In this monograph we have considered a class of autoregressive models whose coefficients are random. The models have special appeal among the non-linear models so far considered in the statistical literature, in that their analysis is quite tractable. It has been possible to find conditions for stationarity and stability, to derive estimates of the unknown parameters, to establish asymptotic properties of these estimates and to obtain tests of certain hypotheses of interest. We are grateful to many colleagues in both Departments of Statistics at the Australian National University and in the Department of Mathematics at the University of Wo110ngong. Their constructive criticism has aided in the presentation of this monograph. We would also like to thank Dr M. A. Ward of the Department of Mathematics, Australian National University whose program produced, after minor modifications, the "three dimensional" graphs of the log-likelihood functions which appear on pages 83-86. Finally we would like to thank J. Radley, H. Patrikka and D. Hewson for their contributions towards the typing of a difficult manuscript. IV CONTENTS CHAPTER 1 INTRODUCTION 1. 1 Introduction 1 Appendix 1. 1 11 Appendix 1. 2 14 CHAPTER 2 STATIONARITY AND STABILITY 15 2. 1 Introduction 15 2. 2 Singly-Infinite Stationarity 16 2. 3 Doubly-Infinite Stationarity 19 2. 4 The Case of a Unit Eigenvalue 31 2. 5 Stability of RCA Models 33 2. 6 Strict Stationarity 37 Appendix 2. 1 38 CHAPTER 3 LEAST SQUARES ESTIMATION OF SCALAR MODELS 40 3.

Inhalt
1 Introduction.- 1.1 Introduction.- 2 Stationarity and Stability.- 2.1 Introduction.- 2.2 Singly-Infinite Stationarity.- 2.3 Doubly-Infinite Stationarity.- 2.4 The Case of a Unit Eigenvalue.- 2.5 Stability of RCA Models.- 2.6 Strict Stationarity 37 Appendix 2.1.- 3 Least Squares Estimation of Scalar Models.- 3.1 Introduction.- 3.2 The Estimation Procedure.- 3.3 Strong Consistency and the Central Limit Theorem.- 3.4 The Consistent Estimation of the Covariance Matrix of the Estimates.- 4 Maximum Likelihood Estimation of Scalar Models.- 4.1 Introduction.- 4.2 The Maximum Likelihood Procedure.- 4.3 The Strong Consistency of the Estimates.- 4.4 The Central Limit Theorem.- 4.5 Some Practical Aspects.- 5 A Monte Carlo Study.- 5.1 Simulation and Estimation Procedures.- 5.2 First and Second Order Random Coefficient Autoregressions.- 5.3 Summary.- 6 Testing the Randomness of the Coefficients.- 6.1 Introduction.- 6.2 The Score Test.- 6.3 An Alternative Test.- 6.4 Power Comparisons 108 Appendix 6.1.- 7 The Estimation of Multivariate Models.- 7.1 Preliminary.- 7.2 The Least Squares Estimation Procedure.- 7.3 The Asymptotic Properties of the Estimates.- 7.4 Maximum Likelihood Estimation.- 7.5 Conclusion.- 8 An Application.- 8.1 Introduction.- 8.2 A Non-Linear Model for the Lynx Data.- References.- Author And Subject Index.