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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

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

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)
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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

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

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

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
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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
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
iii

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
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
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
7 Appendix 193
7.1 Technical lemmas 193
7.2 Asymptotic normality and expansions 198
References 209
Author Index 230
Subject Index 235

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

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

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

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

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

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

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

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.

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)

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)

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.

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

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TODAY
Today
we leave my mother's hogan
my mother's winter hogan.
We leave the shelter of its
rounded walls.
We leave its friendly center fire.
We drive our sheep to the mountains.
For the sheep,
there is grass and shade
and water,
flowing water
and water standing still,
in the mountains.
There is no wind.
There is no sand
up there.

PACKING
My mother's possessions
we tie on the pack horses,
her loom parts
and her wool yarns,
her cooking pots,
her blanket
and my blanket
and the water jug,
white sacks filled with food,
cans of food,
cornmeal and wheat flour,
coffee and sugar.
My mother's possessions,
we tie them all on the
pack horses.
The packs must be steady.
The ropes must be tight.
The knots must be strong.
I cannot pack the horses,
I am too little,
but I can bring the possessions
to my father and my uncle.
I am big enough for that.

GOODBYE TO MY HOGAN
My mother's hogan,
I feel safe
with your rounded walls
about me.
But now I must leave you.
I must leave your fire
and your door.
The sheep need me.
I must go with them
to a place they know,
but that is strange to me.
I put my moccasins,
my precious moccasins,
by your fireplace, my hogan,
so you will not be lonely
while I am gone.

GOODBYE
Land
around my mother's hogan
and sheep trail
and arroyo
and waterhole,
sleep in the sun
this summer.
Rest well
for my sheep
will not be here
to deepen the trail and arroyo
with their little sharp feet.
They will not be here
to eat the short grass,
to drink the stored water.
Sleep,
rest well,
and be ready for our return.

My mother scatters the ashes
from her cooking fire.
She sweeps the hogan floor
with her rabbit-brush broom.
My father lays the bough
across the door
to show that we have gone.
The dogs bark.
They run around the sheep corral
telling the sheep
we are ready to go.
The young corn in the field
hangs its tasseled heads.
Young corn,
my grandmother is staying
at home.
She will take care of you.
My father mounts his horse.
He drives the pack horses before him.
My uncle mounts his horse.
They ride away together,
singing,
across the empty sand.

GOODBYE GRAY CAT
Gray Cat,
I am telling you goodbye.
Today I go to the mountains.
I take my sheep to summer range,
but you, Gray Cat,
you have no sheep
so you must stay at home.
Stay here with my grandmother,
Gray Cat.
She will feed you.
Goodbye, Goodbye.

ACROSS THE SAND
My mother lets down the bars
of the sheep corral.
The flock crowds around her.
The goats look at me.
I think they are saying,
We know where we are going.
The little lambs
walk close by their mothers.
They are like me,
they do not know
if they will like this place
where we are going.
My mother and I,
we drive our sheep
across the sand.
My grandmother
stands at her door
looking after us.

GOODBYE TO GRANDMOTHER
My grandmother,
my little grandmother,
now I am leaving you.
Last year I was too small
to go to the mountains.
I stayed with you,
but this year I am big,
I am almost tall
so I must help drive the sheep
to summer range.
My grandmother,
my little grandmother,
do not be lonely.
I will come back again.

RIDING
Riding,
riding,
riding on my horse
to herd the sheep
across the yellow sand.
Yellow sand is around me.
Yellow sun is above me.
I ride in the middle
of a sand and sun filled world.
Riding,
riding,
riding on my horse
to herd the sheep
across the yellow sand.
Sun heat
and sheep smell
and sand dust
wrap around me
like a blanket
as I ride through the sand
with my sheep.

NOON IN THE SAGEBRUSH
At noon
we reach the sagebrush flats.
Gray-green sagebrush scents the air.
Gray-green sagebrush softens
the yellows of the land.
My mother makes a little fire
no bigger than her coffee pot.
Food is good
and rest is good
at noon
in the sagebrush.

NIGHT CAMP
At night we make camp
in the juniper covered hills.
My father is waiting for us there.
The moon looks down
on the restless sheep
on the hobbled horses.
The moon looks down
on a shooting star.
But I am too tired
to look at anything.
I sleep.

Morning sunrise sees us climbing
up and up
on the mountain trail.
There are pine trees
standing straight and tall.
Brown pine needles
and green grass
cover the ground.
Shadows play with the sunlight.
There is no yellow sand.
The sheep hurry upward,
climbing and pushing
in the narrow trail.
I ride after the sheep.
My horse breathes fast.
His feet stumble
in the narrow trail.
All day long
the sheep climb upward.
They want to eat
and I am hungry, too,
but my mother says,
No.
All day long we ride
to herd the sheep.
Night is almost with us

SUMMER RANGE
Summer range in the mountains
is on a high mesa,
a steep, high mesa,
a flat-topped mesa,
with tall growing pine trees,
with short growing green grass,
with little, winding rivers
and rain filled lakes.
This is summer range for our sheep.

Between the trees
I see water standing
in a bowl of green rushes.
The water is quiet.
It is still
and blue
and cold.
It is a lake
with land all around it.
It is a lake.
The sheep drink
long and steadily.
They stand in the shallow water
at the edges of the lake.
Their little pointed feet
dig deep into the mud
of the lake banks.
I see colored fish
beneath the water
swimming in a rainbow line.
I throw stones into the lake.
The water pushes back in circles
to take the stones.
The dogs swim far out
into the cold waters.
They are thirsty and hot.

I have never seen a lake before.
Gentle rain pools I have seen
and angry flood waters,
but never before
a still, blue lake.
It is beautiful.
A lake is beautiful.

SHELTER
Beneath the trees
I see our summer shelter.
My father and my uncle
have made a shade
to shelter us from night rains
and from the cold
of near-by snow peaks.
They have made us a shade
of cottonwood boughs
and juniper bark.
It has the clean smell
that trees give.

THE SHEEP CORRAL
My father and my uncle
made a sheep corral
while they were waiting
for the sheep and for us
to come up the trail.
They made the sheep corral
of branches,
a circle of branches,
a circle of dark colored boughs.
The sheep stay safe
in their corral tonight
and I sleep
beneath the cottonwood shade.

This morning
when I opened my eyes from
sleeping I could not remember
what place this is.
I thought I was in
my mother's winter hogan.
Now I remember.
This is summer camp.
Tall trees stretch above me.
In the darkness
they look blacker than the night.
As I lie here,
safe and warm beneath
my blanket,
all around me turns to gray mist,
all around me turns to silver.
Darkness is gone,
but it made no sound.
It left no footprints.
The world is still asleep.
Through the pine trees
day comes up
light comes up.

In the pine trees
bird wings are stirring,
bird songs are stirring.
I hear them.
I hear them.
The grass beside my blanket
is wet with night rain.
Morning mist is on the leaves
and in my hair.
I put one toe out,
one brown toe out.
It is hard to get up
when it is cold.
Blue smoke from my mother's fire
curls upward in a thin blue line.
The sheep move inside their corral.
I come out from under my blanket,
from under my warm blanket.
Like the other things around me,
I come out
to greet the day.

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