Testing for Random Walk Coefficients in Regression and State Space Models

325 Pages·1998·21.529 MB·English

by  Dr. Martin Moryson (auth.)

Contributions to Statistics
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Martin Moryson
Testing for Random
Walk Coefficients in
Regression and
State Space Models
With 38 Figures
and 72 Tables
Physica-Verlag
A Springer-Verlag Company
Series Editors
Werner A. MUller
Martina Bihn
Author
Dr. Martin Moryson
Dreiweidenstr. 10
D-65195 Wiesbaden
Germany
ISBN-13: 978-3-7908-1132-2 e-ISBN-13: 978-3-642-99799-0
DOl: 10.1007/978-3-642-99799-0
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Moryson, Martin: Testing for random wane coefficients in regression and state space mod
els: with 72 tahlesIMartin Moryson. - Heidelberg: Physica-Verl., 1998
(Contributions to statistics)
Zugl.: Berlin, Humboldt-Univ., Diss., 1998
ISBN-13: 978-3-7908-1132-2
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Acknowledgements
While I was writing this thesis at the Institute of Statistics and Econo
metrics and the Sonderforschungsbereich 373 at the Humboldt-Universitiit
zu Berlin, I received support and advice from many people. In partic
ular, I am indebted to my supervisor, Prof. Dr. H. Liitkepohl, for his
very helpful comments on many preliminary versions, his patience and
especially his continuous encouragement. Moreover, I like to thank my
co-supervisor, Prof. Dr. J. Wolters, for his critical, but benevolent com
ments and support in the final stage of my dissertation.
I was very lucky in experiencing a very positive working atmosphere
at the institute and in receiving many valuable comments at the joint
seminars with the Freie Universitiit Berlin. Representing this atmosphere
I would like to thank Jorg Breitung, Helmut Herwartz, Kirstin Hubrich
and Thomas Kotter for various kinds of support.
Berlin, March 1998 Martin Moryson
Contents
1 Introduction 1
2 The Linear State Space Model 7
2.1 The Model Set-up. 8
2.2 Some Basic Results 10
2.3 Interpretation of the State Space Model. 13
2.3.1 Regression Models with Time Varying Coefficients 13
2.3.1.1 Hildreth-Houck Model . . . 14
2.3.1.2 Random Walk Coefficients . 16
2.3.1.3 The Return to Normalcy Model . 20
2.3.2 The Noise Model ..... 22
2.4 The Kalman Filter and Smoother 23
2.5 Estimation of the Hyperparameters 31
2.5.1 Setting up the Likelihood 31
2.5.2 The Scoring Algorithm .. 34
2.5.3 The EM Algorithm .... 37
2.5.4 Interplay of the Scoring and EM Algorithm 41
2.5.5 Identification and Consistency . . . . . . . 42
2.5.6 Simulation Study on Consistency ..... 46
2.5.7 Properties of Kalman Filter and Smoother
with Estimated Parameters 51
2.6 An Illustrative Example 53
2.7 Forecasting ...... 56
Vll
Vlll
3 Exact Tests for Univariate Random Walk Coefficients 61
3.1 The Testing Problem . . . . . . 62
3.2 An Exact F -Test . . . . . . . . 68
3.3 A Point Optimal Invariant Test 71
3.4 The Locally Best Invariant Test 76
3.5 Simulation Study . . . . . . . . 81
3.6 Appendix: Determination of Critical Values 95
4 Asymptotic Tests for Univariate Random Walk
Coefficients in Models with Stationary Regressors 101
4.1 Introduction . . . . . . . . . . . . . . . . . . . 101
4.2 Asymptotic Distribution of the LM/LBI Test. 103
4.3 The Hansen Test .......... 106
4.4 The Modified Hansen Test ..... 110
4.5 The Test of Leybourne & McCabe. 111
4.6 Simulation Study .......... 112
5 Asymptotic Tests for Univariate Random Walk
Coefficients in Models with Non-Stationary Regressors 131
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 132
5.2 The Model and the Estimators. . . . . . . . . . . . 137
5.3 Asymptotic Distribution of the LM/LEI Test in the
Presence of 1(1) Regressors .......... 144
5.3.1 Regression Model Without a Trend 145
5.3.2 Regression Model Including a Trend. 147
5.3.3 Some Simulation Results . . . . . . . 147
5.4 Asymptotic Distribution of Test Statistics Based on OLS
Estimators . . . . . . . . . . . . . . 148
5.4.1 Model Without Constant. . . . . 149
5.4.2 Model Including a Constant ... 150
5.4.3 Model with Constant and Trend . 151
5.4.4 Model Including Stationary Regressors 153
5.5 Asymptotic Distribution of Test Statistics Based on
Asymptotically Efficient Estimators 154
5.5.1 Model Without Constant . . . . . . . . . . . 155
IX
5.5.2 Model Including Stationary Regressors 156
5.5.3 Model Including a Constant ...... 157
5.5.4 Model Including a Trend . . . . . . . . 158
5.5.5 Model Including Further 1(1) Regressors 158
5.6 Testing the Constancy of the Intercept 161
5.7 Simulation Study ......... 162
5.8 Tests with Polynomial Regressors 176
6 Testing Trend Stationarity Against Difference
Stationarity in Time Series 183
6.1 Introduction.. 183
6.2 The KPSS Test 184
6.3 The Test of Leybourne & McCabe. 188
6.4 The Choi Test . 190
6.5 The Tsay Test. 192
6.6 POI and LBI Tests 193
6.7 Simulation Study . 195
7 Testing for Multivariate Random Walk Coefficients in
Regression Models 201
7.1 The Testing Problem . . . . . . 202
7.2 Exact Tests . . . . . . . . . . . 203
7.3 Simulation Study: Exact Tests. 211
7.4 Asymptotic Tests in Models with Stationary Regressors 220
7.5 Simulation Study: Stationary Regressors . . . . . . .. 223
7.6 Asymptotic Tests in Models with Integrated Regressors 229
7.6.1 Asymptotic Distribution of the LMjLBI Test in
the Presence of I( 1) Regressors ......... 229
7.6.2 Asymptotic Distribution of Test Statistics Based
on OLS Estimators . . . . . . . . . . . . . . . .. 232
7.6.3 Asymptotic Distribution of Test Statistics Based
on Asymptotically Efficient Estimators . . . . .. 234
7.6.4 Testing the Constancy of All Coefficients
Simultaneously . . . . . . . . . . 242
7.7 Simulation Study: Integrated Regressors . . . . 242
x
8 Testing for Random Walk Coefficients in the Presence
of Varying Coefficients Under Ho 255
8.1 The Testing Problem 255
8.2 Asymptotic Tests . . . 258
8.3 Simulation Study . . . 261
8.3.1 Univariate Case 261
8.3.2 Multivariate Case . 271
9 The Term Structure of German Interest Rates
- Testing the Expectations Hypothesis 277
9.1 The Data 278
9.2 Tests 279
9.2.1 Unit Root Tests. 279
9.2.2 Testing for Co-Integration 281
9.2.3 Testing the Constancy of the Regression
Coefficients . . . . . . . . 283
9.3 Estimation of State Space Models 287
9.4 Conclusions . . . . . 289
10 Resume and Prospects 297
References 301
List of Figures
2.1 Empirical densities of 01,11 and 01,12, model 2 49
0 0
2.2 Empirical densities of 1,11 and 1,12, model 3 50
2.3 Timepaths of example series 53
2.4 Timepaths of fi1,tlT . . . . . . . . . . . . . . . 54
2.5 Timepaths of fi2,tlT .............. . 54
2.6 Timepath of fi2,tIT, based on misspecified model 56
3.1 Power functions of exact tests: models 1 and 2 . 91
3.2 Power functions of exact tests: models 3 and 4 . 92
4.1 Power functions: Xl,t = 1, AR errors. . . . 125
4.2 Power functions: X1,t = 1, MA errors . . . 126
4.3 Power functions: X1,t = N(5, 1), AR errors 127
4.4 Power functions: X1,t = N(5, 1), MA errors 128
= + +
4.5 Power functions: 1 AR errors 129
X1,t 0.8Xt-1 Vt,
4.6 Power functions: X1,t = 1 + 0.8Xt-1 + Vt, MA errors 130
5.1 Timepaths of Yt and tlYt ........ . 133
5.2 Empirical density functions of t-statistics 135
5.3 Power function, model 1a . 172
5.4 Power function, model 1b 173
5.5 Power function, model 2a . 174
5.6 Power function, model 2b 175
7.1 Power functions of exact tests, r = 0 and 0.3 218
7.2 Power functions of exact tests, r = 0.8 '" 219
7.3 Power functions of asymptotic tests: p = 0 227
7.4 Power functions of asymptotic tests: p = 0.3 228
=
7.5 Power functions, 1(1) regressors: modell, p 0 251