Garry Phillips

The Estimation of Regression Models with Time-Varying Parameters

In analysing time series data, the assumption that the coefficients in a regression model are constant over time may not always be reasonable. One way of handling this problem is to allow the parameters to vary over time according to a particular stochastic process. The parameters in models of this type are said to be dynamic, and they represent a generalization of models in which the parameters are random, in that they are independent of each other in different time periods; see, for example, Theil [1971, 622–627].

The independence of tests for structural change in regression models

Abstract It is shown that the common tests for stability of regression coefficients and of the disturbance variance in linear regression models are independent. This enables the size of the joint test procedure to be controlled exactly.

Testing for heteroscedasticity in simultaneous equation models

Abstract In this paper a number of exact and approximate tests for heteroscedasticity in a simultaneous equation model are developed. The exact tests are analogous to those which are available single equation regression models and the approximate tests include a Lagrangian multiplier test. Type 1 error probabilities of approximate tests and the powers of exact and approximate tests are examined using Monte Carlo experiments for several simple cases of heteroscedasticity.

Testing strategies for model specification

A crucial element in the development of econometric methodology during the past decade has been the concern with testing as opposed to estimating econometric models. In this paper we discuss-especially for the econometric analysis of time series-the main types of test procedures, and we also investigate the opportunities to uphold the Neyman-Pearson theory in the context of thorough model specification testing. In applied work it is quite usual to carry out several tests on the same set of sample data. We consider an extension of the Neyman-Pearson framework to the case of such repeated testing, and examine situations where the various hypotheses under test have a particular nesting structure. For the case where a sequence of superposed alternatives is tested by so-called marginal tests, we prove that the various test statistics are asymptotically independent under a common null hypothesis if the statistics are based on either the likelihood-ratio, or the Wald, or the Lagrange-multiplier approach. Testing a particular null hypothesis against a series of juxtaposed alternatives appears to lead to independent test statistics only in specific circumstances. It is shown how independence of test statistics enables the control over the overall Type I error probability, which is an essential element in the Neyman-Pearson theory. Using the notions of constructive hypotheses and auxiliary hypotheses, we can draw a clear distinction between specification tests and misspecification tests. Next an overview is given of approaches to and examples of specification and misspecification testing. With respect to the former, attention is paid to the problem determining the order of dynamics and discriminating between system dynamics and error dynamics. The misspecification testing is reviewed for specification error, nonconstancy of coefficients, heteroscedasticity, serial dependence, and nonnormality of disturbances. Also the problem of testing for several misspecifications jointly or sequentially is considered. Finally we discuss the options and associated difficulties in implementing the various tests in an overall testing strategy.

Testing for serial correlation in simultaneous equation models: Some further results

Abstract In this paper approximate counterparts of the exact tests earlier proposed by the authors are examined. Type 1 error probabilities and test powers are estimated and compared using Monte Carlo experiments. The effect on the Type 1 error probabilities of the misspecification which results when serial correlation is present elsewhere in the system is also investigated.

Testing for Structural Change in Simultaneous Equation Models

Tests for changes in the coefficients of linear regression models, particularly the analysis of covariance and the Chow tests, are well known to econometricians and they are widely used. This paper demonstrates that analogous tests can also be constructed in static simultaneous equation models when equations are estimated by common k-class estimators, e.g., OLS, 2SLS, and LIML. The tests are based on the residuals obtained when the estimated endogenous part of a simultaneous equation is regressed on all the exogenous variables in the system. The tests have many of the characteristics of the regression based tests although the nature of the residuals used makes it more difficult to analyse their power properties.

A note on estimating and testing exogenous variable coefficient estimators in simultaneous equation models

Abstract The exact distribution of 2SLS estimators of exogenous variable parameters is derived conditional on the endogenous parameter estimates. An approximate signific ance test is then developed for which lower bounds for the critical values are found from the F distribution.