Des F. Nicholls

Random coefficient autoregressive models: an introduction

1 Introduction.- 1.1 Introduction.- Appendix 1.1.- Appendix 1.2.- 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.- Appendix 3.1.- Appendix 3.2.- 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.- Appendix 4.1.- Appendix 4.2.- 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.- 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.- Appendix 7.1.- 8 An Application.- 8.1 Introduction.- 8.2 A Non-Linear Model for the Lynx Data.- References.- Author And Subject Index.

Modelling the Accounting Information Requirements of Small Businesses

Studies in various locations have indicated that practising accountants are an important source of advice and information to the small business sector. However, prior research has concentrated on establishing a relationship between the two parties and in ascertaining the extent of services provided. This paper highlights the limited acquisition or preparation of detailed accounting information by Australian small business owner/managers. Operating and environmental variables which influence the preparation or acquisition of detailed accounting information are established and logistic regression techniques used to estimate an appropriate explanatory model, from which estimates of the probability that a firm, with particular attributes which are reflected in the explanatory variables in the model, will prepare or acquire a given level of accounting information.

The Estimation of the Prediction Error Variance

Abstract Spectral methods are used to construct an estimate of the variance of the prediction error for a normal, stationary process. The estimate obtained is shown to be strongly consistent and asymptotically normally distributed. Some aspects of the computations with respect to the fast Fourier transform are considered. The latter half of the article consists of a number of simulations, based on both generated and real data, which illustrate the results obtained. The relation between the estimate and that obtained from a high order autoregression is discussed.

Multiple autoregressive models with random coefficients

This paper derives conditions for the stationarity of a class of multiple autoregressive models with random coefficients. The models considered include as special cases those previously discussed by Andel (Ann. Math. Statist.42 (1971), 755-759; Math. Operationsforsch. Statist.7 (1976), 735-741).

The estimation of multivariate random coefficient autoregressive models

Using a two stage regression procedure estimates of the unknown parameters of a class of multivariate random coefficient autoregressive models are obtained. The estimates are shown, under fairly general conditions, to be strongly consistent and to have a distribution which converges to that of a normally distributed random vector.

Stationarity and Stability

In the case of the scalar RCA model, that is the model with p = 1, Andel (1976) has obtained conditions for the existence of a singly infinite process {X(t); t = 1-n,…,0,1,…} satisfying (1.1.1) which is second order stationary. In this chapter we shall extend the results of Andel to the multivariate RCA model and also obtain conditions for the existence of a doubly infinite process {X(t); t = 0,± 1,± 2,…} which is second order stationary and satisfies (1.1.1) for all t.

A Monte Carlo Study

In order to illustrate the procedures introduced in chapters 3 and 4 a number of simulations were performed with first and second order univariate RCA models for several sets of data of different sizes. While the simulations performed have been by no means exhaustive, as we shall see the results do conform with the asymptotic theory developed in the last two chapters.

Least Squares Estimation of Scalar Models

In chapter 2, conditions were found for the existence of stationary solutions to equations of the form (1.1.1). In practice, however, given that a stationary time series {X(t)} satisfies such an equation, it is necessary to estimate the unknown parameters in order to provide predictors of X(t) given past values of the process. Estimation procedures for fixed coefficient autoregressions are well established, and the asymptotic properties of these estimates are well known (see, for example, chapter 6 of Hannan (1970)). Random coefficient autoregressions are, however, non-linear in nature, and any foreseeable maximum likelihood type estimation method would be an iterative procedure. Such a procedure is discussed in Chapter 4, where the asymptotic properties of the estimates obtained are determined. Iteration must, nevertheless, commence at some point, and since the likelihood will be non-linear and its domain will be of relatively high dimensions, it is likely that there will be local extrema. Hence it is desirable that iterations commence close to the global maximum of the likelihood function for otherwise convergence might be toward a local extremum. ID. this chapter, a least squares estimation procedure is proposed for univariate random coefficient autoregressions which, under certain conditions, is shown to give strongly consistent estimates of the true parameters. The estimates are also shown to obey a central limit theorem.

The Estimation of Multivariate Models

From the point of view of estimation, to date we have only considered scalar RCA models. In this chapter we shall give a brief theoretical discussion of the estimation of multivariate RCA models. While the least squares estimation procedure of chapter 3 will be seen to extend readily to the multivariate situation, the extension of the maximum likelihood procedure is not as straightforward.

Testing the Randomness of the Coefficients

As indicated in the previous chapter, the asymptotic results for the maximum likelihood estimates may be used to test certain hypotheses of interest. The condition (cii), however, which was assumed so as to obtain a standard central limit theorem for the maximum likelihood estimates, precludes the use of the theory derived in chapter 4 to test what is perhaps the most relevant hypothesis, namely that Σ = 0, that is, that the data come from a fixed coefficient autoregression. This chapter examines the testing of hypotheses in general, and in particular, two tests for the hypothesis that Σ = 0.