A new bootstrap procedure to obtain prediction densities of returns and volatilities of GARCH processes is proposed. Financial market participants have shown an increasing interest in prediction intervals as measures of uncertainty. Furthermore, accurate predictions of volatilities are critical for many financial models. The advantages of the proposed method are that it allows incorporation of parameter uncertainty and does not rely on distributional assumptions. The finite sample properties are analyzed by an extensive Monte Carlo simulation. Finally, the technique is applied to the Madrid Stock Market index, IBEX-35.

In this study, we propose a new bootstrap strategy to obtain prediction intervals for autoregressive integrated moving-average processes. Its main advantage over other bootstrap methods previously proposed for autoregressive integrated processes is that variability due to parameter estimation can be incorporated into prediction intervals without requiring the backward representation of the process. Consequently, the procedure is very flexible and can be extended to processes even if their backward representation is not available. Furthermore, its implementation is very simple. The asymptotic properties of the bootstrap prediction densities are obtained. Extensive finitesample Monte Carlo experiments are carried out to compare the performance of the proposed strategy vs. alternative procedures. The behaviour of our proposal equals or outperforms the alternatives in most of the cases. Furthermore, our bootstrap strategy is also applied for the first time to obtain the prediction density of processes with movingaverage components.

a b s t r a c tIn this paper, we establish the asymptotic validity and analyse the finite sample performance of a simple bootstrap procedure for constructing multi-step multivariate forecast densities in the context of non-Gaussian unrestricted VAR models. This bootstrap procedure avoids the backward representation, and, as a consequence, can be used to obtain multivariate forecast densities in, for example, VARMA or VAR-GARCH models. In the context of bivariate stationary VAR(p) models, we show that its finite sample properties are comparable to those of alternatives based on the backward representation. The bootstrap procedure is also implemented in a VAR-DCC model which lacks a backward representation. Finally, joint forecast densities of US quarterly inflation, unemployment and GDP growth are obtained.

It is well known that time series of returns are characterized by volatility clustering and excess kurtosis. Therefore, when modelling the dynamic behavior of returns, inference and prediction methods, based on independent and=or Gaussian observations may be inadequate. As bootstrap methods are not, in general, based on any particular assumption on the distribution of the data, they are well suited for the analysis of returns. This paper reviews the application of bootstrap procedures for inference and prediction of financial time series. In relation to inference, bootstrap techniques have been applied to obtain the sample distribution of statistics for testing, for example, autoregressive dynamics in the conditional mean and variance, unit roots in the mean, fractional integration in volatility and the predictive ability of technical trading rules. On the other hand, bootstrap procedures have been used to estimate the distribution of returns which is of interest, for example, for Value at Risk (VaR) models or for prediction purposes. Although the application of bootstrap techniques to the empirical analysis of financial time series is very broad, there are few analytical results on the statistical properties of these techniques when applied to heteroscedastic time series. Furthermore, there are quite a few papers where the bootstrap procedures used are not adequate.

We use a bootstrap procedure to study the impact of parameter estimation on prediction densities, focusing on seasonal ARIMA processes with possibly non normal innovations. We compare prediction densities obtained using the Box and Jenkins approach with bootstrap densities which may be constructed either taking into account parameter estimation variability or using parameter estimates as if they were known parameters. By means of Monte Carlo experiments, we show that the average coverage of the intervals is closer to the nominal value when intervals are constructed incorporating parameter uncertainty. The effects of parameter estimation are particularly important for small sample sizes and when the error distribution is not Gaussian. We also analyze the effect of the estimation method on the shape of prediction densities comparing prediction densities constructed when the parameters are estimated by Ordinary Least Squares (OLS) and by Least Absolute Deviations (LAD). We show how, when the error distribution is not Gaussian, the average coverage and length of intervals based on LAD estimates are closer to nominal values than those based on OLS estimates. Finally, the performance of the bootstrap intervals is illustrated with two empirical examples.