Large-sample tests of homogeneity for time series models

SUMMARY Limit distributions both under the null and a sequence of alternative hypotheses are derived for the likelihood ratio and related statistics for testing the homogeneity of several autoregressive processes. These results are extended to mixed autoregressive and moving average processes. It is often required to test whether two or more sets of data can be considered as coming from a common time series model. If the hypothesis of homogeneity is accepted one can obviously achieve parsimony in model specification and also obtain better estimates of the model parameters by pooling the data sets. On the other hand, if there is a significant difference between the sets one may be interested in alternative models which may vary between sets but are still close to each other for large samples. When dealing with regression models the homogeneity tests usually lead to analysis of variance techniques. In this note we consider the comparison of autoregressive and also of mixed autoregressive-moving average models. A unified method of deriving limit distributions of the likelihood ratio and related test statistics is suggested in ? 2. The method is illustrated in ? 3 for autoregressive processes augmented by regression variables, and for mixed autoregressive-moving average processes.