![CHAPTER 1
Introduction
1.1 Preliminaries
This monograph basically discusses semiparametric methods in model
estimation, specification testing and selection of nonlinear time series
data. We use the term semiparametric for models which are semipara-
metric partially linear models or other semiparametric regression models
as discussed in Chapters 2–6, in particular Chapters 2 and 5. We also
use the word semiparametric for methods which are semiparametric es-
timation and testing methods as discussed in Chapters 2–6, particularly
in Chapters 3 and 6. Meanwhile, we also use the term nonparametric
for models and methods which are either nonparametric models or non-
parametric methods or both as considered in Chapters 2–5.
1.2 Examples and models
Let (Y, X) be a d + 1–dimensional vector of time series variables with Y
being the response variable and X the vector of d–dimensional covari-
ates. We assume that both X and Y are continuous random variables
with π(x) as the marginal density function of X, f(y|x) being the condi-
tional density function of Y given X = x and f(x, y) as the joint density
function. Let m(x) = E[Y |X = x] denote the conditional mean of Y
given X = x. Let {(Yt, Xt) : 1 ≤ t ≤ T} be a sequence of observa-
tions drawn from the joint density function f(x, y). We first consider a
partially linear model of the form
Yt = E[Yt|Xt] + et = m(Xt) + et = Uτ
t β + g(Vt) + et, (1.1)
where Xt = (Uτ
t , V τ
t )τ
, m(Xt) = E[Yt|Xt], and et = Yt − E[Yt|Xt] is the
error process and allowed to depend on Xt. In model (1.1), Ut and Vt are
allowed to be two different vectors of time series variables. In practice, a
crucial problem is how to identify Ut and Vt before applying model (1.1)
to model sets of real data. For some cases, the identification problem can
be solved easily by using empirical studies. For example, when modelling
1
© 2007 by Taylor & Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-17-320.jpg)
![EXAMPLES AND MODELS 3
series of the form
Yt = Uτ
t β + g
t
T
+ et, (1.3)
where {Yt} is the mean temperature series of interest, Ut = (Ut1, · · · , Utq)τ
is a vector of q–explanatory variables, such as the southern oscillation
index (SOI), t is time in years, β is a vector of unknown coefficients for
the explanatory variables, g(·) is an unknown smooth function of time
representing the trend, and {et} represents a sequence of stationary time
series errors with E[et] = 0 and 0 var[et] = σ2
∞. Recently, Gao and
Hawthorne (2006) have considered some estimation and testing problems
for the trend function of the temperature series model (1.3).
Applying an existing method from Härdle, Liang and Gao (2000) to two
global temperature series (http://www.cru.uea.ac.uk/cru/data/), Gao
and Hawthorne (2006) have shown that a nonlinear trend looks feasible
for each of the temperature series. Figure 1 of Gao and Hawthorne (2006)
shows the annual mean series of the global temperature series from 1867–
1993 and then from 1867–2001.
1860 1880 1900 1920 1940 1960 1980
−0.4
−0.2
0.0
0.2
Years
Temperature
Figure 1.1 The light line is the global temperature series for 1867–1993, while
the solid curve is the estimated trend.
Figure 1.1 shows that the trend estimate appears to be distinctly non-
linear. Figure 1.2 displays the partially linear model fitting to the data
set. The inclusion of the linear SOI component is warranted by the in-
terannual fluctuations of the temperature series. Figures 1.1 and 1.2 also
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-19-320.jpg)
![EXAMPLES AND MODELS 9
Truong and Stone (1994) considered a nonparametric regression model
with a linear autoregressive error model of the form
Yt = g(Xt) + ut with ut = θut−1 + t, (1.14)
where {(Xt, Yt)} is a bivariate stationary time series, θ, satisfying |θ| 1,
is an unknown parameter, g(·) is an unknown function, and {t} is a
sequence of independent errors with zero mean and finite variance 0
σ2
∞. Truong and Stone (1994) proposed a semiparametric estimation
procedure for model (1.14).
Example 1.7 (Partially linear autoregressive conditional heteroscedasti-
city (ARCH) models): For the case where d = 1, {Yt} is a time series,
Xt = Yt−1, and {et} depends on Yt−1, model (1.2) is a partially linear
ARCH model of the form
Yt = βYt−1 + g(Yt−1) + et, (1.15)
where {et} is assumed to be stationary, both β and g are identifiable, and
σ2
(y) = E[e2
t |Yt−1 = y] is a smooth function of y. Hjellvik and Tjøstheim
(1995), and Hjellvik, Yao and Tjøstheim (1998), Li (1999), and Gao and
King (2005) all considered testing for linearity in model (1.15). Granger,
Inoue and Morin (1997) have considered some estimation problems for
the case of β = 1 in model (1.15).
Example 1.8 (Nonlinear and nonstationary time series models): This
example considers two classes of nonlinear and nonstationary time series
models. The first class of models is given as follows:
Yt = m(Xt) + et with Xt = Xt−1 + t, (1.16)
where {t} is a sequence of stationary errors. The second class of models
is defined by
Yt = Yt−1 + g(Yt−1) + et. (1.17)
Recently, Granger, Inoue and Morin (1997) considered the case where
g(·) of (1.17) belongs to a class of parametric nonlinear functions and
then discussed applications in economics and finance. In nonparametric
kernel estimation of m(·) in (1.16) and g(·) of (1.17), existing studies
include Karlsen and Tjøstheim (1998), Phillips and Park (1998), Karlsen
and Tjøstheim (2001), and Karlsen, Myklebust and Tjøstheim (2006).
The last paper provides a class of nonparametric versions of some of
those parametric models proposed in Engle and Granger (1987). Model
(1.16) corresponds to a class of parametric nonlinear models discussed
in Park and Phillips (2001).
Compared with nonparametric kernel estimation, nonparametric spe-
cification testing problems for models (1.16) and (1.17) have just been
considered in Gao et al. (2006). Specifically, the authors have proposed
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-25-320.jpg)
![CHAPTER 2
Estimation in Nonlinear Time Series
2.1 Introduction
This chapter considers semiparametric modelling of nonlinear time series
data. We first propose an additive partially linear modelling method.
A semiparametric single–index modelling procedure is then considered.
Both new estimation methods and implementation procedures are dis-
cussed in some detail. The main ideas are to use either a partially linear
form or a semiparametric single–index form to approximate the condi-
tional mean function rather than directly assuming that the true condi-
tional mean function is of either a partially linear form or a semipara-
metric single–index form.
2.1.1 Partially linear time series models
In time series regression, nonparametric methods have been very pop-
ular both for prediction and characterizing nonlinear dependence. Let
{Yt} and {Xt} be the one–dimensional and d–dimensional time series
data, respectively. For a vector of time series data {Yt, Xt}, the condi-
tional mean function E[Yt|Xt = x] of Yt on Xt = x may be estimated
nonparametrically by the Nadaraya–Watson (NW) estimator when the
dimensionality d is less than three. When d is greater than three, the
conditional mean can still be estimated using the NW estimator, and
an asymptotic theory can be constructed. In practice, however, because
of the so–called curse of dimensionality, this may not be recommended
unless the number of data points is extremely large.
There are several ways of circumventing the curse of dimensionality in
time series regression. Perhaps the two most commonly used are semi-
parametric additive models and single–index models. In time series re-
gression, semiparametric additive fitting can be thought of as an ap-
proximation of conditional quantities such as E[Yt|Yt−1, . . . , Yt−d], and
sometimes (Sperlich, Tjøstheim and Yang 2002) interaction terms are in-
cluded to improve this approximation. An advantage of using the semi-
15
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-31-320.jpg)
![16 ESTIMATION IN NONLINEAR TIME SERIES
parametric additive approach is that a priori information concerning
possible linearity of some of the components can be included in the
model. More specifically, we will look at approximating the conditional
mean function m(Xt) = m(Ut, Vt) = E[Yt|Ut, Vt] by a semiparametric
(partially linear) function of the form
m1(Ut, Vt) = µ + Uτ
t β + g(Vt) (2.1)
such that E [Yt − m1(Ut, Vt)]
2
is minimized over a class of semipara-
metric functions of the form m1(Ut, Vt) subject to E[g(Vt)] = 0 for
the identifiability of m1(Ut, Vt), where µ is an unknown parameter,
β = (β1, . . . , βq)τ
is a vector of unknown parameters, g(·) is an unknown
function over Rp
, both Ut = (Ut1, . . . , Utq)τ
and Vt = (Vt1, . . . , Vtp)τ
may be vectors of time series variables.
Motivation for using the form (2.1) for independent data analysis can be
found in Härdle, Liang and Gao (2000). As for the independent data case,
estimating g(·) in model (2.1) may suffer from the curse of dimensionality
when g(·) is not necessarily additive and p ≥ 3. Thus, this chapter
proposes two different estimation methods. The first estimation method
deals with the case where m(x) is itself an additive partially linear form
and each of the nonparametric components is approximated by a series
of orthogonal functions. For the independent data case, the orthogonal
series estimation method has been used as an alternative to some other
nonparametric estimation methods, such as the kernel method. Recent
monographs include Eubank (1999). As shown in Gao, Tong and Wolff
(2002a), this method provides some natural parametric approximations
to additive partially linear forms.
2.1.2 Semiparametric additive time series models
The main ideas of proposing the second method are taken from Gao, Lu
and Tjøstheim (2006), who have established an estimation procedure for
semiparametric spatial regression. The second method applies to the case
where m(x) is approximated by (2.1) and then proposes approximating
g(·) by ga(·), an additive marginal integration projector as detailed in the
following section. When g(·) itself is additive, i.e., g(x) =
Pp
i=1 gi(xi),
the form of m1(Ut, Vt) can be written as
m1(Ut, Vt) = µ + Uτ
t β +
p
X
i=1
gi(Vti) (2.2)
subject to E [gi(Vti)] = 0 for all 1 ≤ i ≤ p for the identifiability of
m1(Ut, Vt) in (2.2), where gi(·) for 1 ≤ i ≤ p are all unknown one–
dimensional functions over R1
.
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-32-320.jpg)
![INTRODUCTION 17
Our method of estimating g(·) or ga(·) is based on an additive marginal
integration projection on the set of additive functions, but where unlike
the backfitting case, the projection is taken with the product measure of
Vtl for l = 1, · · · , p (Nielsen and Linton 1998). This contrasts with the
smoothed backfitting approach of Mammen, Linton and Nielsen (1999)
to the nonparametric regression case. Marginal integration, although
inferior to backfitting in asymptotic efficiency for purely additive models,
seems well suited to the framework of partially linear estimation. In fact,
in previous work (Fan, Härdle and Mammen 1998; Fan and Li 2003, for
example) in the independent regression case marginal integration has
been used, and we do not know of any work extending the backfitting
theory to the partially linear case. Marginal integration techniques are
also applicable to the case where interactions are allowed between the
the Vtl–variables (cf. also the use of marginal integration for estimating
interactions in ordinary regression problems).
2.1.3 Semiparametric single–index models
As an alternative to (2.2), we assume that m(x) = E[Yt|Xt = x] =
m2(Xt) is given by the semiparametric single–index form
m2(Xt) = Xτ
t θ + ψ(Xτ
t η). (2.3)
When we partition Xt = (Uτ
t , V τ
t )τ
and take θ = (βτ
, 0, · · · , 0)τ
and
η = (0, · · · , 0, ατ
)τ
, form (2.3) becomes the generalized partially linear
form
m2(Xt) = Uτ
t β + ψ(V τ
t α). (2.4)
Various versions of (2.3) and (2.4) have been discussed in the econo-
metrics and statistics literature. Recent studies include Härdle, Hall and
Ichimura (1993), Carroll et al. (1997), Gao and Liang (1997), Xia, Tong
and Li (1999), and Gao and King (2005).
In Sections 2.2 and 2.3 below, some detailed estimation procedures for
m1(Ut, Vt) and m2(Xt) are proposed and discussed extensively. Sec-
tion 2.2 first assumes that the true conditional mean function m(x) =
E[Yt|Xt = x] is of the form (2.2) and develops an orthogonal series es-
timation method for the additive form. Section 2.3 then proposes an
additive marginal integration projection method to estimate form (2.1)
without necessarily assuming the additivity in (2.2).
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-33-320.jpg)
![18 ESTIMATION IN NONLINEAR TIME SERIES
2.2 Semiparametric series estimation
In this section, we employ the orthogonal series method to estimate
each nonparametric function in (2.2). By approximating each gi(·) by
an orthogonal series
Pni
j=1 fij(·)θij with {fij(·)} being a sequence of
orthogonal functions and {ni} being a sequence of positive integers, we
have an approximate model of the form
Yt = µ + Uτ
t β +
p
X
i=1
ni
X
j=1
fij(Vti)θij + et, (2.5)
which covers some natural parametric time series models. For example,
when Utl = Ut−l and Vti = Yt−i, model (2.5) becomes a parametric
nonlinear additive time series model of the form
Yt = µ +
q
X
l=1
Ut−lβl +
p
X
i=1
ni
X
j=1
fij(Yt−i)θij + et. (2.6)
To estimate the parameters involved in (2.5), we need to introduce the
following symbols. For 1 ≤ i ≤ p, let
θi = (θi1, · · · , θini
)τ
, θ = (θτ
1 , · · · , θτ
p )τ
,
Fi = Fini
= (Fi(V1i), . . . , Fi(VT i))τ
, F = (F1, F2, . . . , Fp),
U =
1
T
T
X
t=1
Ut, e
U = U1 − U, · · · , UT − U
τ
,
Y =
1
T
T
X
t=1
Yt, e
Y = Y1 − Y , · · · , YT − Y
τ
,
P = F (Fτ
F)
+
Fτ
, b
U = (I − P)e
U, b
Y = (I − P)e
Y , (2.7)
and n = (n1, · · · , np)τ
and A+
denotes the Moore–Penrose inverse of A.
Using the approximate model (2.6), we define the least squares (LS)
estimators of (β, θ, µ) by
b
β = b
β(n) =
b
Uτ b
U
+
b
Uτ b
Y ,
b
θ = (Fτ
F)
+
Fτ
e
Y − e
U b
β
,
b
µ = Y − U
τ
b
β. (2.8)
Equation (2.8) suggests estimating the conditional mean function m(Xt) =
E[Yt|Xt] by
b
m(Xt; n) = b
µ + Uτ
t
b
β +
p
X
i=1
Fi(Vti)τ b
θi(n), (2.9)
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-34-320.jpg)
![SEMIPARAMETRIC SERIES ESTIMATION 19
where b
θi(n) is the corresponding estimator of θi.
It follows from (2.9) that the prediction equation depends on not only
the series functions {fij : 1 ≤ j ≤ ni, 1 ≤ i ≤ p} but also n, the
vector of truncation parameters. It is mentioned that the choice of the
series functions is much less critical than that of the vector of truncation
parameters. The series functions used in this chapter need to satisfy
Assumptions 2.2 and 2.3 in Section 2.5. The assumptions hold when each
fij belongs to a class of trigonometric series used by Gao, Tong and Wolff
(2002a). Therefore, a crucial problem is how to select k practically. Li
(1985, 1986, 1987) discussed the asymptotic optimality of a generalized
cross–validation (GCV) criterion as well as other model selection criteria.
Wahba (1990) provided a recently published survey of nonparametric
smoothing spline literature up to 1990. Gao (1998) applied a generalized
cross–validation criterion to choose smoothing truncation parameters
for the time series case. In this section, we apply a generalized cross–
validation method to choose k and then determine the estimates in (2.9).
In order to select n, we introduce the following mean squared error:
b
D(n) =
1
T
T
X
t=1
{ b
m(Xt; n) − m(Xt)}
2
. (2.10)
Let g
(mi)
i be the mi–order derivative of the function gi and M0i be a
constant,
Gmi (Si) =
n
g : g
(mi)
i (s) − g
(mi)
i (s0
) ≤ M0i|s − s0
|, s, s0
∈ Si ⊂ R1
o
,
where each mi ≥ 1 is an integer, 0 M0i ∞ and each Si is a compact
subset of R1
. Let also NiT = {piT , piT + 1, . . . , qiT }, in which piT =
aiTdi
, qiT = [biTci
], 0 ai bi ∞, 0 di ci 1
2(mi+1) are
constants, and [x] ≤ x denotes the largest integer part of x.
Definition 2.1. A data-driven estimator b
n = (b
n1, . . . , b
np)τ
is asymptot-
ically optimal if
b
D(b
n)
infn∈NT
b
D(n)
→p 1,
where n ∈ NT = {n = (n1, . . . , np)τ
: ni ∈ NiT }.
Definition 2.2. Select n, denoted by b
nG = (b
n1G, . . . , b
npG)τ
, that achieves
GCV(b
nG) = inf
n∈NT
GCV(n) = inf
n∈NT
b
σ2
(n)
1 − 1
T
Pp
i=1 ni
2 ,
where b
σ2
(n) = 1
T
PT
t=1 {Yt − b
m(Xt; n)}
2
.
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-35-320.jpg)
![20 ESTIMATION IN NONLINEAR TIME SERIES
We now have the following asymptotic properties for b
D(n) and b
nG.
Theorem 2.1. (i) Assume that Assumptions 2.1–2.2(i), 2.3 and 2.4
listed in Section 2.5 hold. Then
b
D(n) =
σ2
T
p
X
i=1
ni +
1
T
E [∆τ
∆] + op
b
D(n)
, (2.11)
where ∆ =
Pp
i=1 [Fiθi − Gi], Gi = (gi(V1i), . . . , gi(VT i))τ
and {Fi(·)} is
as defined before.
(ii) In addition, if Assumption 2.2(ii) holds, then we have
b
D(n) =
σ2
T
p
X
i=1
ni +
p
X
i=1
Cin
−2(mi+1)
i + op( b
D(n)) (2.12)
uniformly over n ∈ NT , where σ2
= E[e2
t ] ∞ and each mi is the
smoothness order of gi.
Theorem 2.2. (i) Under the conditions of Theorem 2.1(i), b
nG is asymp-
totically optimal.
(ii) Under the conditions of Theorem 2.1(ii), we have
b
D(b
nG)
D̂(n̂D)
− 1 = op(T−τ
) (2.13)
and
p
X
i=1
b
niG
b
niD
− 1 = op T−τ
, (2.14)
where b
niD is the i–th component of b
nD = (b
n1D, . . . , b
npD)τ
that minimises
b
D(n) over NT , 0 τ = min(τ1 − 1, τ2 − 2), in which τ1 = 1
2 dmin, τ2 =
1
2 − 2cmax, both 1 and 2 satisfying 0 1 τ1 and 0 2 τ2 are
arbitrarily small, dmin = min1≤i≤p di and cmax = max1≤i≤p ci.
The proofs of Theorems 2.1 and 2.2 are relegated to Section 2.5.
We now define the adaptive and simultaneous estimation procedure as
follows:
(i) solve the LS estimator b
θ(n);
(ii) define the prediction equation by (2.9);
(iii) solve the GCV-based b
nG; and
(iv) define the following adaptive and simultaneous prediction equation
b
m (Xt; b
nG).
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-36-320.jpg)
![22 ESTIMATION IN NONLINEAR TIME SERIES
model the data set. In practice, before applying the estimation procedure
to model the data, a crucial problem is how to test the additivity. Some
related results for additive nonparametric regression have been given by
some authors. See, for example, Gao, Tong and Wolff (2002b).
To illustrate the above estimation procedure, we now include two simu-
lated and real examples for a special case of model (2.2) with µ = β = 0.
Let Vt = (Vt1, Vt2, Vt3)τ
= (Yt−1, Yt−2, Wt)τ
, where {Wt} is to be spe-
cified below.
Example 2.1: Consider the model given by
Yt = 0.25Yt−1 + 0.25
Yt−2
1 + Y 2
t−2
+
1
8π
W2
t + et, t = 3, 4, ..., T, (2.15)
where {et} is uniformly distributed over (−0.5π, 0.5π), Y1 and Y2 are
mutually independent and uniformly distributed over
1
128 , 2π − 1
128
,
(Y1, Y2) is independent of {et : t ≥ 3},
Wt = 0.25Wt−1 − 0.25Wt−2 + t, (2.16)
in which {t} is uniformly distributed over (−0.5π, 0.5π), X1 and X2 are
mutually independent and uniformly distributed over
1
128 , 2π − 1
128
,
and (X1, X2) is independent of {t : t ≥ 3}.
First, it follows from Lemma 3.1 of Masry and Tjøstheim (1997) that
both the stationarity and the mixing condition are met. See also Chapter
4 of Tong (1990), §2.4 of Tjøstheim (1994) and §2.4 of Doukhan (1995).
Thus, Assumption 2.1(i) holds. Second, it follows from (2.15) and (2.16)
that Assumption 2.1(ii) holds immediately. Third, let
g1(x) = 0.25x,
g2(x) = 0.25
x
1 + x2
,
g3(x) =
1
8π
x2
. (2.17)
Since {gi : 1 ≤ i ≤ 3} are continuously differentiable on R1
, there
exist three corresponding periodic functions defined on [0, 2π] that are
continuously differentiable on [0, 2π] and coincide with {gi : 1 ≤ i ≤ 3}
correspondingly (see Hong and White 1995, p.1141). Similarly to §3.2 of
Eastwood and Gallant (1991), we can show that there exist the following
three corresponding trigonometric polynomials
g∗
1(x) =
n1
X
j=1
sin(jx)θ1j,
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-38-320.jpg)
![SEMIPARAMETRIC SERIES ESTIMATION 23
g∗
2(x) =
n2
X
j=1
sin(jx)θ2j,
g∗
3(x) =
n3
X
j=1
cos(jx)θ3j (2.18)
such that Assumptions 2.2(i) and 2.2(ii) are satisfied and the same con-
vergence rate can be obtained as in the periodic case. Obviously, it fol-
lows from (2.18) that Assumption 2.2(i) holds. Fourth, Assumption 2.3
is satisfied due to (2.18) and the orthogonality of trigonometric series.
Finally, Assumption 2.4 holds due to the fact that supt≥1 |Yt| ≤ 2π.
We now define g∗
1, g∗
2 and g∗
3 as the corresponding approximations of g1,
g2 and g3 with
x ∈ S =
1
128
, 2π −
1
128
and hi ∈ NiT =
[aiTdi
], . . . , [biTci
] ,
(2.19)
in which i = 1, 2, 3,
di =
1
2mi + 3
and ci =
1
2mi + 3
+
2mi − 1
6(2mi + 3)
.
In the following simulation, we consider the case where ai = 1, bi = 2
and mi = 1 for i = 1, 2, 3. Let
F1(x) = (sin(x), sin(2x), . . . , sin(n1x))τ
,
F2(x) = (sin(x), sin(2x), . . . , sin(n2x))τ
,
F3(x) = (cos(x), cos(2x), . . . , cos(n3x))τ
.
For the cases of T = 102, 252, 402, 502, and 752, we then compute b
D(n),
b
σ2
(n), GCV(n) and the following quantities: for i = 1, 2, 3,
di(b
niG, b
niD) =
b
niG
b
niD
− 1, d4(b
nG, b
nD) =
b
D(b
nG)
b
D(b
nD)
− 1,
ASEi(b
nG) =
1
N
N
X
n=1
n
Fib
niG
(Zni)τ b
θi(b
hG) − gi(Zni)
o2
,
ASE4(b
nG) =
1
N
N
X
n=1
( 3
X
i=1
Fib
niG
(Zni)τ b
θib
niG
− gi(Zni)
)2
,
VAR(b
nG) = b
σ2
(b
nG) − σ2
,
where N = T − 2, σ2
= π2
12 = 0.822467, b
nG = (b
n1G, b
n2G, b
n3G)τ
, Zn1 =
Yn+1, Zn2 = Yn and Zn3 = Wn+2.
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-39-320.jpg)
![26 ESTIMATION IN NONLINEAR TIME SERIES
only provide an iterative estimation procedure for each gi since their
approach depends heavily on the Gibbs sampler.
Remark 2.6. Both Examples 2.1 and 2.2 demonstrate that the explicit
estimation procedure can not only provide some additional information
for further diagnostics and statistical inference but also produce mod-
els with better predictive power than is available from linear models.
For example, model (2.22) is more appropriate than a completely linear
model for the lynx data as mentioned in Remark 2.2. Moreover, model
(2.22) not only can be calculated at a new design point with the same
convenience as in linear models, but also provides the individual coeffi-
cients, which can be used to measure whether the individual influence
of each Yt−3+i for i = 0, 1, 2 can be negligible.
This section has assumed that the true conditional mean function is of a
semiparametric additive model of the form (2.2) and then developed the
orthogonal series based estimation procedure. As discussed in the next
section, we may approximate the true conditional mean function by the
additive form (2.2) even if the true conditional mean function may not
be expressed exactly as an additive form.
2.3 Semiparametric kernel estimation
As mentioned above (2.1), we are approximating the mean function
m(Ut, Vt) = E[Yt|Ut, Vt] by minimizing
E [Yt − m1(Ut, Vt)]
2
= E [Yt − µ − Uτ
t β − g(Vt)]
2
(2.23)
over a class of functions of the form m1(Ut, Vt) = µ + Uτ
t β + g(Vt) with
E[g(Vt)] = 0. Such a minimization problem is equivalent to minimizing
E [Yt − µ − Uτ
t β − g(Vt)]
2
= E
h
E
n
(Yt − µ − Uτ
t β − g(Vt))
2
|Vt
oi
over some (µ, β, g). This implies that g(Vt) = E [(Yt − µ − Uτ
t β)|Vt] and
µ = E[Yt − Uτ
t β] with β being given by
β = Σ−1
E [(Ut − E[Ut|Vt]) (Yt − E[Yt|Vt])] (2.24)
provided that the inverse Σ−1
= (E [(Ut − E[Ut|Vt]) (Ut − E[Ut|Vt])
τ
])
−1
exists. This also shows that m1(Ut, Vt) is identifiable under the assump-
tion of E[g(Vt)] = 0.
We now turn to estimation assuming that the data are available for
(Yt, Ut, Vt) for 1 ≤ t ≤ T. Since the definitions of the estimators to
be used later are quite involved notationally, we start by outlining the
main steps in establishing estimators for µ, β and g(·) in (2.1) and then
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-42-320.jpg)
![SEMIPARAMETRIC KERNEL ESTIMATION 27
gl(·), l = 1, 2, · · · , p in (2.2). In the following, we give an outline in three
steps.
Step 1: Estimating µ and g(·) assuming β to be known.
For each fixed β, since µ = E[Yt] − E[Uτ
t β] = µY − µτ
U β, the parameter
µ can be estimated by µ̂(β) = Y − U
τ
β, where µY = E[Yt], µU =
(µ
(1)
U , · · · , µ
(q)
U )τ
= E[Ut], Y = 1
T
PT
t=1 Yt and U = 1
T
PT
t=1 Ut.
Moreover, the conditional expectation
g(x) = g(x, β) = E [(Yt − µ − Uτ
t β)|Vt = x]
= E [(Yt − E[Yt] − (Ut − E[Ut])τ
β)|Vt = x] (2.25)
can be estimated by standard local linear estimation (Fan and Gijbels
1996) with ĝT (x, β) = â0(β) satisfying
(â0(β), â1(β)) = arg min
(a0, a1)∈R1
×Rp
(2.26)
×
T
X
t=1
Ỹt − Ũτ
t β − a0 − aτ
1(Vt − x)
2
Kt(x, b),
where Ỹt = Yt − Y , Ũt = (Ũ
(1)
t , · · · , Ũ
(q)
t )τ
= Ut − U and Kt(x, b) =
Qp
l=1 K
Vtl−xl
bl
, with b = bT = (b1, · · · , bp), bl = bl,T being a sequence
of bandwidths for the l-th covariate variable Vtl, tending to zero as T
tends to infinity, and K(·) is a bounded kernel function on R1
.
Step 2: Marginal integration to obtain g1, · · · , gp of (2.2).
The idea of the marginal integration estimator is best explained if g(·)
is itself additive, that is, if
g(Vt) = g(Vt1, · · · , Vtp) =
p
X
l=1
gl(Vtl).
Then, since E [gl (Vtl)] = 0 for l = 1, · · · , p, for k fixed
gk(xk) = E [g(Vt1, · · · , xk, · · · , Vtp)] .
An estimate of gk is obtained by keeping Vtk fixed at xk and then taking
the average over the remaining variables Vt1, · · · , Vt(k−1), Vt(k+1), · · · , Vtp.
This marginal integration operation can be implemented irrespective
of whether or not g(·) is additive. If the additivity does not hold, the
marginal integration amounts to a projection on the space of additive
functions of Vtl, l = 1, · · · , p taken with respect to the product measure of
Vtl, l = 1, · · · , p, obtaining the approximation ga(x, β) =
Pp
l=1 Pl,ω(Vtl, β),
which will be detailed below with β appearing linearly in the expression.
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-43-320.jpg)
![SEMIPARAMETRIC KERNEL ESTIMATION 29
where is the operation of the component-wise product, i.e., a1 b =
(a11b1, · · · , a1pbp) for a1 = (a11, · · · , a1p) and b = (b1, · · · , bp),
ΛT (β) =
λT,0(β)
ΛT,1(β)
, ΓT =
γT,00 ΓT,01
ΓT,10 ΓT,11
, (2.31)
where ΓT,10 = Γτ
T,01 = (γT,01, · · · , γT,0p)τ
and ΓT,11 is the p × p matrix
defined by γT,l1 l2
with l1, l2 = 1, · · · , p, in (2.27). Moreover, ΛT,1(β) =
(λT,1(β), . . . , λT,p(β))τ
with λT,l(β) as defined in (2.28). Analogously for
ΛT , we may define Λ
(0)
T and Λ(s)
in terms of λ(0)
and λ(s)
. Then taking
the first component with c = (1, 0, · · · , 0)τ
∈ R1+p
,
ĝT (x; β) = cτ
Γ−1
T (x)ΛT (x, β)
= cτ
Γ−1
(x)Λ
(0)
T (x) −
q
X
s=1
βscτ
Γ−1
(x)Λ
(s)
T (x)
= H
(0)
T (x) − βτ
HT (x),
where HT (x) = (H
(1)
T (x), · · · , H
(q)
T (x))τ
with H
(s)
T (x) = cτ
Γ−1
T (x)Λ(s)
(x),
1 ≤ s ≤ q. Clearly, H
(s)
T (x) is the local linear estimator of H(s)
(x) =
E
h
U
(s)
t − µ
(s)
U
|Vt = x
i
, 1 ≤ s ≤ q.
We now define U
(0)
t = Yt and µ
(0)
U = µY such that H(0)
(x) = E[(U
(0)
t −
µ
(0)
U )|Vt = x] = E[Yt − µY |Vt = x] and H(x) = (H(1)
(x), · · · , H(q)
(x))τ
=
E[(Ut − µU )|Vt = x]. It follows that g(x, β) = H(0)
(x) − βτ
H(x), which
equals g(x) under (2.1) irrespective of whether g itself is additive.
Step 2: Let w(−k)(·) be a weight function defined on Rp−1
such that
E
h
w(−k)(V
(−k)
t )
i
= 1, and wk(xk) = I[−Lk,Lk](xk) defined on R1
for
some large Lk 0, with
V
(−k)
t = (Vt1, · · · , Vt(k−1), Vt(k+1), · · · , Vtp),
where IA(x) is the conventional indicator function. In addition, we take
Vt(xk) = (Vt1, · · · , Vt(k−1), xk, Vt(k+1), · · · , Vtp).
For a given β, consider the marginal projection
Pk,w(xk, β) = E
h
g(Vt(xk); β)w(−k)
V
(−k)
t
i
wk(xk). (2.32)
It is easily seen that if g is additive as in (2.2), then for −Lk ≤ xk ≤
Lk, Pk,w(xk, β) = gk(xk) up to a constant since it is assumed that
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-45-320.jpg)
![30 ESTIMATION IN NONLINEAR TIME SERIES
E
h
w(−k)(V
(−k)
t )
i
= 1. In general, ga(x, β) =
Pp
l=1 Pl,w(xl, β) is an ad-
ditive marginal projection approximation to g(x) in (2.1) up to a con-
stant in the region x ∈
Qp
l=1[−Ll, Ll]. The quantity Pk,w(xk, β) can then
be estimated by the locally linear marginal integration estimator
b
Pk,w(xk, β) = T−1
T
X
t=1
ĝT (Vt(xk); β) w(−k)
V
(−k)
t
wk(xk)
= P̂
(0)
k,w(xk) −
q
X
s=1
βsP̂
(s)
k,w(xk) = P̂
(0)
k,w(xk) − βτ
P̂U
k,w(xk),
where P̂U
k,w(xk) =
P̂
(1)
k,w(xk), · · · , P̂
(q)
k,w(xk)
τ
, in which
P̂
(s)
k,w(xk) =
1
T
T
X
t=1
H
(s)
T (Vt(xk)) w(−k)
V
(−k)
t
wk(xk)
is the estimator of
P
(s)
k,w(xk) = E
h
H(s)
(Vt(xk))w(−k)
V
(−k)
t
i
wk(xk)
for 0 ≤ s ≤ q and PU
k,w(xk) =
P
(1)
k,w(xk), · · · , P
(q)
k,w(xk)
τ
is estimated
by P̂U
k,w(xk).
We add the weight function wk(xk) = I[−Lk, Lk](xk) in the definition of
P̂
(s)
k,w(xk), since we are interested only in the points of xk ∈ [−Lk, Lk]
for some large Lk. In practice, we may use a sample centered version of
P̂
(s)
k,w(xk) as the estimator of P
(s)
k,w(xk). Clearly, we have
Pk,w(xk, β) = P
(0)
k,w(xk) − βτ
PU
k,w(xk).
Thus, for every β, g(x) = g(x, β) of (2.1) (or rather the approximation
ga(x, β) if (2.2) does not hold) can be estimated by
b
b
g(x, β) =
p
X
l=1
b
Pl,w(xl, β) =
p
X
l=1
P̂
(0)
l,w (xl) − βτ
p
X
l=1
P̂U
l,w(xl). (2.33)
Step 3: We can finally obtain the least squares estimator of β by
b
β = arg min
β∈Rq
T
X
t=1
Ỹt − Ũτ
t β − b
b
g(Vt, β)
2
= arg min
β∈Rq
T
X
t=1
b
Y ∗
t −
b
U∗
t
τ
β
2
, (2.34)
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-46-320.jpg)
![SEMIPARAMETRIC KERNEL ESTIMATION 31
where b
Y ∗
t = Ỹt−
Pp
l=1 P̂
(0)
l,w (Vtl) and b
U∗
t = Ũt−
Pp
l=1 P̂U
l,w(Vtl). Therefore,
b
β =
T
X
t=1
b
U∗
t
b
U∗
t
τ
!−1 T
X
t=1
b
Y ∗
t
b
U∗
t
!
and b
µ = Y − b
βτ
U. (2.35)
We then insert b
β in â0(β) = ĝm,n(x, β) to obtain â0(b
β) = ĝm,n(x, b
β).
In view of this, the locally linear projection estimator of Pk(xk) can be
defined by
b
b
Pk,w(xk) =
1
T
T
X
t=1
ĝT (Vt(xk); b
β) w(−k)
V
(−k)
t
(2.36)
and for xk ∈ [−Lk, Lk] this would estimate gk(xk) up to a constant when
(2.2) holds. To ensure E[gk(Vtk)] = 0, we may rewrite
b
gk(xk) =
b
b
Pk,w(xk) − b
µP (k)
for the estimate of gk(xk) in (2.2), where b
µP (k) = 1
T
PT
t=1
b
b
Pk,w(Vtk).
For the proposed estimators, b
β, and
b
b
Pk,w(·), we establish some asymp-
totic distributions in Theorems 2.3 and 2.4 below under certain technical
conditions. To avoid introducing more mathematical details and symbols
before we state the main results, we relegate such conditions and their
justifications to Section 2.5 of this chapter.
We can now state the asymptotic properties of the marginal integration
estimators for both the parametric and nonparametric components. Let
U∗
t = Ut − µU −
Pp
l=1 PU
l,w(Vtl), Y ∗
t = Yt − µY −
Pp
l=1 P
(0)
l,w (Vtl) and
Rt = U∗
t (Y ∗
t − U∗
t
τ
β).
Theorem 2.3. Assume that Assumptions 2.5–2.9 listed in Section 2.5
hold. Then as T → ∞
√
T
h
(b
β − β) − µβ
i
→D N(0, Σβ) (2.37)
with µβ = BUU
−1
µB and Σβ = BUU
−1
ΣB
BUU
−1
τ
, where
BUU
= E [U∗
1 U∗
1
τ
], µB = E[R0] and ΣB = E [(R0 − µB) (R0 − µB)
τ
].
Furthermore, when (2.2) holds, we have
µβ = 0 and Σβ = BUU
−1
ΣB
BUU
−1
τ
, (2.38)
where ΣB = E [R0Rτ
0 ] with Rt = U∗
t εt, and εt = Yt − m1(Ut, Vt) =
Yt − µ − Uτ
t β − g(Vt).
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-47-320.jpg)
![32 ESTIMATION IN NONLINEAR TIME SERIES
Remark 2.7. Note that
p
X
l=1
P
(0)
l,w (Vtl) − βτ
p
X
l=1
PU
l,w(Vtl) =
p
X
l=1
P
(0)
l,w (Vtl) − βτ
PU
l,w(Vtl)
=
p
X
l=1
Pl,w(Vtl, β) ≡ ga(Vt, β).
Therefore Y ∗
t − U∗
t
τ
β = εt + g(Vt) − ga(Vt, β), where g(Vt) − ga(Vt, β)
is the residual due to the additive approximation. When (2.2) holds, it
means that g(Vt) in (2.1) has the expressions
g(Vt) =
p
X
l=1
gl(Vtl) =
p
X
l=1
Pl,w(Vtl, β) = ga(Vt, β)
and H(Vt) =
Pp
l=1 PU
l,w(Vtl), and hence Y ∗
t − U∗
t
τ
β = εt. As β min-
imizes L(β) = E [Yt − m1(Ut, Vt)]
2
, we have L0
(β) = 0 and E [tU∗
t ] =
E [ij (Ut − E[Ut|Vt])] = 0 when (2.2) holds. This implies E [Rt] = 0
and hence µβ = 0 in (2.37) when the marginal integration estimation
procedure is employed for the additive form of g(·).
In both theory and practice, we need to test whether H0 : β = β0 holds
for a given β0. The case where β0 ≡ 0 is an important one. Before we
state the next theorem, some additional notation is needed. Let
b
BUU
=
1
T
T
X
t=1
b
U∗
t (b
U∗
t )τ
, b
Z∗
t = Z̃t −
p
X
l=1
P̂U
l,w(Vtl),
b
µB =
1
T
T
X
t=1
b
Rt, b
Rt = b
U∗
t
Ŷ ∗
t −
b
U∗
t
τ
b
β
,
b
µβ =
b
BUU
−1
b
µB, b
Σβ =
b
BUU
−1
b
ΣB
b
BUU
−1
τ
,
in which b
ΣB is a consistent estimator of ΣB, defined simply by
b
ΣB =
(
1
T
PT
t=1( b
Rt − b
µB)( b
Rt − b
µB)τ
if (2.1) holds,
1
T
PT
t=1
b
Rt
b
Rτ
t if (2.2) holds.
It can be shown that both b
µβ and b
Σβ are consistent estimators of µβ
and Σβ, respectively.
We now state a corollary of Theorem 2.3 to test hypotheses about β.
Corollary 2.2. Assume that the conditions of Theorem 2.3 hold. Then
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-48-320.jpg)
![SEMIPARAMETRIC KERNEL ESTIMATION 33
as T → ∞
b
Σ
−1/2
β
√
T
h
(b
β − β) − b
µβ
i
→D N(0, Iq),
T
h
(b
β − β) − b
µβ
iτ
b
Σ−1
β
h
(b
β − β) − b
µβ
i
→D χ2
q.
Furthermore, when (2.2) holds, we have as T → ∞,
b
Σ
−1/2
β
√
T
b
β − β
→D N(0, Iq),
√
T(b
β − β)
τ
b
Σ−1
β
√
T(b
β − β)
→D χ2
q.
The proof of Theorem 2.3 is relegated to Section 2.5 while the proof of
Corollary 2.2 is straightforward and therefore omitted.
Next we state the following theorem for the nonparametric component.
Theorem 2.4. Assume that Assumptions 2.5–2.9 listed in Section 2.5
hold. Then for xk ∈ [−Lk, Lk],
p
T bk
b
b
Pk,w(xk) − Pk,w(xk) − bias1k
→D N(0, var1k), (2.39)
where
bias1k =
1
2
b2
k µ2(K)
Z
w(−k)(x(−k)
)f(−k)(x(−k)
)
∂2
g(x, β)
∂x2
k
dx(−k)
and
var1k = J
Z
V (x, β)
[w(−k)(x(−k)
)f(−k)(x(−k)
)]2
f(x)
dx(−k)
with J =
R
K2
(u)du, µ2(K) =
R
u2
K(u)du,
g(x, β) = E
Yij − µ − Zτ
ijβ
|Xij = x
,
and V (x, β) = E
h
Yij − µ − Zτ
ijβ − g(x, β)
2
|Xij = x
i
.
Furthermore, assume that (2.2) holds and that E
h
w(−k)(X
(−k)
ij )
i
= 1.
Then as T → ∞
p
T bk (b
gk(xk) − gk(xk) − bias2k) →D N(0, var2k), (2.40)
where
bias2k =
1
2
b2
k µ2(K)
∂2
gk(xk)
∂x2
k
,
var2k = J
Z
V (x, β)
[w(−k)(x(−k)
)f(−k)(x(−k)
)]2
f(x)
dx(−k)
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-49-320.jpg)
![34 ESTIMATION IN NONLINEAR TIME SERIES
with V (x, β) = E
h
Yij − µ − Zτ
ijβ −
Pp
k=1 gk(xk)
2
|Xij = x
i
.
The proof of Theorem 2.4 is relegated to Section 2.5. Theorems 2.3
and 2.4 may be applied to estimate various additive models such as
model (2.2). In the following example, we apply the proposed estimation
procedure to determine whether a partially linear time series model is
more appropriate than either a completely linear time series model or a
purely nonparametric time series model for a given set of real data.
Example 2.3: In this example, we continue analyzing the Canadian lynx
data with yt = log10{number of lynx trapped in the year (1820 + t)} for
t = 1, 2, ..., 114 (T = 114). Let q = p = 1, Ut = yt−1, Vt = yt−2 and
Yt = yt in model (2.1). We then select yt as the present observation and
both yt−1 and yt−2 as the candidates of the regressors.
Model (2.1) reduces to a partially linear time series model of the form
yt = βyt−1 + g(yt−2) + et. (2.41)
In addition to estimating β and g(·), we also propose to choose a suitable
bandwidth h based on a nonparametric cross–validation (CV) selection
criterion. For i = 1, 2, define
b
gi,t(·) = b
gi,t(·, h)
=
1
T − 3
PT
s=3,s6=t K
·−ys−2
h
ys+1−i
b
πh,t(·)
, (2.42)
where b
πh,t(·) = 1
T −3
PT
s=3,s6=t K
·−ys−2
h
.
We now define a new LS estimate e
β(h) of β by minimizing
T −3
X
t=1
{yt − βyt−1 − b
g1,t(yt−2) − βb
g2,t(yt−2)}
2
.
The CV selection function is then defined by
CV (h) =
1
T − 3
T
X
t=3
n
yt −
h
e
β(h)yt−1 + b
g1,t(yt−2) − e
β(h)b
g2,t(yt−2)
io2
.
(2.43)
For Example 2.3, we choose
K(x) =
1
√
2π
e− x2
2 and H114 = [0.3 · 114− 7
30 , 1.1 · 114− 1
6 ].
Before selecting the bandwidth interval H114, we actually calculated the
© 2007 by Taylor Francis Group, LLC](https://image.slidesharecdn.com/91477-250228000017-55380d3b/85/Nonlinear-time-series-semiparametric-and-nonparametric-methods-1st-Edition-Jiti-Gao-50-320.jpg)
Nonlinear time series semiparametric and nonparametric methods 1st Edition Jiti Gao
- 1. Visit https://ebookultra.com to download the full version and explore more ebooks Nonlinear time series semiparametric and nonparametric methods 1st Edition Jiti Gao _____ Click the link below to download _____ https://ebookultra.com/download/nonlinear-time-series- semiparametric-and-nonparametric-methods-1st-edition- jiti-gao/ Explore and download more ebooks at ebookultra.com
- 2. Here are some suggested products you might be interested in. Click the link to download Nonlinear Modeling of Economic and Financial Time Series 1st Edition Fredj Jawadi https://ebookultra.com/download/nonlinear-modeling-of-economic-and- financial-time-series-1st-edition-fredj-jawadi/ Applied Nonlinear Time Series Analysis Applications in Physics Physiology and Finance 1st Edition Michael Small https://ebookultra.com/download/applied-nonlinear-time-series- analysis-applications-in-physics-physiology-and-finance-1st-edition- michael-small/ Long Memory Time Series Theory and Methods 1st Edition Wilfredo Palma https://ebookultra.com/download/long-memory-time-series-theory-and- methods-1st-edition-wilfredo-palma/ Nonlinear Spatio Temporal Dynamics and Chaos in Semiconductors Cambridge Nonlinear Science Series 1st Edition Eckehard Scholl https://ebookultra.com/download/nonlinear-spatio-temporal-dynamics- and-chaos-in-semiconductors-cambridge-nonlinear-science-series-1st- edition-eckehard-scholl/
- 3. Semiparametric regression 1st Edition David Ruppert https://ebookultra.com/download/semiparametric-regression-1st-edition- david-ruppert/ Solutions Manual to Nonparametric Methods in Statistics SAS Applications 1st Edition Olga Korosteleva https://ebookultra.com/download/solutions-manual-to-nonparametric- methods-in-statistics-sas-applications-1st-edition-olga-korosteleva/ Nonlinear Dynamics with Polymers Fundamentals Methods and Applications 1st Edition John A. Pojman https://ebookultra.com/download/nonlinear-dynamics-with-polymers- fundamentals-methods-and-applications-1st-edition-john-a-pojman/ Iterative regularization methods for nonlinear ill posed problems 1st Edition Kaltenbacher https://ebookultra.com/download/iterative-regularization-methods-for- nonlinear-ill-posed-problems-1st-edition-kaltenbacher/ Methods in Brain Connectivity Inference through Multivariate Time Series Analysis 1 Har/Cdr Edition Koichi Sameshima https://ebookultra.com/download/methods-in-brain-connectivity- inference-through-multivariate-time-series-analysis-1-har-cdr-edition- koichi-sameshima/
- 5. Nonlinear time series semiparametric and nonparametric methods 1st Edition Jiti Gao Digital Instant Download Author(s): Jiti Gao ISBN(s): 9781584886136, 1584886137 Edition: 1 File Details: PDF, 4.63 MB Year: 2007 Language: english
- 6. Nonlinear Time Series Semiparametric and Nonparametric Methods Monographs on Statistics and Applied Probability 108 C6137_FM.indd 1 2/6/07 2:09:49 PM © 2007 by Taylor & Francis Group, LLC
- 7. MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY General Editors V. Isham, N. Keiding, T. Louis, S. Murphy, R. L. Smith, and H. Tong 1 Stochastic Population Models in Ecology and Epidemiology M.S. Barlett (1960) 2 Queues D.R. Cox and W.L. Smith (1961) 3 Monte Carlo Methods J.M. Hammersley and D.C. Handscomb (1964) 4 The Statistical Analysis of Series of Events D.R. Cox and P.A.W. Lewis (1966) 5 Population Genetics W.J. Ewens (1969) 6 Probability, Statistics and Time M.S. Barlett (1975) 7 Statistical Inference S.D. Silvey (1975) 8 The Analysis of Contingency Tables B.S. Everitt (1977) 9 Multivariate Analysis in Behavioural Research A.E. Maxwell (1977) 10 Stochastic Abundance Models S. Engen (1978) 11 Some Basic Theory for Statistical Inference E.J.G. Pitman (1979) 12 Point Processes D.R. Cox and V. Isham (1980) 13 Identification of Outliers D.M. Hawkins (1980) 14 Optimal Design S.D. Silvey (1980) 15 Finite Mixture Distributions B.S. Everitt and D.J. Hand (1981) 16 Classification A.D. Gordon (1981) 17 Distribution-Free Statistical Methods, 2nd edition J.S. Maritz (1995) 18 Residuals and Influence in Regression R.D. Cook and S. Weisberg (1982) 19 Applications of Queueing Theory, 2nd edition G.F. Newell (1982) 20 Risk Theory, 3rd edition R.E. Beard, T. Pentikäinen and E. Pesonen (1984) 21 Analysis of Survival Data D.R. Cox and D. Oakes (1984) 22 An Introduction to Latent Variable Models B.S. Everitt (1984) 23 Bandit Problems D.A. Berry and B. Fristedt (1985) 24 Stochastic Modelling and Control M.H.A. Davis and R. Vinter (1985) 25 The Statistical Analysis of Composition Data J. Aitchison (1986) 26 Density Estimation for Statistics and Data Analysis B.W. Silverman (1986) 27 Regression Analysis with Applications G.B. Wetherill (1986) 28 Sequential Methods in Statistics, 3rd edition G.B. Wetherill and K.D. Glazebrook (1986) 29 Tensor Methods in Statistics P. McCullagh (1987) 30 Transformation and Weighting in Regression R.J. Carroll and D. Ruppert (1988) 31 Asymptotic Techniques for Use in Statistics O.E. Bandorff-Nielsen and D.R. Cox (1989) 32 Analysis of Binary Data, 2nd edition D.R. Cox and E.J. Snell (1989) 33 Analysis of Infectious Disease Data N.G. Becker (1989) 34 Design and Analysis of Cross-Over Trials B. Jones and M.G. Kenward (1989) 35 Empirical Bayes Methods, 2nd edition J.S. Maritz and T. Lwin (1989) 36 Symmetric Multivariate and Related Distributions K.T. Fang, S. Kotz and K.W. Ng (1990) 37 Generalized Linear Models, 2nd edition P. McCullagh and J.A. Nelder (1989) 38 Cyclic and Computer Generated Designs, 2nd edition J.A. John and E.R. Williams (1995) 39 Analog Estimation Methods in Econometrics C.F. Manski (1988) 40 Subset Selection in Regression A.J. Miller (1990) 41 Analysis of Repeated Measures M.J. Crowder and D.J. Hand (1990) 42 Statistical Reasoning with Imprecise Probabilities P. Walley (1991) 43 Generalized Additive Models T.J. Hastie and R.J. Tibshirani (1990) C6137_FM.indd 2 2/6/07 2:09:49 PM © 2007 by Taylor & Francis Group, LLC
- 8. 44 Inspection Errors for Attributes in Quality Control N.L. Johnson, S. Kotz and X. Wu (1991) 45 The Analysis of Contingency Tables, 2nd edition B.S. Everitt (1992) 46 The Analysis of Quantal Response Data B.J.T. Morgan (1992) 47 Longitudinal Data with Serial Correlation—A State-Space Approach R.H. Jones (1993) 48 Differential Geometry and Statistics M.K. Murray and J.W. Rice (1993) 49 Markov Models and Optimization M.H.A. Davis (1993) 50 Networks and Chaos—Statistical and Probabilistic Aspects O.E. Barndorff-Nielsen, J.L. Jensen and W.S. Kendall (1993) 51 Number-Theoretic Methods in Statistics K.-T. Fang and Y. Wang (1994) 52 Inference and Asymptotics O.E. Barndorff-Nielsen and D.R. Cox (1994) 53 Practical Risk Theory for Actuaries C.D. Daykin, T. Pentikäinen and M. Pesonen (1994) 54 Biplots J.C. Gower and D.J. Hand (1996) 55 Predictive Inference—An Introduction S. Geisser (1993) 56 Model-Free Curve Estimation M.E. Tarter and M.D. Lock (1993) 57 An Introduction to the Bootstrap B. Efron and R.J. Tibshirani (1993) 58 Nonparametric Regression and Generalized Linear Models P.J. Green and B.W. Silverman (1994) 59 Multidimensional Scaling T.F. Cox and M.A.A. Cox (1994) 60 Kernel Smoothing M.P. Wand and M.C. Jones (1995) 61 Statistics for Long Memory Processes J. Beran (1995) 62 Nonlinear Models for Repeated Measurement Data M. Davidian and D.M. Giltinan (1995) 63 Measurement Error in Nonlinear Models R.J. Carroll, D. Rupert and L.A. Stefanski (1995) 64 Analyzing and Modeling Rank Data J.J. Marden (1995) 65 Time Series Models—In Econometrics, Finance and Other Fields D.R. Cox, D.V. Hinkley and O.E. Barndorff-Nielsen (1996) 66 Local Polynomial Modeling and its Applications J. Fan and I. Gijbels (1996) 67 Multivariate Dependencies—Models, Analysis and Interpretation D.R. Cox and N. Wermuth (1996) 68 Statistical Inference—Based on the Likelihood A. Azzalini (1996) 69 Bayes and Empirical Bayes Methods for Data Analysis B.P. Carlin and T.A Louis (1996) 70 Hidden Markov and Other Models for Discrete-Valued Time Series I.L. Macdonald and W. Zucchini (1997) 71 Statistical Evidence—A Likelihood Paradigm R. Royall (1997) 72 Analysis of Incomplete Multivariate Data J.L. Schafer (1997) 73 Multivariate Models and Dependence Concepts H. Joe (1997) 74 Theory of Sample Surveys M.E. Thompson (1997) 75 Retrial Queues G. Falin and J.G.C. Templeton (1997) 76 Theory of Dispersion Models B. Jørgensen (1997) 77 Mixed Poisson Processes J. Grandell (1997) 78 Variance Components Estimation—Mixed Models, Methodologies and Applications P.S.R.S. Rao (1997) 79 Bayesian Methods for Finite Population Sampling G. Meeden and M. Ghosh (1997) 80 Stochastic Geometry—Likelihood and computation O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout (1998) 81 Computer-Assisted Analysis of Mixtures and Applications— Meta-analysis, Disease Mapping and Others D. Böhning (1999) 82 Classification, 2nd edition A.D. Gordon (1999) C6137_FM.indd 3 2/6/07 2:09:49 PM © 2007 by Taylor & Francis Group, LLC
- 9. 83 Semimartingales and their Statistical Inference B.L.S. Prakasa Rao (1999) 84 Statistical Aspects of BSE and vCJD—Models for Epidemics C.A. Donnelly and N.M. Ferguson (1999) 85 Set-Indexed Martingales G. Ivanoff and E. Merzbach (2000) 86 The Theory of the Design of Experiments D.R. Cox and N. Reid (2000) 87 Complex Stochastic Systems O.E. Barndorff-Nielsen, D.R. Cox and C. Klüppelberg (2001) 88 Multidimensional Scaling, 2nd edition T.F. Cox and M.A.A. Cox (2001) 89 Algebraic Statistics—Computational Commutative Algebra in Statistics G. Pistone, E. Riccomagno and H.P. Wynn (2001) 90 Analysis of Time Series Structure—SSA and Related Techniques N. Golyandina, V. Nekrutkin and A.A. Zhigljavsky (2001) 91 Subjective Probability Models for Lifetimes Fabio Spizzichino (2001) 92 Empirical Likelihood Art B. Owen (2001) 93 Statistics in the 21st Century Adrian E. Raftery, Martin A. Tanner, and Martin T. Wells (2001) 94 Accelerated Life Models: Modeling and Statistical Analysis Vilijandas Bagdonavicius and Mikhail Nikulin (2001) 95 Subset Selection in Regression, Second Edition Alan Miller (2002) 96 Topics in Modelling of Clustered Data Marc Aerts, Helena Geys, Geert Molenberghs, and Louise M. Ryan (2002) 97 Components of Variance D.R. Cox and P.J. Solomon (2002) 98 Design and Analysis of Cross-Over Trials, 2nd Edition Byron Jones and Michael G. Kenward (2003) 99 Extreme Values in Finance, Telecommunications, and the Environment Bärbel Finkenstädt and Holger Rootzén (2003) 100 Statistical Inference and Simulation for Spatial Point Processes Jesper Møller and Rasmus Plenge Waagepetersen (2004) 101 Hierarchical Modeling and Analysis for Spatial Data Sudipto Banerjee, Bradley P. Carlin, and Alan E. Gelfand (2004) 102 Diagnostic Checks in Time Series Wai Keung Li (2004) 103 Stereology for Statisticians Adrian Baddeley and Eva B. Vedel Jensen (2004) 104 Gaussian Markov Random Fields: Theory and Applications H avard Rue and Leonhard Held (2005) 105 Measurement Error in Nonlinear Models: A Modern Perspective, Second Edition Raymond J. Carroll, David Ruppert, Leonard A. Stefanski, and Ciprian M. Crainiceanu (2006) 106 Generalized Linear Models with Random Effects: Unified Analysis via H-likelihood Youngjo Lee, John A. Nelder, and Yudi Pawitan (2006) 107 Statistical Methods for Spatio-Temporal Systems Bärbel Finkenstädt, Leonhard Held, and Valerie Isham (2007) 108 Nonlinear Time Series: Semiparametric and Nonparametric Methods Jiti Gao (2007) C6137_FM.indd 4 2/6/07 2:09:49 PM © 2007 by Taylor & Francis Group, LLC
- 10. Jiti Gao The University of Western Australia Perth, Australia Nonlinear Time Series Semiparametric and Nonparametric Methods Monographs on Statistics and Applied Probability 108 Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business C6137_FM.indd 5 2/6/07 2:09:49 PM © 2007 by Taylor & Francis Group, LLC
- 11. Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2007 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑10: 1‑58488‑613‑7 (Hardcover) International Standard Book Number‑13: 978‑1‑58488‑613‑6 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com C6137_FM.indd 6 2/6/07 2:09:50 PM © 2007 by Taylor & Francis Group, LLC
- 12. Contents Preface v 1 Introduction 1 1.1 Preliminaries 1 1.2 Examples and models 1 1.3 Bibliographical notes 14 2 Estimation in Nonlinear Time Series 15 2.1 Introduction 15 2.2 Semiparametric series estimation 18 2.3 Semiparametric kernel estimation 26 2.4 Semiparametric single–index estimation 35 2.5 Technical notes 39 2.6 Bibliographical notes 47 3 Nonlinear Time Series Specification 49 3.1 Introduction 49 3.2 Testing for parametric mean models 50 3.3 Testing for semiparametric variance models 65 3.4 Testing for other semiparametric models 68 3.5 Technical notes 72 3.6 Bibliographical notes 80 iii © 2007 by Taylor & Francis Group, LLC
- 13. iv CONTENTS 4 Model Selection in Nonlinear Time Series 83 4.1 Introduction 83 4.2 Semiparametric cross–validation method 86 4.3 Semiparametric penalty function method 92 4.4 Examples and applications 95 4.5 Technical notes 105 4.6 Bibliographical notes 110 5 Continuous–Time Diffusion Models 111 5.1 Introduction 111 5.2 Nonparametric and semiparametric estimation 116 5.3 Semiparametric specification 123 5.4 Empirical comparisons 130 5.5 Technical notes 146 5.6 Bibliographical notes 156 6 Long–Range Dependent Time Series 157 6.1 Introductory results 157 6.2 Gaussian semiparametric estimation 159 6.3 Simultaneous semiparametric estimation 161 6.4 LRD stochastic volatility models 169 6.5 Technical notes 189 6.6 Bibliographical notes 191 7 Appendix 193 7.1 Technical lemmas 193 7.2 Asymptotic normality and expansions 198 References 209 Author Index 230 Subject Index 235 © 2007 by Taylor & Francis Group, LLC
- 14. Preface During the past two decades or so, there has been a lot of interest in both theoretical and empirical analysis of nonlinear time series data. Models and methods used have been based initially on parametric non- linear or nonparametric time series models. Such parametric nonlinear models and related methods may be too restrictive in many cases. This leads to various nonparametric techniques being used to model nonlinear time series data. The main advantage of using nonparametric methods is that the data may be allowed to speak for themselves in the sense of determining the form of mathematical relationships between time series variables. In modelling nonlinear time series data one of the tasks is to study the structural relationship between the present observation and the history of the data set. The problem then is to fit a high dimensional surface to a nonlinear time series data set. While nonparametric tech- niques appear to be feasible and flexible, there is a serious problem: the so-called curse of dimensionality. For the independent and identically distributed case, this problem has been discussed and illustrated in the literature. Since about twenty years ago, various semiparametric methods and mod- els have been proposed and studied extensively in the economics and statistics literature. Several books and many papers have devoted their attention on semiparametric modelling of either independent or depend- ent time series data. The concentration has also been mainly on esti- mation and testing of both the parametric and nonparametric compon- ents in a semiparametric model. Interest also focuses on estimation and testing of conditional distributions using semiparametric methods. Im- portant and useful applications include estimation and specification of conditional moments in continuous–time diffusion models. In addition, recent studies show that semiparametric methods and models may be ap- plied to solve dimensionality reduction problems arising from using fully nonparametric models and methods. These include: (i) semiparametric single–index and projection pursuit modelling; (ii) semiparametric ad- ditive modelling; (iii) partially linear time series regression modelling; and (iv) semiparametric time series variable selection. v © 2007 by Taylor & Francis Group, LLC
- 15. vi PREFACE Although semiparametric methods in time series have recently been men- tioned in several books, this monograph hopes to bring an up–to–date description of the recent development in semiparametric estimation, spe- cification and selection of time series data as discussed in Chapters 1–4. In addition, semiparametric estimation and specification methods dis- cussed in Chapters 2 and 3 are applied to a class of nonlinear continuous– time models with real data analysis in Chapter 5. Chapter 6 examines some newly proposed semiparametric estimation procedures for time series data with long–range dependence. While this monograph involves only climatological and financial data in Chapters 1 and 4–6, the newly proposed estimation and specifications methods are applicable to model sets of real data in many disciplines. This monograph can be used to serve as a textbook to senior undergraduate and postgraduate students as well as other researchers who are interested in the field of nonlinear time series using semiparametric methods. This monograph concentrates on various semiparametric methods in model estimation, specification testing and selection of nonlinear time series data. The structure of this monograph is organized as follows: (a) Chapter 2 systematically studies estimation problems of various param- eters and functions involved in semiparametric models. (b) Chapter 3 dis- cusses parametric or semiparametric specification of various conditional moments. (c) As an alternative to model specification, Chapter 4 exam- ines the proposed parametric, nonparametric and semiparametric model selection criteria to show how a time series data should be modelled using the best available model among all possible models. (d) Chapter 5 considers some of the latest results about semiparametric methods in model estimation and specification testing of continuous–time models. (e) Chapter 6 gives a short summary of recent semiparametric estima- tion methods for long–range dependent time series and then discusses some of the latest theoretical and empirical results using a so–called simultaneous semiparametric estimation method. While the author of this monograph has tried his best to reflect the research work of many researchers in the field, some other closely re- lated studies may be inevitably omitted in this monograph. The author therefore apologizes for any omissions. I would like to thank anyone who has encouraged and supported me to finish the monograph. In particular, I would like to thank Vo Anh, Isa- bel Casas, Songxi Chen, Iréne Gijbels, Chris Heyde, Yongmiao Hong, Maxwell King, Qi Li, Zudi Lu, Peter Phillips, Peter Robinson, Dag Tjøstheim, Howell Tong and Qiying Wang for many helpful and stimu- lating discussions. Thanks also go to Manuel Arapis, Isabel Casas, Chao- hua Dong, Kim Hawthorne and Jiying Yin for computing assistance as © 2007 by Taylor & Francis Group, LLC
- 16. PREFACE vii well as to Isabel Casas and Jiying Yin for editorial assistance. I would also like to acknowledge the generous support and inspiration of my col- leagues in the School of Mathematics and Statistics at The University of Western Australia. Since the beginning of 2002, my research in the field has been supported financially by the Australian Research Council Discovery Grants Program. My final thanks go to my wife, Mrs Qun Jiang, who unselfishly put my interest in the top priority while sacrificing hers in the process, for her constant support and understanding, and two lovely sons, Robert and Thomas, for their cooperation. Without such support and cooperation, it would not be possible for me to finish the writing of this monograph. Jiti Gao Perth, Australia 30 September 2006 © 2007 by Taylor & Francis Group, LLC
- 17. CHAPTER 1 Introduction 1.1 Preliminaries This monograph basically discusses semiparametric methods in model estimation, specification testing and selection of nonlinear time series data. We use the term semiparametric for models which are semipara- metric partially linear models or other semiparametric regression models as discussed in Chapters 2–6, in particular Chapters 2 and 5. We also use the word semiparametric for methods which are semiparametric es- timation and testing methods as discussed in Chapters 2–6, particularly in Chapters 3 and 6. Meanwhile, we also use the term nonparametric for models and methods which are either nonparametric models or non- parametric methods or both as considered in Chapters 2–5. 1.2 Examples and models Let (Y, X) be a d + 1–dimensional vector of time series variables with Y being the response variable and X the vector of d–dimensional covari- ates. We assume that both X and Y are continuous random variables with π(x) as the marginal density function of X, f(y|x) being the condi- tional density function of Y given X = x and f(x, y) as the joint density function. Let m(x) = E[Y |X = x] denote the conditional mean of Y given X = x. Let {(Yt, Xt) : 1 ≤ t ≤ T} be a sequence of observa- tions drawn from the joint density function f(x, y). We first consider a partially linear model of the form Yt = E[Yt|Xt] + et = m(Xt) + et = Uτ t β + g(Vt) + et, (1.1) where Xt = (Uτ t , V τ t )τ , m(Xt) = E[Yt|Xt], and et = Yt − E[Yt|Xt] is the error process and allowed to depend on Xt. In model (1.1), Ut and Vt are allowed to be two different vectors of time series variables. In practice, a crucial problem is how to identify Ut and Vt before applying model (1.1) to model sets of real data. For some cases, the identification problem can be solved easily by using empirical studies. For example, when modelling 1 © 2007 by Taylor & Francis Group, LLC
- 18. 2 INTRODUCTION electricity sales, it is natural to assume the impact of temperature on electricity consumption to be nonlinear, as both high and low temper- atures lead to increased consumption, whereas a linear relationship may be assumed for other regressors. See Engle et al. (1986). Similarly, when modelling the dependence of earnings on qualification and labour market experience variables, existing studies (see Härdle, Liang and Gao 2000) show that the impact of qualification on earnings to be linear, while the dependence of earnings on labour market experience appears to be non- linear. For many other cases, however, the identification problem should be solved theoretically before using model (1.1) and will be discussed in detail in Chapter 4. Existing studies show that although partially linear time series modelling may not be capable of reducing the nonparametric time series regression into a sum of one-dimensional nonparametric functions of individual lags, they can reduce the dimensionality significantly for some cases. Moreover, a feature of partially linear time series modelling is that it takes the true structure of the time series data into account and avoids neglecting some existing information on the linearity of the data. We then consider a different partially linear model of the form Yt = Xτ t β + g(Xt) + et, (1.2) where Xt = (Xt1, · · · , Xtd)τ is a vector of time series, β = (β1, · · · , βd)τ is a vector of unknown parameters, g(·) is an unknown function and can be viewed as a misspecification error, and {et} is a sequence of either dependent errors or independent and identically distributed (i.i.d.) errors. In model (1.2), the error process {et} is allowed to depend on {Xt}. Obviously, model (1.2) may not be viewed as a special form of model (1.1). The main motivation for systematically studying model (1.2) is that partially linear model (1.2) can play a significant role in modelling some nonlinear problems when the linear regression normally fails to appropriately model nonlinear phenomena. We therefore suggest using partially linear model (1.2) to model nonlinear phenomena, and then determine whether the nonlinearity is significant for a given data set (Xt, Yt). In addition, some special cases of model (1.2) have already been considered in the econometrics and statistics literature. We show that several special forms of models (1.1) and (1.2) have some important applications. We present some interesting examples and models, which are either spe- cial forms or extended forms of models (1.1) and (1.2). Example 1.1 (Partially linear time series error models): Consider a partially linear model for trend detection in an annual mean temperature © 2007 by Taylor & Francis Group, LLC
- 19. EXAMPLES AND MODELS 3 series of the form Yt = Uτ t β + g t T + et, (1.3) where {Yt} is the mean temperature series of interest, Ut = (Ut1, · · · , Utq)τ is a vector of q–explanatory variables, such as the southern oscillation index (SOI), t is time in years, β is a vector of unknown coefficients for the explanatory variables, g(·) is an unknown smooth function of time representing the trend, and {et} represents a sequence of stationary time series errors with E[et] = 0 and 0 var[et] = σ2 ∞. Recently, Gao and Hawthorne (2006) have considered some estimation and testing problems for the trend function of the temperature series model (1.3). Applying an existing method from Härdle, Liang and Gao (2000) to two global temperature series (http://www.cru.uea.ac.uk/cru/data/), Gao and Hawthorne (2006) have shown that a nonlinear trend looks feasible for each of the temperature series. Figure 1 of Gao and Hawthorne (2006) shows the annual mean series of the global temperature series from 1867– 1993 and then from 1867–2001. 1860 1880 1900 1920 1940 1960 1980 −0.4 −0.2 0.0 0.2 Years Temperature Figure 1.1 The light line is the global temperature series for 1867–1993, while the solid curve is the estimated trend. Figure 1.1 shows that the trend estimate appears to be distinctly non- linear. Figure 1.2 displays the partially linear model fitting to the data set. The inclusion of the linear SOI component is warranted by the in- terannual fluctuations of the temperature series. Figures 1.1 and 1.2 also © 2007 by Taylor Francis Group, LLC
- 20. 4 INTRODUCTION 1880 1900 1920 1940 1960 1980 −0.4 −0.2 0.0 0.2 0.4 Figure 1.2 The solid line is the global temperature series for 1867–1993, while the dashed line is the estimated series. show that the smooth trend component captures the nonlinear complex- ity inherent in the long term underlying trend. The mean function fitted to the data is displayed in Figure 1.3. The estimated series for the up- dated series is similar in stucture to that for the truncated series from 1867–1993. The hottest year on record, 1998, is represented reasonably. Similar to Figures 1.1 and 1.2, a kind of nonlinear complexity inherent in the long term trend is captured in Figure 1.3. In addition, model (1.3) may be used to model long–range dependent (LRD) and nonstationary data. Existing studies show that there are both LRD and nonstationary properties inherited in some financial and environmental data (see Anh et al. 1999; Mikosch and Starica 2004) for example. Standard Poor’s 500 is a market–value weighted price of 500 stocks. The values in Figure 1.4 are from January 2, 1958 to July 29, 2005. The key findings of such existing studies suggest that in order to avoid misrepresenting the mean function or the conditional mean function of a long–range dependent data, we should let the data ‘speak’ for them- selves in terms of specifying the true form of the mean function or the conditional mean function. This is particularly important for data with © 2007 by Taylor Francis Group, LLC
- 21. EXAMPLES AND MODELS 5 1880 1900 1920 1940 1960 1980 2000 −0.4 −0.2 0.0 0.2 0.4 Figure 1.3 The solid line is the global temperature series for 1867–2001, while the broken line is the estimated series. Figure 1.4 SP 500: January 2, 1958 to July 29, 2005. © 2007 by Taylor Francis Group, LLC
- 22. 6 INTRODUCTION long–range dependence, because unnecessary nonlinearity or complexity in mean functions may cause erroneous LRD. Such issues may be ad- dressed using a general model specification procedure to be discussed in Chapter 3 below. Example 1.2 (Partially linear autoregressive models): Let {ut} be a sequence of time series variables, Yt = ut, Ut = (ut−1, . . . , ut−q)τ , and Vt = (vt1, . . . , vtp)τ be a vector of time series variables. Now model (1.1) is a partially linear autoregressive model of the form ut = q X i=1 βiut−i + g(vt1, . . . , vtp) + et. (1.4) When {vt} is a sequence of time series variables, Vt = (vt−1, . . . , vt−p)τ , Yt = vt, and Ut = (ut1, . . . , utq)τ be a vector of time series variables, model (1.1) is a partially nonlinear autoregressive model of the form vt = q X i=1 αiuti + g(vt−1, . . . , vt−p) + et. (1.5) In theory, various estimation and testing problems for models (1.4) and (1.5) have already been discussed in the literature. See for example, Robinson (1988), Tjøstheim (1994), Teräsvirta, Tjøstheim and Granger (1994), Gao and Liang (1995), Härdle, Lütkepohl and Chen (1997), Gao (1998), Härdle, Liang and Gao (2000), Gao and Yee (2000), and Gao, Tong and Wolff (2002a, 2002b), Gao and King (2005), and Li and Racine (2006). In practice, models (1.4) and (1.5) have various applications. For ex- ample, Fisheries Western Australia (WA) manages commercial fishing in WA. Simple Catch and Effort statistics are often used in regulating the amount of fish that can be caught and the number of boats that are licensed to catch them. The establishment of the relationship between the Catch (in kilograms) and Effort (the number of days the fishing ves- sels spent at sea) is very important both commerically and ecologically. This example considers using a time series model to fit the relationship between catch and effort. The historical monthly fishing data set from January 1976 to December 1999 available to us comes from the Fisheries WA Catch and Effort Statistics (CAES) database. Existing studies from the Fisheries suggest that the relationship between the catch and the effort does not look linear while the dependence of the current catch on the past catch appears to be linear. This suggests using a partially linear model of the form Ct = β1Ct−1 + . . . + βqCt−q + g(Et, Et−1, . . . , Et−p+1) + et, (1.6) © 2007 by Taylor Francis Group, LLC
- 23. EXAMPLES AND MODELS 7 where {et} is a sequence of random errors, Ct and Et represent the catch and the effort at time t, respectively, and g(·) is a nonlinear function. In the detailed computation, we use the transformed data Yt = log10(Ct) and Xt = log10(Et) satisfying the following model Yt+r = β1Yt+r−1 + . . . + βqYt+r−q + g(Xt+r, . . . , Xt+r−p+1) + et, (1.7) where r = max(p, q) and {et} is a random error with zero mean and finite variance. Gao and Tong (2004) proposed a semiparametric variable selection pro- cedure for model (1.1) and then applied the proposed semiparametric selection method to produce the corresponding plots in Figure 1 of their paper. Model (1.1) also covers the following important classes of partially linear time series models as given in Example 1.3 below. Example 1.3 (Population biology model): Consider a partially linear time series model of the form Yt = βYt−1 + g(Yt−τ ) + et, (1.8) where |β| 1 is an unknown parameter, g(·) is a smooth function such that {Yt} is strictly stationary, τ ≥ 2 is an integer, and {et} is a sequence of strictly stationary errors. When g(x) = bx 1+xk , we have a population biology model of the form Yt = βYt−1 + bYt−τ 1 + Y k t−τ + et, (1.9) where 0 β 1, b 0, τ 1 and k ≥ 1 are parameters. The motivation for studying this model stems from the research of population biology model and the Mackey–Glass system. The idea of a threshold is very natural to the study of population biology because the production of eggs (young) per adult per season is generally a saturation–type function of the available food and food supply is generally limited. Here {Yt} denotes the number of adult flies in day t, a is the daily adult survival rate, d is the time delay between birth and maturation, and bYt−τ 1+Y k t−τ accounts for the recruitment of new adults due to births d years in the past, which is nonlinear because of decreased fecundity at higher population levels. Such a class of models have been discussed in Gao (1998) and Gao and Yee (2000). Example 1.4 (Environmetric model): Consider a partially linear model of the form Yt = q X i=1 βiYt−i + g(Vt) + et, (1.10) © 2007 by Taylor Francis Group, LLC
- 24. 8 INTRODUCTION where {Yt} denotes the air quality time series at t period, and {Vt} rep- resents a vector of many important factors such as wind speed and tem- perature. When choosing a suitable vector for {Vt}, we need to take all possible factors into consideration on the one hand but to avoid the com- putational difficulty caused by the spareness of the data and to provide more precise predictions on the other hand. Thus, for this case only wind speed, temperature and one or two other factors are often selected as the most significant factors. Such issues are to be addressed in Chapter 4 below. When the dimension of {Vt} is greater than three, we may suggest using a partially linear additive model of the form Yt = q X i=1 βiYt−i + p X j=1 gj(Vtj) + et, (1.11) where each gj(·) is an unknown function defined over R1 = (−∞, ∞). Model estimation, specification and selection for models in Examples 1.1–1.4 are to be discussed in Chapters 2–4 below. Example 1.5 (Semiparametric single–index model): Consider a gener- alized partially linear time series model of the form Yt = Xτ t θ + ψ(Xτ t η) + et, (1.12) where (θ, η) are vectors of unknown parameters, ψ(·) is an unknown function over R1 , and {et} is a sequence of errors. The parameters and function are chosen such that model (1.12) is identifiable. While model (1.12) imposes certain additivity conditions on both the parametric and nonparametric components, it has been shown to be quite efficient for modelling high–dimensional time series data. Recent studies include Car- roll et al. (1997), Gao and Liang (1997), Xia, Tong and Li (1999), Xia et al. (2004), and Gao and King (2005). In recent years, some other semiparametric time series models have also been discussed as given below. Example 1.6 (Semiparametric regression models): Consider a linear model with a nonparametric error model of the form Yt = Xτ t β + ut with ut = g(ut−1) + t, (1.13) where Xt and β are p–dimensional column vectors, {Xt} is stationary with finite second moments, Yt and ut are scalars, g(·) is an unknown function and possibly nonlinear, and is such that {ut} is at least station- ary with zero mean and finite variance i.i.d. innovations t. Model (1.13) was proposed by Hidalgo (1992) and then estimated by a kernel-based procedure. © 2007 by Taylor Francis Group, LLC
- 25. EXAMPLES AND MODELS 9 Truong and Stone (1994) considered a nonparametric regression model with a linear autoregressive error model of the form Yt = g(Xt) + ut with ut = θut−1 + t, (1.14) where {(Xt, Yt)} is a bivariate stationary time series, θ, satisfying |θ| 1, is an unknown parameter, g(·) is an unknown function, and {t} is a sequence of independent errors with zero mean and finite variance 0 σ2 ∞. Truong and Stone (1994) proposed a semiparametric estimation procedure for model (1.14). Example 1.7 (Partially linear autoregressive conditional heteroscedasti- city (ARCH) models): For the case where d = 1, {Yt} is a time series, Xt = Yt−1, and {et} depends on Yt−1, model (1.2) is a partially linear ARCH model of the form Yt = βYt−1 + g(Yt−1) + et, (1.15) where {et} is assumed to be stationary, both β and g are identifiable, and σ2 (y) = E[e2 t |Yt−1 = y] is a smooth function of y. Hjellvik and Tjøstheim (1995), and Hjellvik, Yao and Tjøstheim (1998), Li (1999), and Gao and King (2005) all considered testing for linearity in model (1.15). Granger, Inoue and Morin (1997) have considered some estimation problems for the case of β = 1 in model (1.15). Example 1.8 (Nonlinear and nonstationary time series models): This example considers two classes of nonlinear and nonstationary time series models. The first class of models is given as follows: Yt = m(Xt) + et with Xt = Xt−1 + t, (1.16) where {t} is a sequence of stationary errors. The second class of models is defined by Yt = Yt−1 + g(Yt−1) + et. (1.17) Recently, Granger, Inoue and Morin (1997) considered the case where g(·) of (1.17) belongs to a class of parametric nonlinear functions and then discussed applications in economics and finance. In nonparametric kernel estimation of m(·) in (1.16) and g(·) of (1.17), existing studies include Karlsen and Tjøstheim (1998), Phillips and Park (1998), Karlsen and Tjøstheim (2001), and Karlsen, Myklebust and Tjøstheim (2006). The last paper provides a class of nonparametric versions of some of those parametric models proposed in Engle and Granger (1987). Model (1.16) corresponds to a class of parametric nonlinear models discussed in Park and Phillips (2001). Compared with nonparametric kernel estimation, nonparametric spe- cification testing problems for models (1.16) and (1.17) have just been considered in Gao et al. (2006). Specifically, the authors have proposed © 2007 by Taylor Francis Group, LLC
- 26. 10 INTRODUCTION a novel unit root test procedure for stationarity in a nonlinear time series setting. Such a test procedure can initially avoid misspecification through the need to specify a linear conditional mean. In other words, the authors have considered estimating the form of the conditional mean and testing for stationarity simultaneously. Such a test procedure may also be viewed as a nonparametric counterpart of those tests proposed in Dickey and Fuller (1979), Phillips (1987) and many others in the parametric linear time series case. Example 1.9 (Semiparametric diffusion models): This example involves using model (1.2) to approximate a continuous-time process of the form drt = µ(rt)dt + σ(rt)dBt, (1.18) where µ(·) and σ(·) are respectively the drift and volatility functions of the process, and Bt is standard Brownian motion. Since there are incon- sistency issues for the case where both µ(·) and σ(·) are nonparametric, we are mainly interested in the case where one of the functions is para- metric. The first case is where µ(r, θ) is a known parametric function indexed by a vector of unknown parameters, θ ∈ Θ (a parameter space), and σ(r) is an unknown but sufficiently smooth function. The main motivation for considering such a class of semiparametric dif- fusion models is due to: (a) most empirical studies suggest using a simple form for the drift function, such as a polynomial function; (b) when the form of the drift function is unknown and sufficiently smooth, it may be well–approximated by a parametric form, such as by a suitable polyno- mial function; (c) the drift function may be treated as a constant function or even zero when interest is on studying the stochastic volatility of {rt}; and (d) the precise form of the diffusion function is very crucial, but it is quite problematic to assume a known form for the diffusion function due to the fact that the instantaneous volatility is normally unobserv- able. The second case is where σ(r, ϑ) is a positive parametric function indexed by a vector of unknown parameters, ϑ ∈ Θ (a parameter space), and µ(r) is an unknown but sufficiently smooth function. As pointed out in existing studies, such as Kristensen (2004), there is some evid- ence that the assumption of a parametric form for the diffusion function is also reasonable in such cases where the diffusion function is already pre–specified, the main interest is, for example, to specify whether the drift function should be linear or quadratic. Model (1.18) has been applied to model various economic and financial data sets, including the two popular interest rate data sets given in Figures 1.5 and 1.6. Recently, Arapis and Gao (2006) have proposed some new estimation © 2007 by Taylor Francis Group, LLC
- 27. EXAMPLES AND MODELS 11 1965 1972 1980 1988 1995 5 10 15 Year Federal Funds Rate Figure 1.5 Three-month T-Bill rate, January 1963 to December 1998. and testing procedures for model (1.18) using semiparametric methods. Such details, along with some other recent developments, are discussed in Chapter 5 below. Example 1.10 (Continuous–time models with long–range dependence): Recent studies show that the standard Brownian motion involved in (1.18) needs to be replaced by a fractional Brownian motion when data exhibit long–range dependence. Comte and Renault (1996, 1998) pro- posed using a continuous–time model of the form dZ(t) = −αZ(t)dt + σdBβ(t), Z(0) = 0, t ∈ (0, ∞), (1.19) where Bβ(t) is general fractional Brownian motion given by Bβ(t) = R t 0 (t−s)β Γ(1+β) dB(s), and Γ(x) is the usual Γ function. Gao (2004) then dis- cussed some estimation problems for the parameters involved. More re- cently, Casas and Gao (2006) have systematically established both large and finite sample results for such estimation problems. Some of these results are discussed in Chapter 6 below. More recently, Casas and Gao (2006) have proposed a so–called simul- © 2007 by Taylor Francis Group, LLC
- 28. 12 INTRODUCTION 1973 1977 1981 1985 1990 1994 0.05 0.10 0.15 0.20 0.25 Year Eurodollar Interest Rate Figure 1.6 Seven-Day Eurodollar Deposit rate, June 1, 1973 to February 25, 1995. taneous semiparametric estimation procedure for a class of stochastic volatility models of the form dY (t) = V (t)dB1(t) and dZ(t) = −αZ(t)dt + σdBβ(t), (1.20) where V (t) = eZ(t) , Y (t) = ln(S(t)) with S(t) being the return process, B1(t) is a standard Brownian motion and independent of B(t). The paper by Casas and Gao (2006) has established some asymptotic theory for the proposed estimation procedure. Both the proposed theory and the estimation procedure are illustrated using simulated and real data sets, including the SP 500 data. To show why the SP 500 data may show some kind of long–range dependence, Table 1.1 provides autocorrelation values for several versions of the compounded returns of the SP 500 data. Chapter 6 below discusses some details about both the estimation and implementation of model (1.20). © 2007 by Taylor Francis Group, LLC
- 29. EXAMPLES AND MODELS 13 data lag1 2 5 10 20 40 70 100 T = 500 Wt 0.0734 -0.0458 0.0250 0.0559 -0.0320 -0.0255 0.0047 0.0215 |Wt|1/2 -0.0004 0.1165 0.1307 0.0844 0.0605 -0.0128 0.0430 -0.0052 |Wt| 0.0325 0.1671 0.1575 0.1293 0.092 -0.0141 0.0061 -0.0004 |Wt|2 0.0784 0.2433 0.1699 0.1573 0.1117 -0.0094 -0.0283 0.0225 T = 2000 Wt 0.0494 -0.0057 -0.0090 0.0142 0.0012 -0.0209 0.0263 0.0177 |Wt|1/2 -0.0214 -0.0072 0.0826 0.0222 0.0280 -0.0040 0.0359 0.0001 |Wt| -0.0029 0.0187 0.0997 0.0258 0.0505 0.0036 0.0422 -0.0020 |Wt|2 0.0401 0.0562 0.1153 0.0275 0.0668 0.0018 0.0376 -0.0045 T = 10000 Wt 0.1580 -0.0224 0.0122 0.0125 0.0036 0.0079 0.0028 0.0071 |Wt|1/2 0.1161 0.0813 0.1196 0.0867 0.0789 0.0601 0.0775 0.0550 |Wt| 0.1223 0.0986 0.1326 0.0989 0.0944 0.0702 0.0879 0.0622 |Wt|2 0.1065 0.1044 0.1281 0.0937 0.0988 0.0698 0.0847 0.0559 T = 16127 Wt 0.0971 -0.0362 0.0054 0.0180 0.0036 0.0222 -0.0061 0.0041 |Wt|1/2 0.1783 0.1674 0.1879 0.1581 0.1567 0.1371 0.1252 0.1293 |Wt| 0.2044 0.2012 0.2215 0.1831 0.1835 0.1596 0.1439 0.1464 |Wt|2 0.1864 0.2018 0.2220 0.1684 0.1709 0.1510 0.1303 0.1321 Table 1.1 Autocorrelation of Wt, |W|ρ for ρ = 1 2 , 1, 2 for the SP 500 where Wt = ln St St−1 with {St} be the SP 500 daily values. © 2007 by Taylor Francis Group, LLC
- 30. 14 INTRODUCTION 1.3 Bibliographical notes Recent books on parametric linear and nonlinear time series include Tong (1990), Granger and Teräsvirta (1993), Tanaka (1996), Franses and Van Dijk (2000), Galka (2000), Chan (2002), Fan and Yao (2003), Kantz and Schreiber (2004), Tsay (2005), and Granger, Teräsvirta and Tjøstheim (2006). In addition, nonparametric methods have been applied to model both independent and dependent time series data as discussed in Fan and Gijbels (1996), Hart (1997), Eubank (1999), Pagan and Ullah (1999), Fan and Yao (2003), Granger, Teräsvirta and Tjøstheim (2006), and Li and Racine (2006). Applications of semiparametric methods and models to time series data have been discussed in Fan and Gijbels (1996), Pagan and Ullah (1999), Härdle, Liang and Gao (2000), Fan and Yao (2003), Ruppert, Wand and Carroll (2003), Granger, Teräsvirta and Tjøstheim (2006), and Li and Racine (2006). © 2007 by Taylor Francis Group, LLC
- 31. CHAPTER 2 Estimation in Nonlinear Time Series 2.1 Introduction This chapter considers semiparametric modelling of nonlinear time series data. We first propose an additive partially linear modelling method. A semiparametric single–index modelling procedure is then considered. Both new estimation methods and implementation procedures are dis- cussed in some detail. The main ideas are to use either a partially linear form or a semiparametric single–index form to approximate the condi- tional mean function rather than directly assuming that the true condi- tional mean function is of either a partially linear form or a semipara- metric single–index form. 2.1.1 Partially linear time series models In time series regression, nonparametric methods have been very pop- ular both for prediction and characterizing nonlinear dependence. Let {Yt} and {Xt} be the one–dimensional and d–dimensional time series data, respectively. For a vector of time series data {Yt, Xt}, the condi- tional mean function E[Yt|Xt = x] of Yt on Xt = x may be estimated nonparametrically by the Nadaraya–Watson (NW) estimator when the dimensionality d is less than three. When d is greater than three, the conditional mean can still be estimated using the NW estimator, and an asymptotic theory can be constructed. In practice, however, because of the so–called curse of dimensionality, this may not be recommended unless the number of data points is extremely large. There are several ways of circumventing the curse of dimensionality in time series regression. Perhaps the two most commonly used are semi- parametric additive models and single–index models. In time series re- gression, semiparametric additive fitting can be thought of as an ap- proximation of conditional quantities such as E[Yt|Yt−1, . . . , Yt−d], and sometimes (Sperlich, Tjøstheim and Yang 2002) interaction terms are in- cluded to improve this approximation. An advantage of using the semi- 15 © 2007 by Taylor Francis Group, LLC
- 32. 16 ESTIMATION IN NONLINEAR TIME SERIES parametric additive approach is that a priori information concerning possible linearity of some of the components can be included in the model. More specifically, we will look at approximating the conditional mean function m(Xt) = m(Ut, Vt) = E[Yt|Ut, Vt] by a semiparametric (partially linear) function of the form m1(Ut, Vt) = µ + Uτ t β + g(Vt) (2.1) such that E [Yt − m1(Ut, Vt)] 2 is minimized over a class of semipara- metric functions of the form m1(Ut, Vt) subject to E[g(Vt)] = 0 for the identifiability of m1(Ut, Vt), where µ is an unknown parameter, β = (β1, . . . , βq)τ is a vector of unknown parameters, g(·) is an unknown function over Rp , both Ut = (Ut1, . . . , Utq)τ and Vt = (Vt1, . . . , Vtp)τ may be vectors of time series variables. Motivation for using the form (2.1) for independent data analysis can be found in Härdle, Liang and Gao (2000). As for the independent data case, estimating g(·) in model (2.1) may suffer from the curse of dimensionality when g(·) is not necessarily additive and p ≥ 3. Thus, this chapter proposes two different estimation methods. The first estimation method deals with the case where m(x) is itself an additive partially linear form and each of the nonparametric components is approximated by a series of orthogonal functions. For the independent data case, the orthogonal series estimation method has been used as an alternative to some other nonparametric estimation methods, such as the kernel method. Recent monographs include Eubank (1999). As shown in Gao, Tong and Wolff (2002a), this method provides some natural parametric approximations to additive partially linear forms. 2.1.2 Semiparametric additive time series models The main ideas of proposing the second method are taken from Gao, Lu and Tjøstheim (2006), who have established an estimation procedure for semiparametric spatial regression. The second method applies to the case where m(x) is approximated by (2.1) and then proposes approximating g(·) by ga(·), an additive marginal integration projector as detailed in the following section. When g(·) itself is additive, i.e., g(x) = Pp i=1 gi(xi), the form of m1(Ut, Vt) can be written as m1(Ut, Vt) = µ + Uτ t β + p X i=1 gi(Vti) (2.2) subject to E [gi(Vti)] = 0 for all 1 ≤ i ≤ p for the identifiability of m1(Ut, Vt) in (2.2), where gi(·) for 1 ≤ i ≤ p are all unknown one– dimensional functions over R1 . © 2007 by Taylor Francis Group, LLC
- 33. INTRODUCTION 17 Our method of estimating g(·) or ga(·) is based on an additive marginal integration projection on the set of additive functions, but where unlike the backfitting case, the projection is taken with the product measure of Vtl for l = 1, · · · , p (Nielsen and Linton 1998). This contrasts with the smoothed backfitting approach of Mammen, Linton and Nielsen (1999) to the nonparametric regression case. Marginal integration, although inferior to backfitting in asymptotic efficiency for purely additive models, seems well suited to the framework of partially linear estimation. In fact, in previous work (Fan, Härdle and Mammen 1998; Fan and Li 2003, for example) in the independent regression case marginal integration has been used, and we do not know of any work extending the backfitting theory to the partially linear case. Marginal integration techniques are also applicable to the case where interactions are allowed between the the Vtl–variables (cf. also the use of marginal integration for estimating interactions in ordinary regression problems). 2.1.3 Semiparametric single–index models As an alternative to (2.2), we assume that m(x) = E[Yt|Xt = x] = m2(Xt) is given by the semiparametric single–index form m2(Xt) = Xτ t θ + ψ(Xτ t η). (2.3) When we partition Xt = (Uτ t , V τ t )τ and take θ = (βτ , 0, · · · , 0)τ and η = (0, · · · , 0, ατ )τ , form (2.3) becomes the generalized partially linear form m2(Xt) = Uτ t β + ψ(V τ t α). (2.4) Various versions of (2.3) and (2.4) have been discussed in the econo- metrics and statistics literature. Recent studies include Härdle, Hall and Ichimura (1993), Carroll et al. (1997), Gao and Liang (1997), Xia, Tong and Li (1999), and Gao and King (2005). In Sections 2.2 and 2.3 below, some detailed estimation procedures for m1(Ut, Vt) and m2(Xt) are proposed and discussed extensively. Sec- tion 2.2 first assumes that the true conditional mean function m(x) = E[Yt|Xt = x] is of the form (2.2) and develops an orthogonal series es- timation method for the additive form. Section 2.3 then proposes an additive marginal integration projection method to estimate form (2.1) without necessarily assuming the additivity in (2.2). © 2007 by Taylor Francis Group, LLC
- 34. 18 ESTIMATION IN NONLINEAR TIME SERIES 2.2 Semiparametric series estimation In this section, we employ the orthogonal series method to estimate each nonparametric function in (2.2). By approximating each gi(·) by an orthogonal series Pni j=1 fij(·)θij with {fij(·)} being a sequence of orthogonal functions and {ni} being a sequence of positive integers, we have an approximate model of the form Yt = µ + Uτ t β + p X i=1 ni X j=1 fij(Vti)θij + et, (2.5) which covers some natural parametric time series models. For example, when Utl = Ut−l and Vti = Yt−i, model (2.5) becomes a parametric nonlinear additive time series model of the form Yt = µ + q X l=1 Ut−lβl + p X i=1 ni X j=1 fij(Yt−i)θij + et. (2.6) To estimate the parameters involved in (2.5), we need to introduce the following symbols. For 1 ≤ i ≤ p, let θi = (θi1, · · · , θini )τ , θ = (θτ 1 , · · · , θτ p )τ , Fi = Fini = (Fi(V1i), . . . , Fi(VT i))τ , F = (F1, F2, . . . , Fp), U = 1 T T X t=1 Ut, e U = U1 − U, · · · , UT − U τ , Y = 1 T T X t=1 Yt, e Y = Y1 − Y , · · · , YT − Y τ , P = F (Fτ F) + Fτ , b U = (I − P)e U, b Y = (I − P)e Y , (2.7) and n = (n1, · · · , np)τ and A+ denotes the Moore–Penrose inverse of A. Using the approximate model (2.6), we define the least squares (LS) estimators of (β, θ, µ) by b β = b β(n) = b Uτ b U + b Uτ b Y , b θ = (Fτ F) + Fτ e Y − e U b β , b µ = Y − U τ b β. (2.8) Equation (2.8) suggests estimating the conditional mean function m(Xt) = E[Yt|Xt] by b m(Xt; n) = b µ + Uτ t b β + p X i=1 Fi(Vti)τ b θi(n), (2.9) © 2007 by Taylor Francis Group, LLC
- 35. SEMIPARAMETRIC SERIES ESTIMATION 19 where b θi(n) is the corresponding estimator of θi. It follows from (2.9) that the prediction equation depends on not only the series functions {fij : 1 ≤ j ≤ ni, 1 ≤ i ≤ p} but also n, the vector of truncation parameters. It is mentioned that the choice of the series functions is much less critical than that of the vector of truncation parameters. The series functions used in this chapter need to satisfy Assumptions 2.2 and 2.3 in Section 2.5. The assumptions hold when each fij belongs to a class of trigonometric series used by Gao, Tong and Wolff (2002a). Therefore, a crucial problem is how to select k practically. Li (1985, 1986, 1987) discussed the asymptotic optimality of a generalized cross–validation (GCV) criterion as well as other model selection criteria. Wahba (1990) provided a recently published survey of nonparametric smoothing spline literature up to 1990. Gao (1998) applied a generalized cross–validation criterion to choose smoothing truncation parameters for the time series case. In this section, we apply a generalized cross– validation method to choose k and then determine the estimates in (2.9). In order to select n, we introduce the following mean squared error: b D(n) = 1 T T X t=1 { b m(Xt; n) − m(Xt)} 2 . (2.10) Let g (mi) i be the mi–order derivative of the function gi and M0i be a constant, Gmi (Si) = n g : g (mi) i (s) − g (mi) i (s0 ) ≤ M0i|s − s0 |, s, s0 ∈ Si ⊂ R1 o , where each mi ≥ 1 is an integer, 0 M0i ∞ and each Si is a compact subset of R1 . Let also NiT = {piT , piT + 1, . . . , qiT }, in which piT = aiTdi , qiT = [biTci ], 0 ai bi ∞, 0 di ci 1 2(mi+1) are constants, and [x] ≤ x denotes the largest integer part of x. Definition 2.1. A data-driven estimator b n = (b n1, . . . , b np)τ is asymptot- ically optimal if b D(b n) infn∈NT b D(n) →p 1, where n ∈ NT = {n = (n1, . . . , np)τ : ni ∈ NiT }. Definition 2.2. Select n, denoted by b nG = (b n1G, . . . , b npG)τ , that achieves GCV(b nG) = inf n∈NT GCV(n) = inf n∈NT b σ2 (n) 1 − 1 T Pp i=1 ni 2 , where b σ2 (n) = 1 T PT t=1 {Yt − b m(Xt; n)} 2 . © 2007 by Taylor Francis Group, LLC
- 36. 20 ESTIMATION IN NONLINEAR TIME SERIES We now have the following asymptotic properties for b D(n) and b nG. Theorem 2.1. (i) Assume that Assumptions 2.1–2.2(i), 2.3 and 2.4 listed in Section 2.5 hold. Then b D(n) = σ2 T p X i=1 ni + 1 T E [∆τ ∆] + op b D(n) , (2.11) where ∆ = Pp i=1 [Fiθi − Gi], Gi = (gi(V1i), . . . , gi(VT i))τ and {Fi(·)} is as defined before. (ii) In addition, if Assumption 2.2(ii) holds, then we have b D(n) = σ2 T p X i=1 ni + p X i=1 Cin −2(mi+1) i + op( b D(n)) (2.12) uniformly over n ∈ NT , where σ2 = E[e2 t ] ∞ and each mi is the smoothness order of gi. Theorem 2.2. (i) Under the conditions of Theorem 2.1(i), b nG is asymp- totically optimal. (ii) Under the conditions of Theorem 2.1(ii), we have b D(b nG) D̂(n̂D) − 1 = op(T−τ ) (2.13) and p X i=1 b niG b niD − 1 = op T−τ , (2.14) where b niD is the i–th component of b nD = (b n1D, . . . , b npD)τ that minimises b D(n) over NT , 0 τ = min(τ1 − 1, τ2 − 2), in which τ1 = 1 2 dmin, τ2 = 1 2 − 2cmax, both 1 and 2 satisfying 0 1 τ1 and 0 2 τ2 are arbitrarily small, dmin = min1≤i≤p di and cmax = max1≤i≤p ci. The proofs of Theorems 2.1 and 2.2 are relegated to Section 2.5. We now define the adaptive and simultaneous estimation procedure as follows: (i) solve the LS estimator b θ(n); (ii) define the prediction equation by (2.9); (iii) solve the GCV-based b nG; and (iv) define the following adaptive and simultaneous prediction equation b m (Xt; b nG). © 2007 by Taylor Francis Group, LLC
- 37. SEMIPARAMETRIC SERIES ESTIMATION 21 If σ2 is unknown, it will be estimated by b σ2 (b nG). Furthermore, we have the following asymptotic normality. Corollary 2.1. Under the conditions of Theorem 2.1(i), we have as T → ∞ √ T b σ2 (b nG) − σ2 → N 0, var(e2 1) . The proof of Corollary 2.1 is relegated to Section 2.5. Remark 2.1. Theorem 2.1 provides asymptotic representations for the average squared error b D(n). See Härdle, Hall and Marron (1988) for an equivalent result in nonparametric kernel regression. In addition, The- orem 2.2(i) shows that the GCV based b nG is asymptotically optimal. This conclusion is equivalent to Corollary 3.1 of Li (1987) in the model selection problem. However, the fundamental difference between our dis- cussion in this section and Li (1987) is that we use the GCV method to determine how many terms are required to ensure that each nonpara- metric function can be approximated optimally, while Li (1987) sugges- ted using the GCV selection criterion to determine how many variables should be employed in a linear model. Due to the different objectives, our conditions and conclusions are different from those of Li (1987), although there are some similarities. Remark 2.2. Theorem 2.2(ii) not only establishes the asymptotic op- timality but also provides the rate of convergence. This rate of con- vergence is equivalent to that of bandwidth estimates in nonparametric kernel regression. See Härdle, Hall and Marron (1992). More recently, Hurvich and Tsai (1995) have established a similar result for a lin- ear model selection. Moreover, it follows from Theorem 2.2(ii) that the rate of convergence depends heavily on di and ci. Let di = 1 2mi+3 and ci = 1 2mi+3 +ηi for arbitrarily small ηi 0. Then the rate of convergence will be of order min min 1≤i≤p 1 2(2mi + 3) , max 1≤i≤p 2mi − 1 2(2mi + 3) − for some arbitrarily small 0. Obviously, if each gi is continuously differentiable, then the rate of convergence will be close to 1 10 − . This is equivalent to Theorem of Hurvich and Tsai (1995). As a result of the Theorem, the rate of convergence can be close to 1 2 . See also Theorem 1 and Remark 2 of Härdle, Hall and Marron (1992). Remark 2.3. In this chapter, we assume that the data set {(Yt, Xt) : t ≥ 1} satisfies model (2.2) and then propose the orthogonal series method to © 2007 by Taylor Francis Group, LLC
- 38. 22 ESTIMATION IN NONLINEAR TIME SERIES model the data set. In practice, before applying the estimation procedure to model the data, a crucial problem is how to test the additivity. Some related results for additive nonparametric regression have been given by some authors. See, for example, Gao, Tong and Wolff (2002b). To illustrate the above estimation procedure, we now include two simu- lated and real examples for a special case of model (2.2) with µ = β = 0. Let Vt = (Vt1, Vt2, Vt3)τ = (Yt−1, Yt−2, Wt)τ , where {Wt} is to be spe- cified below. Example 2.1: Consider the model given by Yt = 0.25Yt−1 + 0.25 Yt−2 1 + Y 2 t−2 + 1 8π W2 t + et, t = 3, 4, ..., T, (2.15) where {et} is uniformly distributed over (−0.5π, 0.5π), Y1 and Y2 are mutually independent and uniformly distributed over 1 128 , 2π − 1 128 , (Y1, Y2) is independent of {et : t ≥ 3}, Wt = 0.25Wt−1 − 0.25Wt−2 + t, (2.16) in which {t} is uniformly distributed over (−0.5π, 0.5π), X1 and X2 are mutually independent and uniformly distributed over 1 128 , 2π − 1 128 , and (X1, X2) is independent of {t : t ≥ 3}. First, it follows from Lemma 3.1 of Masry and Tjøstheim (1997) that both the stationarity and the mixing condition are met. See also Chapter 4 of Tong (1990), §2.4 of Tjøstheim (1994) and §2.4 of Doukhan (1995). Thus, Assumption 2.1(i) holds. Second, it follows from (2.15) and (2.16) that Assumption 2.1(ii) holds immediately. Third, let g1(x) = 0.25x, g2(x) = 0.25 x 1 + x2 , g3(x) = 1 8π x2 . (2.17) Since {gi : 1 ≤ i ≤ 3} are continuously differentiable on R1 , there exist three corresponding periodic functions defined on [0, 2π] that are continuously differentiable on [0, 2π] and coincide with {gi : 1 ≤ i ≤ 3} correspondingly (see Hong and White 1995, p.1141). Similarly to §3.2 of Eastwood and Gallant (1991), we can show that there exist the following three corresponding trigonometric polynomials g∗ 1(x) = n1 X j=1 sin(jx)θ1j, © 2007 by Taylor Francis Group, LLC
- 39. SEMIPARAMETRIC SERIES ESTIMATION 23 g∗ 2(x) = n2 X j=1 sin(jx)θ2j, g∗ 3(x) = n3 X j=1 cos(jx)θ3j (2.18) such that Assumptions 2.2(i) and 2.2(ii) are satisfied and the same con- vergence rate can be obtained as in the periodic case. Obviously, it fol- lows from (2.18) that Assumption 2.2(i) holds. Fourth, Assumption 2.3 is satisfied due to (2.18) and the orthogonality of trigonometric series. Finally, Assumption 2.4 holds due to the fact that supt≥1 |Yt| ≤ 2π. We now define g∗ 1, g∗ 2 and g∗ 3 as the corresponding approximations of g1, g2 and g3 with x ∈ S = 1 128 , 2π − 1 128 and hi ∈ NiT = [aiTdi ], . . . , [biTci ] , (2.19) in which i = 1, 2, 3, di = 1 2mi + 3 and ci = 1 2mi + 3 + 2mi − 1 6(2mi + 3) . In the following simulation, we consider the case where ai = 1, bi = 2 and mi = 1 for i = 1, 2, 3. Let F1(x) = (sin(x), sin(2x), . . . , sin(n1x))τ , F2(x) = (sin(x), sin(2x), . . . , sin(n2x))τ , F3(x) = (cos(x), cos(2x), . . . , cos(n3x))τ . For the cases of T = 102, 252, 402, 502, and 752, we then compute b D(n), b σ2 (n), GCV(n) and the following quantities: for i = 1, 2, 3, di(b niG, b niD) = b niG b niD − 1, d4(b nG, b nD) = b D(b nG) b D(b nD) − 1, ASEi(b nG) = 1 N N X n=1 n Fib niG (Zni)τ b θi(b hG) − gi(Zni) o2 , ASE4(b nG) = 1 N N X n=1 ( 3 X i=1 Fib niG (Zni)τ b θib niG − gi(Zni) )2 , VAR(b nG) = b σ2 (b nG) − σ2 , where N = T − 2, σ2 = π2 12 = 0.822467, b nG = (b n1G, b n2G, b n3G)τ , Zn1 = Yn+1, Zn2 = Yn and Zn3 = Wn+2. © 2007 by Taylor Francis Group, LLC
- 40. 24 ESTIMATION IN NONLINEAR TIME SERIES The simulation results below were performed 1000 times and the means are tabulated in Table 2.1 below. Table 2.1. Simulation Results for Example 2.1 N 100 250 400 500 750 NiT {1,. . . ,5} {1,. . . ,6} {1,. . . ,6} {1,. . . ,6} {1,. . . ,7} d1(b n1G, b n1D) 0.10485 0.08755 0.09098 0.08143 0.07943 d2(b n2G, b n2D) 0.11391 0.07716 0.08478 0.08964 0.07983 d3(b n3G, b n3D) 0.09978 0.08155 0.08173 0.08021 0.08371 d4(b nG, b nD) 0.32441 0.22844 0.24108 0.22416 0.22084 ASE1(b nG) 0.03537 0.01755 0.01123 0.00782 0.00612 ASE2(b nG) 0.02543 0.01431 0.00861 0.00609 0.00465 ASE3(b nG) 0.02507 0.01348 0.00795 0.00577 0.00449 ASE4(b nG) 0.06067 0.03472 0.02131 0.01559 0.01214 VAR(b nG) 0.05201 0.03361 0.01979 0.01322 0.01086 Remark 2.4. Both Theorem 2.2(ii) and Table 2.1 demonstrate that the rate of convergence of the GCV based di for 1 ≤ i ≤ 4 is of order T− 1 10 . In addition, the simulation results for ASEi(b nG) given in Table 2.1 show that when ni is of order T 1 5 , the rate of convergence of each ASEi is of order T− 4 5 . Example 2.2: In this example, we consider the Canadian lynx data. This data set is the annual record of the number of Canadian lynx trapped in the MacKenzie River district of North–West Canada for the years 1821 to 1934. Tong (1976) fitted an eleventh-order linear Gaussian autoregressive model to Yt = log10{number of lynx trapped in the year (1820 + t)} for t = 1, 2, ..., 114 (T = 114). It follows from the definition of {Yt, 1 ≤ t ≤ 114} that all the transformed values {Yt : t ≥ 1} are bounded. We apply the above estimation procedure to fit the real data set listed in Example 2.2 by the following third–order additive autoregressive model of the form Yt = g1(Yt−1) + g2(Yt−2) + g3(Yt−3) + et, t = 4, 5, . . . , T, (2.20) where {gi : i = 1, 2, 3} are unknown functions, and {et} is a sequence of independent random errors with zero mean and finite variance. © 2007 by Taylor Francis Group, LLC
- 41. SEMIPARAMETRIC SERIES ESTIMATION 25 Similarly, we approximate g1, g2 and g3 by g∗ 1(u) = n1 X j=1 f1j(u)θ1j, g∗ 2(v) = n2 X j=1 f2j(v)θ2j, g∗ 3(w) = n3 X j=1 f3j(w)θ3j, (2.21) respectively, where f1j(u) = sin(ju) for 1 ≤ j ≤ n1, f2j(v) = sin(jv) for 1 ≤ j ≤ n2, f3j(w) = cos(jw) for 1 ≤ j ≤ n3, and hj ∈ NjT = n T0.2 , . . . , h 2T 7 30 io . Our simulation suggests using the following polynomial prediction b Yt = b n1G X j=1 sin(jYt−1)θ1j + b n2G X j=1 sin(jYt−2)θ2j + b n3G X j=1 cos(jYt−3)θ3j, (2.22) where b n1G = 5, b n2G = b n3G = 6, and the coefficients are given in the following Table 2.2. Table 2.2. Coefficients for Equation (2.22) θ1 = (θ11, . . . , θ15)τ θ2 = (θ21, . . . , θ26)τ θ3 = (θ31, . . . , θ36)τ 11.877 -2.9211 -6.8698 18.015 -5.4998 -7.8529 10.807 -4.9084 -7.1952 4.1541 -3.1189 -4.8019 0.7997 -1.2744 -2.0529 -0.2838 -0.4392 The estimator of the error variance was 0.0418. Some plots for Example 2.2 are given in Figure 2.1 of Gao, Tong and Wolff (2002a). Remark 2.5. For the Canadian lynx data, Tong (1976) fitted an eleventh– order linear Gaussian autoregressive model to the data, and the estimate of the error variance was 0.0437. Figure 2.1 shows that when using equa- tion (2.20) to fit the real data set, the estimator of g1 is almost linear while the estimators of both g2 and g3 appear to be nonlinear. This find- ing is the same as the conclusion reached by Wong and Kohn (1996), who used a Bayesian based iterative procedure to fit the real data set. Their estimator of the error variance was 0.0421, which is comparable with our variance estimator of 0.0418. Moreover, our estimation proced- ure provides the explicit equation (2.22) and the CPU time for Example 2.2 just took about 2 minutes. By contrast, Wong and Kohn (1996) can © 2007 by Taylor Francis Group, LLC
- 42. 26 ESTIMATION IN NONLINEAR TIME SERIES only provide an iterative estimation procedure for each gi since their approach depends heavily on the Gibbs sampler. Remark 2.6. Both Examples 2.1 and 2.2 demonstrate that the explicit estimation procedure can not only provide some additional information for further diagnostics and statistical inference but also produce mod- els with better predictive power than is available from linear models. For example, model (2.22) is more appropriate than a completely linear model for the lynx data as mentioned in Remark 2.2. Moreover, model (2.22) not only can be calculated at a new design point with the same convenience as in linear models, but also provides the individual coeffi- cients, which can be used to measure whether the individual influence of each Yt−3+i for i = 0, 1, 2 can be negligible. This section has assumed that the true conditional mean function is of a semiparametric additive model of the form (2.2) and then developed the orthogonal series based estimation procedure. As discussed in the next section, we may approximate the true conditional mean function by the additive form (2.2) even if the true conditional mean function may not be expressed exactly as an additive form. 2.3 Semiparametric kernel estimation As mentioned above (2.1), we are approximating the mean function m(Ut, Vt) = E[Yt|Ut, Vt] by minimizing E [Yt − m1(Ut, Vt)] 2 = E [Yt − µ − Uτ t β − g(Vt)] 2 (2.23) over a class of functions of the form m1(Ut, Vt) = µ + Uτ t β + g(Vt) with E[g(Vt)] = 0. Such a minimization problem is equivalent to minimizing E [Yt − µ − Uτ t β − g(Vt)] 2 = E h E n (Yt − µ − Uτ t β − g(Vt)) 2 |Vt oi over some (µ, β, g). This implies that g(Vt) = E [(Yt − µ − Uτ t β)|Vt] and µ = E[Yt − Uτ t β] with β being given by β = Σ−1 E [(Ut − E[Ut|Vt]) (Yt − E[Yt|Vt])] (2.24) provided that the inverse Σ−1 = (E [(Ut − E[Ut|Vt]) (Ut − E[Ut|Vt]) τ ]) −1 exists. This also shows that m1(Ut, Vt) is identifiable under the assump- tion of E[g(Vt)] = 0. We now turn to estimation assuming that the data are available for (Yt, Ut, Vt) for 1 ≤ t ≤ T. Since the definitions of the estimators to be used later are quite involved notationally, we start by outlining the main steps in establishing estimators for µ, β and g(·) in (2.1) and then © 2007 by Taylor Francis Group, LLC
- 43. SEMIPARAMETRIC KERNEL ESTIMATION 27 gl(·), l = 1, 2, · · · , p in (2.2). In the following, we give an outline in three steps. Step 1: Estimating µ and g(·) assuming β to be known. For each fixed β, since µ = E[Yt] − E[Uτ t β] = µY − µτ U β, the parameter µ can be estimated by µ̂(β) = Y − U τ β, where µY = E[Yt], µU = (µ (1) U , · · · , µ (q) U )τ = E[Ut], Y = 1 T PT t=1 Yt and U = 1 T PT t=1 Ut. Moreover, the conditional expectation g(x) = g(x, β) = E [(Yt − µ − Uτ t β)|Vt = x] = E [(Yt − E[Yt] − (Ut − E[Ut])τ β)|Vt = x] (2.25) can be estimated by standard local linear estimation (Fan and Gijbels 1996) with ĝT (x, β) = â0(β) satisfying (â0(β), â1(β)) = arg min (a0, a1)∈R1 ×Rp (2.26) × T X t=1 Ỹt − Ũτ t β − a0 − aτ 1(Vt − x) 2 Kt(x, b), where Ỹt = Yt − Y , Ũt = (Ũ (1) t , · · · , Ũ (q) t )τ = Ut − U and Kt(x, b) = Qp l=1 K Vtl−xl bl , with b = bT = (b1, · · · , bp), bl = bl,T being a sequence of bandwidths for the l-th covariate variable Vtl, tending to zero as T tends to infinity, and K(·) is a bounded kernel function on R1 . Step 2: Marginal integration to obtain g1, · · · , gp of (2.2). The idea of the marginal integration estimator is best explained if g(·) is itself additive, that is, if g(Vt) = g(Vt1, · · · , Vtp) = p X l=1 gl(Vtl). Then, since E [gl (Vtl)] = 0 for l = 1, · · · , p, for k fixed gk(xk) = E [g(Vt1, · · · , xk, · · · , Vtp)] . An estimate of gk is obtained by keeping Vtk fixed at xk and then taking the average over the remaining variables Vt1, · · · , Vt(k−1), Vt(k+1), · · · , Vtp. This marginal integration operation can be implemented irrespective of whether or not g(·) is additive. If the additivity does not hold, the marginal integration amounts to a projection on the space of additive functions of Vtl, l = 1, · · · , p taken with respect to the product measure of Vtl, l = 1, · · · , p, obtaining the approximation ga(x, β) = Pp l=1 Pl,ω(Vtl, β), which will be detailed below with β appearing linearly in the expression. © 2007 by Taylor Francis Group, LLC
- 44. 28 ESTIMATION IN NONLINEAR TIME SERIES In addition, it has been found convenient to introduce a pair of weight functions (wk, w(−k)) in the estimation of each component, hence the index w in Pl,w. The details are given in Equations (2.32)–(2.36) below. Step 3: Estimating β. The last step consists in estimating β. This is done by weighted least squares, and it is easy since β enters linearly in our expressions. In fact, using the expression of g(x, β) in Step 1, we obtain the weighted least squares estimator β̂ of β in (2.34) below. Finally, this is re–introduced in the expressions for µ̂ and P̂ resulting in the estimates in (2.35) and (2.36) below. In the following, steps 1–3 are written correspondingly in more detail. Step 1: To write our expression for (â0(β), â1(β)) in (2.26), we need to introduce some more notation. Xt = Xt(x, b) = (Vt1 − x1) b1 , · · · , (Vtp − xp) bp τ , and let bπ = Qp l=1 bl. We define for 0 ≤ l1, l2 ≤ p, γT,l1l2 = (Tbπ)−1 T X t=1 (Xt(x, b))l1 (Xt(x, b))l2 Kt(x, b), (2.27) where (Xt(x, b))l = (Vtl−xl) bl for 1 ≤ l ≤ p. We then let (Xt(x, b))0 ≡ 1 and define λT,l(β) = (Tbπ)−1 T X t=1 Ỹt − Ũτ t β (Xt(x, b))l Kt(x, b) (2.28) and where, as before, Ỹt = Yt − Ȳ and Ũt = Ut − Ū. Note that λT,l(β) can be decomposed as λT,l(β) = λ (0) T,l − q X s=1 βsλ (s) T,l, for l = 0, 1, · · · , p, (2.29) in which λ (0) T,l = λ (0) T,l(x, b) = (Tbπ)−1 PT t=1 Ỹt (Xt(x, b))l Kt(x, b), λ (s) T,l = λ (s) T,l(x, b) = (Tbπ)−1 T X t=1 Ũts (Xt(x, b))l Kt(x, b), 1 ≤ s ≤ q. We can then express the local linear estimates in (2.26) as (â0(β), â1(β) b) τ = Γ−1 T ΛT (β), (2.30) © 2007 by Taylor Francis Group, LLC
- 45. SEMIPARAMETRIC KERNEL ESTIMATION 29 where is the operation of the component-wise product, i.e., a1 b = (a11b1, · · · , a1pbp) for a1 = (a11, · · · , a1p) and b = (b1, · · · , bp), ΛT (β) = λT,0(β) ΛT,1(β) , ΓT = γT,00 ΓT,01 ΓT,10 ΓT,11 , (2.31) where ΓT,10 = Γτ T,01 = (γT,01, · · · , γT,0p)τ and ΓT,11 is the p × p matrix defined by γT,l1 l2 with l1, l2 = 1, · · · , p, in (2.27). Moreover, ΛT,1(β) = (λT,1(β), . . . , λT,p(β))τ with λT,l(β) as defined in (2.28). Analogously for ΛT , we may define Λ (0) T and Λ(s) in terms of λ(0) and λ(s) . Then taking the first component with c = (1, 0, · · · , 0)τ ∈ R1+p , ĝT (x; β) = cτ Γ−1 T (x)ΛT (x, β) = cτ Γ−1 (x)Λ (0) T (x) − q X s=1 βscτ Γ−1 (x)Λ (s) T (x) = H (0) T (x) − βτ HT (x), where HT (x) = (H (1) T (x), · · · , H (q) T (x))τ with H (s) T (x) = cτ Γ−1 T (x)Λ(s) (x), 1 ≤ s ≤ q. Clearly, H (s) T (x) is the local linear estimator of H(s) (x) = E h U (s) t − µ (s) U |Vt = x i , 1 ≤ s ≤ q. We now define U (0) t = Yt and µ (0) U = µY such that H(0) (x) = E[(U (0) t − µ (0) U )|Vt = x] = E[Yt − µY |Vt = x] and H(x) = (H(1) (x), · · · , H(q) (x))τ = E[(Ut − µU )|Vt = x]. It follows that g(x, β) = H(0) (x) − βτ H(x), which equals g(x) under (2.1) irrespective of whether g itself is additive. Step 2: Let w(−k)(·) be a weight function defined on Rp−1 such that E h w(−k)(V (−k) t ) i = 1, and wk(xk) = I[−Lk,Lk](xk) defined on R1 for some large Lk 0, with V (−k) t = (Vt1, · · · , Vt(k−1), Vt(k+1), · · · , Vtp), where IA(x) is the conventional indicator function. In addition, we take Vt(xk) = (Vt1, · · · , Vt(k−1), xk, Vt(k+1), · · · , Vtp). For a given β, consider the marginal projection Pk,w(xk, β) = E h g(Vt(xk); β)w(−k) V (−k) t i wk(xk). (2.32) It is easily seen that if g is additive as in (2.2), then for −Lk ≤ xk ≤ Lk, Pk,w(xk, β) = gk(xk) up to a constant since it is assumed that © 2007 by Taylor Francis Group, LLC
- 46. 30 ESTIMATION IN NONLINEAR TIME SERIES E h w(−k)(V (−k) t ) i = 1. In general, ga(x, β) = Pp l=1 Pl,w(xl, β) is an ad- ditive marginal projection approximation to g(x) in (2.1) up to a con- stant in the region x ∈ Qp l=1[−Ll, Ll]. The quantity Pk,w(xk, β) can then be estimated by the locally linear marginal integration estimator b Pk,w(xk, β) = T−1 T X t=1 ĝT (Vt(xk); β) w(−k) V (−k) t wk(xk) = P̂ (0) k,w(xk) − q X s=1 βsP̂ (s) k,w(xk) = P̂ (0) k,w(xk) − βτ P̂U k,w(xk), where P̂U k,w(xk) = P̂ (1) k,w(xk), · · · , P̂ (q) k,w(xk) τ , in which P̂ (s) k,w(xk) = 1 T T X t=1 H (s) T (Vt(xk)) w(−k) V (−k) t wk(xk) is the estimator of P (s) k,w(xk) = E h H(s) (Vt(xk))w(−k) V (−k) t i wk(xk) for 0 ≤ s ≤ q and PU k,w(xk) = P (1) k,w(xk), · · · , P (q) k,w(xk) τ is estimated by P̂U k,w(xk). We add the weight function wk(xk) = I[−Lk, Lk](xk) in the definition of P̂ (s) k,w(xk), since we are interested only in the points of xk ∈ [−Lk, Lk] for some large Lk. In practice, we may use a sample centered version of P̂ (s) k,w(xk) as the estimator of P (s) k,w(xk). Clearly, we have Pk,w(xk, β) = P (0) k,w(xk) − βτ PU k,w(xk). Thus, for every β, g(x) = g(x, β) of (2.1) (or rather the approximation ga(x, β) if (2.2) does not hold) can be estimated by b b g(x, β) = p X l=1 b Pl,w(xl, β) = p X l=1 P̂ (0) l,w (xl) − βτ p X l=1 P̂U l,w(xl). (2.33) Step 3: We can finally obtain the least squares estimator of β by b β = arg min β∈Rq T X t=1 Ỹt − Ũτ t β − b b g(Vt, β) 2 = arg min β∈Rq T X t=1 b Y ∗ t − b U∗ t τ β 2 , (2.34) © 2007 by Taylor Francis Group, LLC
- 47. SEMIPARAMETRIC KERNEL ESTIMATION 31 where b Y ∗ t = Ỹt− Pp l=1 P̂ (0) l,w (Vtl) and b U∗ t = Ũt− Pp l=1 P̂U l,w(Vtl). Therefore, b β = T X t=1 b U∗ t b U∗ t τ !−1 T X t=1 b Y ∗ t b U∗ t ! and b µ = Y − b βτ U. (2.35) We then insert b β in â0(β) = ĝm,n(x, β) to obtain â0(b β) = ĝm,n(x, b β). In view of this, the locally linear projection estimator of Pk(xk) can be defined by b b Pk,w(xk) = 1 T T X t=1 ĝT (Vt(xk); b β) w(−k) V (−k) t (2.36) and for xk ∈ [−Lk, Lk] this would estimate gk(xk) up to a constant when (2.2) holds. To ensure E[gk(Vtk)] = 0, we may rewrite b gk(xk) = b b Pk,w(xk) − b µP (k) for the estimate of gk(xk) in (2.2), where b µP (k) = 1 T PT t=1 b b Pk,w(Vtk). For the proposed estimators, b β, and b b Pk,w(·), we establish some asymp- totic distributions in Theorems 2.3 and 2.4 below under certain technical conditions. To avoid introducing more mathematical details and symbols before we state the main results, we relegate such conditions and their justifications to Section 2.5 of this chapter. We can now state the asymptotic properties of the marginal integration estimators for both the parametric and nonparametric components. Let U∗ t = Ut − µU − Pp l=1 PU l,w(Vtl), Y ∗ t = Yt − µY − Pp l=1 P (0) l,w (Vtl) and Rt = U∗ t (Y ∗ t − U∗ t τ β). Theorem 2.3. Assume that Assumptions 2.5–2.9 listed in Section 2.5 hold. Then as T → ∞ √ T h (b β − β) − µβ i →D N(0, Σβ) (2.37) with µβ = BUU −1 µB and Σβ = BUU −1 ΣB BUU −1 τ , where BUU = E [U∗ 1 U∗ 1 τ ], µB = E[R0] and ΣB = E [(R0 − µB) (R0 − µB) τ ]. Furthermore, when (2.2) holds, we have µβ = 0 and Σβ = BUU −1 ΣB BUU −1 τ , (2.38) where ΣB = E [R0Rτ 0 ] with Rt = U∗ t εt, and εt = Yt − m1(Ut, Vt) = Yt − µ − Uτ t β − g(Vt). © 2007 by Taylor Francis Group, LLC
- 48. 32 ESTIMATION IN NONLINEAR TIME SERIES Remark 2.7. Note that p X l=1 P (0) l,w (Vtl) − βτ p X l=1 PU l,w(Vtl) = p X l=1 P (0) l,w (Vtl) − βτ PU l,w(Vtl) = p X l=1 Pl,w(Vtl, β) ≡ ga(Vt, β). Therefore Y ∗ t − U∗ t τ β = εt + g(Vt) − ga(Vt, β), where g(Vt) − ga(Vt, β) is the residual due to the additive approximation. When (2.2) holds, it means that g(Vt) in (2.1) has the expressions g(Vt) = p X l=1 gl(Vtl) = p X l=1 Pl,w(Vtl, β) = ga(Vt, β) and H(Vt) = Pp l=1 PU l,w(Vtl), and hence Y ∗ t − U∗ t τ β = εt. As β min- imizes L(β) = E [Yt − m1(Ut, Vt)] 2 , we have L0 (β) = 0 and E [tU∗ t ] = E [ij (Ut − E[Ut|Vt])] = 0 when (2.2) holds. This implies E [Rt] = 0 and hence µβ = 0 in (2.37) when the marginal integration estimation procedure is employed for the additive form of g(·). In both theory and practice, we need to test whether H0 : β = β0 holds for a given β0. The case where β0 ≡ 0 is an important one. Before we state the next theorem, some additional notation is needed. Let b BUU = 1 T T X t=1 b U∗ t (b U∗ t )τ , b Z∗ t = Z̃t − p X l=1 P̂U l,w(Vtl), b µB = 1 T T X t=1 b Rt, b Rt = b U∗ t Ŷ ∗ t − b U∗ t τ b β , b µβ = b BUU −1 b µB, b Σβ = b BUU −1 b ΣB b BUU −1 τ , in which b ΣB is a consistent estimator of ΣB, defined simply by b ΣB = ( 1 T PT t=1( b Rt − b µB)( b Rt − b µB)τ if (2.1) holds, 1 T PT t=1 b Rt b Rτ t if (2.2) holds. It can be shown that both b µβ and b Σβ are consistent estimators of µβ and Σβ, respectively. We now state a corollary of Theorem 2.3 to test hypotheses about β. Corollary 2.2. Assume that the conditions of Theorem 2.3 hold. Then © 2007 by Taylor Francis Group, LLC
- 49. SEMIPARAMETRIC KERNEL ESTIMATION 33 as T → ∞ b Σ −1/2 β √ T h (b β − β) − b µβ i →D N(0, Iq), T h (b β − β) − b µβ iτ b Σ−1 β h (b β − β) − b µβ i →D χ2 q. Furthermore, when (2.2) holds, we have as T → ∞, b Σ −1/2 β √ T b β − β →D N(0, Iq), √ T(b β − β) τ b Σ−1 β √ T(b β − β) →D χ2 q. The proof of Theorem 2.3 is relegated to Section 2.5 while the proof of Corollary 2.2 is straightforward and therefore omitted. Next we state the following theorem for the nonparametric component. Theorem 2.4. Assume that Assumptions 2.5–2.9 listed in Section 2.5 hold. Then for xk ∈ [−Lk, Lk], p T bk b b Pk,w(xk) − Pk,w(xk) − bias1k →D N(0, var1k), (2.39) where bias1k = 1 2 b2 k µ2(K) Z w(−k)(x(−k) )f(−k)(x(−k) ) ∂2 g(x, β) ∂x2 k dx(−k) and var1k = J Z V (x, β) [w(−k)(x(−k) )f(−k)(x(−k) )]2 f(x) dx(−k) with J = R K2 (u)du, µ2(K) = R u2 K(u)du, g(x, β) = E Yij − µ − Zτ ijβ |Xij = x , and V (x, β) = E h Yij − µ − Zτ ijβ − g(x, β) 2 |Xij = x i . Furthermore, assume that (2.2) holds and that E h w(−k)(X (−k) ij ) i = 1. Then as T → ∞ p T bk (b gk(xk) − gk(xk) − bias2k) →D N(0, var2k), (2.40) where bias2k = 1 2 b2 k µ2(K) ∂2 gk(xk) ∂x2 k , var2k = J Z V (x, β) [w(−k)(x(−k) )f(−k)(x(−k) )]2 f(x) dx(−k) © 2007 by Taylor Francis Group, LLC
- 50. 34 ESTIMATION IN NONLINEAR TIME SERIES with V (x, β) = E h Yij − µ − Zτ ijβ − Pp k=1 gk(xk) 2 |Xij = x i . The proof of Theorem 2.4 is relegated to Section 2.5. Theorems 2.3 and 2.4 may be applied to estimate various additive models such as model (2.2). In the following example, we apply the proposed estimation procedure to determine whether a partially linear time series model is more appropriate than either a completely linear time series model or a purely nonparametric time series model for a given set of real data. Example 2.3: In this example, we continue analyzing the Canadian lynx data with yt = log10{number of lynx trapped in the year (1820 + t)} for t = 1, 2, ..., 114 (T = 114). Let q = p = 1, Ut = yt−1, Vt = yt−2 and Yt = yt in model (2.1). We then select yt as the present observation and both yt−1 and yt−2 as the candidates of the regressors. Model (2.1) reduces to a partially linear time series model of the form yt = βyt−1 + g(yt−2) + et. (2.41) In addition to estimating β and g(·), we also propose to choose a suitable bandwidth h based on a nonparametric cross–validation (CV) selection criterion. For i = 1, 2, define b gi,t(·) = b gi,t(·, h) = 1 T − 3 PT s=3,s6=t K ·−ys−2 h ys+1−i b πh,t(·) , (2.42) where b πh,t(·) = 1 T −3 PT s=3,s6=t K ·−ys−2 h . We now define a new LS estimate e β(h) of β by minimizing T −3 X t=1 {yt − βyt−1 − b g1,t(yt−2) − βb g2,t(yt−2)} 2 . The CV selection function is then defined by CV (h) = 1 T − 3 T X t=3 n yt − h e β(h)yt−1 + b g1,t(yt−2) − e β(h)b g2,t(yt−2) io2 . (2.43) For Example 2.3, we choose K(x) = 1 √ 2π e− x2 2 and H114 = [0.3 · 114− 7 30 , 1.1 · 114− 1 6 ]. Before selecting the bandwidth interval H114, we actually calculated the © 2007 by Taylor Francis Group, LLC
- 51. Another Random Scribd Document with Unrelated Content
- 52. Fig. 573. the valve discs, thus compensating for the expansion and contraction of the metal and insuring a perfectly tight valve, regardless of the temperature of the steam. The Holyoke Improved Speed Governor for water wheels is shown in Figs. 574 and 575. The following is a description of the two figures where the same letters are used to designate the parts appearing in both illustrations: The pulley, A, is the receiving pulley, and is designed to run at 400 revolutions per minute, receiving its power from the water-wheel shaft, or countershaft belted from the same. Contained in the pulley, A, are the two governing weights, BB, of which the centrifugal forces are overcome by the springs, CC. The varying motions of the governing weights, BB, are transmitted through racks and pinions in the hub of pulley, A, to levers, K and L, which operate the valve, N, admitting water under a light pressure to the cylinder, O. The water is admitted to the cylinder, O, through ports at either end, causing the piston to move forward or backward, governed by the movement of the governing weights, BB.
- 53. Fig. 574.
- 54. Fig. 575. The pulley, A, is keyed to the main shaft, and at the opposite end is keyed a bevel pinion running in mesh with a bevel gear on either side, all of which are contained in the gear-case, P. These gears cause the clutch discs, D, to run in opposite directions. In each disc is a clutch, E, keyed to a shaft, transmitting power to the pinion, S, running in mesh with the spur gear, R, which is loose on the shaft, J, and transmits its power through the pin clutch, T, to gate shaft, J. The gate shaft, J, is connected by a pair of bevel gears to the shaft and hand wheel, Q. The motion of the piston rod, I, caused by the movement of piston in cylinder, O, is carried by the lever, G, to the clutch shaft, F, by means of the pivoted nut at V. The clutch shaft, F, operates either clutch, E, corresponding to the movement of the governing weights,
- 55. BB, caused by the variation in speed. From the clutch thus engaged, the power is carried by the clutch shaft, F, through the gears, S and R, and the pin clutch, T, to the gate shaft, J. Fig. 576. The makers of the machine here described, say: “In the year 1902 our attention was called to a new governor invented by Nathaniel Lombard, and after finding by actual tests that this governor possessed advantages over all others then in use, we were induced to make arrangements for its manufacture and sale. Two years have been spent in improving and perfecting this machine, hence the name ‘The Improved Governor.’” The governor is provided with a steadying device operated by the chain, H. The gate shaft, J, is designed to make four, six or eight turns to open the gate, four being the regular number.
- 56. The receiving pulley and governor gate shaft may revolve in either direction, as desired. The receiving pulley is designed to run at 400 revolutions per minute, and is driven by a 4-inch double belt. The governor gate shaft may be arranged to open the gates in four, six or eight turns, and may be extended on either or both sides of the governor to meet the necessary requirements. The governor is capable of exerting a pressure ranging from 25,000 to 50,000 foot pounds on the governor gate shaft. The advantages claimed for this improvement on the Lombard governor are thus stated: 1. It requires only a light water pressure to handle the heaviest gates. 2. It is simple in construction. All parts are easy of access. 3. There are no pumps working under high pressure. 4. There are no dash pots to get out of adjustment, due to the change in temperature of oil, etc. 5. There is but one belt on this machine. 6. All parts which are constantly in motion are equipped with ring- oiling bearings. Fig. 576 is an illustration of the mechanism necessary to raise and lower the head gates which are used to admit and regulate, also to shut off the water supply from pond or lake to the flume conveying it to the wheel. In this case there are two head gates having racks upon the upright timbers connecting with the gates. Two shrouded pinions engage these racks, which are keyed upon a shaft having a large spur wheel at its end, as represented. A pinion upon a second shaft engages this spur wheel which in turn has also a spur wheel which engages a pinion upon the crank shaft having two cranks opposite one another. By means of these cranks with two to four men upon each crank the gates are operated very satisfactorily.
- 57. These shafts and gears are mounted upon heavy cast iron brackets bolted to the floor. Altogether it forms a very massive piece of mechanism. The Utility combination pump governor is shown in the figure below. This mechanism may be bolted on any tank or receiver where the water level is to be automatically maintained. It consists of a closed pocket containing a float, A, which rises and falls with the water level inside the tank. When the water rises above the desired level the float opens the throttle valve and starts the pump, and when it subsides the float falls and shuts off the steam.
- 60. CONDENSING APPARATUS. A condenser is an apparatus, separate from the cylinder, in which exhaust steam is condensed by the action of cold water; condensation is the act or process of reducing, by depression of temperature or increase of pressure, etc., to another and denser form, as gas to the condition of a liquid or steam to water. There is an electrical device called “a condenser” which must not be confounded with the hydraulic apparatus of the same name; there is also an optical instrument designated by the same term, which belongs to still another division of practical science. A vacuum is defined very properly as an empty space; a space in which there is neither steam, water or air—the absolute absence of everything. The condenser is the apparatus by which, through the cooling of the steam by means of cold water, a vacuum is obtained. The steam after expelling the air from the condenser fills it with its own volume which is at atmospheric pressure nearly 1700 times that of the same weight of water. Now when a vessel is filled with steam at atmospheric pressure, and this steam is cooled by external application of cold water, it will immediately give up its heat, which will pass off in the cooling water, and the steam will again appear in a liquid state, occupying only 1⁄1700 part of its original volume. But if the vessel be perfectly tight and none of the outside air can enter, the space in the vessel not occupied by the water contains nothing, as before stated. The air exerting a pressure of nearly 15 pounds to the square inch of the surface of the vessel tries to collapse it; now if we take a cylinder fitted with a piston and connect its closed end to this vessel by means of a pipe, the atmospheric pressure will push this piston down. The old low pressure engines were operated almost entirely upon this principle, the steam only served to push the piston up and exhaust the air from the cylinder.
- 61. In Fig. 578 is exhibited the effect of jets of water from a spray nozzle meeting a jet of steam; the latter instead of filling the space with steam is returned to its original condition of water and the space as shown becomes a vacuum. Briefly stated condensation and the production of a vacuum may be used to advantage in the following ways: 1. By increasing the power without increasing the fuel consumption. 2. By saving fuel without reducing the output of power. 3. By saving the boiler feed water required in proportion to the saving of fuel. 4. By furnishing boiler feed water free from lime and other scaling impurities. 5. By preventing the noise of the escaping exhaust steam. 6. By permitting the boiler pressure to be lowered ten to twenty pounds without reducing the power or the economy of the engine. The discovery of the advantages arising from the condensation of steam by direct contact with water was accidental. In the earliest construction of steam-engines the desired vacuum was produced by the circulation of water through a jacket around the cylinder. This was a slow and tedious process, the engine making only seven or eight strokes per minute. “An accidental unusual circumstance pointed out the remedy, and greatly increased the effect. As the engine was at work, the attendants were one day surprised to see it make several strokes much quicker than usual; and upon searching for the cause, they found, says Desaguliers, ‘a hole through the piston which let the cold water (kept upon the piston to prevent the entrance of air at the packing) into the space underneath.’ The water falling through the steam condensed it almost instantaneously, and produced a vacuum with far less water than when applied to the exterior of the cylinder. This led Newcomen to remove the outer cylinder, and to insert the lower end of the
- 62. Fig. 578. water pipe into the bottom of the cylinder, so that on opening a cock a jet of cold water was projected through the vapor. This beautiful device is the origin of the injection pipe with a spray nozzle still used in low- pressure engines.” The apparatus described above is called the jet-condenser and is in use up to the present day in various forms. In the Fig. 577, page 298, the jet is shown at C. It will be understood that steam enters through the cock D and comes in contact with a spray of cold water at the bottom, where it is condensed and passes into the air pump through which it is discharged. By this diagram, Fig. 577, may be understood in a simple yet accurate manner the course of steam from the time it leaves the boiler until it is discharged from the condenser. Referring to the upper section of the plate, a sectional view of a steam cylinder, jet condenser, air pump and exhaust piping is shown. The high pressure steam “aa” is represented by dark shading, and the low pressure or expanded steam “bb” by lighter shading. The steam enters the side “aa,” is cut off, and expansion takes place moving the piston in the direction of the arrow to the end of the stroke. The exhaust valve now opens and the piston starts to return. The low pressure steam instead of passing direct to the atmosphere, as is the case of a high pressure engine, flows into a chamber “C,” and is brought in contact with a spray of cold water. The heat being absorbed by the water, the steam is condensed and
- 63. reduced in volume, thus forming a vacuum. It is, however, necessary to remove the water formed by the condensed steam together with the water admitted to condense the steam, also a small amount of air and vapor. For this purpose, a pump is required, which is called the air pump. Fig. 579. Condensers are classified into surface condensers and jet condensers, both again being divided into direct connected and indirect connected condensers. The surface condenser (see Fig. 579) is mainly used in marine practice because it gives a better vacuum, and keeps the condensed steam separate from the cooling water; it consists of a vessel, of varied shapes, having a number of brass tubes passing from head to head. The ends of this vessel are closed by double heads, the tubes are expanded into the inner one on one end, while their other ends pass through stuffing-boxes in the other inner head. The “admiralty” or rectangular surface condenser is represented in Fig. 579. This form occupies less floor space than the round shell,
- 64. Fig. 580. Fig. 581. and is preferred upon steam yachts and small vessels. Steam is condensed on its introduction at the top of the apparatus where it comes in contact with the cool surfaces of the tubes. Through these water is circulated by a centrifugal pump driven usually by a separate engine.
- 65. Fig. 582. The water of condensation leaves the condenser at the bottom and is drawn off by the vacuum pump. The water from the circulating pump enters at the bottom right-hand end; following the direction indicated by the arrows, it flows through the lower half of the tubes towards the left whence it returns through the upper half of the tubes towards the right and escapes overboard through the water outlet pipe. It will be observed that the coolest water encounters the lowest temperature of steam at the bottom, hence the best results are reached. There is also a baffle plate just above the upper row of
- 66. Fig. 583. tubes to compel a uniform distribution of exhaust steam among the tubes, as shown in the engraving. These tubes are usually small—1⁄2″ outside diameter—of brass and coated with tin inside and outside to prevent galvanic action which is liable to attack the brass tubes and cause them to corrode. Fig. 581 shows an end view of the right-hand head of the surface condenser here described. A single tube is shown in detail in Fig. 580. One end of the tube is drawn sufficiently thick to chase upon it deep screw threads, while a slot facilitates its removal by a screw-driving tool. The other end is packed and held in place by a screw gland, which is also provided with a slot. In this way the tube is firmly held in one head, and, though tightly fitted in the other, is free to move longitudinally under the influence of expansion or contraction, due to the varying heat. In some cases engineers prefer the ordinary arrangement of screw glands at both ends of the tubes, with the usual wick packing. The mechanism illustrated in Figs. 582 and 583 shows a combined condenser and feed-water heater. A compact and efficient method of heating the feed-water from the hot well is of great importance; this is the case in cold weather when the circulating water is at a low temperature. The Volz apparatus is a combined condenser and feed-water heater; the shell or exhaust steam chamber contains a set of tubes, through which the feed-water passes, while the lower part contains
- 67. the condensing tubes, both parts being in proper communication with their respective water chambers. The heater tubes being located immediately adjacent to the exhaust inlet, are exposed to the hottest steam, and the feed-water becomes nearly as high temperature as that of the vacuum. Pages 304 and 305 show the sectional and outside views. The enclosing shell containing the combined heater and condenser is a well ribbed cylindrical iron casting; free and independent access is provided to either set of tubes by removing corresponding heads. The illustration, Fig. 584, is a longitudinal section of one side of the condenser pump, and also a section of the condenser cone, spray pipe, exhaust elbow and injection elbow. “A” is the exhaust to which is connected the pipe that conducts to the apparatus the steam or vapor that is to be condensed. The injection water is conveyed by a pipe attached to the injection opening at “B.” “C” is the spray pipe, and has, at its lower extremity, a number of vertical slits through which the injection water passes and spreads out into thin sheets. The spray cone “D” scatters the water passing over it, and thus ensures a rapid intermixture with the steam. This spray cone is adjustable by means of a stem passing through a stuffing-box at the top of the condenser, and is operated by the handle “E.” The cone should be left far enough down to pass the quantity of water needed for condensation. All regulation of the injection water must be done by an injection valve placed in the injection pipe at a convenient point. Note.—The surface condensers, Figs. 579-581, are made by the Wheeler Condenser and Engineering Co., New York, as are also the Volz combined surface condenser and feed water heater, shown in Figs. 582 and 583. The operation of this condensing apparatus is as follows: steam being admitted to the cylinders “K,” so as to set the pump in motion, a vacuum is formed in the condenser, the engine cylinder, the connecting exhaust pipe, and the injection pipe. This causes the
- 68. injection water to enter through the injection pipe attached at “B” and spray pipe “C” into the condenser cone “F.” The main engine being started, the exhaust steam enters through the exhaust pipe at “A,” and, coming in contact with the cold water, is rapidly condensed. The velocity of the steam is communicated to the water, and the whole passes through the cone “F” into the pump “G” at a high velocity, carrying with it, in a comingled condition the air or uncondensable vapor which enters the condenser with the steam. The mingled air and water is discharged by the pump through the valves and pipe at “J” before sufficient time or space has been allowed for separation to occur. Fig. 584.
- 69. Fig. 585. The exhaust steam induction condenser is based upon the same principle heretofore explained under the section relating to injectors. See Fig. 585. The exhaust steam enters through the nozzle, A. The injection water surrounds this nozzle and issues downward through the annular space between the nozzle and the main casting. The steam meeting the water is condensed, and by virtue of its weight and of the momentum which it has acquired in flowing into the vacuum the resulting water continues downward, its velocity being further increased, and the column solidified by the contraction of the nozzle shown. The air is in this way carried along with the water and it is impossible for it to get back against the rapidly flowing steam in the contracted neck. The condenser will lift its own water twenty feet or so. When water can be had under sufficient head to thus feed itself into the system, and the hot-well can at the same time be so situated as to drain itself, it makes a remarkably simple and efficient arrangement. In case the elevation is so great that a pump has to be used to force the injection, the pump has to do less work than the ordinary air pump, and its exhaust can be used to heat the feed water. The Bulkley “Injector” condenser is shown in Fig. 586, arranged so that the condensing water is supplied by a pump. The condenser is connected to a vertical exhaust pipe from the engine, at a height of about 34 feet above the level of the “hot-well.” An air-tight discharge pipe extends from the condenser nearly to the bottom of the “hot- well,” as shown in the engraving.
- 70. The condenser is supplied by a pump as shown, or from a tank, or from a natural “head” of water; the action is continuous, the water being delivered into the “hot-well” below. The area of the contracted “neck” of the condenser is greater than that of the annular water inlet described above, and the height of the water column overcomes the pressure of the atmosphere without. Fig. 586.
- 71. The supply pump delivers cool water only, and is therefore but one-third of the size of the air-pump. The pressure of the atmosphere elevates the water about 26 feet to the condenser. The accompanying diagrams, Figs. 587 and 588, are worthy of study. They represent a condenser plant designed by the Schutte Koerting Co., Philadelphia, and placed on steam-vessels plying on fresh water. In these drawings the parts are designed by descriptive lettering instead the ordinary way of reference figures; this adds to the convenience of the student in considering this novel application of the condenser-injector, the action of which is described in the following paragraphs. Fig. 587.
- 72. Fig. 588. For steamers plying on fresh water lakes, bays and rivers it is unnecessary to go to the expense of installing surface condensers such as are used in salt water; keel condensers, however, are used in both cases. The keel condenser consists of two copper or brass pipes running parallel and close to the keel, one on each side united by a return bend at the stern post. The forward ends are connected, one to the exhaust pipe of the engine while the other end is attached to the suction of the air pump. In other cases both forward ends are attached to the exhaust pipe of the steam engine while the water of condensation is drawn through a smaller pipe connected with the return bend at the stern post which is the lowest part of the keel condenser. Fig. 587 is much used for vessels running in fresh water. The illustration is a two-thirds midship section of a vessel with pipe connections to the bilge—bottom injection—side injection into the centrifugal pump, thence upward through suction pipe into the ejector condenser where it meets and condenses the exhaust steam from the engine and so on through the discharge pipe overboard. The plan of piping with valves, drain pipes and heater are shown in Fig. 588.
- 73. In case of the failure of any of the details of this mechanism to perform their respective functions a free exhaust valve and pipe is provided which may be brought instantly into use. The discharge pipe has a “kink” in it to form a water seal, as represented with a plug underneath to drain in case of frost, or in laying up the vessel in winter. A pipe leads from globe valve (under discharge elbow) to feed pump for hot water. Condensing Surface Required. In the early days of the surface condenser it was thought necessary to provide a cooling surface in the condenser equal to the heating surface in the boilers, the idea being that it would take as much surface to transfer the heat from a pound of steam to the cooling water and condense the steam as it would to transfer the heat from the hot gases to the water in the boiler and convert it into steam. The difference in temperature, too, between the hot gases and the water in the boiler is considerably greater than that between the steam in the condenser and the cooling water.
- 74. Note.—The following list gives the numbers with the corresponding names of the parts of the surface condenser, shown in the above outline sketch: 1, condenser shell; 2, outside heads; 3, exhaust inlet; 4, exhaust outlet; 5, water inlet; 6, water outlet; 7, peep holes; 8, tube heads; 9, partition; 10, rib; 11, tubes; 12, stuffing-boxes.
- 75. Note.—The numbers and names of parts in the above figure, representing in outline a jet condenser, are as follows: 1, condenser body; 2, exhaust inlet; 3, discharge; 4, injection valve; 5, spray pipe; 6, spraying device. Steam, however, gives up its heat to a relatively cool surface much more readily than do the hot furnace gases, and the positively circulated cooling water takes up that heat and keeps the temperature of the surface down, while in a boiler the absorption depends in a great measure upon the ability of the water by natural circulation to get into contact with the surface and take up the heat by evaporization. It has been found, therefore, that a much smaller surface will suffice in a condenser than in the boilers which it serves. The Wheeler Condenser and Engineering Company, who make a specialty of surface condensers, say that one square foot of cooling surface is usually allowed to each 10 pounds of steam to be condensed per hour, with the condensing water at a normal temperature not exceeding 75°. This figure seems to be generally used for average conditions. Special cases require special treatment.
- 76. For service in the tropics the cooling surface should be at least ten per cent. greater than this estimate. Where there is an abundance of circulating water the surface may be much less, as with a keel condenser, where 50 pounds of steam is sometimes condensed per hour per square foot of surface; or a water works engine, where all the water pumped is discharged through the condenser and not appreciably raised in temperature, probably condensing 20 to 40 pounds of steam per hour per square foot of surface. Under the division of this volume devoted to “air and vacuum pumps,” much information has been given relating to the principles of the condensation of steam and also some illustrations of working machines. Still it may be well to say this, in addition, that— All questions in regard to a vacuum become plain when we consider that the atmosphere itself exerts a pressure of nearly 15 pounds, and measure everything from an absolute zero, 15 pounds below the atmospheric pressure. We live at the bottom of an ocean of air. The winds are its currents; we can heat it, cool it, breathe and handle it, weigh it, and pump it as we would water. The depth of this atmospheric ocean cannot be determined as positively as could one of liquid, for the air is elastic and expands as the pressure decreases in the upper layers. Its depth is variously estimated at from 20 to 212 miles. We can, however, determine very simply how much pressure it exerts per square inch.
- 78. UTILITIES AND ATTACHMENTS Working Ship Pumps by Ropes.
- 79. UTILITIES AND ATTACHMENTS. Utility is a Latin word meaning the same as the Saxon word usefulness, hence a utility is something to be used to advantage. An attachment is that by which one thing is connected to another; some adjunct attached to a machine or instrument to enable it to do a special work; these are too numerous to be described in this work; moreover their number is being so constantly added to that it would be vain to make the attempt. A few examples only follow. The Receiver is one of the most important and useful parts or connections of a steam pump. This apparatus, frequently called “Pump and Governor,” and illustrated in Figs. 589, 590 and 591, is designed to automatically drain heating systems and machines or appliances used in manufacturing which depend upon a free circulation of steam for their efficiency. It furthermore is arranged to automatically pump the water of condensation drained from such systems back to the boilers without loss of heat. By this operation it serves a double purpose: first to automatically relieve the system of the water of condensation constantly collecting therein, thus insuring a free and unobstructed circulation, and, incidentally, preventing snapping and hammering in the piping, which in many cases is due to entrained water; and second, to automatically deliver this water, which in many cases is at the boiling point, directly to the boilers without the intervention of tanks or other devices commonly used. Not only does it relieve the system of a troublesome factor, but it introduces a supply of feed water to the boiler at a temperature impossible otherwise without the use of a special water heater. The economy resulting from its use is unquestionable, and the satisfactory and increasing use of this machine leaves no doubt as to its efficiency.
- 80. As will be seen by the illustrations, the apparatus consists of a cylinder or oval closed receiver, which, together with the pump, is mounted upon and secured to a substantial base, making the whole machine compact and self-contained. The automatic action of the pump and its speed are controlled by a float in the receiver operating directly, without the use of intervening levers, cranks and stuffing boxes, to open or close a governor valve in the steam supply pipe to the pump, thus making the action of the pump conditional upon the rise and fall of the float in the receiver. Fig. 589. In each of the three receivers shown there is a ball float which appears through the side of the receiver, Fig. 590; these depend upon the principle of specific gravity for their operation. The lever fastened to the ball float operates the throttle valve of the pump; as the vessel fills with water the float rises opens the throttle valve, and starts the pump.
- 81. In Fig. 589 is shown the Deane automatic duplex steam pump and receiver fitted with valves for hot water; it is also provided with three separate inlets for convenience in connecting the returns. In placing the apparatus, it is only necessary to so locate it that all returns will drain naturally towards receiver and that there are no pockets in the piping. When it is desired to use the automatic receiver as the sole means of feeding the boilers, it will be necessary to introduce a small supply of water from some outside source to equalize the loss which occurs. It is desirable that this water should flow into receiver rather than into discharge pipe. Fig. 590.
- 82. Welcome to our website – the ideal destination for book lovers and knowledge seekers. With a mission to inspire endlessly, we offer a vast collection of books, ranging from classic literary works to specialized publications, self-development books, and children's literature. Each book is a new journey of discovery, expanding knowledge and enriching the soul of the reade Our website is not just a platform for buying books, but a bridge connecting readers to the timeless values of culture and wisdom. With an elegant, user-friendly interface and an intelligent search system, we are committed to providing a quick and convenient shopping experience. Additionally, our special promotions and home delivery services ensure that you save time and fully enjoy the joy of reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookultra.com