Recursive prediction and likelihood evaluation for periodic ARMA models (Q2742774)

Recursive prediction and likelihood evaluation techniques for periodic autoregressive moving-average (PARMA) time series models are investigated. A time series \(\{ X_t\}\) with finite second moments is said to be a PARMA series with period \(T\geq 1\) if it is a solution to the periodic linear difference equation NEWLINE\[NEWLINE (X_{nT +\nu} - \mu_{\nu}) - \sum_{k=1}^{p(\nu)} \phi_k(\nu) (X_{nT +\nu -k} - \mu_{\nu -k}) = \varepsilon_{nT +\nu} + \sum_{k=1}^{q(\nu)} \theta_k(\nu) \varepsilon_{nT +\nu -k}. NEWLINE\]NEWLINE The error sequence \(\{ \varepsilon_t\}\) is mean-zero periodic white noise with var\((\varepsilon_{nT +\nu}) = \sigma^2 (\nu).\) The authors find the best linear predictor \(\hat X_{nT +\nu}.\) For example, NEWLINE\[NEWLINE \hat X_{nT +\nu} =\sum_{k=1}^p \phi(\nu) X_{nT +\nu -k} + \sum_{k=1}^q \theta_{nT +\nu -1,k} (X_{nT +\nu -k} - \hat X_{nT +\nu -k}),\quad nT + \nu >m,NEWLINE\]NEWLINE where \(\theta_{t,j}\) are some coefficients. The asymptotic forms of these recursions are explored and monotonicity properties of the mean squared prediction errors are derived. \(h\)-step-ahead predictors for \(h \geq 2\) are considered, too. The authors also show how the likelihood of a vector ARMA model can be evaluated by writing the vector ARMA model in PARMA form. Two examples with explicit calculations for PARMA series and general periodic autoregression are presented.