Portmanteau test statistics for seasonal serial correlation in time series models.

Esam Mahdi

doi:10.1186/s40064-016-3167-4

Abstract

The seasonal autoregressive moving average SARMA models have been widely adopted for modeling many time series encountered in economic, hydrology, meteorological, and environmental studies which exhibited strong seasonal behavior with a period s. If the model is adequate, the autocorrelations in the errors at the seasonal and the nonseasonal lags will be zero. Despite the popularity uses of the portmanteau tests for the SARMA models, the diagnostic checking at the seasonal lags 1s,2s,3s,ldots ,ms, where m is the largest lag considered for autocorrelation and s is the seasonal period, has not yet received as much attention as it deserves. In this paper, we devise seasonal portmanteau test statistics to test whether the seasonal autocorrelations at multiple lags s of time series are different from zero. Simulation studies are performed to assess the performance of the asymptotic distribution results of the proposed statistics in finite samples. Results suggest to use the proposed tests as complementary to those classical tests found in literature. An illustrative application is given to demonstrate the usefulness of this test.

Highlights

The proposed goodness-of-fit tests modify those statistics given in Box and Pierce (1970), Ljung and Box (1978), Fisher and Gallagher (2012) and Mahdi and McLeod (2012) to the SARMA class, respectively, as follows m
Where φ(B) = 1 − φ1B1 − · · · φpBp and θ (B) = 1 − θ1B1 − · · · θqBq are polynomials in B of degrees p and q respectively, whereas (Bs) = 1 − 1Bs − · · · φps Bsps and (Bs) = 1 − 1Bs − · · · qs Bsqs are polynomials in Bs of degrees ps and qs respectively, p and q ≥ 0 are the order of the non-seasonal autoregressive, AR, model and moving average, MA, model respectively, whereas ps and qs ≥ 0 are the order of the seasonal autoregressive, SAR, model and seasonal moving average, SMA, model respectively, B is the backshift operator on t, and s > 0 is the length of the seasonal period
Despite the popularity of the SARMA models in various economic and financial data, the goodness-of-fit portmanteau tests at multiple period lags 1s, 2s, 3s, . . . , ms, where m is the largest lag considered for autocorrelation and s is the seasonal period, has not yet received as much attention as it should deserve

Summary

Introduction

The proposed goodness-of-fit tests modify those statistics given in Box and Pierce (1970), Ljung and Box (1978), Fisher and Gallagher (2012) and Mahdi and McLeod (2012) to the SARMA class, respectively, as follows m Comparison of type I error rates The empirical type I error rates at nominal levels 1, 5, and 10 % for the portmanteau seasonal test statistics using the approximation distributions based on 104 simulations have been evaluated under the Gaussian SAR (1)s models where s = 4, 12.

Results

Conclusion