Vector Auto-Regressive model Part one.pdf

Econ 423 Lecture Notes:
Additional Topics in Time Series1
John C. Chao
April 25, 2017
1These notes are based in large part on Chapter 16 of Stock and Watson (2011).
They are for instructional purposes only and are not to be distributed outside of the
classroom.
John C. Chao () April 25, 2017 1 / 34

Vector Autoregression (VAR)
Motivation: One may be interested in forecasting two or more
variables; such as rate of in‡ation, rate of unemployment, growth rate
of GDP, and interest rates. In this case, it is bene…cial to develop a
single model that allows you to forecast all these variables in a
systemic approach.
De…nition: A VAR (p), i.e., a vector autoregression of order p, a set
of m time series regressions, in which the regressors are the p lagged
values of the m time series variables.
Example: m = 2 case
Y1t = β10 + β11Y1t 1 + + β1pY1t p
+γ11Y2t 1 + + γ1pY2t p + u1t ,
Y2t = β20 + β21Y1t 1 + + β2pY1t p
+γ21Y2t 1 + + γ2pY2t p + u2t .
John C. Chao () April 25, 2017 2 / 34

Estimation and Inference
Algebraically, the VAR model is simply a system of m linear
regressions; or, to put it another way, it is a multivariate linear
regression model.
The coe¢ cients of the VAR can be estimated by estimating each
equation by OLS.
Under appropriate conditions, the OLS estimators are consistent and
have a joint normal distribution in large samples in the stationary case.
In consequence, in the stationary case, inference can proceed in the
usual way; for example, 95% con…dence interval on coe¢ cients can be
constructed based on the usual rule:
estimated coe¢ cients 1.96 standard errors.
John C. Chao () April 25, 2017 3 / 34

Estimation and Inference (con’
t)
An Advantage of the VAR: By modeling the dynamics of m
variables as a system, one can test joint hypotheses that involve
restrictions across multiple equations.
Example: In a two-varlable VAR (1), one might be interested in
testing the null hypothesis
H0 : β11 β21 = 0
on the unrestricted model
Y1t = β10 + β11Y1t 1 + γ11Y2t 1 + u1t ,
Y2t = β20 + β21Y1t 1 + γ21Y2t 1 + u2t .
Since the estimated coe¢ cients have a jointly normal large sample
distribution, the restrictions on these coe¢ cients can be tested by
computing the t- or the F-statistic.
Importantly, many hypotheses of interest to economists can be
formulated as cross-equation restrictions.
John C. Chao () April 25, 2017 4 / 34

Modeling Issues
How many variables should be included in a VAR?
(i) Having more variables leads to having more coe¢ cients to estimate
which, in turn, increases estimation error and can result in a
deterioration of forecast accuracy.
(ii) Concrete example: A VAR with 5 variables and 4 lags will have 21
coe¢ cients to estimate in each equation, leading to a total of 105
coe¢ cients that must be estimated.
(iii) A preferred strategy is to keep m relatively small and to make sure that
the variables included are plausibly related to each other, so that they
will be useful in forecasting one another.
(iv) For example, economic theory suggests that the in‡ation rate, the
unemployment rate, and the short-term interest rate are related to one
another, suggesting that it would be useful to model these variables
together in a VAR system.
John C. Chao () April 25, 2017 5 / 34

Modeling Issues (con’
t)
Determining the lag order in VAR’
s
(i) We can estimate the lag length of a VAR using either the F-test or an
information criterion, but the latter is preferred as it trades o¤ between
goodness of …t and the dimension of the model, whereas the F-test
does not.
(ii) BIC in the vector case:
BIC (p) = ln
h
det b
Σu
i
+ m (mp + 1)
ln T
T
,
where b
Σu is an estimate of Σu, the m m covariance matrix of the
VAR errors. Let b
uit and b
ujt be, respectively, the OLS residual for the
ith and jth equations, respectively, and note that the (i, j)th
element of
b
Σu is given by
b
Σu (i, j) =
1
T
T
∑
t=1
b
uit b
ujt ,
i.e., an estimate of Cov uit , ujt . Moreover, det b
Σu denotes the
determinant of the matrix b
Σu.
John C. Chao () April 25, 2017 6 / 34

Modeling Issues (con’
t)
Determining the lag order in VAR’
s (con’
t)
(iii) AIC in the vector case: Analogous to BIC,
AIC (p) = ln
h
det b
Σu
i
+ m (mp + 1)
2
T
.
(iv) Note that "penalty term" for both AIC and BIC involves the factor
m (mp + 1) ,
whih is the total number of coe¢ cients in an m-variable VAR(p)
model, as there are m equations each having an intercept as well as p
lags of the m variables.
(v) Note also that increasing the lag order by one leads to an additional
m2 coe¢ cients to estimate.
(vi) Let P = f0, 1, 2, ...., pg. As in the univariate case, we can select the
lag order based on BIC or AIC using the following estimation rule
b
pBIC = arg min
p2P
BIC (p) ,
b
pAIC = arg min
p2P
AIC (p) .
John C. Chao () April 25, 2017 7 / 34

Empirical Example: A VAR Model of the Rates of In‡ation
and Unemployment
Estimating a VAR (4) model for ∆Inft and Unempt using data from
1982:I to 2004:IV gives the following result:
[
∆Inf t = 1.47
(0.55)
0.64
(0.12)
∆Inft 1 0.64
(0.10)
∆Inft 2 0.13
(0.11)
∆Inft 3
0.13
(0.09)
∆Inft 4 3.49
(0.58)
Unempt 1 + 2.80
(0.94)
Unempt 2
+2.44
(1.07)
Unempt 3 2.03
(0.55)
Unempt 4, R
2
= 0.44 ;

Unempt = 0.22
(0.12)
+ 0.005
(0.017)
∆Inft 1 0.004
(0.018)
∆Inft 2 0.007
(0.018)
∆Inft 3
0.003
(0.014)
∆Inft 4 + 1.52
(0.11)
Unempt 1 0.29
(0.18)
Unempt 2
0.43
(0.21)
Unempt 3 + 0.16
(0.11)
Unempt 4, R
2
= 0.982 .
John C. Chao () April 25, 2017 8 / 34

Empirical Example (con’
t)
Granger Causality Tests
(a) Write the VAR (4) model for ∆Inft and Unempt as
∆Inft = β10 + β11∆Inft 1 + β12∆Inft 2 + β13∆Inft 3
+β14∆Inft 4 + γ11Unempt 1 + γ12Unempt 2
+γ13Unempt 3 + γ14Unempt 4 + u1t
Unempt = β20 + β21∆Inft 1 + β22∆Inft 2 + β23∆Inft 3
+β24∆Inft 4 + γ21Unempt 1 + γ22Unempt 2
+γ23Unempt 3 + γ24Unempt 4 + u2t .
(b) Test Granger non-causality of lagged unemployment rates on changes
in in‡ation, i.e.,
H0 : γ11 = γ12 = γ13 = γ14 = 0.
In this case, F = 11.04 with p-value = 0.001, so that H0 is rejected.
John C. Chao () April 25, 2017 9 / 34

Empirical Example (con’
t)
Granger Causality Tests (con’
t)
(c) In this case, we conclude that unemployment rate is a useful predictor
for changes in in‡ation, given lags in in‡ation.
(d) Test Granger non-causality of lagged changes in in‡ation rates on the
unemployment rate, i.e.,
H0 : β21 = β22 = β23 = β24 = 0.
Here, F = 0.16 with p-value = 0.96, so that H0 is not rejected in this
case.
John C. Chao () April 25, 2017 10 / 34

Cointegration
Intuitive Notion of Common Stochastic Trend:
It is possible that two or more time series with stochastic trends can
move together so closely over the long run that they appear to have
the same trend component. In this case, they are said to share a
common stochastic trend.
Orders of Integration, Di¤erencing, and Stationarity
1 If Yt is integrated of order one (denoted Yt I (1)); then, its …rst
di¤erence ∆Yt is stationary, i.e., ∆Yt I (0). In this case, Yt has a
unit autoregressive root.
2 If Yt is integrated of order two (denoted Yt I (2)); then, its second
di¤erence ∆2Yt is stationary. In this case, ∆Yt I (1).
3 If Yt is integrated of order d (denoted Yt I (d)); then, ∆d Yt is
stationary, i.e., Yt must be di¤erenced d times in order to produce a
series that is stationary.
John C. Chao () April 25, 2017 11 / 34

Cointegration (con’
t)
De…nition of Cointegration:
Suppose that Xt and Yt are integrated of order one. If, for some
coe¢ cient θ, Zt = Yt θXt is integrated of order zero; then, Xt and
Yt are said to be cointegrated. The coe¢ cient θ is called the
cointegrating coe¢ cient.
Remark:
If Xt and Yt are cointegrated, then they have the same, or common,
stochastic trend. Computing the di¤erence Yt θXt then eliminates
this common stochastic trend.
John C. Chao () April 25, 2017 12 / 34

Deciding If Variables Are Cointegrated
Three ways to decide whether two variables is cointegrated:
1 Use expert knowledge and economic theory.
2 Graph the series and see whether they appear to have a common
stochastic trend.
3 Perform statistical test for cointegration.
John C. Chao () April 25, 2017 13 / 34

Testing for Cointegration
Some Observations: Let Yt and Xt be two time series such that
Yt I (1) and Xt I (1).
1 If Yt and Xt are cointegrated with cointegrating coe¢ cient θ, then
Yt θXt I (0).
2 On the other hand, if Yt and Xt are not cointegrated, then
Yt θXt I (1).
3 1. and 2. suggest that we can test for the presence of cointegration by
testing
H0 : Yt θXt I (1) versus H1 : Yt θXt I (0)
Two Cases
1 θ is known, i.e., a value for θ is suggested by expert knowledge or by
economic theory. In this case, one can simply construct the time series
Zt = Yt θXt
and test the null hypothesis H0 : Yt θXt I (1) using the
augmented Dickey-Fuller test.
John C. Chao () April 25, 2017 14 / 34

Testing for Cointegration (con’
t)
2. θ is unknown: In this case, perhaps the easiest approach is to adopt a
two-step procedure
1 Step 1: Estimate the cointegrating coe¢ cient θ by OLS estimation of
the regression
Yt = α + θXt + Zt
and obtain the residual series b
Zt = Yt b
α b
θXt .
2 Step 2: Apply a unit root test, such as the augmented Dickey-Fuller
test, to test whether the residual series b
Zt is an I (1) process. (Engle
and Granger, 1987, and Phillips and Ouliaris, 1990).
John C. Chao () April 25, 2017 15 / 34

t)
3. Remark: A complication which arises when θ is unknown is that,
under H0, b
Zt I (1), so that the regression of Yt on Xt is a spurious
regression, which implies, in particular, that b
θ is not a consistent
estimator. As a result, we cannot use the same critical values which
apply in Case 1 discussed earlier.
4. The two-step procedure can be extended in a straightforward manner
to cases with more than one regressor (e.g., the case with k regressors
X1t , ..., Xkt ) by running the multiple regression
Yt = α + θ1X1t + + θk Xkt + Zt
and testing the residual process b
Zt = Yt b
α b
θ1X1t
b
θk Xkt
for the presence of a unit root. Critical values for the residual-based
cointegration test do depend on the number of regressors, however.
John C. Chao () April 25, 2017 16 / 34

t)
Table: Critical Values for Residual-
Based Tests for Cointegration
# of X’
s in the regression 10% 5% 1%
1 3.12 3.41 3.96
2 3.52 3.80 4.36
3 3.84 4.16 4.73
4 4.20 4.49 5.07
John C. Chao () April 25, 2017 17 / 34

Vector Error Correction Model
Suppose that Xt I (1) and Yt I (1), and suppose that Xt and Yt
are cointegrated. Then, it turns out that a bivariate VAR model in
terms of the …rst di¤erences ∆Xt and ∆Yt is misspeci…ed.
The correct model will include the term Yt 1 θXt 1 in addition to
the lagged values of ∆Xt and ∆Yt .
More speci…cally, the correct model is of the form
∆Yt = β10 + β11∆Yt 1 + + β1p ∆Yt p
+γ11∆Xt 1 + + γ1p ∆Xt p
+α1 (Yt 1 θXt 1) + u1t ,
∆Xt = β20 + β21∆Yt 1 + + β2p ∆Yt p
+γ21∆Xt 1 + + γ2p ∆Xt p
+α2 (Yt 1 θXt 1) + u2t .
This model is known as the vector error correction model (VECM),
and the term Yt 1 θXt 1 is called the error correction term.
John C. Chao () April 25, 2017 18 / 34

Vector Error Correction Model (con’
t)
Remarks:
1 In a VECM, past values of the error correction term Yt θXt help to
predict future values of ∆Yt and/or ∆Xt .
2 Note also that a VAR model in …rst di¤erences is misspeci…ed in this
case precisely because it omits the error correction term.
In the case where θ is known; set Zt 1 = Yt 1 θXt 1, and we have
∆Yt = β10 + β11∆Yt 1 + + β1p ∆Yt p
+γ11∆Xt 1 + + γ1p ∆Xt p
+α1Zt 1 + u1t ,
∆Xt = β20 + β21∆Yt 1 + + β2p ∆Yt p
+γ21∆Xt 1 + + γ2p ∆Xt p
+α2Zt 1 + u2t ,
so that the parameters of the VECM can be estimated by linear least
squares in this case.
John C. Chao () April 25, 2017 19 / 34

t)
In the case where θ is unknown; then, the VECM is nonlinear in
parameters, so that one cannot directly apply linear least squares.
In this case, there are a few di¤erent approaches to estimating the
parameters of a VECM.
1 Approach 1: Two-step procedure.
(i) Step 1: Estimate θ by a preliminary OLS regression
Yt = α + θXt + Zt
and obtain the residual b
Zt 1 = Yt 1
b
θXt 1.
John C. Chao () April 25, 2017 20 / 34

t)
(ii) Step 2: Plug b
Zt 1 into the VECM speci…cation to obtain
∆Yt = β10 + β11∆Yt 1 + + β1p ∆Yt p
+γ11∆Xt 1 + + γ1p ∆Xt p
+α1
b
Zt 1 + b
u1t ,
∆Xt = β20 + β21∆Yt 1 + + β2p ∆Yt p
+γ21∆Xt 1 + + γ2p ∆Xt p
+α2
b
Zt 1 + b
u2t ,
The remaining parameters of the VECM can then be estimated by
linear least squares. Note that this approach exploits the fact b
θ is a
consistent estimator of θ if the assumption of cointegration is correct.
Moreover, rate of convergence for this estimator is T which is faster
than the usual
p
T convergence rate.
John C. Chao () April 25, 2017 21 / 34

t)
2. Approach 2: A more e¢ cient approach is to estimate all the
parameters θ, β10, ..., β1p, β20, ..β2p , γ11, ..., γ1p, γ21, ..., γ2p ,
and (α1, α2) in the model
∆Yt = β10 + β11∆Yt 1 + + β1p ∆Yt p
+γ11∆Xt 1 + + γ1p ∆Xt p
+α1 (Yt 1 θXt 1) + u1t ,
∆Xt = β20 + β21∆Yt 1 + + β2p ∆Yt p
+γ21∆Xt 1 + + γ2p ∆Xt p
+α2 (Yt 1 θXt 1) + u2t .
jointly by full system maximum likelihood. This is the approach that
has been developed by Soren Johansen (see Johansen 1988, 1991).
John C. Chao () April 25, 2017 22 / 34

Models of Conditional Heteroskedasticity - Motivation
Consider again the AR (1) model
Yt = βYt 1 + ut ,
where jβj < 1 and fut g i.i.d. 0, σ2 .
Note that for this model
E [Yt+1] = 0
but
E [Yt+1jYt , Yt 1, ...] = E [Yt+1jYt ] = βYt ,
so that by using information about current and past values of Yt , this
model allows one to improve on ones forecast of the mean-level of
Yt+1 over that which can be obtained when this information is not
used.
John C. Chao () April 25, 2017 23 / 34

Models of Conditional Heteroskedasticity - Motivation
Shortcoming of this model: The same improvement is not achieved
when forecasting the error variance with this model since
E u2
t+1jYt , Yt 1, ... = E u2
t+1 = σ2
Observation: This model is not rich enough to allow for better
prediction of the error variance based on past information. In
particular, the independence assumption on the errors precludes any
forecast improvement.
On the other hand, many …nancial and macroeconomic time series
exhibit "volatility clustering." Volatility clustering suggests the
possible presence of time dependent variance or time-varying
heteroskedasticty that may be forecastable. Interestingly, this can
occur even if the time series itself is close to being serially
uncorrelated so that the mean-level is di¢ cult to forecast.
John C. Chao () April 25, 2017 24 / 34

Models of Conditional Heteroskedasticity - Empirical
Motivation
John C. Chao () April 25, 2017 25 / 34

Why would there be interest in forecasting variance?
First, in …nance, the variance of the return to an asset is a measure of
the risk of owning that asset. Hence, investors, particularly those who
are risk averse, would naturally be interested in predicting return
variances.
Secondly, the value of some …nancial derivatives, such as options,
depends on the variance of the underlying assets. Thus, an options
trader would want to obtain good forecasts of future volatility to help
her or him decide on the price at which to buy or sell options.
Thirdly, being able to forecast variance could allow one to have more
accurate forecast intervals that adapt to changing economic
conditions.
John C. Chao () April 25, 2017 26 / 34

AutoRegressive Conditional Heteroskedasticity (ARCH)
Models
Here, we will discuss two frequently used models of time-varying
heteroskedasticity: the autoregressive conditional
heteroskedasticity (ARCH) model and its extension, the
generalized ARCH (or GARCH) model.
ARCH(1) process: Consider the ADL(1,1) regression
Yt = β0 + β1Yt 1 + γ1Xt 1 + ut .
Instead of modeling fut g as an independent sequence of random
variables, as we have before, the ARCH(1) process takes
ut = εt α0 + α1u2
t 1
1/2
,
where α0 > 0, 0 < α1 < 1, and fεt g i.i.d.N (0, 1) .
Remark: We have described here a ADL(1,1) model with ARCH
errors; but, in principle, an ARCH process can be applied to model
the error variance for any time series regression.
John C. Chao () April 25, 2017 27 / 34

ARCH Models
Some Moment Calculations
(i) Conditional Mean:
E [ut jut 1, ut 2, ...] =
h
α0 + α1u2
t 1
i1/2
E [εt jut 1, ut 2, ...]
=
h
α0 + α1u2
t 1
i1/2
E [εt ]
= 0
(ii) Unconditional Mean:
E [ut ] = E (E [ut jut 1, ut 2, ...])
(by law of iterated expectations)
= E [0]
= 0.
John C. Chao () April 25, 2017 28 / 34

ARCH Models
(iii) Conditional Variance:
E
h
u2
t jut 1, ut 2, ...
i
=
h
α0 + α1u2
t 1
i
E
h
ε2
t jut 1, ut 2, ...
i
=
h
α0 + α1u2
t 1
i
E
h
ε2
t
i
=
h
α0 + α1u2
t 1
i
(iv) Autocovariances: Let j be any positive integer, and note that
E ut ut j = E ut j E [ut jut 1, ut 2, ...]
(by law of iterated expectations)
= E ut j 0
= 0.
Remark: Interestingly, an ARCH process is serially uncorrelated but
not independent. These features are important for the modeling of
asset returns.
John C. Chao () April 25, 2017 29 / 34

ARCH Models
More Moments: It can also be shown that
Var (ut ) = E u2
t =
α0
1 α1
,
E u4
t =
"
3α2
0
(1 α1)2
#
1 α2
1
1 3α2
1
.
(Note: we assume that α0 > 0 and 0 < α1 < 1).
John C. Chao () April 25, 2017 30 / 34

ARCH Models
Remark: Note that since
1 α2
1
1 3α2
1
> 1,
we have that
E u4
t =
"
3α2
0
(1 α1)2
#
1 α2
1
1 3α2
1
>
3α2
0
(1 α1)2
= 3 E u2
t
2
.
On the other hand, if ut had been normally distributed, say
fut g i.i.d.N 0, σ2 ; then, we would have
E u4
t = 3 E u2
t
2
= 3σ4. Hence, the ARCH error process has
“fatter-tails" than that implied by the normal distribution.
John C. Chao () April 25, 2017 31 / 34

ARCH Models
ARCH(p) process: A straightforward extension of the ARCH(1)
model is the p-th order ARCH process given by
ut = εt α0 + α1u2
t 1 + + αpu2
t p
1/2
,
where fεt g i.i.d.N (0, 1); αi > 0 for i = 0, 1, ..., p; and
α1 + + αp < 1.
John C. Chao () April 25, 2017 32 / 34

GARCH Models
GARCH(p,q) process: A useful generalization of the ARCH model
is the following GARCH model due to Bollerslev (1986).
ut = h1/2
t εt ,
where
ht = α0 + α1u2
t 1 + + αpu2
t p + δ1ht 1 + + δqht q.
Assumptions:
(i) fεt g i.i.d.N (0, 1);
(ii) α0 > 0 and αi 0 for i = 1, ..., p;
(iii) δj 0 for j = 1, ..., q.
Remark: Note that even a GARCH(1,1) model will allow ht to
depend on u2
t from the distant past. Thus, GARCH provides a clever
way of capturing slowly changing variances without having to specify
a model that has a lot of parameters to estimate.
Remark: Both ARCH and GARCH can be estimated using the
method of maximum likelihood.
John C. Chao () April 25, 2017 33 / 34

Empirical Illustration
A simple model of stock return with time-varying volatility is the
following
Rt = µ + ut
where fut g follows a GARCH(1,1) process, i.e.,
ut = h1/2
t εt ,
ht = α0 + α1u2
t 1 + δ1ht 1.
The textbook provides empirical results of …tting this model to daily
percentage changes in the NYSE index using data on all trading days
from January 2, 1990 to November 11, 2005. The results are
b
Rt = b
µ = 0.049
(0.012)
b
ht = 0.0079
(0.0014)
+ 0.072
(0.005)
u2
t 1 + 0.919
(0.006)
ht 1
John C. Chao () April 25, 2017 34 / 34

Vector Auto-Regressive model Part one.pdf

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