tailieunhanh - Handbook of Economic Forecasting part 15

Handbook of Economic Forecasting part 15. 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 | 114 . West Then P 1 2 T r f ft 1 -Eft is asymptotically normal with variance-covariance matrix V V fh FBSf h SfhB F khhFVpF . V is the long run variance of P-1 2 J2T r ft 1 f - Eft and is the same object as V defined in XhhFVfF is the long run variance of F P R r 2 BRi 2H and X fh FBSf h SfhB F is the covariance between the two. This completes the statement of the general result. To illustrate the expansion and the asymptotic variance I will temporarily switch from my example of comparison of MSPEs to one in which one is looking at mean prediction error. The variable ft is thus redefined to equal the prediction error ft et and Eft is the moment of interest. I will further use a trivial example in which the only predictor is the constant term yt f et. Let us assume as well as in the Hoffman and Pagan 1989 and Ghysels and Hall 1990 analyses of predictive tests of instrument-residual orthogonality that the fixed scheme is used and predictions are made using a single estimate of f . This single estimate is the least squares estimate on the sample running from 1 to R fR R-1 R 1 ys. Now et 1 et 1 - r - f et 1 - R Y R 1 es. So T T R P-1 2 et 1 P-1 2 et 1 - P R 1 2l R-1 2 4 t R t R s 1 This is in the form or with F -1 R-1 2 XR 1 es Op 1 terms due to the sequence of estimates of f B 1 H R- Y. r 1 es and the op 1 term identically zero. If et is well behaved say . with finite variance a2 the bivariate vector P-l 2 f t r et 1 R-l 2 f r 1 es is asymptotically normal with variance covariance matrix a 2I2. It follows that t i R P-1 2 E et 1 - P R 1 2 i R-1 2 E ej A N 0 1 n a2 . The variance in the normal distribution is in the form with Xfh 0 Xhh n V FVfF a2. Thus use of fR rather than f in predictions inflates the asymptotic variance of the estimator of mean prediction error by a factor of 1 n. In general when uncertainty about f matters asymptotically the adjustment to the standard error that would be appropriate if predictions .