Weak Convergence of Marked Empirical Processes for Focused Inference on AR(p) vs AR(p + 1) Stationary Time Series

The technique applied by the authors to construct consistent and focused tests of fit for i. i. d. samples and regression models is extended to AR models for stationary time series. This approach leads to construct a consistent goodness-of-fit test for the null hypothesis that a stationary series is governed by an autoregressive model of a given order p. In addition of the consistency, the test is focused to detect efficiently the alternative of an AR(p + 1) model. The basic functional statistic conveying the information provided by the series is the process of accumulated sums of the residuals computed under the model of the null hypothesis of fit, reordered as concomitants of the conveniently delayed process. This process is transformed in order to obtain a new process with the same limiting Gaussian law encountered in earlier applications of the technique. Therefore, a Watson type quadratic statistic computed from this process has the same asymptotic laws under the null hypothesis of fit, and also under the alternatives of focusing, than the test statistics used in those applications. As a consequence, the resulting test has the same desirable performance as the tests previously developed by applying the same kind of transformations of processes.