Spline smoothing in small area trend estimation and forecasting

Semiparametric models combining both non-parametric trends and small area random effects are now currently being investigated in small area estimation (SAE). These models can prevent bias when the functional form of the relationship between the response and the covariates is unknown. Furthermore, penalized spline regression can be a good tool to incorporate non-parametric regression models into the SAE techniques, as it can be represented as a mixed effects model. A penalized spline model is considered to analyze trends in small areas and to forecast future values of the response. The prediction mean squared error (MSE) for the fitted and the predicted values, together with estimators for those quantities, are derived. The procedure is illustrated with real data consisting of average prices per squared meter of used dwellings in nine neighborhoods of the city of Vitoria, Spain, during the period 1993–2007. Dwelling prices for the next five years are also forecast. A simulation study is conducted to assess the performance of both the small area trend estimator and the prediction MSE estimators. The results confirm a good behavior of the proposed estimators in terms of bias and variability.