A Multi-Step Forecast Density

This paper makes two contribution to the literature on density forecasts. First, we propose a novel bootstrap approach to estimate forecasting densities based on nonparametric techniques. The method is based on the Markov Bootstrap that is suitable to resample dependent data. The combination of nonparametric and bootstrap methods delivers density forecasts that are flexible in capturing markovian dependence (linear and nonlinear) occurring in any moment of the distri-bution. Second, we improve the testing approach to evaluate density forecasts by considering a set of tests for dynamical misspecification such as autocorrelation, heteroskedasticity and neglected nonlinearity. The approach is useful because rejections of the tests give insights into ways to im-prove the forecasting model. By Monte Carlo simulations we show that the proposed evaluation strategy has much higher power to detect misspecification of the density forecasts compared to previous analysis. The proposed nonparametric-bootstrap forecasting method exhibits the ability to capture correctly the dynamics of linear and nonlinear time series models. We also investigate the performance at higher orders and propose methods to deal with the “curse of dimensionality”. Finally, we empirically investigate the relevance of the method in out-of-sample forecasting the density of 3 business cycles variables for the US: real GDP, the Coincident Indicator and Industrial Production. The results indicate that the method gives reliable density forecasts for all variables and performs better compared to parametric forecasting methods.