Conventional estimation methods for the the spatial autoregressive (SAR) model rely on the key assumption that the spatial lag parameter is time-invariant for the entire study period. This strong assumption is likely be violated in many economic situations where spillovers may change over time. At the other extreme, a time-varying model where the spatial lag coefficient changes every period might be unnecessary. This paper specifies a periodic spatial autoregressive model with fixed effects and develops three estimation methods: two-stage instrumental variable (2SLS) method, quasi-maximum likelihood estimation (QMLE) approach and generalized method of moments (GMM). The Monte Carlo study investigates the small-sample properties of the proposed estimators under various scenarios and evaluates the costs of misspecification of the Data Generating Process, pointing to the usefulness of the periodic spatial autoregressive model from an applied perspective.