Kybernetika 54 no. 2, 375-399, 2018

QMLE of periodic bilinear models and of PARMA models with periodic bilinear innovations

Abdelouahab Bibi and Ahmed GhezalDOI: 10.14736/kyb-2018-2-0375

Abstract:

This paper develops an asymptotic inference theory for bilinear $\left( BL\right) $ time series models with periodic coefficients $\left( PBL\text{ for short}\right) $. \ For this purpose, we establish firstly a necessary and sufficient conditions for such models to have a unique stationary and ergodic solutions (in periodic\ sense). Secondly, we examine the consistency and the asymptotic normality of the quasi-maximum likelihood estimator $\left( QMLE\right) $ under very mild moment condition for the innovation errors. As a result, it is shown that whenever the model is strictly stationary, the moment of some positive order of $PBL$ model exists and is finite, under which the strong consistency and asymptotic normality of $QMLE$ for $PBL$ are proved. Moreover, we consider also the periodic $ARMA$ $\left( PARMA\right) $ models with $PBL$ innovations and we prove the consistency and the asymptotic normality of its $QMLE.$

Keywords:

asymptotic normality, periodic bilinear model, periodic $ARMA$ model, strict and second-order periodic stationarity, strong consistency

Classification:

2M10, 62M15

paper.pdf

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