This paper develops an asymptotic inference theory for bilinear $\left(BL\right)$ time series models with periodic coefficients $\left(PBL\text{for}\text{short}\right)$. 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...