A Bootstrap-Based Nonparametric Forecast Density

The interest in density forecasts (as opposed to solely modeling the conditional mean) arises from the possibility of dynamics in higher moments of a time series as well as, in some applications, the interest in forecasting the probability of future events. By combining the idea of Markov bootstrapping with kernel density estimation, this paper presents a simple nonparametric method for estimating out-of-sample multi-step density forecasts. The paper also considers a host of evaluation tests to examine dynamical misspecification of estimated density forecasts by targeting autocorrelation, heteroskedasticity and neglected nonlinearity. These tests are useful as rejections of the tests give insights into ways to improve a particular forecasting model. In an extensive Monte Carlo analysis involving a range of commonly used linear and nonlinear time series processes, the nonparametric method is shown to work reasonably well across the simulated models for a suitable choice of bandwidth (smoothing parameter). Furthermore, the application of the method to the US Industrial Production series provides multi-step density forecasts that show no sign of dynamic misspecification