Handbook of Economic Forecasting part 88. 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 | 844 T. G. Andersen et al. and covariance matrix fàt t-i- The log-likelihood function is given by the sum of the corresponding T logarithmic conditional normal densities logL G Yt . Yi TN _ log 2n 1 - 2 log t t-1 G - Yt - Mt t-1 0 Qt t-1 G -1 Yt - Mt t-1 G t 1 where we have highlighted the explicit dependence on the parameter vector G. Provided that the assumption of conditional normality is true and the parametric models for the mean and covariance matrices are correctly specified the resulting estimates say GT will satisfy the usual optimality conditions associated with maximum likelihood. Moreover even if the conditional normality assumption is violated the resulting estimates may still be given a QMLE interpretation with robust parameter inference based on the sandwich-form of the covariance matrix estimator as discussed in Section . Meanwhile as discussed in Section 2 when constructing interval or VaR type forecasts the whole conditional distribution becomes important. Thus in parallel to the discussion in Sections and other multivariate conditional distributions may be used in place of the multivariate normal distributions underlying the likelihood function in . Different multivariate generalizations of the univariate fat-tailed Student t distribution in have proved quite successful for many daily and weekly financial rate of returns. The likelihood function in or generalizations allowing for conditionally nonnormal innovations may in principle be maximized by any of a number of different numerical optimization techniques. However even for moderate values of N say N 5 the dimensionality of the problem for the general model in or the diagonal vech model in renders the computations hopelessly demanding from a practical perspective. As previously noted this lack of tractability motivates the more parsimonious parametric specifications discussed below. An alternative approach for circumventing the curse-of-dimensionality .
