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Handbook of Economic Forecasting part 43

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Handbook of Economic Forecasting part 43. 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 | 394 A. Harvey The general filtering expressions may be difficult to solve analytically. Linear Gaussian models are an obvious exception and tractable solutions are possible in a number of other cases. Of particular importance is the class of conditionally Gaussian models described in the next subsection and the conjugate filters for count and qualitative observations developed in the subsection afterwards. Where an analytic solution is not available Kitagawa 1987 has suggested using numerical methods to evaluate the various densities. The main drawback with this approach is the computational requirement this can be considerable if a reasonable degree of accuracy is to be achieved. 9.2. Conditionally Gaussian models A conditionally Gaussian state space model may be written as yt Zt Yt-i at dAYt-i et et Y _i N 0 Ht Yt-i 169 at Tt Yt_i at-1 Ct Y-i Rt Yt-i nt nt Yt-i - N 0 Qt Yt-i i70 with a0 N a0 P0 . Even though the system matrices may depend on observations up to and including yt-i they may be regarded as being fixed once we are at time t - i. Hence the derivation of the Kalman filter goes through exactly as in the linear model with at t-i and Pt t-i now interpreted as the mean and covariance matrix of the distribution of at conditional on the information at time t - i. However since the conditional mean of at will no longer be a linear function of the observations it will be denoted by at t-i rather than by at t-i. When at t-i is viewed as an estimator of at then Pt t-i can be regarded as its conditional error covariance or MSE matrix. Since Pt t-i will now depend on the particular realization of observations in the sample it is no longer an unconditional error covariance matrix as it was in the linear case. The system matrices will usually contain unknown parameters ÿ. However since the distribution of yt conditional on Yt-i is normal for all t i . T the likelihood function can be constructed from the predictive errors as in 95 . The predictive distribution of yT