to be a normal white noise with unit variance. We set
the initial condition as x10 = 5. We modify the MATLAB code of the preceding
example as follows:
x0= [5 0 0 0]
a= [1.025 0 0 0 ]
A= compn(a)
y= dlsim(A, C, G, D, w, x0)
Figure 2.7.4.a displays {x1t}. Notice the tendency for the randomness to die
out, in the sense that the one-step ahead prediction error variance remains unity
while the mean level of the process is growing exponentially at rate 1.025 per
period.
0
50
100
150
200
250
300
0 20 40 60 80 100 120 140 160
Fig. 2.7.4.a. Growth with homoskedas-
tic noise.
0
50
100
150
200
250
0 20 40 60 80 100 120 140 160
Fig. 2.7.4.b. Growth with heteroskedas-
tic noise.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-46-320.jpg)
![32 Stochastic Linear Difference Equations
2.7.7. Growth with heteroskedastic noise
To arrest the tendency of the relative uncertainty to die out in the previous
example, we modify it by setting
x1t+1 = 1.025x1t + w∗
t+1
where
w∗
t = 1.025t/2
wt
and where {wt} continues to be a normal white noise with unit variance. This
specification makes the variance of w∗
t equal to (2.025)t
.
To simulate this model, we modify the code of the previous example as
follows:
n= 150
t= [1 : n]0
t= t./2
g= (1.025). ∧ t
wg= w. ∗ g
y= dlsim (A, C, G, D, wg, x0)
Figure 2.7.4.b displays {x1t}. Notice that the randomness now fails to die out.
2.7.8. Second order vector autoregression
We want to simulate the second order vector autoregression
z1t+1 = .9z1t + .05z1t−1 + .05z2t + .01z2t−1 + w1t+1
z2t+1 = −.04z1t − .06z1t−1 + .75z2t − .1z2t−1 + w2t+1
where wt = [w1t, w2t]0
is a normally distributed vector white noise with identity
covariance matrix. To simulate this system, we define
xt =
z1t
z1t−1
z2t
z2t−1
](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-47-320.jpg)
![Computer Examples 33
We use the MATLAB code
A= [.9 .05 .05 .01; 1 0 0 0; −.04 − .06 .75 − .1; 0 0 1 0]
C= [1 0 0 0; 0 0 1 0]
G= zeros (2, 4)
G(1, 1)= 1
G(2, 3)= 1
w= randn (150, 2)
D= zeros (2, 2)
x0= zeros (4, 1)
y= dlsim (A, C, G, D, w, x0)
In figure Fig. 2.7.5, we plot the first and third columns of y, which equal
{z1t,z2t}.
-6
-4
-2
0
2
4
6
0 20 40 60 80 100 120 140 160
Figure 2.7.5: Simulation of second-order vector autoregres-
sion.
We can use the MATLAB command dimpulse to compute the impulse
response of this system in response to each innovation (w1t, w2t). We employ
the following code:
i1 = dimpulse(A, C, G, D, 1, 20)](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-48-320.jpg)
![34 Stochastic Linear Difference Equations
This creates the response over twenty periods of the two variables z1t and z2t
to the first innovation w1t . We also use
i2 = dimpulse(A, C, G, D, 2, 20)
This creates the response over twenty periods of the two variables z1t and z2t
to the second innovation w2t . We display these impulse response functions in
figure 2.7.6.
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18
response of z1
response of z2
Fig. 2.7.6.a. Response to first innova-
tion.
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18
response of z1
response of z2
Fig. 2.7.6.b. Response to second inno-
vation.
We can use the MATLAB file evardec.m to compute the decomposition of
j -step ahead prediction error variances. If we want to compute this decomposi-
tion for horizons j extending from 1 to 20, we use the code:
[tab1, tab2] = evardec(A, C, G, −20, −1, eye(2))
The output in tab1 is a 20 × (1 + 2) table. The first column records the
horizon j . For i = 1, 2, the (i + 1)th
column records the diagonal element of
vj corresponding to the ith
variable zit . An orthogonal decomposition of these
variances into the parts attributable to w1 and w2 is contained in tab2. The
first column of tab2 records the horizon j , followed by two columns giving the](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-49-320.jpg)
![Computer Examples 35
diagonal element of the matrix vj,1 defined by (2.3.8) as the jth
row element.
Then j is repeated in the fourth column, followed by two columns giving the
diagonal element of the matrix vj,2 .
2.7.9. A rational expectations model
We now indicate how to simulate the model described by equations (2.5.33).
We let the variable mt be generated by
mt+1 = 1.2mt − .3mt−1 + wt+1.
To implement this we set
A= compn ([1.2 − .3])
C= [1 0]0
G1= [1 0]
D= zeros(2, 1)
We set γ = .5 and λ = .9. To implement formula (2.5.33) we set
G2 = .5 ∗ G1/(eye (2) − .7 ∗ A)
Then we set
G = [G1; G2]
To simulate the system we set
x0= [1 0]0
w= rand(150, 1)
y= dlsim(A, C, G, D, w, x0)
The first column of y is the simulation for m, while the second is the simulation
for p. We plot these in figure 2.7.7.a.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-50-320.jpg)
![36 Stochastic Linear Difference Equations
-35
-30
-25
-20
-15
-10
-5
0
5
10
15
0 20 40 60 80 100 120 140 160
m
p
Fig. 2.7.7.a. Simulation of m and p.
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0 2 4 6 8 10 12 14 16 18
response of m
response of p
Fig. 2.7.7.b. Impulse Response to in-
novation in m.
Figure 2.7.7.b gives the response of pt and mt to an innovation in money
w1 . We compute this by using
y = dimpulse(A, C, G, D, 1, 20)
To obtain a representation of the solution (2.5.33) in the vector arma form
d(L)
pt
mt
=
n1(L)
n2(L)
wt,
we use the command
[n, d] = ss2tf(A, C, G, D, 1)
For our example, we obtain the output
n =
0 1 0
0 3.0675 −.8282
d = [ 1 −1.2 .3 ]
This output implies that our system has the representation
(1 − 1.2L + .3L2
)
mt
pt
=
1
3.0675 −.8282L
wt.
Notice that the first row of this representation agrees with the process for mt
that we assumed.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-51-320.jpg)
![Then they hanged the Jew upon the gallows. As for Honest Heinrich,
he continued his wanderings in the world, and soon made his
fortune. When he had become rich, he went home again to his
village, and courted his neighbour's daughter, who had formerly
jilted him when he was poor, but who loved him now dearly, not
because he was rich (she said) but on account of his former poverty.
Soon they married, and were happy ever after.
THE POPE'S WIFE.
There are several modifications current of the story of the Jew in the
Thicket just told. A similar story which in olden time was popular in
England, is given under the heading 'A Mery Geste of the Frere and
the Boye,' in Ritson's Pieces of Ancient Popular Poetry, London,
1791. Again, a somewhat similar story is current in Greece. A lad has
a flute given to him by some superhuman being. He goes to the
market-place of the town, where piles of crockery are exhibited for
sale. As soon as he begins to play, all the pots, jugs and basins fly
about in the air and clash against each other until they are broken to
pieces. The personage whom he compels to dance in the thorns is a
priest.[74]
Perhaps the most tragic incident of this kind is the sad fate of the
Pope's wife, related by the Wallachians. It need scarcely be said that
it does not concern the Pope of Rome, who, as everyone knows, has
no wife. But in Wallachia the common village priest of the Greek
Church is called Pope, and may marry. He generally avails himself of
the permission.
As regards Bakâla, whose music, as we shall presently see, killed the
Pope's wife, various tricks of his are on record, which clearly show
that he was a great fool, somewhat resembling the German Till
Eulenspiegel, who had perhaps more happy ideas than many
persons who have passed for wise.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-58-320.jpg)
![Convince me? cried the Pope's wife: Fudge! I to be frightened by
a bagpipe? Let him come on!
Then the Pope thought that it was time to withdraw for the sake of
quietness. But in the evening he took Bakâla aside, and desired him
just to serenade their mistress for a little while under the window.
Before Bakâla commenced playing the Pope sat down on the ground
and bound two heavy stones to his feet by way of precaution, while
his wife busied herself in the upper story of the house. No sooner
had Bakâla begun his performance than she danced so furiously that
she made the whole house shake. Bakâla played faster and faster;
her stamping grew louder and louder. She danced until she had
actually stamped a hole in the floor, through which she descended
into the lower story. The Pope peeped into the room; and when he
saw what had happened he felt sorry, and he beckoned Bakâla to
leave off playing. But, alas! he beckoned too late! The poor lady had
danced herself to death.
Now, one might have thought the Pope would have dismissed
Bakâla, telling him that his services were not any further required.
But this is just precisely what he did not do. On the contrary, he kept
Bakâla in his service, and treated him even better than before.[75]
THE TWO HUNCHBACKS.
The story of the two Hunchbacks is widely diffused. It is told in
Ireland as well as in Germany and Italy; moreover it is said to be
also current in Spain. There are, of course, many varieties of it in
these countries. Compare, for instance, the Irish narrative of
Lusmore, in 'Fairy Legends and Traditions of the South of Ireland, by
T. Crofton Croker,' with the one given here, which has been obtained
from the country people in Rhenish Prussia.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-60-320.jpg)
![On St. Matthew's day, in the year 1549, a poor hump-backed
musician was returning late at night to Aachen[76] from a village
where he had been playing at a wedding. Being in a half drowsy
state, he took but little heed of time or place, and so he passed the
Minster without concerning himself about anything particularly, just
as the large clock in the tower boomed midnight. The sound startled
him, especially as at the same time there arose in the air a strange
whirring like the unearthly sound of owls and bats on the wing. It
now occurred to him that this was the night of quarter-day, and he
quickened his steps to escape the terrors of the ghost's hour and of
apparitions. Nervously he turned into the Schmiedstrasse (Smith-
street) as the nearest way to his home, which was in the
Jakobstrasse (James-street). But on reaching the Fish Market,—what
did he see! All the stalls glistened with innumerable lights, and about
them were seated a large party of richly-dressed ladies, feasting on
dainty viands served in golden and silver dishes, and drinking
sparkling wine from crystal goblets. The musician, much frightened,
endeavoured to hide himself in a corner; for, he had not the least
doubt that he saw an assemblage of witches. But it was too late;
one of the ladies nearest him had already observed him, and she
conducted him to the table.
Don't be frightened! said the lady to the musician, who stood
before her with chattering teeth and trembling knees: Don't be
frightened; but, play us some merry tunes, and thou shalt be paid
for it.
The poor hunchback had no choice but to take up his violin, and to
amuse the strange company as long as they pleased. Having quickly
set aside the stalls with everything upon them, the witches—among
whom the poor hunchback thought he recognised several ladies of
high position from the town—whirled round in pairs to the sound of
his fiddle. But the strangest thing was that the longer the fellow
continued to play, the finer and fuller his performance appeared to
him; so that he really thought he must be either dreaming, or there](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-61-320.jpg)
![With these words he threw off his jacket and turned round. The lady
quickly uncovered a silver dish, from which she took the hump of the
former musician, and before the vain virtuoso was aware of it, she
had pressed it on his back beside the other hump.
The clock had struck One, and the witches were already on their
broom-sticks riding through the air homewards, when the musician
recovered from his shock. He slowly put his hand to his back, hoping
that perchance he might only have had a bad dream. But no! it was
all right,—or rather all wrong. There remained now nothing for him
to do but to take up his jacket and make the best of his way home.
But the jacket felt so unusually heavy;—could there, perhaps, be
gold in it to make up in some measure for the cruel infliction?
Eagerly he rummaged the pockets; but what should he find? A few
heavy stones and rubbish.[77]
THE PARSON'S ADVICE.
This tale of the Manx people is almost literally copied from 'The
History and Description of the Isle of Man, by George Waldron,
London, 1744.'
A man, one day, was led by invisible musicians for several miles
together; and not being able to resist the harmony, followed till it
conducted him to a large common, where a great number of people
were sitting round a table, and eating and drinking in a very jovial
manner. Among them were some faces which he thought he had
formerly seen; but he forbore taking any notice, or they to him; till,
the little people offering him drink, one of them whose features
seemed not unknown to him, plucked him by the coat, and forbade
him, whatever he did, to taste anything he saw before him. 'For, if
you do,' added he, 'you will be as I am, and return no more to your
family.'](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-64-320.jpg)
![with a remarkably fine-flavoured Schiedam, from a bottle so oddly-
shaped that he had never seen anything like it, so funny it was. He
could not help smiling whenever he looked at the bottle.
Gradually his violin became heavier—of course, that was from the
golden ducats which the dancers continually threw into it. But also
his arm became heavier, and at last old Kartof felt altogether too
heavy, sank softly down, and fell asleep.
How long he lay in this state no one knows, nor is ever likely to
know. But, thus much is certain, when old Kartof awoke the day was
already far advanced, and the sun shone brightly upon his face. He
rubbed his eyes and looked about, doubtful whether he was a man
of property or whether he had only dreamt of golden ducats. There
was the violin lying in the grass near his feet. He hastily took it up;—
it felt as light as usual. He shook it;—no rattling of ducats. He held it
before his face and peeped into the sound holes;—to be sure, there
was something in it, yellow and glittering like gold. He shook it out
on the grass;—what should it be?—a score or two of decayed yellow
birch-leaves.
Disappointed, old Kartof rose to his feet to look around whether he
could not find the place where the fire had been.
Yes, there it was! Some embers were still glimmering in the ashes.
This appeared to him more odd than anything else he had
experienced. But old Kartof, after all, took the matter quietly
enough. He lighted his pipe, and taking up his violin set out on his
way home, resolving as he went never to go to that confounded
place again after twelve o'clock at midnight.[78]
GIPSIES.
There prevails in popular traditions much mystery respecting gipsies.
No wonder that this should be the case, since these strange](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-69-320.jpg)
![well as any of us.
The boy took the powder and swallowed it. In a moment his feet
became so light that he found it impossible to keep them on the
ground. The slightest movement which he made raised him into the
air. He danced upon the ears of the growing corn, on the tops of the
trees,—yea, even on the weather-cock of the church-tower. The
people of the village thought this suspicious, and shook their heads,
especially when they furthermore observed a disinclination in the
boy to attend church. They, therefore, consulted with the parson
about the boy. The parson sent for him, and got him all right on his
legs again by means of exorcism; but it was a hard struggle to
banish the potent effects of the gipsy's powder.[79]
The gipsies were formerly supposed to be descendants of the
ancient Egyptians. The German peasants call them Taters,[80] a
name indicating an Asiatic origin; and it has been ascertained that
they migrated from Western India. The roving Nautch-people in
Hindustan are similarly musical and mysterious.
THE NAUTCH-PEOPLE.
The Nautch-people in Hindustan are not only singers and dancers
who exhibit their skill before those who care to admire and to
reward them; but they possess also dangerous charms.
In a popular story of the Hindus, called 'Chandra's Vengeance' we
are told of a youth who, on hearing the music of the Nautch-people
at a great distance, is irresistibly compelled to traverse the jungle in
search of them. When, after twelve days' anxious endeavour to
reach them, he discovers their encampment, Moulee, the daughter
of the chief Nautch-woman, approaches him singing and dancing,
and throws to him the garland of flowers which she wears on her
head. He feels spell-bound, and the Nautch-people offer him a drink
which, as soon as he has tasted it, makes him totally forget his](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-71-320.jpg)
![reseated himself on the throne, he governed as if nothing had ever
occurred to interrupt his reign.[81]
THE MONK OF AFFLIGHEM.
The aim of the present series of popular stories demands that some
notice should now be taken of such musical legends as breathe a
thorough Christian spirit. Several of these are, as might be expected,
very beautiful; but they are familiar to most readers. One or two
which are less well known may, however, find a place here.
The legend of the Monk of Afflighem bears some resemblance to the
beautiful tradition of the Seven Sleepers. If it fails to interest the
reader, the cause must be assigned to the simple manner in which it
is told rather than to the subject itself.
Towards the end of the eleventh century occurred in the Abbey of
Afflighem, in Dendermonde, East Flanders, a most wonderful event,
the pious Fulgentius being at that time the Abbot of the monastery.
One day, a monk of very venerable appearance, whom no one
remembered to have seen before, knocked at the door of the
monastery, announcing himself as one of the brotherhood. The pious
Abbot Fulgentius asked him his name, and from what country he
had come. Whereupon the monk looked at the Abbot with surprise,
and said that he belonged to the house. Being further questioned,
he replied that he had only been away for a few hours. He had been
singing the Matins, he said, in the morning of the same day in the
choir with the other brothers. When, in chanting, they came to the
verse of the ninetieth psalm, which says: For, a thousand years in
thy sight are but as yesterday! he pondered upon it so deeply that
he did not perceive when the singers left the choir, and he remained
sitting alone, absorbed by the words. After he had been a while in
this state of reflection, he heard heavenly strains of music, and on
looking up he saw a little bird which sang with a voice so](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-73-320.jpg)
![enchantingly melodious that he arose in ecstasy. The little bird flew
to the neighbouring wood, whither he followed it. He had been only
a little while in the wood listening to the heavenly song of the bird;
and now, in coming back he felt bewildered,—the appearance of the
neighbourhood was so changed he scarcely knew it again.
When the pious Abbot Fulgentius heard the monk speak thus, he
asked of him the name of the Abbot, and also the name of the King
who governed the country. And after the monk had answered him
and mentioned the names, it was found to the astonishment of all
that these were the names of the Abbot and the King who had lived
three hundred years ago. The monk startled, lifted up his eyes, and
said: Now indeed I see that a thousand years are but as one day
before the Lord. Whereupon he asked the pious Abbot Fulgentius to
administer to him the Holy Sacraments; and having devoutly
received them, he expired.[82]
THE PLAGUE IN GOLDBERG.
The inhabitants of Goldberg, a town in Germany, observe an old
custom of inaugurating Christmas, which is peculiar to themselves.
Having attended divine service, which commences at midnight on
Christmas Eve, they assemble at two o'clock to form a procession to
the Niederring, a hill situated close to the town. When the
procession has arrived at the top of the Niederring, old and young
unite in singing the Chorale Uns ist ein Kindlein heut geboren (For
us this day a child is born). As soon as this impressive act of
devotion is concluded, the town band stationed in the tower of the
old parish church performs on brass instruments the noble Chorale
Allein Gott in der Höh sei Ehr (All glory be to God on High), which
in the stillness of the night is heard over the whole town, and even
in the neighbouring villages.
The origin of this annual observance dates from the time when the
town of Goldberg was visited by a deadly plague called Der schwarze](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-74-320.jpg)
![[Listen]
Uns ist ein Kind-lein heut ge-born,
Gott mit
uns! Von ein'r Jung-frau aus-er-
korn. Gott mit
uns! Gott mit uns! Wer will seyn
wi-der uns!
The Christmas Chorale sung in the refreshing mountain air
wonderfully invigorated their desponding spirits. They arose and
solemnly vowed henceforth to unite in Christian fellowship, with
reliance upon the wisdom of the divine ordinances. The next day they
buried their dead; and when their vow became known in the
neighbourhood, many good people were drawn to Goldberg. The
town soon revived, and prospered more than ever.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-77-320.jpg)
![The inhabitants have not forgotten the visitation which befel their
forefathers, but remember it in humiliation; and this is a lasting
blessing.[83]
FICTIONS AND FACTS.
Knowledge is, of course, to superstition as light is to darkness; still,
some nations endowed with a lively imagination, although they are
much advanced in mental development, cling to the superstitions of
their forefathers, since the superstitions accord with their poetical
conceptions, or are endeared to them by associations which
pleasantly engage the imaginative faculties.
Besides, in countries where the inhabitants frequently witness grand
and awful natural phenomena, their poetical conceptions are likely to
be more or less nourished by these impressive occurrences, however
well acquainted they may be with their natural causes.
It is therefore not surprising that many superstitious notions, such as
have been recorded in the preceding stories, should be found in
civilized nations.
Moreover, in some countries, a more careful research into the old
traditions harbouring among the uneducated classes of the people
has been made, than in other countries. It would, therefore, be
hasty, from the sources at present accessible, to judge of the degree
of mental development attained by individual nations. The Germans
are not less rational than the English; nevertheless, a far greater
number of Fairy Tales have been collected in Germany than in
England.
An enquiry into the musical traditions of the different European races
is likely to increase in interest the more we turn to the mythological
conceptions originally derived from Central Asia, and dispersed
throughout Europe at a period on which history is silent, but upon](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-78-320.jpg)
![wooer.[84] The same may be said of the Spanish Bolero and
Fandango.
The excitement of the chase appears to be another cause of the
origin of dramatic music. The savage, in pursuing the animals which
he requires for his subsistence, experiences successes and
disappointments which are to him highly interesting, and the
recollection of which he enjoys. He naturally feels proud of results
which he could not have achieved without agility and shrewdness,
and he delights in showing to his friends how he proceeded in
accomplishing his feat. Besides, savages have a strong instinct for
imitation, almost like monkeys. Hence their fancy for counterfeiting
the habits of certain animals which they chase and with the
peculiarities of which they are generally well acquainted.
The aborigines of Australia have a dance in which they imitate the
movements of the Kangaroo. The women sing, and produce a
rhythmical accompaniment by beating two pieces of wood together;
while the men, who represent the Kangaroos, produce sounds
peculiar to these animals. The North American Indians have an Eagle
Dance, a Bear Dance, and even a Dog Dance. The natives of
Kamtschatka have a dance in which they cleverly imitate, not only
the attitudes and tricks of the Bear, but also its voice. The peasants
in Finland, in the beginning of the present century, still occasionally
performed a similar dance, or rather action. The Aleutian Islanders,
who have various pantomimic dances executed with masks frightfully
ugly, have also a favourite representation in which a sportsman
shoots a beautiful bird, and afterwards cries for grief at having killed
it; when, suddenly, the beautiful bird revives, changed into a
beautiful woman. The sportsman, of course, falls over head and ears
in love with her, and thus all ends well.[85] This story is enacted with
recitations accompanied by some musical instruments.
Next to love and the chase, it is probably war which elicited the first
attempts at dramatic music. To recall to the memory by a lively
description with gesticulations, the valiant deeds, clever stratagems,
and glorious achievements of the warriors after the battle, must have](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-81-320.jpg)
![comedies, in which the habits of any foreigners with whom they have
become acquainted, are cleverly mimicked and ridiculed.
The missionary W. Ellis remarks of the Polynesian Islanders that they
had songs which, when recited on public occasions, were
accompanied with gestures and actions corresponding to the events
and scenes described, and which assumed in this respect a histrionic
character. In some cases, and on public occasions, the action
represented a kind of pantomime.[86] Other travellers have given
more detailed accounts of these performances. During Captain Cook's
first voyage round the world, Banks and Solander, who accompanied
him, witnessed in one of the Society Islands, in the year 1769, a
comedy with music and dancing, performed by the natives, the
subject of which was the adroitness of a thief, and his subsequent
capture. At Cook's second circumnavigation, during the years 1772-
75, he was treated by the Society Islanders with a somewhat similar
comic opera called Teto (i.e. The Thief). G. Forster, who was with
Cook, remarks that the dialogue, which of course he was unable to
understand, seemed to be closely connected with their actions. One
of them kneeled down, and another beat him and plucked him by the
beard. Then two others were treated by the torturer in the same
unceremonious manner; until one of them seized a stick and gave
him a sound thrashing in return. This formed the conclusion of the
first act, and the players withdrew. The commencement of the
second act was announced by the musicians beating their drums.
There were actresses as well as actors engaged in the performance.
[87] A more detailed account of the dramatic attempts of the
Polynesian Islanders is given by W. Mariner, who, during his sojourn
with the natives, had the best opportunity of becoming acquainted
with their customs and amusements. His observations, which refer
especially to the Tonga Islanders, show that the actors recite
sentences which are answered by a chorus of singers. There is a
great variety in their movements and groupings. Occasionally they
sing slowly, and afterwards quickly for about a quarter of an hour.
Sometimes they form a semi-circle, assume a bending position, and](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-83-320.jpg)
![sing in a subdued tone of voice a soft air; which is soon again
followed by a loud and vehement recitation.[88]
Grotesque dresses and adornments are, of course, an essential
attribute in these entertainments. Neither are buffoons wanting.
According to B. Seeman, the entertainment called Kalau Rere, which
he witnessed in the Fiji Islands, with its high poles, streamers,
evergreens, masquerading, trumpet-shells, chants and other wild
music, is the nearest approach to dramatic representation the Fijians
seem to have made, and it is with them what private theatricals are
with us. They are also on other occasions very fond of dressing
themselves in fantastic, often very ridiculous costume; and in nearly
every large assembly there are buffoons. Court fools, in many
instances hunchbacks, are attached to the chief's establishment.[89]
Also the Negroes in Senegambia and Upper Guinea have buffoons,
who delight the people with their antics and acting in processions
and public festivities. Buffoons are popular even in Mohammedan
countries, where dramatic performances are generally considered
objectionable. Morier states that in Persia the princes, governors of
provinces, etc., as well as the King, have a band of Looties, or
buffoons, in their pay, who are looked upon as a necessary part of
Persian state. They attend at merry-makings and public festivals, and
some of them are endowed with great natural wit. This was, for
instance, the case with a certain buffoon named Looti Bashee. His
dress, when he came to the ambassador, was composed of a felt hat,
the crown of which was made like ours, but with two long ears
projecting before, and two behind. Others of his troop were dressed
in the same way; all looked grotesque, and I conjectured that
nothing could give one a better idea of Satyrs and Bacchanalians,
particularly as they were attended by a suite of monkeys headed by a
large ape, which were educated to perform all sorts of tricks. They
carried copper drums slung under the arm, which they beat with their
fingers, making a noise like castanets; others played the tambourine;
and when all this was put into motion, with their voices roaring in
loud chorus, the scene was unique.[90]](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-84-320.jpg)
![Sir Robert Ker Porter witnessed at Bagdad, in the beginning of the
present century, a kind of musical drama performed by men and
boys, the latter being dressed like females. This amusement, he
remarks, is the only one of a theatrical complexion known among
the people. It is often called for by the female part of the inhabitants;
but I am told that with the men it is now very rare, the Pasha so
setting his face against it as to forbid the avowed existence of hirable
dancing-boys in his capital.[91] There is a Turkish theatre at Pera in
which Turkish plays, adapted from the Italian, are acted by Turkish
actors, and Turkish women appear unveiled upon the stage.[92] The
women in the hareem, who in their diversions are only permitted to
employ slaves of their own sex, occasionally make them act
melodramas, the subject of which is generally a love story.
The Indians in Mexico have some characteristic dances in which
scenes are pantomimically enacted referring to Montezuma and to
the conquest of Mexico by the Spaniards.
In most of the entertainments, of which examples have just been
given, the music must necessarily partake of a dramatic character.
Generally, the tunes are not selected at pleasure, but certain tunes
belong to certain representations. The dramatic effect of the music
depends, however, chiefly upon its execution, which naturally
changes according to the action which it accompanies. Thus, if the
actors represent a sentimental or heart-rending scene, their vocal
effusions will naturally be in a subdued tone, and the sympathizing
musicians will touch their instruments delicately and slowly. If, on the
other hand, the actors represent some exciting or heart-stirring
scene, they will naturally raise their voices, and the musicians will
play louder and faster as a matter of course. In fact, when their pulse
beats quicker, the rhythmical flow of their music, however rude and
inartistic it may be, becomes more animated unpremeditatedly. Such
is the most primitive condition, or the commencement of the
development of dramatic music. Let us now examine it in a
somewhat more advanced stage of cultivation.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-85-320.jpg)
![The Javanese, who among the islanders of the Indian Archipelago
are renowned for their skill in the dramatic art, generally use fabulous
traditions from their own history, or Hindu legends, as subjects for
their performances, which are acted exclusively by men. A full band
of musicians generally accompanies the drama. The instruments
mostly belong to the class called Instruments of Percussion, but
several of them are constructed with plates of metal which produce a
series of sweet tones, arranged according to the pentatonic scale.
Some of the Javanese airs, which have been collected by Europeans,
are very expressive, and it might be instructive to musical enquirers,
if some really musical European visitor in Java would faithfully
commit to notation the orchestral accompaniments of some of the
most popular Javanese dramas. Madame Ida Pfeiffer relates that she
was treated in the house of a Rajah, at Bandong, with a kind of
pantomime in three acts, the third of which represented a combat.
The music that accompanied the combat, she remarks, was very
noisy and discordant; but, on the defeat of the one party, a soft
plaintive melody arose at some distance off. The whole performance
was really pretty and expressive.[93] Sir Stamford Raffles, and other
travellers, give similar descriptions, and have besides much to say
about the clever puppet-shows of the Javanese, in which the
characters of dramas are represented by puppets, or by their
shadows.
The Siamese are fond of theatrical performances. According to
Turpin's history of Siam, published in the year 1771, whenever they
burn the body of a minister or great man, a theatre is erected on the
side of a river, where the actors appear habited according to their
parts; and during three days they never quit the scene from eight in
the morning till seven at night. De La Loubère, who visited Siam in
the year 1687, says that the subjects of the dramas are historical, in
verse, serious, and sung by several actors who are always present,
and who only sing reciprocally. One of them sings the historian's part,
and the rest sing those of the personages which the history makes to
speak; but they are all men that sing, and no women. About a
century ago it appears to have been the custom to employ only men](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-86-320.jpg)
![as actors, although there were female dancers. But, at the present
day there are actresses, at any rate in the palace of the King, where
Sir John Bowring saw them perform on several occasions. In one of
these entertainments the actors were all females, almost all girls. A
few matrons, however, took the part of warriors, monkeys, priests;
and the three manageresses, or prompteresses, were not only old
and ugly, but seemed very spiteful, and on several occasions scolded
and slapped the ladies who required correction. One of them had the
drama written on black sheets in white letters before her, from which
she prompted the singers of the recitative. The story began by the
appearance of a monster monkey in a forest, which is visited by a
number of ladies of rank, one of whom, after an unsuccessful
struggle, the others having managed to escape, the monster monkey
contrives to carry off. She is redeemed by the interference of a priest,
whose temple is in the forest. Afterwards we are introduced to a
sovereign Court, where all the ceremonies are observed which are
practised in daily life, the dresses being those ordinarily worn, and
most gorgeous they are.... There is a battle, and rewards to the
victors, and a crowning of a king's son in recompense for his valour,
and offerings to Buddha, and a great feast, etc.[94] The principal
performers act, but do not speak. The tale is told in recitative by a
body of singers, accompanied by various instruments. The band
assisting generally consists of about twenty members who play on
wind instruments of the oboe kind, gongs, large castanets above a
foot in length, and several sonorous instruments of percussion
constructed with slabs of wood, or plates of metal, somewhat similar
to those of the Javanese before mentioned.
The Cochin-Chinese are remarkably fond of dramatic entertainments,
which are generally of an operatic character commemorating
historical events. An English gentleman who witnessed the
performance of some of these plays remarks of the actors: Their
singing is good, when the ear has become accustomed to it; and the
modulation of voice of the females is really captivating.[95] Sir
George Staunton was evidently surprised to find that a kind of
historical opera, which he heard in the town of Turon (called by the](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-87-320.jpg)
![natives Hansán) contained recitatives, airs, and choruses, which
were, he says, as regular as upon the Italian stage. He adds:
Some of the female performers were by no means despicable
singers. They all observed time accurately, not only with their voices,
but every joint of their hands and feet was obedient to the regular
movement of the instruments.[96] The band consisted of stringed
instruments, wind instruments, and instruments of percussion. Sir
John Barrow describes the theatre at Turon as a shed of bamboo.
He relates: In the farther division of the building a party of
comedians was engaged in the midst of an historical drama when we
entered; but, on our being seated they broke off, and, coming
forward, made before us an obeisance of nine genuflexions and
prostrations, after which they returned to their labours, keeping up
an incessant noise and bustle during our stay. The heat of the day,
the thermometer in the shade standing at 81 deg. in the open air,
and at least 10 deg. higher in the building, the crowds that thronged
to see the strangers, the horrible crash of the gongs, kettle-drums,
trumpets, and squalling flutes, were so stunning and oppressive that
nothing but the novelty of the scene could possibly have detained us
for a moment. The most entertaining, as well as the least noisy part
of the theatrical exhibition, was a sort of Interlude, performed by
three young women for the amusement, it would seem, of the
principal actress, who sat as a spectator in the dress and character of
some ancient Queen, whilst an old eunuch, very whimsically dressed,
played his antic tricks like a scaramouch or buffoon in a Harlequin
entertainment. The dialogue in this part differed entirely from the
querulous and nearly monotonous recitation of the Chinese, being
light and comic, and occasionally interrupted by cheerful airs which
generally concluded with a chorus. These airs, rude and unpolished
as they were, appeared to be regular compositions, and were sung in
exactly measured time. One in particular attracted our attention,
whose slow melancholy movement breathed the kind of plaintiveness
so peculiar to the native airs of the Scotch, to which indeed it bore a
close resemblance.](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-88-320.jpg)
![Probably the air was founded on the pentatonic scale, which is
common in the music of the Chinese and Javanese, and of which
traces are to be found in the Scotch popular tunes.
The voices of the women are shrill and warbling, but some of their
cadences were not without melody. The instruments at each pause
gave a few short flourishes, till the music gradually increased in
loudness by the swelling and deafening gong. Knowing nothing of the
language, we were of course as ignorant of the subject as the
majority of an English audience is of an Italian opera.[97]
A curious mode of paying the actors, which prevails in Cochin-China,
may be mentioned here. An Englishman who was present at a
theatrical performance in the town of Kangwarting, relates that the
Quong, or governor of the province, bore the expense of the
entertainment. The musical drama was performed in a large shed
before a great concourse of spectators. The Quong was there
squatted on a raised platform in front of the actors with a small drum
before him, supported in a diagonal position, on which he would
strike a tap every time any part of the performance pleased him;
which also was a signal for his purse-bearer to throw a small string of
about twenty cash to the actors. To my taste, this spoiled the effect
of the piece; for, every time the cash fell among them there would be
a silence, and the next moment a scramble for the money; and it fell
so frequently as almost to keep time with the discordant music of the
orchestra. The actors were engaged by the day, and in this manner
received their payment, the amount of which entirely depended upon
the approbation of the Quong and the number of times he encored
them by tapping his drum. I could see that many of them paid far
more attention to the drum than they did to their performance;
though I suppose, the amount thrown to them is equally divided.
Sometimes the string on which the cash was tied, unluckily broke,
and the money flew in all directions; by which some of the
bystanders profited, not being honourable enough to hand it up to
the poor actors.[98]](https://image.slidesharecdn.com/11190614-250607024141-8370f813/85/Recursive-Models-Of-Dynamic-Linear-Economies-Lars-Hansen-Thomas-J-Sargent-89-320.jpg)
Recursive Models Of Dynamic Linear Economies Lars Hansen Thomas J Sargent
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- 5. Recursive Models of Dynamic Linear Economies
- 6. Recursive Models of Dynamic Linear Economies Lars Hansen University of Chicago Thomas J. Sargent New York University and Hoover Institution c Lars Peter Hansen and Thomas J. Sargent 21 March 2005
- 7. Contents Acknowledgements xii Preface xiii Part I: Components of an economy 1. Introduction 3 1.1. Introduction. 1.2. Computer Programs. 1.3. Organization. 2. Stochastic Linear Difference Equations 9 2.1. Introduction. 2.2. Notation and Basic Assumptions. 2.3. Predic- tion Theory. 2.4. Transforming Variables to Uncouple Dynamics. 2.5. Examples. 2.5.1. Deterministic seasonals. 2.5.2. Indeterministic season- als. 2.5.3. Univariate autoregressive processes. 2.5.4. Vector autoregres- sions. 2.5.5. Polynomial time trends. 2.5.6. Martingales with drift. 2.5.7. Covariance stationary processes. 2.5.8. Multivariate ARMA processes. 2.5.9. Prediction of a univariate first order ARMA. 2.5.10. Growth. 2.5.11. A rational expectations model. 2.6. The Spectral Density Ma- trix. 2.7. Computer Examples. 2.7.1. Deterministic seasonal. 2.7.2. Indeterministic seasonal, unit root. 2.7.3. Indeterministic seasonal, no unit root. 2.7.4. First order autoregression. 2.7.5. Second order autore- gression. 2.7.6. Growth with homoskedastic noise. 2.7.7. Growth with heteroskedastic noise. 2.7.8. Second order vector autoregression. 2.7.9. A rational expectations model. 2.8. Conclusion. 3. The Economic Environment 39 3.1. Information. 3.2. Taste and Technology Shocks. 3.3. Technologies. 3.4. Examples of Technologies. 3.4.1. Other technologies. 3.5. Prefer- ences and Household Technologies. 3.6. Examples of Household Tech- nology Preference Structures. 3.7. Constraints to Keep the Solutions “Square Summable”. 3.8. Summary. – v –
- 8. vi Contents 4. Optimal Resource Allocation 57 4.1. Planning problem. 4.2. Lagrange Mmultipliers. 4.3. Dynamic pro- gramming. 4.4. Lagrange multipliers as gradients of value function. 4.5. Planning problem as linear regulator. 4.6. Solutions for five economies. 4.6.1. Preferences. 4.6.2. Technology. 4.6.3. Information. 4.6.4. Brock- Mirman model. 4.6.5. A growth economy fueled by habit persistence. 4.6.6. Lucas’s pure exchange economy. 4.6.7. An economy with a durable consumption good. 4.7. Hall’s model. 4.8. Higher Adjustment Costs. 4.9. Altered ‘growth condition’. 4.10. A Jones-Manuelli economy. 4.11. Durable consumption goods. 4.12. Summary. A. Synthesizing the linear regulator. B. A Brock-Mirman model. 4.B.1. Uncertainty. 4.B.2. Optimal Stationary States. 5. The Commodity Space 105 5.1. Valuation. 5.2. Price systems as linear functionals. 5.3. A one period model under certainty. 5.4. One period under uncertainty. 5.5. An infinite number of periods and uncertainty. 5.5.1. Conditioning in- formation. 5.6. Lagrange multipliers. 5.7. Summary. A. Appendix. 6. A Competitive Economy 113 6.1. Introduction. 6.2. The Problems of Households and Firms. 6.2.1. Households. 6.2.2. Firms of type I. 6.2.3. Firms of type II. 6.3. Compet- itive Equilibrium. 6.4. Lagrangians. 6.4.1. Households. 6.4.2. Firms of type I. 6.4.3. Firms of type II. 6.5. Equilibrium Price System. 6.6. Asset Pricing. 6.7. Term Structure of Interest Rates. 6.8. Re-opening Mar- kets. 6.8.1. Recursive price system. 6.8.2. Non-Gaussian asset prices. 6.9. Summary of Pricing Formulas. 6.10. Asset Pricing Example. 6.10.1. Preferences. 6.10.2. Technology. 6.10.3. Information. 6.11. Exercises. 7. Applications 139 7.1. Introduction. 7.2. Partial Equilibrium Interpretation. 7.2.1. Par- tial equilibrium investment under uncertainty. 7.3. Introduction. 7.4. A Housing Model. 7.4.1. Demand. 7.4.2. House producers. 7.5. Cattle Cycles. 7.5.1. Mapping cattle farms into our framework. 7.5.2. Pref- erences. 7.5.3. Technology. 7.6. Models of Occupational Choice and Pay. 7.6.1. A one-occupation model. 7.6.2. Skilled and unskilled work- ers. 7.7. A Cash-in-Advance Model. 7.7.1. Reinterpreting the household technology. 7.8. Taxation in a Vintage Capital Model. A. Decentralizing the Household.
- 9. Contents vii 8. Efficient Computations 157 8.1. Introduction. 8.2. The Optimal Linear Regulator Problem. 8.3. Transformations to eliminate discounting and cross-products. 8.4. Sta- bility Conditions. 8.5. Invariant Subspace Methods. 8.5.1. Px as La- grange multiplier. 8.5.2. Invariant subspace methods. 8.5.3. Distorted Economies. 8.5.4. Transition Dynamics. 8.6. The Doubling Algorithm. 8.7. Partitioning the State Vector. 8.8. The Periodic Optimal Linear Regulator. 8.9. A Periodic Doubling Algorithm. 8.9.1. Partitioning the state vector. 8.10. Linear Exponential Quadratic Gaussian Control. 8.10.1. Doubling algorithm. A. Concepts of Linear Control Theory. B. Symplectic Matrices. C. Alternative forms of Riccati equation.
- 10. viii Contents Part II: Representations and Properties 9. Representation and Estimation 187 9.1. The Kalman Filter. 9.2. Innovations Representation. 9.3. Conver- gence results. 9.3.1. Time-Invariant Innovations Representation. 9.4. Serially Correlated Measurement Errors. 9.5. Combined System. 9.6. Recursive Formulation of Likelihood Function. 9.6.1. Initialization. 9.6.2. Non-existence of a stationary distribution. 9.6.3. Serially cor- related measurement errors. 9.7. Wold Representation. 9.8. Vector Au- toregression for {yt}. 9.8.1. The factorization identity. 9.8.2. Location of zeros of characteristic polynomial. 9.8.3. Wold and autoregressive representations (white measurement errors). 9.8.4. Serially correlated measurement errors. 9.9. Innovations in yt+1 as Functions of Innova- tions wt+1 and ηt+1 . 9.10. Innovations in the yt ’s and the wt ’s in a Permanent Income Model. 9.10.1. Preferences. 9.10.2. Technology. 9.10.3. Information. 9.11. Frequency Domain Estimation. 9.12. Ap- proximation Theory. 9.13. Aggregation Over Time. 9.14. Simulation Estimators. A. Initialization of the Kalman Filter. 10. Semiparametric Estimation with Limited Information 227 10.1. Introduction. 10.2. Underlying Economic Model. 10.3. Econome- trician’s information and the implied orthogonality conditions. 10.4. An Adjustment Cost Example. 10.5. A Slightly Simpler Estimation Prob- lem. 10.5.1. Scalar Parameterizations of B. 10.6. Multidimensional Parameterizations of B. 10.7. Nonparametric Estimation of B. 10.8. Back to the Adjustment Cost Model. 11. Representation of Demand 239 11.1. Introduction. 11.2. Canonical Representations of Services. 11.3. Dynamic Demand Functions for Consumption Goods. 11.3.1. The mul- tiplier µw 0 . 11.3.2. Dynamic Demand System. 11.3.3. Foreshadow of Gorman aggregation. 11.4. Computing Canonical Representations. 11.4.1. Heuristics. 11.4.2. An auxiliary problem that induces a canonical rep- resentation. 11.5. Operator Identities. 11.6. Becker-Murphy Model of Rational Addiction. A. Fourier transforms. 11.A.1. Primer on trans- forms. 11.A.2. Time reversal and Parseval’s formula. 11.A.3. One sided
- 11. Contents ix sequences. 11.A.4. Useful properties. 11.A.5. One sided transforms. 11.A.6. Discounting. 11.A.7. Fourier transforms. 11.A.8. Verifying Equivalent Valuations. 11.A.9. Equivalent representations of prefer- ences. 11.A.10. First term: factorization identity. 11.A.11. Second term. 11.A.12. Third term. 12. Gorman Heterogeneous Households 265 12.1. Introduction. 12.2. A Digression on Gorman Aggregation. 12.3. An Economy with Heterogeneous Consumers. 12.4. Allocations. 12.4.1. Consumption sharing rules. 12.5. Risk Sharing Implications. 12.6. Im- plementing the Allocation Rule with Limited Markets. 12.7. A Com- puter Example. 12.8. Exercises. 12.8.1. Part one. 12.8.2. Part two. 12.9. Economic integration. 12.9.1. Preferences:. 12.9.2. Technology. 12.9.3. Information. 13. Permanent Income Models 287 13.1. Technology. 13.2. Two Implications. 13.3. Solution. 13.4. Deter- ministic Steady States. 13.5. Cointegration. 13.6. Constant Marginal Utility of Income. 13.7. Consumption Externalities. 13.8. Tax Smooth- ing Models. 14. Non-Gorman Heterogeneity Among Households 307 14.1. Introduction. 14.2. Households’ Preferences. 14.2.1. Technol- ogy. 14.3. A Pareto Problem. 14.4. Competitive Equilibrium. 14.4.1. Households. 14.4.2. Firms of type I and II. 14.4.3. Definition of compet- itive equilibrium. 14.5. Computation of Equilibrium. 14.5.1. Candidate equilibrium prices. 14.5.2. A Negishi algorithm. 14.6. Mongrel Aggrega- tion. 14.6.1. Static demand. 14.6.2. Frequency domain representation of preferences. 14.7. A Programming Problem for Mongrel Aggregation. 14.7.1. Factoring S0 S . 14.8. Summary of Findings. 14.9. The Mon- grel Preference Shock Process. 14.9.1. Interpretation of ŝt component. 14.10. Choice of Initial Conditions.
- 12. x Contents Part III: Extensions 15. Equilibria with Distortions 331 15.1. Introduction. 15.2. A Representative Agent Economy with Dis- tortions. 15.2.1. a. Consumption externalities. 15.2.2. b. Production externalities. 15.2.3. c. Taxes. 15.3. Households. 15.4. Firms. 15.5. Information. 15.6. Equilibrium. 15.7. Heterogeneous Households with Distortions. 15.7.1. Households. 15.7.2. Firms of type I. 15.7.3. Firms of type II. 15.7.4. Government. 15.7.5. Definition of equilibrium. 15.7.6. Equilibrium computation. 15.8. Government Deficits and Debt. 15.9. Examples. 15.9.1. A production externality. 15.9.2. Consumption tax only. 15.9.3. Machinery investment subsidy. 15.9.4. ‘Personal’ habit persistence. 15.9.5. ‘Social’ habit persistence. 15.10. Conclusions. A. Invariant subspace equations for first specification. 15.A.1. Household’s Lagrangian. 15.A.2. Firm’s first order conditions. 15.A.3. Representa- tiveness conditions. B. Invariant subspace equations for heterogeneous agent model. 16. Recursive Risk Sensitive Control 367 16.1. Introduction. 16.2. A Control Problem. 16.3. Pessimistic Inter- pretation. 16.4. Recursive Preferences. 16.4.1. Endowment economy. 16.5. Asset Pricing. 16.6. Characterizing the Pricing Expectations Op- erator. 16.7. Production Economies. 16.8. Risk-Sensitive Investment under Uncertainty. 16.9. Equilibrium Prices in the Adjustment Cost Economies. 17. Periodic Models of Seasonality 385 17.1. Introduction. 17.2. A Periodic Economy. 17.3. Asset Pricing. 17.4. Prediction Theory. 17.5. The Term Structure of Interest Rates. 17.6. Conditional Covariograms. 17.7. The Stacked and Skip-Sampled System. 17.8. Covariances of the Stacked, Skip Sampled Process. 17.9. The Tiao-Grupe Formula. 17.9.1. A state space realization of the Tiao- Grupe formulation. 17.10. Some Calculations with a Periodic Hall Model. 17.11. Periodic Innovations Representations for the Periodic Model. A. A Model of Disguised Periodicity. 17.13. A1. Two Illustra- tions of Disguised Periodicity. 17.14. A2. Mathematical Formulation of Disguised Periodicity.
- 13. Contents xi Part IV: Economies as Objects 18. Introduction to Objects 425 18.1. Matlab Objects. 18.1.1. Definitions. 18.1.2. Matlab Specifics. 18.1.3. How to Define a Matlab Class. 18.2. Summary. 19. Economies as Matlab Objects 431 19.1. Introduction. 19.2. Parent Classes: Information. 19.2.1. Struc- ture. 19.2.2. Functions. 19.3. Parent Classes: Technology. 19.3.1. Struc- ture. 19.3.2. Functions. 19.4. Parent Classes: Preferences. 19.4.1. Structure. 19.4.2. Functions. 19.5. Child Class: Economy. 19.5.1. Struc- ture. 19.5.2. Fields containing the history of the economy. 19.5.3. Functions. 19.5.4. Constructing the object and changing parameters. 19.5.5. Analyzing the economy. 19.6. Working with economies. 19.6.1. The built-in economies. 19.6.2. Mixing and matching built-in parent objects. 19.6.3. Building your own economy. 19.7. Tutorial. 20. MATLAB Programs 441 20.1. Matlab programs. 21. References 495 22. Index 509 23. Author Index 513 24. Matlab Index 515
- 15. Preface – xiii –
- 16. Part I Components of an economy
- 18. Chapter 1 Introduction 1.1. Introduction This book views many apparently disparate dynamic economic models as ex- amples of a single class of models that can be adapted and specialized to study diverse economic phenomena. The class of models was created by using recent advances in (i) the theory of recursive dynamic competitive economies;1 (ii) methods for estimating and interpreting vector autoregression;2 (iii) linear op- timal control theory;3 and (iv) computer languages for rapidly manipulating linear optimal control systems.4 We combine these elements to build a class of models for which the competitive equilibria are vector autoregressions that can be swiftly computed, represented, and simulated using the methods of linear optimal control theory. We use the computer language MATLAB to implement the computations. This language has a powerful vocabulary and a convenient structure that liberate time and energy from programming, and thereby spur creative application of linear control theory. Our goal has been to create a class of models that merge recursive economic theory and with dynamic econometrics. Systems of autoregressions and of mixed autogregressive, moving average processes are a dominant setting for dynamic econometrics. We constructed our economic models by adopting a version of recursive competitive theory in which an outcome of theorizing is a vector autoregression. We formulated this class of models because practical difficulties of comput- ing and estimating recursive equilibrium models still limit their use as a tool for thinking about applied problems in economic dynamics. Recursive competi- tive equilibria were themselves developed as a special case of the Arrow-Debreu competitive equilibrium, both to restrict the range of outcomes possible in the 1 This work is summarized by Harris (1987) and Stokey, Lucas, and Prescott (1989). 2 See Sims (1980), Hansen and Sargent (1980, 1981, 1990). 3 For example, see Kwakernaak and Sivan (1972), and Anderson and Moore (1979). 4 See the MATLAB manual. – 3 –
- 19. 4 Introduction Arrow-Debreu setting and to create a framework for studying applied problems in dynamic economies of long duration. Relative to the general Arrow-Debreu setting, the great advantage of the recursive competitive equilibrium formulation is that equilibria can be computed by solving a discounted dynamic program- ming problem. Further, under particular additional conditions, an equilibrium can be represented as a Markov process in the state variables. When that Markov process has an invariant distribution to which the process converges, there exists a vector autoregressive representation. Thus, the theory of recur- sive competitive equilibria holds out the promise of making closer contact with econometric theory than did previous formulations of equilibrium theory. Two computational difficulties have left much of this promise unrealized. The first is Bellman’s “curse of dimensionality” which usually makes dynamic programming a costly procedure for systems with even small numbers of state variables. The second problem is that after a dynamic program has been solved and the equilibrium Markov process computed, the vector autoregression implied by the theory has to be computed by applying classic projection formulas to a large number of second moments of the stationary distribution associated with that Markov process. Typically, each of these computational problems can be solved only approximately. Good research along a number of lines is now being directed at evaluating alternative ways of making these approximations.5 The need to make these approximations originates in the fact that for gen- eral functional forms for objective functions and constraints, even one iteration on the functional equation of Richard Bellman cannot be performed analytically. It so happens that the functional forms economists would most like to use have been of this general class for which Bellman’s equation cannot be iterated upon analytically. Linear control theory studies the most important special class of prob- lems for which iterations on Bellman’s equation can be performed analytically: problems with a quadratic objective function and a linear transition function. Application of dynamic programming leads to a system of well understood and rapidly solvable equations known as the matrix Riccati equation. The philosophy of this book is to swallow hard and to accept up front as primitive descriptions of tastes, technology, and information specifications that satisfy the assumptions of linear optimal control theory. This approach 5 See Marcet (1989) and Judd (1990). Also see Coleman (1990) and Tauchen (1990).
- 20. Organization 5 purchases the ability rapidly to compute equilibria together with a form of equi- librium that is automatically in the form of a vector autoregression. A cost of the approach is that it does not accommodate many specifications that we would like to be able to analyze. The purpose of this book is to display the versatility and tractability of our class of models. Versions of a wide range of models from modern capital theory and asset pricing theory can be represented within our framework. The equilibria of these models can be computed so easily that we hope that the reader will soon be thinking of improvements to our specifications. We provide formulas and software for the reader to experiment. 1.2. Computer Programs In writing this book, we put ourselves under a restriction that we should supply the reader with a computer program that implements every equilibrium concept and mathematical representation that we describe. The programs are written in MATLAB, and are described throughout the book. When a MATLAB program is referred to in the text, we place it in typewriter font. Similarly, all computer code is placed in typewriter font.6 You will get much more out of this book if you use and modify our programs as you read. 1.3. Organization This book is organized as follows. Chapter 10 describes the first order lin- ear vector stochastic difference equation, and shows how special cases of it are formed by a variety of models of time series processes that have been studied by economists. This difference equation will be used to represent the information flowing to economic agents within our models. It will also be used to represent the equilibrium of the model. Chapter 3 defines an economic environment in terms of the preferences of a representative agent, the technology for producing goods, stochastic processes 6 To run our programs, you will need MATLAB’s Control Toolkit in addition to the basic MATLAB software.
- 21. 6 Introduction disturbing preferences and the technology, and the information structure of the economy. The stochastic processes fit into the model introduced in chapter 10, while the preferences, technology, and information structure are specified with an eye toward making the competitive equilibrium one that can be computed by the application of linear control theory. Chapter 4 describes a social planning problem associated with the equilib- rium of the model. The problem is formulated in two ways, first as a variational problem using stochastic Lagrange multipliers, and then as a dynamic program- ming problem. We describe how to compute the solution of the dynamic pro- gramming problem using formulas from linear control theory. The solution of the social planning problem is a first order vector stochastic difference equation of the form studied in chapter 10. We also show how to use the value function for the social planning problem to compute the Lagrange multipliers associated with the planning problem. These multipliers are later used in chapter 6 to compute the equilibrium price system. Chapter 5 describes the price system and the commodity space that sup- port a competitive equilibrium. We use a formulation that lets the values that appear in agents’ budget constraints and objective functions be represented as conditional expectations of geometric sums of streams of future “prices” times quantities. Chapter 5 relates these prices to Arrow-Debreu state contingent prices. Chapter 6 describes a decentralized version of our economy, and defines and computes a competitive equilibrium. Competitive equilibrium quantities solve a social planning problem. The price system can be deduced from the stochastic Lagrange multipliers associated with the social planning problem. Chapter 7 describes versions of several dynamic models from the literature that fit easily within our class of models. Chapter 9 describes the links between our theoretical equilibrium and au- toregressive representations of time series of observables. We show how to obtain an autoregressive representation for a list of observable variables that are linear functions of the state variables of the model. The autoregressive representation is naturally affiliated with a recursive representation of the likelihood function for the observable variables. In describing how to deduce the autoregressive representation from the parameters determining the equilibrium of the model, and possibly also from parameters of measurement error processes, we are com- pleting a key step needed to permit econometric estimation of the model’s free
- 22. Organization 7 parameters. Chapter 9 also treats two other topics intimately related to econo- metric implementation of the models; aggregation over time, and the theory of approximation of one model by another. Chapter 8 describes fast methods to compute equilibria. We describe how doubling algorithms can speed the computation of expectations of geometric sums of quadratic forms, and help to solve dynamic programming problems. Chapter 11 describes alternative ways to represent demand. It identifies an equivalence class of preference specifications that imply the same demand functions, and characterizes a special subset of them as canonical household preferences. Canonical representations of preferences are useful for describing economies with heterogeneity among household’s preferences. Chapter 12 describes a version of our economy with the type of heterogene- ity among households allowed when preferences aggregate in a sense introduced by Terrance Gorman . In this setting, affine Engle curves of common slope pre- vail and give rise to a representative consumer. This representative consumer is ‘easy to find,’ and from the point of view of equilibrium computation of prices and aggregate quantities, adequately stands in for the household of chapters 3–6. The allocations to individual consumers require additional computations, which this chapter describes. Chapter 13 uses our model of preferences to represent multiple goods ver- sions of permanent income models along the lines of Robert Hall’s (1978). We retain Hall’s specification of the ‘storage’ technology for accumulating physical assets, and also the restriction on the discount factor, depreciation rate, and gross return on capital that delivered to Hall a martingale for the marginal utility of consumption. Adopting Hall’s specification of the storage technology imparts a martingale characterization to the model, but it is hidden away in an ‘index’ whose increments drive the behavior of consumption demands for various goods, which themselves are not martingales. This model forms a convenient laboratory for thinking about the sources in economic theory of ‘unit roots’ and ‘co-integrating vectors.’ Chapter 14 describes a setting in which there is more heterogeneity among households’ preferences, causing the conditions for Gorman aggregation to fail. Households’ Engle curves are still affine, but dispersion of their slopes arrests Gorman aggregation. There is another sense, originating with Negishi, in which there is a representative household whose preferences represent a complicated kind of average over the preferences of different types of households. We show
- 23. 8 Introduction how to compute and interpret this preference ordering over economy-wide aggre- gates. This average preference ordering cannot be computed before one knows the distribution of wealth evaluated at equilibrium prices. Chapter 15 describes economies with production and consumption exter- nalities and also distortions due to a government’s imposing distorting flat rate taxes. Equilibria of these economies has to be computed by a direct attack on Euler equations and budget constraints, rather than via dynamic programming for an artificial social planning problem. Chapter 16 describes a recursive version of Jacobson’s and Whittle’s ‘risk sensitive’ preferences. This preference specification has the features that, al- though it violates certainty equivalence – so that the conditional covariance of forecast error distributions impinge on equilibrium decision rules – it does so in a way that preserves linear equilibrium laws of motion, and retains calculation of equilibria and asset prices via simple modifications of our standard formulas. These preferences are a version of those studied by Epstein and Zin ( ) and Weil ( ). Chapter 17 describes how to adapt our setup to include features of the periodic models of seasonality that have been studied by Osborne (1988), Todd (1990), and Ghysels (1993). Chapter 20 is a manual of the MATLAB programs that we have prepared to implement the calculations described in this book. The design is consistent with other MATLAB manuals. The notion of duality and the ‘factorization identity’ from recursive lin- ear optimal control theory are used repeatedly in Chapter 9 (on representing equilibria econometrically), and chapters 11, 12, and 14 (on representing and aggregating preferences). ‘Duality’ is the observation that recursive filtering problems (Kalman filtering) have the same mathematical structure as recursive formulations of linear optimal control problems (leading to Riccati equations via dynamic programming). That duality applies so often in our settings in effect ‘halves’ the mathematical apparatus that we require.
- 24. Chapter 2 Stochastic Linear Difference Equations 2.1. Introduction This chapter introduces the first-order vector linear stochastic difference equa- tion, which we use in two important ways. We use it first to represent the information flowing to economic agents, then again to represent equilibria of our models. The first-order linear stochastic difference equation is associated with a tidy theory of prediction and a host of procedures for econometric appli- cation. Their ease of analysis has prompted us to adopt economic specifications that cause our equilibria to have representations in terms of a first-order linear stochastic difference equation. The first order vector stochastic difference equation is recursive because it expresses next period’s vector of state variables as a linear function of this period’s state vector and a vector of new disturbances to the system. These disturbances form a “martingale difference sequence,” and are the basic building block out of which the time series are created. Martingale difference sequences are easy to forecast, a fact that delivers convenient recursive formulas for optimal predictions. 2.2. Notation and Basic Assumptions Let {xt : t = 1, 2, . . .} be a sequence of n-dimensional random vectors, i.e. an n-dimensional stochastic process. The vector xt contains variables observed by economic agents at time t. Let {wt : t = 1, 2, . . .} be a sequence of N - dimensional random vectors. The vectors {wt} will be treated as building blocks for {xt : t = 1, 2, . . .}, in the sense that we shall be able to express xt as the sum of two terms. The first is a moving average of past wt ’s. The second describes the effects of an initial condition. The {wt} process is used to generate a sequence of information sets {Jt : t = 0, 1, . . .}. Let J0 be generated by x0 and Jt be generated by x0, w1, . . . , wt , which means that Jt consists of the set – 9 –
- 25. 10 Stochastic Linear Difference Equations of all measurable functions of {x0, w1, . . . , wt}.1 The building block process is assumed to be a martingale difference sequence adapted to this sequence of information sets. We explain what this means by advancing the following Definition 1: The sequence {wt : t = 1, 2, . . .} is said to be a martingale difference sequence adapted to {Jt : t = 0, 1, . . .} if E(wt+1|Jt) = 0 for t = 0, 1, . . . . In addition, we assume that the building block process is conditionally ho- moskedastic, a phrase whose meaning is conveyed by Definition 2: The sequence {wt : t = 1, 2, . . .} is said to be conditionally homoskedastic if E(wt+1w0 t+1 | Jt) = I for t = 0, 1, . . . . It is an implication of the law of iterated expectations that {wt : t = 1, 2, . . .} is a sequence of (unconditional) mean zero, serially uncorrelated random vectors.2 In addition, the entries of wt are assumed to be mutually uncorrelated. The process {xt : t = 1, 2, . . .} is constructed recursively using an initial random vector x0 and a time invariant law of motion: xt+1 = Axt + Cwt+1 , for t = 0, 1, . . . , (2.2.1) where A is an n by n matrix and C is an n by N matrix. Representation (2.2.1) will be a workhorse in this book. First, we will use (2.2.1) to model the information upon which economic agents base their decisions. Information will consist of variables that drive shocks to preferences and to technologies. Second, we shall specify the economic problems faced by the agents in our models and the economic process through which agents’ decisions 1 The phrase “J0 is generated by x0 ” means that J0 can be expressed as a measurable function of x0 . 2 Where φ1 and φ2 are information sets with φ1 ⊂ φ2 , and x is a random variable, the law of iterated expectations states that E (x | φ1) = E (E (x | φ2) | φ1) . Letting φ1 be the information set corresponding to no observations on any random variables, letting φ2 = Jt , and applying this law to the process {wt}, we obtain E wt+1 = E E wt+1 | Jt = E (0) = 0.
- 26. Prediction Theory 11 are coordinated (competitive equilibrium) so that the state of the economy has a representation of the form (2.2.1). 2.3. Prediction Theory A tractable theory of prediction is associated with (2.2.1). This theory is used extensively both in computing the equilibrium of the model and in repre- senting that equilibrium in the form of (2.2.1). The optimal forecast of xt+1 given current information is E (xt+1 | Jt) = Axt, (2.3.1) and the one-step-ahead forecast error is xt+1 − E (xt+1 | Jt) = Cwt+1. (2.3.2) The covariance matrix of xt+1 conditioned on Jt is just CC0 : E (xt+1 − E (xt+1 | Jt)) (xt+1 − E (xt+1 | Jt)) 0 = CC0 . (2.3.3) Sometimes we use a nonrecursive expression for xt as a function of x0, w1, w2, . . . , wt . Using (2.2.1) repeatedly, we obtain xt = Axt−1 + Cwt = A2 xt−2 + ACwt−1 + Cwt = ht−1 X τ=0 Aτ Cwt−τ i + At x0. (2.3.4) Representation (2.3.4) is one type of moving-average representation. It ex- presses {xt : t = 1, 2, . . .} as a linear function of current and past values of the building block process {wt : t = 1, 2, . . .} and an initial condition x0 . 3 3 Slutsky (1937) argued that business cycle fluctuations could be well modelled by moving average processes. Sims (1980) showed that a fruitful way to summarize correlations between time series is to calculate an impulse response function. In chapter 8, we study the relationship between the impulse response functions calculated by Sims (1980) and the impulse response function associated with (2.3.4).
- 27. 12 Stochastic Linear Difference Equations The moving average piece of representation (2.3.4) is often called an impulse response function. An impulse response function depicts the response of current and future values of {xt} to an imposition of a random shock wt . In represen- tation (2.3.4), the impulse response function is given by entries of the vector sequence {Aτ C : τ = 0, 1, . . .}.4 Shift (2.3.4) forward in time: xt+j = j−1 X s=0 As Cwt+j−s + Aj xt. (2.3.5) Projecting both sides of (2.3.5) on the information set {x0, wt, wt−1, . . . , w1} gives5 Etxt+j = Aj xt. (2.3.6) where Et(·) ≡ E[(·) | x0, wt, wt−1, . . . , w1] = E(·) | Jt , and xt is in Jt . Equation (2.3.6) gives the optimal j step ahead prediction. It is useful to obtain the covariance matrix of the j -step ahead prediction error xt+j − Etxt+j = j−1 X s=0 As Cwt−s+j (2.3.7) We have E(xt+j − Etxt+j) (xt+j − Etxt+j)0 = j−1 X k=0 Ak CC0 Ak0 ≡ vj (2.3.8a) Note that vj defined in (2.3.8a) can be calculated recursively via v1 = CC0 vj = CC0 + Avj−1A0 , j ≥ 2. (2.3.8b) The matrix vj is the covariance matrix of the errors in forecasting xt+j on the basis of time t information xt . To decompose these covariances into parts attributable to the individual components of wt , we let iτ be an N -dimensional 4 Given matrices A and C , the impulse response function can be calculated using the MATLAB program dimpulse.m. 5 For an elementary discussion of linear least squares projections, see Sargent (1987b, chapter IX).
- 28. Transforming Variables to Uncouple Dynamics 13 column vector of zeroes except in position τ , where there is a one. Define a matrix υj,τ by υj,τ = j−1 X k=0 Ak Ciτ i0 τ C0 A 0 k . (2.3.8c) Note that PN τ=1 iτ i0 τ = I , so that from (2.3.8a) and (2.3.8c) we have N X τ=1 υj,τ = υj. Evidently, the matrices {υj,τ , τ = 1, . . . , N} give an orthogonal decomposition of the covariance matrix of j -step ahead prediction errors into the parts at- tributable to each of the components τ = 1, . . . , N .6 The “innovation accounting” methods of Sims (1980) are based on (2.3.8). Sims recommends computing the matrices vj,τ in (2.3.8) for a sequence j = 0, 1, 2, . . .. This sequence represents the effects of components of the shock process wt on the covariance of j -step ahead prediction errors for each series in xt . 2.4. Transforming Variables to Uncouple Dynamics A convenient analytical device for the analysis of linear system (2.2.1) is to uncouple the dynamics using the distinct eigenvalues of the matrix A. We use the Jordan decomposition of the matrix A: A = TDT−1 , (2.4.1) where T is a nonsingular matrix and D is a matrix constructed as follows. Recall that the eigenvalues of A are the zeroes of the polynomial det (ζI − A). This polynomial has n zeroes because A is n by n. Not all of these zeroes are necessarily distinct, however.7 Suppose that there are m ≤ n distinct zeroes 6 For given matrices A and C , the matrices vj,τ and vj are calculated by the MATLAB program evardec.m. 7 In the case in which the eigenvalues of A are distinct, D is taken to be the diagonal matrix whose entries are the eigenvalues and T is the matrix of eigenvectors corresponding to those eigenvalues.
- 29. 14 Stochastic Linear Difference Equations of this polynomial, denoted δ1, δ2, . . . , δm . For each δj , we construct a matrix Dj that has the same dimension as the number of zeroes of det (ζI − A) that are equal to δj . The diagonal entries of Dj are δj and the entries in the single diagonal row above the main diagonal are all either zero or one. The remaining entries of Dj are zero. Then the matrix D is block diagonal with Dj in the jth diagonal block. Transform the state vector xt as follows: x∗ t = T−1 xt. (2.4.2) Substituting into (2.2.1), we have that x∗ t+1 = Dx∗ t + T−1 Cwt+1. (2.4.3) Since D is block diagonal, we can partition x∗ t according to the diagonal blocks of D or, equivalently, according to the distinct eigenvalues of A. In the law of motion (2.4.3), partition j of x∗ t+1 is linked only to partition j of x∗ t . In this sense, the dynamics of system (2.4.3) are uncoupled. To calculate multi-period forecasts and dynamic multipliers, we must raise the matrix A to integer powers (see (2.3.6)). It is straightforward to verify that Aτ = T(Dτ )T−1 . (2.4.4) Since D is block diagonal, Dτ is also block diagonal, where block j is just (Dj)τ . The matrix (Dj)τ is upper triangular with δτ j on the diagonal, with all entries of the kth upper right diagonal given by (δj)τ−k τ!/[k!(τ − k)!] for 0 ≤ k ≤ τ, (2.4.5) and zeroes elsewhere. Consequently, raising D to an integer power involves raising the eigenvalues to integer powers. Some of the eigenvalues of A may be complex. In this case, it is convenient to use the polar decomposition of the eigenvalues. Write eigenvalue δj in polar form as δj = ρj exp(iθj) = ρj[cos(θj) + i sin(θj)] (2.4.6) where ρj =| δj |. Then δτ j = (ρj)τ exp(iτθj) = (ρj)τ [cos(τθj) + i sin(τθj)]. (2.4.7)
- 30. Examples 15 We shall often assume that ρj is less than or equal to one, which rules out instability in the dynamics. Whenever ρj is strictly less than one, the term (ρj)τ decays to zero as τ → ∞. When θj is different from zero, eigenvalue j induces an oscillatory component with period (2π/ | θj |). 2.5. Examples Next we consider some examples of processes that can be accommodated by (2.2.1). 2.5.1. Deterministic seasonals We use (2.2.1) to represent the model yt = yt−4 . Let n = 4, C = 0, xt = (yt, yt−1, yt−2, yt−3)0 , x0 = (0 0 0 1)0 , A = 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 , C = 0 0 0 0 . (2.5.1) In this case the A matrix has four distinct eigenvalues and the absolute values of each of these eigenvalues is one. Two eigenvalues are real (1, −1) and two eigenvalues are imaginary (i, −i), and so have period four. The resulting sequence {xt : t = 1, 2, . . .} oscillates deterministically with period four. It can be used to model deterministic seasonals in quarterly time series.
- 31. 16 Stochastic Linear Difference Equations 2.5.2. Indeterministic seasonals We want to use (2.2.1) to represent the model yt = α4yt−4 + wt, (2.5.2) where wt is a martingale difference sequence and | α4 |≤ 1. We define xt = [yt, yt−1, yt−2, yt−3]0 , n = 4, A = 0 0 0 α4 1 0 0 0 0 1 0 0 0 0 1 0 , C = 1 0 0 0 . With these definitions, (2.2.1) represents (2.5.2). This model displays an “in- deterministic” seasonal. Realizations of (2.5.2) display recurrent, but aperiodic, seasonal fluctuations. 2.5.3. Univariate autoregressive processes We can use (2.2.1) to represent the model yt = α1yt−1 + α2yt−2 + α3yt−3 + α4yt−4 + wt, (2.5.3) where wt is a martingale difference sequence. We set n = 4, xt = [yt yt−1 yt−2 yt−3]0 , A = α1 α2 α3 α4 1 0 0 0 0 1 0 0 0 0 1 0 , C = 1 0 0 0 . The matrix A has the form of the companion matrix to the vector [α1 α2 α3 α4].
- 32. Examples 17 2.5.4. Vector autoregressions Reinterpret (2.5.3) as a vector process in which yt is a (k × 1) vector, αj a (k × k) matrix, and wt a k × 1 martingale difference sequence. Then (2.5.3) is termed a vector autoregression. To map this into (2.2.1), we set n = k · 4, A = α1 α2 α3 α4 I 0 0 0 0 I 0 0 0 0 I 0 , C = I 0 0 0 where I is the (k × k) identity matrix. 2.5.5. Polynomial time trends Let n = 2, x0 = [0 1]0 , and A = 1 1 0 1 , C = 0 0 . (2.5.4) Notice that D = A in the Jordan decomposition of A. It follows from (2.4.5) that At = 1 t 0 1 . (2.5.5) Hence xt = (t, 1)0 , so that the first component of xt is a linear time trend and the second component is a constant. It is also possible to use (2.2.1) to represent polynomial trends of any order. For instance, let n = 3, C = 0, x0 = (0, 0, 1)0 , and A = 1 1 0 0 1 1 0 0 1 . (2.5.6) Again, A = D in the Jordan decomposition of A. It follows from (2.4.5) that At = 1 t t(t − 1)/2 0 1 t 0 0 1 . (2.5.7) Then x0 t = [t(t−1)/2, t, 1], so that xt contains linear and quadratic time trends.
- 33. 18 Stochastic Linear Difference Equations 2.5.6. Martingales with drift We modify the linear time trend example by making C nonzero. Suppose that N is one and C0 = [1 0]. Since A = 1 1 0 1 and At = 1 t 0 1 , it follows that Aτ C = 1 0 . (2.5.8) Substituting into the moving-average representation (2.3.4), we obtain (2.25) x1t = t−1 X τ=0 wt−τ + [1 t]x0 where x1t is the first entry of xt . The first term on the right-hand side of the preceding equation is a cumulated sum of martingale differences, and is called a martingale, while the second term is a translated linear function of time. 2.5.7. Covariance stationary processes Next we consider specifications of x0 and A which imply that the first two moments of {xt : t = 1, 2, . . .} are replicated over time. Let A satisfy A = A11 A12 0 1 , (2.5.9) where A11 is an (n − 1) × (n − 1) matrix with eigenvalues that have moduli strictly less than one and A12 is an (n − 1) × 1 column vector. In addition, let C0 = [C0 1 0]. We partition x0 t = [x0 1t x0 2t] where x1t has n − 1 entries. It follows from (2.2.1) that x1t+1 = A11x1t + A12x2t + C1wt+1 (2.5.10) x2t+1 = x2t. (2.5.11) By construction, the second component, x2t , simply replicates itself over time. For convenience, take x20 = 1 so that x2t = 1 for t = 1, 2, . . .. We can use (2.5.10) to compute the first two moments of x1t . Let µt = Ex1t . Taking unconditional expectations on both sides of (2.5.10) gives µt+1 = A11µt + A12. (2.5.12)
- 34. Examples 19 We can solve the nonstochastic difference equation (2.5.12) for the stationary value of µt . Define µ as the stationary value of µt , and substitute µ for µt and µt+1 in (2.5.12). Solving for µ gives µ = (I − A11)−1 A12 . Therefore, if Ex10 = (I − A11)−1 A12, (2.5.13) then Ex1t will be constant over time and equal to the value on the right side of (2.5.13). Further, if the eigenvalues of A11 are less than unity in modulus, then starting from any initial value of µ0 , µt will converge to the stationary value (I − A11)−1 A12 . Next we use (2.5.10) to compute the unconditional covariances of xt . Sub- tracting (2.5.12) from (2.5.10) gives (x1t+1 − µt+1) = A11(x1t − µt) + C1wt+1 (2.5.14) From (2.5.14) it follows that (x1t+1 − µt+1)(x1t+1 − µt+1)0 = A11(x1t − µt)(x1t − µt)0 A0 11 + C1wt+1w0 t+1C0 1 + C1wt+1(x1t − µt)0 A0 11 + A11(x1t − µt)w0 t+1C0 1. The law of iterated expectations implies that wt+1 is orthogonal to (x1t − µt). Therefore, taking expectations on both sides of the above equation gives Vt+1 = A11VtA0 11 + C1C0 1, where Vt ≡ E(x1t − µt)(x1t − µt)0 . Evidently, the stationary value V of the covariance matrix Vt must satisfy V = A11V A0 11 + C1C0 1. (2.5.15) It is straightforward to verify that V is a solution of (2.5.15) if and only if V = ∞ X j=0 Aj 11C1C0 1Aj0 11. (2.5.16) The infinite sum (2.5.16) converges under the condition that the eigenvalues of A11 are less in modulus than unity.8 If the covariance matrix of x10 is V and 8 Equation (2.5.15) is known as the discrete Lyapunov equation. Given the matrices A11 and C1 , this equation is solved by the MATLAB program dlyap.m.
- 35. 20 Stochastic Linear Difference Equations the mean of x10 is (I −A11)−1 A12 , then the covariance and mean of x1t remain constant over time. In this case, the process is said to be covariance stationary. If the eigenvalues of A11 are all less than unity in modulus, then Vt → V as t → ∞, starting from any initial value V0 . From (2.3.8) and (2.5.16), notice that if all of the eigenvalues of A11 are less than unity in modulus, then limj→∞ vj = V . That is, the covariance matrix of j -step ahead forecast errors converges to the unconditional covariance matrix of x as the horizon j goes to infinity.9 The matrix V can be decomposed according to the contributions of each entry of the process {wt}. Let ιτ be an N -dimensional column vector of zeroes except in position τ , where there is a one. Then I = N X τ=1 ιτ ι0 τ . (2.5.17) Define a matrix Ṽτ Ṽτ ≡ ∞ X j=o (A11)j C1ιτ ι0 τ C0 1(A11)j0 (2.5.18) We have, by analogy to (2.5.15) and (2.5.16), that Ṽτ satisfies Ṽτ = A11Ṽτ A0 11+ C1iτ i0 τ C0 1 . In light of (2.5.17), (2.5.18), and (2.5.16) we have that V = N X τ=1 Ṽτ . (2.5.19) The matrix Ṽτ has the interpretation of being the contribution to V of the τth component of the process {wt : t = 1, 2, . . .}. Hence, (2.5.19) gives a decomposition of the covariance matrix V into the portions attributable to each of the underlying economic shocks. Next, consider the autocovariances of {xt : t = 1, 2, . . .}. From the law of iterated expectations, it follows that E[(x1t+τ − µ)(x1t − µ)0 ] = E{E[(x1t+τ − µ) | Jt](x1t − µ)0 } = E[Aτ 11(x1t − µ)(x1t − µ)0 ] = Aτ 11V. (2.5.20) 9 The doubling algorithm described in chapter 9 can be used to compute the solution of (2.5.15) via iterations that approximate (2.5.16). The algorithm is implemented in the MATLAB programs doublej.m and doublej2.m .
- 36. Examples 21 Notice that this expected cross-product or autocovariance does not depend on calendar time but only on the gap τ between the time indices.10 Indepen- dence of means, covariances, and autocovariances from calendar time defines covariance stationary processes. For the particular class of processes we are considering, if the covariance matrix does not depend on calendar time, then none of the autocovariance matrices does. 2.5.8. Multivariate ARMA processes Specification (2.2.1) assumes that xt contains all the information that is avail- able at time t to forecast xt+1 . In many applications, vector time series are modelled as multivariate autoregressive moving-average (ARMA) processes. Let yt be a vector stochastic process. An ARMA process {yt : t = 1, 2, . . .} has a representation of the form: yt = α1yt−1 + α2yt−2 + · · · + αkyt−k + γ0wt + γ1wt−1 + · · · + γkwt−k. (2.5.21) where E[wt | yt−1, yt−2, · · · yt−k+1, wt−1, wt−2, · · · wt−k+1] = 0. The require- ment that the same number of lags of y enter (2.5.21) as the number of lags of w is not restrictive because some coefficients can be set to zero. Hence we can think of k as being the greater of the two lag lengths. A representation such as (2.5.21) can be shown to satisfy (2.2.1). To see this, we define xt = yt α2yt−1 + α3yt−2 · · · + αkyt−k+1 + γ1wt + γ2wt−1 · · · + γk−1wt−k+2 + γkwt−k+1 α3yt−1 · · · + αkyt−k+2 + γ2wt · · · + γk−1wt−k+3 + γkwt−k+2 . . . αkyt−1 + γk−1wt + γkwt−1 γkwt (2.5.22) 10 Equation (2.5.20) shows that the matrix autocovariogram of x1t (i.e., Γτ ≡ E[(x1t+τ − µ)(x1t − µ)0] taken as a function of τ ) satisfies the nonrandom difference equation Γt+1 = A11Γt with initial condition Γ0 = V .
- 37. 22 Stochastic Linear Difference Equations C = γ0 γ1 . . . γk (2.5.23) and A = α1 I · · · 0 α2 0 · · · 0 . . . . . . ... . . . αk 0 · · · I 0 0 · · · 0 (2.5.24) It is straightforward to verify that the resulting process {xt : t = 1, 2, . . .} satisfies (2.2.1). 2.5.9. Prediction of a univariate first order ARMA Consider the special case of (2.5.21) yt = α1yt−1 + γ0wt + γ1wt−1 (2.5.25) where yt is a scalar stochastic process and wt is a scalar white noise. Assume that | α1 | 1 and that | γ1/γ0 | 1. Applying (2.5.22), we define the state xt as xt = yt γ1wt . Applying (2.5.23) and (2.5.24), we have C = γ0 γ1 , A = α1 1 0 0 . We can apply (2.3.6) to obtain a formula for the optimal j -step ahead prediction of yt . Using (2.3.6) in the present example gives Et yt+j γ1wt+j = αj 1 αj−1 1 0 0 yt γ1wt which implies that Etyt+j = αj 1yt + αj−1 1 γ1wt. (2.5.26)
- 38. Examples 23 We can use (2.5.26) to derive a famous formula of John F. Muth (1960). Assume that the system (2.5.25) has been operating forever, so that the initial time is infinitely far in the past. Then using the lag operator L, express (2.5.25) as (1 − α1L)yt = (γ0 + γ1L)wt. Solving for wt gives wt = γ−1 0 1 − α1L 1 + γ1 γ0 L yt, which expresses wt as a geometric distributed lag of current and past yt ’s. Substituting this expression for wt into (2.5.26) and rearranging gives Etyt+j = αj−1 1 h α1 + γ1 γ0 1 + γ1 γ0 L i yt. In the limiting case as α1 → 1 from below, this formula becomes Etyt+j = h 1 + γ1 γ0 1 + γ1 γ0 L i yt, (2.5.27) which is independent of the forecast horizon j . In the limiting case of α1 = 1, it is optimal to forecast yt for any horizon as a geometric distributed lag of past y’s. This is Muth’s finding that a univariate process whose first difference is a first order moving average is optimally forecast via an “adaptive expectations” scheme (i.e., a geometric distributed lag with the weights adding up to unity). 2.5.10. Growth In much of our analysis, we assume that the eigenvalues of A have absolute values less than or equal to one. We have seen that such a restriction still allows for polynomial growth. Geometric growth can also be accommodated by suitably scaling the state vector. For instance, suppose that {x+ t : t = 1, 2, . . .} satisfies: x+ t+1 = A+ x+ t + Cw+ t+1 (2.5.28) where E(w+ t+1 | Jt) = 0 and E[w+ t+1(w+ t+1)0 | Jt] = (ε)t I . The positive number ε can be bigger than one. The eigenvalues of A+ are assumed to have absolute values that are less than or equal to ε 1 2 , an assumption that we make to assure
- 39. 24 Stochastic Linear Difference Equations that the matrix A to be defined below has eigenvalues with modulus bounded above by unity. We transform variables as follows: xt = (ε)− t 2 x+ t (2.5.29) wt = (ε)− t 2 w+ t . (2.5.30) The transformed process {wt : t = 1, 2, . . .} is now conditionally homoskedastic as required because E[wt+1(wt+1)0 | Jt] = I . Furthermore, the transformed process {xt : t = 1, 2, . . .} satisfies (2.2.1) with A = ε− 1 2 A+ . The matrix A now satisfies the restriction that its eigenvalues are bounded in modulus by unity. The original process {x+ t : t = 1, 2, . . .} is allowed to grow over time at a rate of up to .5 log (ε). 2.5.11. A rational expectations model Consider a model in which a variable pt is related to a variable mt via pt = λEtpt+1 + γmt , 0 λ 1 (2.5.31) where mt = Gxt (2.5.32) and xt is governed by (2.2.1). In (2.5.31), Et(·) denotes E(·) | Jt . This is a rational expectations version of Cagan’s model of hyperinflation (here pt is the log of the price level and mt the log of the money supply) or a version of Le Roy and Porter’s and Shiller’s model of stock prices (here pt is the stock price and mt is the dividend). Recursions on (2.5.31) establish that a solution to (2.5.31) is pt = Etγ P∞ j=0 λj mt+j. Using (2.3.6) and (2.5.32) in this equation gives pt = γG P∞ j=0 λj Aj xt, or pt = γG(I − λA)−1 xt. Collecting our results, we have that (pt, mt) satisfies pt mt = γG(I − λA)−1 G xt xt+1 = Axt + Cwt+1. (2.5.33) System (2.5.33) embodies the cross-equation restrictions associated with ratio- nal expectations models: note that the same parameters in A, G that pin down
- 40. Computer Examples 25 the stochastic process for mt also enter the equation that determines pt as a function of the state xt . It is useful to show how to derive (2.5.33) using the method of undetermined coefficients. Returning to (2.5.31), we guess that a solution for pt is of the form pt = Hxt , where H is a matrix to be determined. Given this guess and (2.2.1), it follows that Etpt+1 = HEtxt+1 = HAxt. Substituting this and (2.5.32) into (2.5.31) gives Hxt = λHAxt + γGxt, which must hold for all realizations xt . This implies that H = λHA + γG or H = γG(I − λA)−1 , which agrees with (2.5.33). 2.6. The Spectral Density Matrix Let the mean vector of xt from the stationary distribution of an {xt} process be denoted µ. Define the autocovariance function of the {xt} process to be Cx(τ) = E[xt − µ] [xt−τ − µ]0 . The spectral density matrix of the {xt} process is defined as Sx(ω) = ∞ X τ=−∞ Cx(τ)e−iωτ . (2.6.1) Consider an {xt} process governed by (2.2.1), in which xt is partitioned as in equations (2.5.10),(2.5.11), so that x2t is the constant term. Then the spectral density can be represented as Sx(ω) = (I − A11e−iω )−1 C1C0 1 (I − A0 11e+iω )−1 . (2.6.2) From Sx(ω),11 the autocovariances can be recovered via the inversion formula Cx(τ) = 1 2π Z π −π Sx(ω)e+iωτ dω. (2.6.3) These formulas enable us to compute the spectral and cross-spectral statis- tics for any of the large variety of models that are special cases of (2.2.1). 11 The MATLAB program spectral.m can be used to compute a spectral density matrix. The program requires that the position of the constant term, denoted nnc, in xt be specified. The program then forms the appropriate matrices A11 and C1 in equations (2.5.10),(2.5.11), and applies formula (2.6.2).
- 41. 26 Stochastic Linear Difference Equations 2.7. Computer Examples We now use some MATLAB programs to generate examples that fit into the framework of this chapter. 2.7.1. Deterministic seasonal We can use the program dlsim.m to simulate the model of the deterministic seasonal described above. In using dlsim.m, we specify four matrices A, C, G, D whose dimensions must be comparable. In particular, we require that A be n × n, that C be n × k, that G be ` × n, and that D be ` × k. For the case of an indeterministic seasonal, we want to create the following matrices: A = 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 , C = 0 0 0 0 G =I, D = 0 0 0 0 To accomplish this, we use the following MATLAB code a= [0 0 0 1] A= compn (a) (This sets A equal to the companion matrix of a.) C = zeros(4, 1) (This sets C equal to a 4 × 1 matrix of zeros.) G = eye (4) (This sets G equal to the 4 × 4 identify matrix.) D = zeros(4, 1)
- 42. Computer Examples 27 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 12 14 16 18 20 * * * * * * * * * * * * * * * * * * * * Fig. 2.7.1.a. Deterministic Seasonal. -2 -1 0 1 2 3 4 0 5 10 15 20 25 30 35 40 Fig. 2.7.1.b. Indeterministic Seasonal with unit root. We want to simulate the system xt+1 = Axt + Cwt+1 yt = Gxt + Dwt+1 with an “input” of wt+1 ≡ 0. We form an input vector w of length 20 by the statement: w = zeros (20, 1) We set the initial condition by x0 = [1 0 0 0]0 To generate the simulation, we set y = dlsim(A, C, G, D, w, x0). This generates the 20 × 4 matrix y, the ith column of which is the time path taken by the ith state variable (remember that G = I and D = 0). We plot the time path of the first component of the state vector in Fig. 2.7.1.a and Fig. 2.7.1.b.
- 43. 28 Stochastic Linear Difference Equations 2.7.2. Indeterministic seasonal, unit root We implement a model of an indeterministic seasonal by altering the pre- ceding example by replacing w with a sequence of i.i.d. normal random variates. We specify that Ewt+1 = 0, Ew2 t+1 = 1. We accomplish this by the MATLAB phrase w = randn(150, 1) We have generated a white noise of length 150. We create the simulation by setting C = [1 0 0 0]0 y = dlsim (A, C, G, D, w, x0). We report the first component of y in figure 2. Note the tendency of the system to display explosive oscillations. We invite the reader to calculate the variance of x1t as a function of t. 2.7.3. Indeterministic seasonal, no unit root We now set a= [0 0 0 .7] A = compn(a) With all other matrices defined as in the preceding example, we form y = dlsim (A, C, G, D, w, x0) We plot the component x1t in figure 2.7.2. Notice that the explosive oscillations that were present in Fig. 2.7.1.b are no longer present.
- 44. Computer Examples 29 -3 -2 -1 0 1 2 3 0 5 10 15 20 25 30 35 40 Figure 2.7.2: Indeterministic seasonal with no unit root. 2.7.4. First order autoregression We want to simulate the first order autoregression x1t+1 = .9x1t + wt+1, where wt+1 is a normally distributed white noise with unit variance. We accom- plish this by modifying the MATLAB code of the previous example as follows: x0= [0 0 0 0]0 a= [.9 0 0 0] A= compn(a) y= dlsim(A, C, G, D, w, x0) Fig. 2.7.3.a graphs the first component of y, which is the process {x1t}.
- 45. 30 Stochastic Linear Difference Equations -5 -4 -3 -2 -1 0 1 2 3 4 5 0 20 40 60 80 100 120 140 160 Fig. 2.7.3.a. Simulation of first-order autoregression. -6 -4 -2 0 2 4 6 0 20 40 60 80 100 120 140 160 Fig. 2.7.3.b Simulation of second-order autoregression. 2.7.5. Second order autoregression We simulate the system x1t+1 = 1.2x1t − .3x1t−1 + wt+1 by modifying the code of the preceding example as follows: a= [1.2 − .3 0 0] A= compn (a) y= dlsim (A, C, G, D, w, x0) Fig. 2.7.3.b displays the output of {x1t}.
- 46. Computer Examples 31 2.7.6. Growth with homoskedastic noise We want to simulate the model x1t+1 = 1.025x1t + wt+1 where {wt+1} continues to be a normal white noise with unit variance. We set the initial condition as x10 = 5. We modify the MATLAB code of the preceding example as follows: x0= [5 0 0 0] a= [1.025 0 0 0 ] A= compn(a) y= dlsim(A, C, G, D, w, x0) Figure 2.7.4.a displays {x1t}. Notice the tendency for the randomness to die out, in the sense that the one-step ahead prediction error variance remains unity while the mean level of the process is growing exponentially at rate 1.025 per period. 0 50 100 150 200 250 300 0 20 40 60 80 100 120 140 160 Fig. 2.7.4.a. Growth with homoskedas- tic noise. 0 50 100 150 200 250 0 20 40 60 80 100 120 140 160 Fig. 2.7.4.b. Growth with heteroskedas- tic noise.
- 47. 32 Stochastic Linear Difference Equations 2.7.7. Growth with heteroskedastic noise To arrest the tendency of the relative uncertainty to die out in the previous example, we modify it by setting x1t+1 = 1.025x1t + w∗ t+1 where w∗ t = 1.025t/2 wt and where {wt} continues to be a normal white noise with unit variance. This specification makes the variance of w∗ t equal to (2.025)t . To simulate this model, we modify the code of the previous example as follows: n= 150 t= [1 : n]0 t= t./2 g= (1.025). ∧ t wg= w. ∗ g y= dlsim (A, C, G, D, wg, x0) Figure 2.7.4.b displays {x1t}. Notice that the randomness now fails to die out. 2.7.8. Second order vector autoregression We want to simulate the second order vector autoregression z1t+1 = .9z1t + .05z1t−1 + .05z2t + .01z2t−1 + w1t+1 z2t+1 = −.04z1t − .06z1t−1 + .75z2t − .1z2t−1 + w2t+1 where wt = [w1t, w2t]0 is a normally distributed vector white noise with identity covariance matrix. To simulate this system, we define xt = z1t z1t−1 z2t z2t−1
- 48. Computer Examples 33 We use the MATLAB code A= [.9 .05 .05 .01; 1 0 0 0; −.04 − .06 .75 − .1; 0 0 1 0] C= [1 0 0 0; 0 0 1 0] G= zeros (2, 4) G(1, 1)= 1 G(2, 3)= 1 w= randn (150, 2) D= zeros (2, 2) x0= zeros (4, 1) y= dlsim (A, C, G, D, w, x0) In figure Fig. 2.7.5, we plot the first and third columns of y, which equal {z1t,z2t}. -6 -4 -2 0 2 4 6 0 20 40 60 80 100 120 140 160 Figure 2.7.5: Simulation of second-order vector autoregres- sion. We can use the MATLAB command dimpulse to compute the impulse response of this system in response to each innovation (w1t, w2t). We employ the following code: i1 = dimpulse(A, C, G, D, 1, 20)
- 49. 34 Stochastic Linear Difference Equations This creates the response over twenty periods of the two variables z1t and z2t to the first innovation w1t . We also use i2 = dimpulse(A, C, G, D, 2, 20) This creates the response over twenty periods of the two variables z1t and z2t to the second innovation w2t . We display these impulse response functions in figure 2.7.6. -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 16 18 response of z1 response of z2 Fig. 2.7.6.a. Response to first innova- tion. -0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 16 18 response of z1 response of z2 Fig. 2.7.6.b. Response to second inno- vation. We can use the MATLAB file evardec.m to compute the decomposition of j -step ahead prediction error variances. If we want to compute this decomposi- tion for horizons j extending from 1 to 20, we use the code: [tab1, tab2] = evardec(A, C, G, −20, −1, eye(2)) The output in tab1 is a 20 × (1 + 2) table. The first column records the horizon j . For i = 1, 2, the (i + 1)th column records the diagonal element of vj corresponding to the ith variable zit . An orthogonal decomposition of these variances into the parts attributable to w1 and w2 is contained in tab2. The first column of tab2 records the horizon j , followed by two columns giving the
- 50. Computer Examples 35 diagonal element of the matrix vj,1 defined by (2.3.8) as the jth row element. Then j is repeated in the fourth column, followed by two columns giving the diagonal element of the matrix vj,2 . 2.7.9. A rational expectations model We now indicate how to simulate the model described by equations (2.5.33). We let the variable mt be generated by mt+1 = 1.2mt − .3mt−1 + wt+1. To implement this we set A= compn ([1.2 − .3]) C= [1 0]0 G1= [1 0] D= zeros(2, 1) We set γ = .5 and λ = .9. To implement formula (2.5.33) we set G2 = .5 ∗ G1/(eye (2) − .7 ∗ A) Then we set G = [G1; G2] To simulate the system we set x0= [1 0]0 w= rand(150, 1) y= dlsim(A, C, G, D, w, x0) The first column of y is the simulation for m, while the second is the simulation for p. We plot these in figure 2.7.7.a.
- 51. 36 Stochastic Linear Difference Equations -35 -30 -25 -20 -15 -10 -5 0 5 10 15 0 20 40 60 80 100 120 140 160 m p Fig. 2.7.7.a. Simulation of m and p. 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 0 2 4 6 8 10 12 14 16 18 response of m response of p Fig. 2.7.7.b. Impulse Response to in- novation in m. Figure 2.7.7.b gives the response of pt and mt to an innovation in money w1 . We compute this by using y = dimpulse(A, C, G, D, 1, 20) To obtain a representation of the solution (2.5.33) in the vector arma form d(L) pt mt = n1(L) n2(L) wt, we use the command [n, d] = ss2tf(A, C, G, D, 1) For our example, we obtain the output n = 0 1 0 0 3.0675 −.8282 d = [ 1 −1.2 .3 ] This output implies that our system has the representation (1 − 1.2L + .3L2 ) mt pt = 1 3.0675 −.8282L wt. Notice that the first row of this representation agrees with the process for mt that we assumed.
- 52. Conclusion 37 2.8. Conclusion In the following chapter we describe a class of economic structures with prices and quantities that can be represented in terms of a vector linear stochastic difference equation. In particular, the state of the economy xt will be repre- sented by a version of (2.2.1), while a vector yt containing various prices and quantities will simply be linear functions of the state, i.e., yt = Gxt . The rest of this book studies how the parameters of the matrices A, C, G can be inter- preted as functions of parameters that determine the preferences, technology, and information flows in the economy.
- 54. Chapter 3 The Economic Environment This chapter describes an economic environment with five key components: a sequence of information sets, laws of motion for taste and technology shocks, a technology for producing consumption goods, a technology for producing ser- vices from consumer durables and consumption purchases, and a preference ordering over consumption services. A particular economy is selected by spec- ifying a set of matrices A22, C2, Ub , and Ud that characterize the motion of information sets and of taste and technology shocks; matrices Φc, Φg, Φi, Γ, ∆k , and Θk that determine the technology for producing consumption goods; and matrices ∆h, Θh, Λ, and Π and a scalar β that determine the preference order- ing over consumption goods. This chapter describes and gives examples of each component of the economic environment. 3.1. Information Agents have a common information set at each date t. We use a vector mar- tingale difference sequence {wt : t = 1, 2, . . .} to construct the sequence of information sets {Jt : t = 0, 1, . . .}. The initial information set J0 is generated by a vector x0 0 = (h0 −1, k0 −1, z0 0) of initial conditions, each component of which will be described subsequently. The time t information set Jt is generated by x0, w1, w2, . . . , wt . We maintain: Assumption 1: E(wt | Jt−1) = 0 and E(wtw0 t | Jt−1) = I for t = 1, 2, . . .. – 39 –
- 55. Exploring the Variety of Random Documents with Different Content
- 56. O ho! said Honest Heinrich, that matter we shall easily settle! Presently he took up his fiddle to try its effect upon the Jew. One stroke of the bow, and the Jew began to wabble;—another stroke, and he lifted up his right leg;—a third stroke, and the dancing began in earnest. O dear me! cried the Jew, leave off that confounded fiddling! The thorns hurt me dreadfully! Upon my honour, I shall be a dead man before I am safely out of the thicket! But, Honest Heinrich was becoming warm with trying his newly-acquired instrument; so he only replied: Never mind the thorns; all right! and struck up a quicker tune. O torture! cried the perspiring dancer, I am a ruined man! Here,—here is my whole bag of money,—all genuine coins,— take it,—only cease that fiddling! Honest Heinrich made what musicians call a brilliant cadence, which caused the Jew to throw a few somersaults, and then gave the finishing stroke, or in other words, the concluding chord. The Jew crept out of the thicket, handed over the bag to the fiddler, and made off as rapidly as he could into the wide world. Honest Heinrich, on the other hand, took the direction towards the town with the intention of restoring the bag of money to its rightful owner. He was soon met by a man dressed in an unpretending kind of uniform, who, seeing the bag, in a friendly and almost playful way, gave Honest Heinrich a little tap on his shoulder, and said: You are wanted; you must come with me to town. Then Honest Heinrich was taken to prison; and when the judge asked him about the bag of money, and he replied, A Jew gave it me, the judge smiled and said, A Jew? you will never make me believe that! In short, Honest Heinrich was found guilty of robbery, and the judge sentenced him to be hanged. There prevailed a strange taste in the town where this occurred. Whenever an execution took place, the people had a kind of festival. Days, nay, even weeks, before the interesting event, the wretched culprit was considered almost as a martyr. Whatever he said was
- 57. carefully recorded, and made publicly known. Men of rank felt honoured when he shook hands with them; and when the awful hour for his execution had arrived, and he stood under the gallows, he would address the throng of people assembled as spectators. The women, of course, relished the exciting scene even more than the men, and cried with all their heart. Now, as Honest Heinrich was innocent, he did not like to have any fuss made about him; so, when he stood under the gallows, he only asked that he might be permitted to play a Last Farewell upon his dear fiddle. The judge said he would not deny the last request of a dying sinner. Pray, your worship! cried the Jew, who had mingled with the spectators, and who rejoiced in his heart at the turn which the money affair had taken, Pray, your worship, do not allow him his fiddle; his music will do us mischief! But the judge took no notice of the Jew, and said, Play, my lad, but make it short; we have not much time to lose. Then Honest Heinrich took his fiddle and played. One stroke with the bow, and all the people began to wabble. Another stroke, and every one lifted up his right leg. A third stroke, and the dancing began in earnest. The judge, the clergyman, the doctor, the hangman, the Jew, women with their babies in their arms, ladies with their smelling-bottles in their hands; in short, every one present, old and young, danced with the utmost exertion. Even the very dogs which had followed their masters, raised themselves upon their hind-legs and danced, profusely perspiring like all the people. Hold! stop! hold! cried the exhausted judge, Thy life is spared; only put aside that dreadful fiddle! As soon as Honest Heinrich heard the judge's promise of acquittal he ceased playing and came down the steps from the gallows. At the foot of the steps he found the Jew lying prostrate on his back. Confess directly, said Honest Heinrich, how you came by the bag of money, or I shall give you a little private performance, with a brilliant cadence at the end, you know! In a moment the alarmed Jew stood upon his legs again, and exclaimed, Upon my honour, I stole it!
- 58. Then they hanged the Jew upon the gallows. As for Honest Heinrich, he continued his wanderings in the world, and soon made his fortune. When he had become rich, he went home again to his village, and courted his neighbour's daughter, who had formerly jilted him when he was poor, but who loved him now dearly, not because he was rich (she said) but on account of his former poverty. Soon they married, and were happy ever after. THE POPE'S WIFE. There are several modifications current of the story of the Jew in the Thicket just told. A similar story which in olden time was popular in England, is given under the heading 'A Mery Geste of the Frere and the Boye,' in Ritson's Pieces of Ancient Popular Poetry, London, 1791. Again, a somewhat similar story is current in Greece. A lad has a flute given to him by some superhuman being. He goes to the market-place of the town, where piles of crockery are exhibited for sale. As soon as he begins to play, all the pots, jugs and basins fly about in the air and clash against each other until they are broken to pieces. The personage whom he compels to dance in the thorns is a priest.[74] Perhaps the most tragic incident of this kind is the sad fate of the Pope's wife, related by the Wallachians. It need scarcely be said that it does not concern the Pope of Rome, who, as everyone knows, has no wife. But in Wallachia the common village priest of the Greek Church is called Pope, and may marry. He generally avails himself of the permission. As regards Bakâla, whose music, as we shall presently see, killed the Pope's wife, various tricks of his are on record, which clearly show that he was a great fool, somewhat resembling the German Till Eulenspiegel, who had perhaps more happy ideas than many persons who have passed for wise.
- 59. Well, Bakâla, one fine day, took it into his head to ascend a high mountain, merely for pleasure, and for the sake of boasting. Arrived at the top of the mountain he was fortunate enough to make the acquaintance of a well-disposed spirit, who offered him a present from the clouds. The articles from which Bakâla was invited to select a keepsake looked mean and shabby, like those which people generally consign to the lumber-room. Bakâla, however, examined them carefully, and chose an old and dusty bagpipe; for he imagined, as some people are apt to do, that he was madly fond of music. Moreover, the sound of the bagpipe—this Bakâla soon discovered—had the power of making everyone dance. When Bakâla had come down from the mountain he engaged himself as shepherd to a village Pope in the valley. Every day he led the sheep into the fields, and blowing his bagpipe he made them caper and jump into the air like grasshoppers. And when, one morning, his master had sneaked out before him into the fields, and had hid himself in some bushes of sloes and dog-roses to watch his servant's strange proceedings, Bakâla made the Pope dance as well as his flock. The Pope was a soft-hearted sort of man. Quietness he loved above all things in the world; for its sake no sacrifice appeared to him too great. As to his wife, she was of a different disposition. To say the truth, she was just the reverse of her husband. She had more courage in her little finger than he had in all his limbs. His Yes was her No, and when he called a thing white she was sure to declare that she had long since found it to be very black indeed. Neither would she believe in the power of Bakâla's bagpipe. When the poor Pope, after his return from the sloes and dog-roses, showed her his tattered clothes and scratched limbs, all the sympathy he got from her was, Tush! tush! nonsense! If I were as soft-hearted as some people are said to be, I might perhaps pity you. Well, my dear, replied the cowed husband, you shall hear him to- night. I want to convince you——
- 60. Convince me? cried the Pope's wife: Fudge! I to be frightened by a bagpipe? Let him come on! Then the Pope thought that it was time to withdraw for the sake of quietness. But in the evening he took Bakâla aside, and desired him just to serenade their mistress for a little while under the window. Before Bakâla commenced playing the Pope sat down on the ground and bound two heavy stones to his feet by way of precaution, while his wife busied herself in the upper story of the house. No sooner had Bakâla begun his performance than she danced so furiously that she made the whole house shake. Bakâla played faster and faster; her stamping grew louder and louder. She danced until she had actually stamped a hole in the floor, through which she descended into the lower story. The Pope peeped into the room; and when he saw what had happened he felt sorry, and he beckoned Bakâla to leave off playing. But, alas! he beckoned too late! The poor lady had danced herself to death. Now, one might have thought the Pope would have dismissed Bakâla, telling him that his services were not any further required. But this is just precisely what he did not do. On the contrary, he kept Bakâla in his service, and treated him even better than before.[75] THE TWO HUNCHBACKS. The story of the two Hunchbacks is widely diffused. It is told in Ireland as well as in Germany and Italy; moreover it is said to be also current in Spain. There are, of course, many varieties of it in these countries. Compare, for instance, the Irish narrative of Lusmore, in 'Fairy Legends and Traditions of the South of Ireland, by T. Crofton Croker,' with the one given here, which has been obtained from the country people in Rhenish Prussia.
- 61. On St. Matthew's day, in the year 1549, a poor hump-backed musician was returning late at night to Aachen[76] from a village where he had been playing at a wedding. Being in a half drowsy state, he took but little heed of time or place, and so he passed the Minster without concerning himself about anything particularly, just as the large clock in the tower boomed midnight. The sound startled him, especially as at the same time there arose in the air a strange whirring like the unearthly sound of owls and bats on the wing. It now occurred to him that this was the night of quarter-day, and he quickened his steps to escape the terrors of the ghost's hour and of apparitions. Nervously he turned into the Schmiedstrasse (Smith- street) as the nearest way to his home, which was in the Jakobstrasse (James-street). But on reaching the Fish Market,—what did he see! All the stalls glistened with innumerable lights, and about them were seated a large party of richly-dressed ladies, feasting on dainty viands served in golden and silver dishes, and drinking sparkling wine from crystal goblets. The musician, much frightened, endeavoured to hide himself in a corner; for, he had not the least doubt that he saw an assemblage of witches. But it was too late; one of the ladies nearest him had already observed him, and she conducted him to the table. Don't be frightened! said the lady to the musician, who stood before her with chattering teeth and trembling knees: Don't be frightened; but, play us some merry tunes, and thou shalt be paid for it. The poor hunchback had no choice but to take up his violin, and to amuse the strange company as long as they pleased. Having quickly set aside the stalls with everything upon them, the witches—among whom the poor hunchback thought he recognised several ladies of high position from the town—whirled round in pairs to the sound of his fiddle. But the strangest thing was that the longer the fellow continued to play, the finer and fuller his performance appeared to him; so that he really thought he must be either dreaming, or there
- 62. must be a whole band of violins and flutes placed behind him which joined in his performance. Now the Minster clock struck a quarter to one; all the dancers instantaneously stopped, visibly exhausted, and everything was reinstated in its former order. Hesitating, the musician looked on, uncertain whether he ought to stay any longer, or whether he might go; when the lady who had engaged his services came up to him and said: 'Brave musician! thou hast done thy work to our content, and shalt now receive thy recompense. While saying the words she pulled off his jacket, and, before he was aware of it, she had slipped behind him, and at one grasp relieved him of his hump. Who so happy as the disburthened fiddler? In thankfulness he was just going to throw himself on his knees before his benefactress,—when the clock struck One, and in a moment, ladies, lights, and dishes were gone, and the musician found himself at dark night standing alone in the middle of the Fish Market. Bewildered, he put his hand to his back, doubting lest the adventure had been merely a confused dream. But, no; it was reality! The hump was gone, and the happy fellow rejoiced in feeling as upright as man can be. Moreover, his joy was still increased when he took up his jacket, which lay before him on the ground. Perceiving it to be unaccountably heavy, and thrusting his hands into the pockets to ascertain the cause, he found that both pockets were filled with money. Doubly happy, he hastened home, and in thankfulness he made the next morning an offering of his fiddle to his Patron Saint, under whose image in the church he hung it as a glorious relic to be venerated by his children and his children's children for ever. Now, the marvellous affair created, as may easily be understood, an immense sensation in the town. People went to the church to look at the fiddle; and whenever the lucky musician showed himself in public, a knot of curious idlers hovered around him, anxious to get a peep at his back. Moreover, his good fortune, as may likewise be easily understood, aroused the envy of his rivals in his profession.
- 63. The most envious of these professional brothers possessed himself a tolerably respectable hump, which annoyed him all the more, since he was not less vain than envious. His estimation of his personal appearance was, however, exceeded by that of his musical accomplishments. How surprised they will be! said he to himself: If that wretched scraper could please them, I am sure I have only to treat them with a few of my inimitable flourishes, and I shall be a straight man and a man of property in no time! It was at midnight of St. Gerhard's day when the vain virtuoso repaired to the Fish Market. The old clock of the Minster had already boomed the last stroke announcing the twelfth hour, when he arrived at the place. He actually found there a large party of ladies, just as he expected, and they invited him to play. Confidently he stepped forward, and having bowed with a smile which he was wont to assume whenever he appeared before the public, he threw his fiddlestick across the strings and extemporized a few rapid passages up and down, to show at once his superior skill. But, how wretchedly provoking! Never in his life had he produced such miserable tones; they sounded so execrably thin and poor, as if the strings had been stretched over a piece of solid wood instead of a violin. Enraged, he renewed his exertions, but only to render the matter worse; for, now he produced a noise so horribly ear-piercing that he thought there must be standing behind him a whole chorus of whistling and screeching sneerers accompanying his performance. Highly exasperated, he tucked his violin under his arm, and walked up to the dancing witches. Then boldly addressing one of the richly- attired ladies, in whom he believed he recognised the wife of the burgomaster of the town, he said:— Ah, Madam! I wonder what your husband, our respected burgomaster would say if he knew of your night-excursions on the broom-stick! But that is your own affair. All I care for is my due reward, if you please.
- 64. With these words he threw off his jacket and turned round. The lady quickly uncovered a silver dish, from which she took the hump of the former musician, and before the vain virtuoso was aware of it, she had pressed it on his back beside the other hump. The clock had struck One, and the witches were already on their broom-sticks riding through the air homewards, when the musician recovered from his shock. He slowly put his hand to his back, hoping that perchance he might only have had a bad dream. But no! it was all right,—or rather all wrong. There remained now nothing for him to do but to take up his jacket and make the best of his way home. But the jacket felt so unusually heavy;—could there, perhaps, be gold in it to make up in some measure for the cruel infliction? Eagerly he rummaged the pockets; but what should he find? A few heavy stones and rubbish.[77] THE PARSON'S ADVICE. This tale of the Manx people is almost literally copied from 'The History and Description of the Isle of Man, by George Waldron, London, 1744.' A man, one day, was led by invisible musicians for several miles together; and not being able to resist the harmony, followed till it conducted him to a large common, where a great number of people were sitting round a table, and eating and drinking in a very jovial manner. Among them were some faces which he thought he had formerly seen; but he forbore taking any notice, or they to him; till, the little people offering him drink, one of them whose features seemed not unknown to him, plucked him by the coat, and forbade him, whatever he did, to taste anything he saw before him. 'For, if you do,' added he, 'you will be as I am, and return no more to your family.'
- 65. The poor man was much affrighted, but resolved to obey the injunction. Accordingly, a large silver cup, filled with some sort of liquor, being put into his hand, he found an opportunity to throw what it contained on the ground. Soon after, the music ceasing, all the company disappeared, leaving the cup in his hand; and he returned home, though much wearied and fatigued. He went the next day and communicated to the minister of the parish all that had happened, and asked his advice how he should dispose of the cup: To which the parson replied, he could not do better than devote it to the service of the church. And this very cup, they say, is that which is now used for the consecrated wine in Kirk Merlugh. RELICS OF THE GOBLINS. The old tradition embodied in the preceding story from the Isle of Man, is also current,—with various modifications,—in the north of Germany, in Denmark, and in Sweden. Afzelius, in his interesting account of Swedish popular superstitions, mentions some curious notions on this subject. The country people in Sweden still preserve an old belief that if a person drinks of the contents of a beaker, offered to him by the goblins inhabiting the mountains, he loses all recollection of the past, and must become one of them. Several cups are said to have been purloined from these mysterious beings by persons who stealthily avoided partaking of the proffered liquor. Some are still shown in churches, to which they were presented by the purloiners; and it is asserted that these oddly-shaped vessels were formerly used in the Communion Service. The goblins in Sweden have their principal meetings at midnight before Christmas, and their amusements consist chiefly in music and dancing. They generally assemble in those isolated spots among the mountains where are found large stones resting on pillars, around which they delight to dance. It is considered decidedly dangerous to encounter them at their pastimes on Christmas Eve.
- 66. Many years ago,—some say it was so far back as in the year 1490,— a farmer's wife in Sweden, whose name was Cissela Ulftand, distinctly heard, on Christmas Eve, the wild music of the goblins who had assembled not far from her house. The farm in which the good woman lived is called Ljungby, and the group of curiously-placed stones around which the goblins had congregated is well known to many people; indeed, almost everyone in Sweden knows the Magle- Stone. Well, when Mistress Ulftand heard the music, she spoke to one of her farm-servants, a strong and daring young fellow, and induced him to saddle a horse and to ride in the direction of the Magle- Stone, that he might learn something about the mysterious people, and tell her afterwards all he had seen. The lad rather liked the adventure; he lost no time in mounting his horse, and was soon galloping towards the scene of the music and rejoicing. In approaching the Magle-Stone, he somewhat slackened his speed; however, he drew quite near to the dancers. After he had been gazing a little while at the strange party, a handsome damsel came up to him and handed him a drinking-horn and a pipe, with the request that he would first drink the health of the King and then blow the pipe. The lad accepted both, the drinking-horn and the pipe; but, as soon as he had them in his hands, he poured out the contents of the horn, and spurring his horse he gallopped off over hedges and ditches straight homewards. The whole company of goblins followed him in the wildest uproar, threatening and imploring him to restore to them their property; but the fellow proved too quick for them, and succeeded in safely reaching the farm, where he delivered up the trophies of his daring enterprise to his mistress. The goblins now promised all manner of good luck to the farmer's wife and her family, if she would return to them the two articles; but she kept them, and they are still preserved in Ljungby as a testimony to the truth of this wonderful narrative.
- 67. The drinking-horn is of a metallic composition, the nature of which has not been exactly ascertained; its ornaments are, however, of brass. The pipe is made of the bone of a horse. Moreover, the possession of these relics, we are told, has been the cause of a series of disasters to the owners of the farm. The lad who brought them to the house died three days after the daring enterprise, and the day following, the horse suddenly fell down and expired. The farm-house has twice burnt down, and the descendants of the farmer's wife have experienced all kinds of misfortunes, which to enumerate would be not less laborious than painful. It is only surprising that they should still keep the unlucky horn and pipe. THE GOLDEN HARVEST. This is a genuine Dutch story. A long time may have elapsed since the hero of the event recorded was gathered to his fathers. Howbeit, his name lives, and his deeds will perhaps be longer retained by the people in pleasant remembrance than the deeds of some heroes who have made more noise in the world. An old village crowder, whose name was Kartof, and who lived in Niederbrakel, happened once, late in the night, to traverse a little wood on his way home from Opbrakel, where he had been playing at a dance during the wake. He had his pockets full of coppers, and felt altogether mighty comfortable and jolly; for the young folks in Opbrakel had treated him well, and the liquor was genuine Old Hollands. But, there is nothing complete in this world, as the saying is, and as old Kartof was presently to experience to his dismay, when he put his hand into his pocket for his match-box. Had he not just filled his old clay pipe in the pleasant expectation, amounting to a certainty, that he should indulge in a comfortable smoke all the way home? And did he not feel, with a certain pride, that he deserved a good smoke after all his exertions with the fiddlestick? But what use
- 68. was it to rummage his pockets for the match-box! It certainly was not there, and must have been lost or left behind somewhere. The deuce! muttered old Kartof, If I had only a bit of fire now to light my pipe, I should not care for anything else in the world, I am sure! Scarcely had he said these words, when he espied a light gleaming through the bushes. He went towards it, but it was much further off than it at first appeared to him; indeed, he had to go more than a hundred yards into the brush-wood before he came up to it. He now saw that it was a large fagot burning, around which a party of men and women, joined hand in hand, were dancing in a circle. How odd! thought old Kartof; but being a man accustomed to genteel society, he was at no loss how to address them politely; so, taking off his hat, he said:— Ladies and Gentlemen! Excuse me. I hope I am not intruding too much if I ask the favour of your permission to help myself to a little fire to light my pipe. He had not even quite finished his speech, when several of the dancers stepped forward and handed him glowing embers in abundance. Now, when approaching him they perceived that he carried a violin under his arm, they importuned him to play for them to dance, intimating that he should be well rewarded for his services. Why not? said old Kartof: It is only about midnight, and I can sleep to-morrow in the day-time; it will not be the first time that I have gone to bed in the morning. While talking in this way, he tuned his instrument; and soon he struck up his best tunes, one after the other. But, though he played ever so much, he could never play enough, the dancers were so insatiable! Whenever his arm sank down from sheer fatigue, they threw a golden ducat into the sound-hole of his violin, which pleased him immensely, and always animated him to renew his exertions, especially also as they did not neglect to refresh him occasionally
- 69. with a remarkably fine-flavoured Schiedam, from a bottle so oddly- shaped that he had never seen anything like it, so funny it was. He could not help smiling whenever he looked at the bottle. Gradually his violin became heavier—of course, that was from the golden ducats which the dancers continually threw into it. But also his arm became heavier, and at last old Kartof felt altogether too heavy, sank softly down, and fell asleep. How long he lay in this state no one knows, nor is ever likely to know. But, thus much is certain, when old Kartof awoke the day was already far advanced, and the sun shone brightly upon his face. He rubbed his eyes and looked about, doubtful whether he was a man of property or whether he had only dreamt of golden ducats. There was the violin lying in the grass near his feet. He hastily took it up;— it felt as light as usual. He shook it;—no rattling of ducats. He held it before his face and peeped into the sound holes;—to be sure, there was something in it, yellow and glittering like gold. He shook it out on the grass;—what should it be?—a score or two of decayed yellow birch-leaves. Disappointed, old Kartof rose to his feet to look around whether he could not find the place where the fire had been. Yes, there it was! Some embers were still glimmering in the ashes. This appeared to him more odd than anything else he had experienced. But old Kartof, after all, took the matter quietly enough. He lighted his pipe, and taking up his violin set out on his way home, resolving as he went never to go to that confounded place again after twelve o'clock at midnight.[78] GIPSIES. There prevails in popular traditions much mystery respecting gipsies. No wonder that this should be the case, since these strange
- 70. vagabonds are in most countries so very different from the inhabitants in their appearance and habits; and their occupations are often so well calculated to appeal to the imagination of superstitious people, that a gipsy is regarded by them almost as a sorcerer. His better-half not unfrequently pretends to be a soothsayer, and he is often a musician. However different the gipsy hordes which rove about in European countries may be from each other in some respects, they are all fond of music, magic, and mysterious pursuits. Among the gipsy bands in Hungary and Transylvania talented instrumental performers are by no means rare; and in Russia, the gipsy singers of Moscow enjoy a wide reputation for their musical accomplishments. It is told,—not as a myth but as a fact,—that when the celebrated Italian singer Signora Catalani heard in Moscow the most accomplished of the gipsy singing-girls of that town, she was so highly delighted with the performance that she took from her shoulders a splendid Cashmere shawl which the Pope had presented to her in admiration of her own talent, and embracing the dear gipsy girl, she insisted on her accepting the shawl, saying that it was intended for the matchless cantatrice which she now found she could not longer regard herself. There is a wildness in the gipsy musical performances, which admirably expresses the characteristic features of these vagrants. Indeed theirs is just the sort of music which people ought to make who encamp in the open air, feed upon hedgehogs and whatever they can lay hand on, and profess to be adepts in sorcery and prophecy. The following event is told by the peasants in the Netherlands as having occurred in Herzeele. A troop of gipsies had arrived in a valley near that place. They stretched a tight rope, on which they danced, springing sometimes into the air so high that all who saw it were greatly astonished. A little boy among the spectators cried: Oh, if I could but do that!— Nothing is easier, said an old gipsy who stood near him: Here is a powder; when you have swallowed it, you will be able to dance as
- 71. well as any of us. The boy took the powder and swallowed it. In a moment his feet became so light that he found it impossible to keep them on the ground. The slightest movement which he made raised him into the air. He danced upon the ears of the growing corn, on the tops of the trees,—yea, even on the weather-cock of the church-tower. The people of the village thought this suspicious, and shook their heads, especially when they furthermore observed a disinclination in the boy to attend church. They, therefore, consulted with the parson about the boy. The parson sent for him, and got him all right on his legs again by means of exorcism; but it was a hard struggle to banish the potent effects of the gipsy's powder.[79] The gipsies were formerly supposed to be descendants of the ancient Egyptians. The German peasants call them Taters,[80] a name indicating an Asiatic origin; and it has been ascertained that they migrated from Western India. The roving Nautch-people in Hindustan are similarly musical and mysterious. THE NAUTCH-PEOPLE. The Nautch-people in Hindustan are not only singers and dancers who exhibit their skill before those who care to admire and to reward them; but they possess also dangerous charms. In a popular story of the Hindus, called 'Chandra's Vengeance' we are told of a youth who, on hearing the music of the Nautch-people at a great distance, is irresistibly compelled to traverse the jungle in search of them. When, after twelve days' anxious endeavour to reach them, he discovers their encampment, Moulee, the daughter of the chief Nautch-woman, approaches him singing and dancing, and throws to him the garland of flowers which she wears on her head. He feels spell-bound, and the Nautch-people offer him a drink which, as soon as he has tasted it, makes him totally forget his
- 72. family and his dear home. So he remains with the Nautch-people, and wanders with them about the country as one of the company. Again, in a Hindu story called 'Panch-Phul Ranee,' a Rajah, or King, is enchanted by the Nautch-people, so that he finds his happiness in roving with them from place to place, and in beating the drum for the dancers. His enchantment is accomplished in this way: He had set out on a journey, leaving his wife and infant son behind. One day he happened to fall in with a gang of Nautch-people, singing and dancing. He was a remarkably handsome man, and the Nautch- people, on seeing him approach, said to each other How well he would look beating the drum for the dancers! The Rajah was hungry and told them that he required some food; whereupon one of the women offered him a little rice, upon which her companions threw a certain powder. He ate it, and the effect was that it made him forget his wife, child, rank, journey, and whatever had happened to him in all his life. He willingly remained with the Nautch-people, and wandered about with them, beating the drum at their performances, full eighteen years. His son, the prince, being now grown up, could no longer be detained from setting out in the world in search of his beloved father. After many fruitless attempts the prince discovered his father among the Nautch people,—a wild, ragged-looking man whose business it was to beat the drum. The joyful prince summoned the wisest doctors in the kingdom to restore the Rajah to his former consciousness; but their exertions did not at first prove at all successful. In vain did they assure the old drummer that he was a Rajah, and that he ought to remember his former greatness and splendour. The old man always answered that he remembered nothing but how to beat the drum; and, to prove his assertion, he treated them on the spot with a tap and roll on his tom-tom. He really believed that he had beaten it all his life. However, through the unabated exertions of the doctors, a slight remembrance came gradually over him; and by-and-by his former mental power returned. He now recollected that he had a wife and a son. He also recognized his old friends and servants. Having
- 73. reseated himself on the throne, he governed as if nothing had ever occurred to interrupt his reign.[81] THE MONK OF AFFLIGHEM. The aim of the present series of popular stories demands that some notice should now be taken of such musical legends as breathe a thorough Christian spirit. Several of these are, as might be expected, very beautiful; but they are familiar to most readers. One or two which are less well known may, however, find a place here. The legend of the Monk of Afflighem bears some resemblance to the beautiful tradition of the Seven Sleepers. If it fails to interest the reader, the cause must be assigned to the simple manner in which it is told rather than to the subject itself. Towards the end of the eleventh century occurred in the Abbey of Afflighem, in Dendermonde, East Flanders, a most wonderful event, the pious Fulgentius being at that time the Abbot of the monastery. One day, a monk of very venerable appearance, whom no one remembered to have seen before, knocked at the door of the monastery, announcing himself as one of the brotherhood. The pious Abbot Fulgentius asked him his name, and from what country he had come. Whereupon the monk looked at the Abbot with surprise, and said that he belonged to the house. Being further questioned, he replied that he had only been away for a few hours. He had been singing the Matins, he said, in the morning of the same day in the choir with the other brothers. When, in chanting, they came to the verse of the ninetieth psalm, which says: For, a thousand years in thy sight are but as yesterday! he pondered upon it so deeply that he did not perceive when the singers left the choir, and he remained sitting alone, absorbed by the words. After he had been a while in this state of reflection, he heard heavenly strains of music, and on looking up he saw a little bird which sang with a voice so
- 74. enchantingly melodious that he arose in ecstasy. The little bird flew to the neighbouring wood, whither he followed it. He had been only a little while in the wood listening to the heavenly song of the bird; and now, in coming back he felt bewildered,—the appearance of the neighbourhood was so changed he scarcely knew it again. When the pious Abbot Fulgentius heard the monk speak thus, he asked of him the name of the Abbot, and also the name of the King who governed the country. And after the monk had answered him and mentioned the names, it was found to the astonishment of all that these were the names of the Abbot and the King who had lived three hundred years ago. The monk startled, lifted up his eyes, and said: Now indeed I see that a thousand years are but as one day before the Lord. Whereupon he asked the pious Abbot Fulgentius to administer to him the Holy Sacraments; and having devoutly received them, he expired.[82] THE PLAGUE IN GOLDBERG. The inhabitants of Goldberg, a town in Germany, observe an old custom of inaugurating Christmas, which is peculiar to themselves. Having attended divine service, which commences at midnight on Christmas Eve, they assemble at two o'clock to form a procession to the Niederring, a hill situated close to the town. When the procession has arrived at the top of the Niederring, old and young unite in singing the Chorale Uns ist ein Kindlein heut geboren (For us this day a child is born). As soon as this impressive act of devotion is concluded, the town band stationed in the tower of the old parish church performs on brass instruments the noble Chorale Allein Gott in der Höh sei Ehr (All glory be to God on High), which in the stillness of the night is heard over the whole town, and even in the neighbouring villages. The origin of this annual observance dates from the time when the town of Goldberg was visited by a deadly plague called Der schwarze
- 75. Tod (The black Death). According to some accounts the awful visitation occurred in the year 1553; at all events this date appears to have been assigned to it on an old slab embedded in the wall of the parish church of Goldberg; but the inscription has become so much obliterated in the course of time, that no one can make out the year with certainty. Thus much, however, is declared by all to be authentic: The plague spread throughout the town with frightful rapidity. The people died in their houses, in the streets, everywhere, at night, and in the day-time. Some, while at their work, suddenly were stricken and fell down dead. Some died while at their meals; others while at prayers; others in their endeavours to escape the scourge by hastening away from the doomed town. Indeed, it was as if the Angel of Death had stretched out his hand over the place, saying Ye are all given up to me! The plague raged for some weeks, and then quietness reigned in Goldberg. The few survivors had shut themselves up solitarily in their houses, not knowing of each other; for, no one now ventured into the street; neither did anyone open a window, fearing the poisonous air; for the corpses were lying about, and there remained none living to bury the dead. Such was the condition of Goldberg in the month of December, just before Christmas. On Christmas Eve one of the solitary survivors, deeply impressed with the import of the holy festival, attained the blessing of a firm trust in the wisdom of the inscrutable decrees of Providence. He thought of the happy time of his childhood when his parents lighted up for him the glorious Christmas tree; and this recalled to his mind the simple and impressive Christmas hymn which his mother had taught him to recite on the occasion. Strengthened by devout contemplation, he ventured to open the window. The night was beautiful, and the air wafted to him so pure and delicious that he resolved to leave his prison. At the second hour after midnight he went out of the house, and bent his steps through the desolated streets towards the Niederring. Arrived at the top of
- 76. the hill he knelt down and sang from the depth of his heart the Christmas hymn. His voice was heard by another solitary survivor, who perceiving that he was not, as he had supposed, the only person still living in Goldberg, gained courage and likewise from his hiding place repaired to the Niederring, and kneeling down joined the singer with sincere devotion. Soon a third person made his appearance, slowly drawing near like one risen from the grave. Then a fourth, a fifth, until the number of them amounted to twenty-five; and these were all the inhabitants of Goldberg who had escaped the ravages of the Black Death.
- 77. [Listen] Uns ist ein Kind-lein heut ge-born, Gott mit uns! Von ein'r Jung-frau aus-er- korn. Gott mit uns! Gott mit uns! Wer will seyn wi-der uns! The Christmas Chorale sung in the refreshing mountain air wonderfully invigorated their desponding spirits. They arose and solemnly vowed henceforth to unite in Christian fellowship, with reliance upon the wisdom of the divine ordinances. The next day they buried their dead; and when their vow became known in the neighbourhood, many good people were drawn to Goldberg. The town soon revived, and prospered more than ever.
- 78. The inhabitants have not forgotten the visitation which befel their forefathers, but remember it in humiliation; and this is a lasting blessing.[83] FICTIONS AND FACTS. Knowledge is, of course, to superstition as light is to darkness; still, some nations endowed with a lively imagination, although they are much advanced in mental development, cling to the superstitions of their forefathers, since the superstitions accord with their poetical conceptions, or are endeared to them by associations which pleasantly engage the imaginative faculties. Besides, in countries where the inhabitants frequently witness grand and awful natural phenomena, their poetical conceptions are likely to be more or less nourished by these impressive occurrences, however well acquainted they may be with their natural causes. It is therefore not surprising that many superstitious notions, such as have been recorded in the preceding stories, should be found in civilized nations. Moreover, in some countries, a more careful research into the old traditions harbouring among the uneducated classes of the people has been made, than in other countries. It would, therefore, be hasty, from the sources at present accessible, to judge of the degree of mental development attained by individual nations. The Germans are not less rational than the English; nevertheless, a far greater number of Fairy Tales have been collected in Germany than in England. An enquiry into the musical traditions of the different European races is likely to increase in interest the more we turn to the mythological conceptions originally derived from Central Asia, and dispersed throughout Europe at a period on which history is silent, but upon
- 79. which some light has been thrown by recent philological and ethnological researches. A word remains to be said on the musical myths of modern date. We read in the biographies of our celebrated musicians facts which would almost certainly be regarded as fictions, were they not well authenticated. On the other hand, it would not be difficult to point out modern myths referring to the art of music. Tempting as it might be to cite the most remarkable examples of this kind, and anecdotes relating to musicians in which fiction is strangely mingled with fact, it is unnecessary to notice them here; for, are they not written in our works on the history of the art and science of music?
- 80. DRAMATIC MUSIC OF UNCIVILIZED RACES. The first music of a dramatic kind originated probably in the passion of love. Savages, unacquainted with any other dramatic performances, not unfrequently have dances representing courtship, and songs to which these dances are executed. However rude the exhibitions may be, and however inartistic the songs may appear,— which, in fact, generally consist merely of short phrases constantly repeated, and perhaps interspersed with some brutish utterances,— they may nevertheless be regarded as representing the germ from which the opera has gradually been developed. Dancing is not necessarily associated with dramatic music; the dances of nations in a low degree of civilization are, however, often representations of desires or events rather than unmeaning jumps and evolutions. Even in the popular dances of nations in an advanced state of civilization love is generally the most attractive subject for exhibition by action and music. The Italian national dances,—the Saltarello, the Monferrino, and several others,—have an unmistakable meaning; or, as Mac Farlane says, there is a story in them which at times is told in a very broad, significant, and unsophistical way. The story is a sort of primitive courtship, varied by the coyness or coquetry of the female dancer, and animated by the passion and impatience of the
- 81. wooer.[84] The same may be said of the Spanish Bolero and Fandango. The excitement of the chase appears to be another cause of the origin of dramatic music. The savage, in pursuing the animals which he requires for his subsistence, experiences successes and disappointments which are to him highly interesting, and the recollection of which he enjoys. He naturally feels proud of results which he could not have achieved without agility and shrewdness, and he delights in showing to his friends how he proceeded in accomplishing his feat. Besides, savages have a strong instinct for imitation, almost like monkeys. Hence their fancy for counterfeiting the habits of certain animals which they chase and with the peculiarities of which they are generally well acquainted. The aborigines of Australia have a dance in which they imitate the movements of the Kangaroo. The women sing, and produce a rhythmical accompaniment by beating two pieces of wood together; while the men, who represent the Kangaroos, produce sounds peculiar to these animals. The North American Indians have an Eagle Dance, a Bear Dance, and even a Dog Dance. The natives of Kamtschatka have a dance in which they cleverly imitate, not only the attitudes and tricks of the Bear, but also its voice. The peasants in Finland, in the beginning of the present century, still occasionally performed a similar dance, or rather action. The Aleutian Islanders, who have various pantomimic dances executed with masks frightfully ugly, have also a favourite representation in which a sportsman shoots a beautiful bird, and afterwards cries for grief at having killed it; when, suddenly, the beautiful bird revives, changed into a beautiful woman. The sportsman, of course, falls over head and ears in love with her, and thus all ends well.[85] This story is enacted with recitations accompanied by some musical instruments. Next to love and the chase, it is probably war which elicited the first attempts at dramatic music. To recall to the memory by a lively description with gesticulations, the valiant deeds, clever stratagems, and glorious achievements of the warriors after the battle, must have
- 82. been always a fascinating entertainment to the victorious combatants. The Dyaks in Borneo, who preserve the heads of their slain enemies suspended near their hearths as ornamental trophies, perform a war-dance in which some of the combatants, gaily decorated, cleverly act a scene by seizing swords and handling them in various expressive ways. The Scalp-Dance of the North American Indians, performed in celebration of a victory, may be described as a kind of histrionic entertainment, which generally takes place at night by torchlight. The singular procedure of the Maori warriors in New Zealand in a certain dance, of projecting all of them their tongues simultaneously at fixed intervals, appears to be a pantomimic expression of defiance or contempt for the enemy. The Corroborie Dance of the natives of Australia had perhaps also originally reference to warlike exploits, although this does not appear at once evident to European witnesses. Twenty or more men paint their naked dark bodies to represent skeletons, which they accomplish by drawing white lines across the body with pipe-clay, to correspond with the ribs, and broader ones on the arms, legs, and the head. Thus prepared they perform the Corroborie at night before a fire. The spectators, placed at some distance from them, see only the white skeletons, which vanish and re-appear whenever the dancers turn round. The wild and ghastly action of the skeletons is accompanied by vocal effusions and some rhythmical noise which a number of hidden bystanders produce by beating their shields in regular time. Traces of dramatic music in its most primitive condition may also be discovered in representations of occurrences and scenes like the following: Wilhelm Steller, in his 'Description of Kamtschatka' (published in the German language in the year 1774), says that the inhabitants of that country possess an astounding talent for imitating the manners and conduct of strangers whom they happen to see. During their long evenings one of their chief amusements consists in acting extempore
- 83. comedies, in which the habits of any foreigners with whom they have become acquainted, are cleverly mimicked and ridiculed. The missionary W. Ellis remarks of the Polynesian Islanders that they had songs which, when recited on public occasions, were accompanied with gestures and actions corresponding to the events and scenes described, and which assumed in this respect a histrionic character. In some cases, and on public occasions, the action represented a kind of pantomime.[86] Other travellers have given more detailed accounts of these performances. During Captain Cook's first voyage round the world, Banks and Solander, who accompanied him, witnessed in one of the Society Islands, in the year 1769, a comedy with music and dancing, performed by the natives, the subject of which was the adroitness of a thief, and his subsequent capture. At Cook's second circumnavigation, during the years 1772- 75, he was treated by the Society Islanders with a somewhat similar comic opera called Teto (i.e. The Thief). G. Forster, who was with Cook, remarks that the dialogue, which of course he was unable to understand, seemed to be closely connected with their actions. One of them kneeled down, and another beat him and plucked him by the beard. Then two others were treated by the torturer in the same unceremonious manner; until one of them seized a stick and gave him a sound thrashing in return. This formed the conclusion of the first act, and the players withdrew. The commencement of the second act was announced by the musicians beating their drums. There were actresses as well as actors engaged in the performance. [87] A more detailed account of the dramatic attempts of the Polynesian Islanders is given by W. Mariner, who, during his sojourn with the natives, had the best opportunity of becoming acquainted with their customs and amusements. His observations, which refer especially to the Tonga Islanders, show that the actors recite sentences which are answered by a chorus of singers. There is a great variety in their movements and groupings. Occasionally they sing slowly, and afterwards quickly for about a quarter of an hour. Sometimes they form a semi-circle, assume a bending position, and
- 84. sing in a subdued tone of voice a soft air; which is soon again followed by a loud and vehement recitation.[88] Grotesque dresses and adornments are, of course, an essential attribute in these entertainments. Neither are buffoons wanting. According to B. Seeman, the entertainment called Kalau Rere, which he witnessed in the Fiji Islands, with its high poles, streamers, evergreens, masquerading, trumpet-shells, chants and other wild music, is the nearest approach to dramatic representation the Fijians seem to have made, and it is with them what private theatricals are with us. They are also on other occasions very fond of dressing themselves in fantastic, often very ridiculous costume; and in nearly every large assembly there are buffoons. Court fools, in many instances hunchbacks, are attached to the chief's establishment.[89] Also the Negroes in Senegambia and Upper Guinea have buffoons, who delight the people with their antics and acting in processions and public festivities. Buffoons are popular even in Mohammedan countries, where dramatic performances are generally considered objectionable. Morier states that in Persia the princes, governors of provinces, etc., as well as the King, have a band of Looties, or buffoons, in their pay, who are looked upon as a necessary part of Persian state. They attend at merry-makings and public festivals, and some of them are endowed with great natural wit. This was, for instance, the case with a certain buffoon named Looti Bashee. His dress, when he came to the ambassador, was composed of a felt hat, the crown of which was made like ours, but with two long ears projecting before, and two behind. Others of his troop were dressed in the same way; all looked grotesque, and I conjectured that nothing could give one a better idea of Satyrs and Bacchanalians, particularly as they were attended by a suite of monkeys headed by a large ape, which were educated to perform all sorts of tricks. They carried copper drums slung under the arm, which they beat with their fingers, making a noise like castanets; others played the tambourine; and when all this was put into motion, with their voices roaring in loud chorus, the scene was unique.[90]
- 85. Sir Robert Ker Porter witnessed at Bagdad, in the beginning of the present century, a kind of musical drama performed by men and boys, the latter being dressed like females. This amusement, he remarks, is the only one of a theatrical complexion known among the people. It is often called for by the female part of the inhabitants; but I am told that with the men it is now very rare, the Pasha so setting his face against it as to forbid the avowed existence of hirable dancing-boys in his capital.[91] There is a Turkish theatre at Pera in which Turkish plays, adapted from the Italian, are acted by Turkish actors, and Turkish women appear unveiled upon the stage.[92] The women in the hareem, who in their diversions are only permitted to employ slaves of their own sex, occasionally make them act melodramas, the subject of which is generally a love story. The Indians in Mexico have some characteristic dances in which scenes are pantomimically enacted referring to Montezuma and to the conquest of Mexico by the Spaniards. In most of the entertainments, of which examples have just been given, the music must necessarily partake of a dramatic character. Generally, the tunes are not selected at pleasure, but certain tunes belong to certain representations. The dramatic effect of the music depends, however, chiefly upon its execution, which naturally changes according to the action which it accompanies. Thus, if the actors represent a sentimental or heart-rending scene, their vocal effusions will naturally be in a subdued tone, and the sympathizing musicians will touch their instruments delicately and slowly. If, on the other hand, the actors represent some exciting or heart-stirring scene, they will naturally raise their voices, and the musicians will play louder and faster as a matter of course. In fact, when their pulse beats quicker, the rhythmical flow of their music, however rude and inartistic it may be, becomes more animated unpremeditatedly. Such is the most primitive condition, or the commencement of the development of dramatic music. Let us now examine it in a somewhat more advanced stage of cultivation.
- 86. The Javanese, who among the islanders of the Indian Archipelago are renowned for their skill in the dramatic art, generally use fabulous traditions from their own history, or Hindu legends, as subjects for their performances, which are acted exclusively by men. A full band of musicians generally accompanies the drama. The instruments mostly belong to the class called Instruments of Percussion, but several of them are constructed with plates of metal which produce a series of sweet tones, arranged according to the pentatonic scale. Some of the Javanese airs, which have been collected by Europeans, are very expressive, and it might be instructive to musical enquirers, if some really musical European visitor in Java would faithfully commit to notation the orchestral accompaniments of some of the most popular Javanese dramas. Madame Ida Pfeiffer relates that she was treated in the house of a Rajah, at Bandong, with a kind of pantomime in three acts, the third of which represented a combat. The music that accompanied the combat, she remarks, was very noisy and discordant; but, on the defeat of the one party, a soft plaintive melody arose at some distance off. The whole performance was really pretty and expressive.[93] Sir Stamford Raffles, and other travellers, give similar descriptions, and have besides much to say about the clever puppet-shows of the Javanese, in which the characters of dramas are represented by puppets, or by their shadows. The Siamese are fond of theatrical performances. According to Turpin's history of Siam, published in the year 1771, whenever they burn the body of a minister or great man, a theatre is erected on the side of a river, where the actors appear habited according to their parts; and during three days they never quit the scene from eight in the morning till seven at night. De La Loubère, who visited Siam in the year 1687, says that the subjects of the dramas are historical, in verse, serious, and sung by several actors who are always present, and who only sing reciprocally. One of them sings the historian's part, and the rest sing those of the personages which the history makes to speak; but they are all men that sing, and no women. About a century ago it appears to have been the custom to employ only men
- 87. as actors, although there were female dancers. But, at the present day there are actresses, at any rate in the palace of the King, where Sir John Bowring saw them perform on several occasions. In one of these entertainments the actors were all females, almost all girls. A few matrons, however, took the part of warriors, monkeys, priests; and the three manageresses, or prompteresses, were not only old and ugly, but seemed very spiteful, and on several occasions scolded and slapped the ladies who required correction. One of them had the drama written on black sheets in white letters before her, from which she prompted the singers of the recitative. The story began by the appearance of a monster monkey in a forest, which is visited by a number of ladies of rank, one of whom, after an unsuccessful struggle, the others having managed to escape, the monster monkey contrives to carry off. She is redeemed by the interference of a priest, whose temple is in the forest. Afterwards we are introduced to a sovereign Court, where all the ceremonies are observed which are practised in daily life, the dresses being those ordinarily worn, and most gorgeous they are.... There is a battle, and rewards to the victors, and a crowning of a king's son in recompense for his valour, and offerings to Buddha, and a great feast, etc.[94] The principal performers act, but do not speak. The tale is told in recitative by a body of singers, accompanied by various instruments. The band assisting generally consists of about twenty members who play on wind instruments of the oboe kind, gongs, large castanets above a foot in length, and several sonorous instruments of percussion constructed with slabs of wood, or plates of metal, somewhat similar to those of the Javanese before mentioned. The Cochin-Chinese are remarkably fond of dramatic entertainments, which are generally of an operatic character commemorating historical events. An English gentleman who witnessed the performance of some of these plays remarks of the actors: Their singing is good, when the ear has become accustomed to it; and the modulation of voice of the females is really captivating.[95] Sir George Staunton was evidently surprised to find that a kind of historical opera, which he heard in the town of Turon (called by the
- 88. natives Hansán) contained recitatives, airs, and choruses, which were, he says, as regular as upon the Italian stage. He adds: Some of the female performers were by no means despicable singers. They all observed time accurately, not only with their voices, but every joint of their hands and feet was obedient to the regular movement of the instruments.[96] The band consisted of stringed instruments, wind instruments, and instruments of percussion. Sir John Barrow describes the theatre at Turon as a shed of bamboo. He relates: In the farther division of the building a party of comedians was engaged in the midst of an historical drama when we entered; but, on our being seated they broke off, and, coming forward, made before us an obeisance of nine genuflexions and prostrations, after which they returned to their labours, keeping up an incessant noise and bustle during our stay. The heat of the day, the thermometer in the shade standing at 81 deg. in the open air, and at least 10 deg. higher in the building, the crowds that thronged to see the strangers, the horrible crash of the gongs, kettle-drums, trumpets, and squalling flutes, were so stunning and oppressive that nothing but the novelty of the scene could possibly have detained us for a moment. The most entertaining, as well as the least noisy part of the theatrical exhibition, was a sort of Interlude, performed by three young women for the amusement, it would seem, of the principal actress, who sat as a spectator in the dress and character of some ancient Queen, whilst an old eunuch, very whimsically dressed, played his antic tricks like a scaramouch or buffoon in a Harlequin entertainment. The dialogue in this part differed entirely from the querulous and nearly monotonous recitation of the Chinese, being light and comic, and occasionally interrupted by cheerful airs which generally concluded with a chorus. These airs, rude and unpolished as they were, appeared to be regular compositions, and were sung in exactly measured time. One in particular attracted our attention, whose slow melancholy movement breathed the kind of plaintiveness so peculiar to the native airs of the Scotch, to which indeed it bore a close resemblance.
- 89. Probably the air was founded on the pentatonic scale, which is common in the music of the Chinese and Javanese, and of which traces are to be found in the Scotch popular tunes. The voices of the women are shrill and warbling, but some of their cadences were not without melody. The instruments at each pause gave a few short flourishes, till the music gradually increased in loudness by the swelling and deafening gong. Knowing nothing of the language, we were of course as ignorant of the subject as the majority of an English audience is of an Italian opera.[97] A curious mode of paying the actors, which prevails in Cochin-China, may be mentioned here. An Englishman who was present at a theatrical performance in the town of Kangwarting, relates that the Quong, or governor of the province, bore the expense of the entertainment. The musical drama was performed in a large shed before a great concourse of spectators. The Quong was there squatted on a raised platform in front of the actors with a small drum before him, supported in a diagonal position, on which he would strike a tap every time any part of the performance pleased him; which also was a signal for his purse-bearer to throw a small string of about twenty cash to the actors. To my taste, this spoiled the effect of the piece; for, every time the cash fell among them there would be a silence, and the next moment a scramble for the money; and it fell so frequently as almost to keep time with the discordant music of the orchestra. The actors were engaged by the day, and in this manner received their payment, the amount of which entirely depended upon the approbation of the Quong and the number of times he encored them by tapping his drum. I could see that many of them paid far more attention to the drum than they did to their performance; though I suppose, the amount thrown to them is equally divided. Sometimes the string on which the cash was tied, unluckily broke, and the money flew in all directions; by which some of the bystanders profited, not being honourable enough to hand it up to the poor actors.[98]
- 90. The Burmese have dramas performed by men, and also comedies represented by means of marionettes, or puppets. In the latter entertainments the figures are cleverly managed by persons situated beneath a stage which is hidden by a coarse curtain. The dialogues between these figures are much relished by the common spectators. At any rate, as they are apt to elicit uproarious mirth, they may be supposed to be often irresistibly comic. The real dramatic performances of the Burmese are acted by professional players, generally in the open air. The principal characters of the piece usually consist of a prince, a princess, a humble lover, a slave, and a buffoon. The female characters are represented by boys dressed in female attire. The dresses are handsome and gorgeous. However, the best theatrical performances take place in a building. On these occasions, there are two musical bands, one being placed on each side of the scene. The principal musical instruments of such an orchestra are of the percussion kind, containing a series of sonorous slabs of wood, or plates of metal, and somewhat resembling the Javanese instruments, but being attuned according to a diatonic order of intervals, instead of the pentatonic order. Also a curious contrivance, consisting of a set of drums suspended in a frame, each drum having a fixed tone, is used on these occasions. Moreover, the Burmese orchestra generally contains several wind instruments of the oboe and trumpet kind, as well as cymbals, large castanets of split bamboo, and other instruments of percussion, which serve to heighten the rhythmical effect of the music. The story of the drama is usually taken from ancient Burmese history. Captain Henry Yule, who has given a more detailed account of the Burmese plays than any previous traveller, remarks that when he was at Amarapoora he procured copies of some of the plays which he saw acted, from which it was evident to him that, while the general plan of the drama, comprising the more dignified and solemn part of the dialogue, was written down at considerable length, the humorous portions were left to the extempore wit of the actors. The following scenes are from a drama commemorating an episode from the life of Oodeinna, King of Kauthambi, a country in India. This drama, which was obtained by
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