Bootstrapping State-Space Models: Distribution-Free Estimation in View of Prediction and Forecasting

Abstract

Linear models, seasonal autoregressive integrated moving average (SARIMA) models, and state-space models have been widely adopted to model and forecast economic data. While modeling using linear models and SARIMA models is well established in the literature, modeling using state-space models has been extended with the proposal of alternative estimation methods to the maximum likelihood. However, maximum likelihood estimation assumes, as a rule, that the errors are normal. This paper suggests implementing the bootstrap methodology, utilizing the model’s innovation representation, to derive distribution-free estimates—both point and interval—of the parameters in the time-varying state-space model. Additionally, it aims to estimate the standard errors of these parameters through the bootstrap methodology. The simulation study demonstrated that the distribution-free estimation, coupled with the bootstrap methodology, yields point forecasts with a lower mean-squared error, particularly for small time series or when dealing with smaller values of the autoregressive parameter in the state equation of state-space models. In this context, distribution-free estimation with the bootstrap methodology serves as an alternative to maximum likelihood estimation, eliminating the need for distributional assumptions. The application of this methodology to real data showed that it performed well when compared to the usual maximum likelihood estimation and even produced prediction intervals with a similar amplitude for the same level of confidence without any distributional assumptions about the errors.