ARMAX model specification testing, with an application to unemployment in the Netherlands
The author presents a new model specification test for testing an ARMAX model specification against the alternative hypothesis that the involved ARMAX specification is false. The \((k+1)\)-variate time series process \(\{(y_ t,x_ t)\}\) with \(x_ t\) as a k-component vector is assumed to be real-valued and the error-process is denoted by \(e_ t\). Under the null it is assumed that for each t the conditional expectation of the error relative to the entire past of the process involved equals zero with probability one. Under the null the test statistic of the test is N(0,1). Given the assumption that the time series are strictly stationary, this test is consistent in the sense that the absolute value of the test statistic converges in probability to infinity if the null is false. This property ensures that asymptotically any sort of misspecification will be detected. To perform the test it is neither required that the errors are Gaussian white noise nor that they are homoscedastic. The weaker requirements imply that standard specification tests such as LR and LM and the Box- Pierce test are not applicable in a straight forward manner. Furthermore, estimation of the parameters of the vector ARMAX model with x as a vector of exogenous variables as given by \textit{E. J. Hannan}, \textit{W. T. M. Dunsmuir} and \textit{M. Deistler}, J. Multivariate Anal. 10, 275- 295 (1980; Zbl 0445.62098), cannot be applied directly here for three reasons: first, x being exogenous implies that the error process is independent of the \((x_ t)\) process so that feedback from \(y_ t\) to \(x_ t\) is excluded. Secondly, incorporating this condition in the test procedure would affect the type-I-error of the test (the true null may be rejected), and thirdly, the ARMAX structure need not hold under the alternative that the null is false. In part two of the paper the author applies this ARMAX specification test to test the rational expectations - natural rate hypothesis for the Netherlands unemployment rate. Part three presents the econometric theory and more technical aspects of the test results. Some theorems are proved in the appendix.
rational expectations - natural rate hypothesis
scientific article; zbMATH DE number 4007538
scientific article