La Trobe University - Department of Statistical Science
A standard result in statistical inference says that the asymptotic distribution of the likelihood ratio statistic (and of several others) under the null hypothesis is a chi-square. For this result to hold, it is important that the parameters defining the model be identified under the null hypothesis. There are many models, particularly those used in econometrics, that are non-standard (i.e. some parameters defining the model are not identified under the null hypothesis); examples include, testing for the number of components in a normal mixture, ARCH-in-Mean, GARCH, random coefficient models, and threshold models. The literature that deals with this type of non-standard models is reasonably large. However, only a very small number of papers have considered the case when the alternative hypothesis also contains inequality constraints (i.e. order restrictions). In this seminar, I will address this particular topic. In particular, I will discuss the main results in Beg, Silvapulle, and Silvapulle (2001).
Initially, I will provide an introduction to some of the basic issues; this literature is quite old now (it goes back to Davies, 1977, Biometrika). However, it would be helpful to briefly recall the main points. Then, I will consider the case when there are parameter constraints in the alternative hypothesis. For simplicity, the presentation will be restricted to the ARCH-in-Mean model, although the principles are quite general. A data example will be used to illustrate the main results.