Parsimonious periodic autoregressive models for time series with evolving trend and seasonality

Abstract

This paper proposes an extension of Periodic AutoRegressive (PAR) modelling for time series with evolving features. The large scale of modern datasets, in fact, implies that the time span may subtend several evolving patterns of the underlying series, affecting also seasonality. The proposed model allows several regimes in time and a possibly different PAR process with a trend term in each regime. The means, autocorrelations and residual variances may change both with the regime and the season, resulting in a very large number of parameters. Therefore as a second step we propose a grouping procedure on the PAR parameters, in order to obtain a more parsimonious and concise model. The model selection procedure is a complex combinatorial problem, and it is solved basing on genetic algorithms that optimize an information criterion. The model is tested in both simulation studies and real data analysis from different fields, proving to be effective for a wide range of series with evolving features, and competitive with respect to more specific models.

Appendix: Asymptotic equivalence of a test of hypothesis and an information criterion

Let us consider two nested models $M_0$ and $M_1$, where $M_0$ has p parameters and $M_1$ has q additional parameters that under the null hypothesis $H_0$ are assumed equal to zero. A standard F test under gaussianity for $H_0$ is obtained by the test statistic:

$$\begin{aligned} F = \frac{(SS_0-SS_1)/q}{SS_1/(N-p-q)} \end{aligned}$$

(4)

where $SS_0$ and $SS_1$ are the residual sum of squares of the model $M_0$ and $M_1$ fitted to N observations.

Under the null hypothesis the statistic (4) follows a F distribution with q and $N-p-q$ degrees of freedom. Let ${\overline{F}} _\alpha$ denote the $(1-\alpha )$-quantile of that distribution, then hypothesis $H_0$ is rejected at level $1-\alpha$ whenever $F>{\overline{F}}_\alpha$, or

$$\begin{aligned} \log (SS_0) - \log (SS_1) > \log \left( 1+ {\overline{F}}_\alpha \frac{q}{N-p-q} \right) . \end{aligned}$$

For N large the right hand side may be approximated by ${\overline{F}}_\alpha \, q/(N-p-q)$ and rejection is equivalent to

$$\begin{aligned} \log (SS_0) - \log (SS_1) > {\overline{F}}_\alpha \frac{q}{N-p-q} \end{aligned}$$

that may be also written $\mathrm{IC}_0^* - \mathrm{IC}_1^* >0$ where $\mathrm{IC}_0^*= \log (SS_0) + \pi ^* \, p$ , $\mathrm{IC}_1^*= \log (SS_1) + \pi ^* \, (p+q)$ and

$$\begin{aligned} \pi ^*= \frac{N}{N-p-q} \, {\overline{F}}_\alpha . \end{aligned}$$

(5)

It follows that choosing a penalizing constant equal to $\pi ^*$ in (5), model $M_1$ will be preferred to model $M_0$ by the information criterion whenever the test rejects the null hypothesis $H_0$.