Testing periodicity in time series models -- A recommendation of bootstrap models
A periodic (seasonal) autoregressive model of order \(p\) is specified as follows: \[ x_t=\nu_s+a_{s,t}x_{t-1}+a_{s,2}x_{t-2}+\dots+a_{s,p}x_{s-p}+u_t, \] where \(s\) is the number of seasons within the year at which \(x_t\) is observed (\(s=1,\dots,S\)). A nonperiodic model can be considered as a restricted case of the periodic one: \(a_{1,j}=a_{2,j}=\dots=a_{S,j}\). A test for nonperiodicity (\(H_0\)) can be based on the log-likelihood ratio \(Q_{LR}\); under the assumption of independent, normally distributed errors \(u_t\) with time invariant variance \(Q_{LR}=T(\ln\hat\sigma_r^2-\ln\sigma_n^2)\), where \(\hat\sigma_r^2\) and \(\sigma_n^2\) denote the MLE of error variances in the restricted and unrestricted model, respectively. Under \(H_0\) \(Q_{LR}\) is asymptotically \(\chi^2\) distributed with \(S-1\) degrees of freedom. The author proposes a bootstrap procedure for the estimation of the \(Q_{LR}\) distribution for a small sample case and a wild bootstrap procedure for the case of time-dependent errors variances. Results of Monte Carlo study and an illustration for macroeconomic data on four states are given.
scientific article; zbMATH DE number 1336630