tailieunhanh - Handbook of Economic Forecasting part 58

Handbook of Economic Forecasting part 58. Research on forecasting methods has made important progress over recent years and these developments are brought together in the Handbook of Economic Forecasting. The handbook covers developments in how forecasts are constructed based on multivariate time-series models, dynamic factor models, nonlinear models and combination methods. The handbook also includes chapters on forecast evaluation, including evaluation of point forecasts and probability forecasts and contains chapters on survey forecasts and volatility forecasts. Areas of applications of forecasts covered in the handbook include economics, finance and marketing | 544 . Stock . Watson known this permits inference about G. In turn the estimator of G can be used in 30 to compute the empirical Bayes estimator. The estimation of the prior can be done either parametrically or nonparametrically. Parametric empirical Bayes The parametric empirical Bayes approach entails specifying a parametric prior distribution Gi X 0 where 0 is an unknown parameter vector that is common to all the priors. Then the marginal distribution of p is mi x 0 f x P dGi 0 . If Gi G for all i then there are n . observations on p from the marginal m x 0 and inference can proceed by maximum likelihood or by method of moments. In the application at hand where the regressors are the principal components one might specify a prior with a spread that declines with i following some parametric structure. In this case p constitute n independent draws from a heteroskedastic marginal distribution with parameterized heteroskedasticity which again permits estimation of 0. Although the discussion has assumed that a2 is known it can be estimated consistently if n T x as long as n T const 1. As a leading case one could adopt the conjugate g-prior An alternative approach to parameterizing Gi is to adopt a hierarchical prior. Clyde and George 2000 take this approach for wavelet transforms as applied to signal compression where the prior is allowed to vary depending on the wavelet level. Nonparametric empirical Bayes The nonparametric empirical Bayes approach treats the prior as an unknown distribution. Suppose that the prior is the same G for all i so that p t for all i. Then the second expression in 30 suggests the estimator N P 31 where f is an estimator of f. The virtue of the estimator 31 is that it does not require direct estimation of G for this reason Maritz and Lwin 1989 refer to it as a simple empirical Bayes estimator. Instead the estimator 31 only requires estimation of the derivative of the log of the marginal likelihood i x dln mi x dx dm x dx m x .