tailieunhanh - Handbook of Economic Forecasting part 41

Handbook of Economic Forecasting part 41. 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 | 374 A. Harvey the mean of Ayt. This is best seen by writing 107 as Ayt A 0 T Ayt-1 - i Ay -r - A 0 1 109 where A is an N x K matrix such that AA 0 so that there are no slopes in the co-integrating vectors. The elements of A are the growth rates of the series. Thus 11 3 I - j A T Vi . 110 Structural time series models have an implied triangular representation as we saw in 104 . The connection with VECMs is not so straightforward. The coefficients of the VECM representation for any UC model with common random walk plus drift trends can be computed numerically by using the algorithm of Koopman and Harvey 2003 . Here we derive analytic expressions for the VECM representation of a local level model 101 noting that in terms of the general state space model Z I II . The coefficient matrices in the VECM depend on the K x N steady-state Kalman gain matrix K as given from the algebraic Riccati equations. Proceeding in this way can give interesting insights into the structure of the VECM. From the vector autoregressive form of the Kalman filter 88 noting that T IK so L IK - KZ we have yt 3 Z Ik - Ik - KZ L -1Kyt- vt Var vt F. 111 Note that F and K depend on Z Yn and Ee via the steady-state covariance matrix P. This representation corresponds to a VAR with vt 1t and F E. The polynomial in the infinite vector autoregression 106 is therefore L In - Z Ik - Ik - KZ L -1KL. The matrix 1 IN - Z KZ -1K 112 has the property that 1 Z 0 and K 1 0. Its rank is easily seen to be R as required by the Granger representation theorem this follows because it is idempotent and so the rank is equal to the trace. The expression linking 3 to i and ft is obtained from 89 as 0 i 3 IN - Z KZ -1K Z KZ -1 t 113 11 If we don t want time trends in the series the growth rates must be set to zero so we must constrain 3 to depend only on the R parameters in i by setting 3 -Ti . In the special case when R N there are no time trends and 3 -Ti is the unconditional mean. Ch. 7 Forecasting with Unobserved .