Bootstrap for correcting the mean square error of prediction and smoothed estimates in structural models

It is well known that the uncertainty in the estimation of parameters produces the underestimation of the mean square error (MSE) both for in-sample and out-of-sample estimation. In the state space framework, this problem can affect confidence intervals for smoothed estimates and forecasts, which are generally built by state vector predictors that use estimated model parameters. In order to correct this problem, this paper proposes and compares parametric and nonparametric bootstrap methods based on procedures usually employed to calculate the MSE in the context of forecasting and smoothing in state space models. The comparisons are performed through an extensive Monte Carlo study which illustrates, empirically, the bias reduction in the estimation of MSE for prediction and smoothed estimates using the bootstrap approaches. The finite sample properties of the bootstrap procedures are analyzed for Gaussian and non-Gaussian assumptions of the error term. The procedures are also applied to real time series, leading to satisfactory results.