Asymptotic expansion approximations and the distributions of various test statistics in dynamic econometric models

In this thesis we examine the derivation of asymptotic expansion approximations to the cumulative distribution functions of asymptotically chi-square test statistics under the null hypothesis being tested and the use of such approximations in the investigation of the properties of testing procedures. We are particularly concerned with how the structure of various test statistics may simplify the derivation of asymptotic expansion approximations to their cumulative distribution functions and also how these approximations can be used in conjunction with other small sample techniques to investigate the properties of testing procedures. In Chapter 1 we briefly review the construction of test statistics based on the Wald testing principle and in Chapter 2 we review the various approaches to finite sample theory which have been adopted in econometrics including asymptotic expansion methods. In Chapter 3 we derive asymptotic expansion approximations to the joint cumulative distribution functions of asymptotically chi-square test statistics making explicit use of certain aspects of the structure of such test statistics. In Chapters 4, 5 and 6 we apply these asymptotic expansion approximations under the null hypothesis, in conjunction with other small sample techniques, to a number of specific testing problems. The test statistics considered in Chapters 4 and 6 are Wald test statistics and those considered in Chapter 5 are predictive failure test statistics. The asymptotic expansion approximations to the cumulative distribution functions of the test statistics under the null hypothesis are evaluated numerically; the Implementation of the algorithm for obtaining asymptotic expansion approximations to the cumulative distribution functions of test statistics is discussed in an Appendix on Computing. Finally, in Chapter 7 we draw overall conclusions from the earlier chapters of the thesis and discuss briefly directions for possible future research