Handbook of Economic Forecasting part 21. 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 | 174 A. Timmermann where e is an N x T matrix of forecast errors. Letting fj be the i j entry of T ef j the i j element of T e and j the i j element of the single factor covariance matrix Tef while j is the i j element of Ee they demonstrate that the optimal shrinkage takes the form a 1 n p where N N n 2 AsyVar ATê j i 1 j 1 N N p o AsyCov VTfij ATdij N N Y Ai ij 2- Hence n measures the scaled sum of asymptotic variances of the sample covariance matrix Êe p measures the scaled sum of asymptotic covariances of the sample covariance matrix jÊe and the single-factor covariance matrix jÊef while y measures the degree of misspecification bias in the single factor model. Ledoit and Wolf propose consistent estimators n p and y under the assumption of IID forecast . Constraints on combination weights Shrinkage bears an interesting relationship to portfolio weight constraints in finance. It is commonplace to consider minimization of portfolio variance subject to a set of equality and inequality constraints on the portfolio weights. Portfolio weights are often constrained to be non-negative due to no short selling and not to exceed certain upper bounds due to limits on ownership in individual stocks . Reflecting this let E be an estimate of the covariance matrix for some cross-section of asset returns with row i column j element E i j and consider the optimization program w argmin w Lw W 2 S-t- WI 1 i 0 Mi â 71 i 1 . N i 1 . N. 13 It is worth pointing out that the assumption that e is IID is unlikely to hold for forecast errors which could share common dynamics in first second or higher order moments or even be serially correlated cf. Diebold 1988 . Ch. 4 Forecast Combinations 175 This gives a set of Kuhn-Tucker conditions 2 T j 8j - 8i 0 1 N j ki 0 and ki 0 if ai 0 8i 0 and 8i 0 if 8 Lagrange multipliers for the lower and upper bounds are collected in the vectors X X1 kNy and 3 81 8N X0 is the Lagrange multiplier for the constraint that the weights sum to one. .
