Handbook of Economic Forecasting part 42. 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 | 384 A. Harvey the discrete time transition equation aT Tr ar-1 nT t 1 . T 129 where T exp AM I A8r 1A2822 1 Ä3ä 130 and nT is a multivariate white-noise disturbance term with zero and covariance matrix Qt f1 eA sT-s RQR eA sT s ds. 131 0 The condition for a t to be stationary is that the real parts of the characteristic roots of A should be negative. This translates into the discrete time condition that the roots of T exp A should lie outside the unit circle. If a t is stationary the mean of a t is zero and the covariance matrix is . . 0 Var a t e-As RQR e-As ds. 132 œ The initial conditions for a t0 are therefore a1 0 0 and P1 0 Var a t . The main structural components are formulated in continuous time in the following way. Trend In the local level model the level component t is defined by d t on dWn t where Wn t is a standard Wiener process and on is a non-negative parameter. Thus the increment d t has mean zero and variance a2 dt. The linear trend component is 0 1 0 0 dp. t dß t p. t dt ß t dt ov dWn t O dWz t 133 where Wn t and W t are mutually independent Wiener processes. Cycle The continuous cycle is d t d t log p r r log p t dt Ÿ t dt Ok dWJt Ok dW t 134 where WK t and W t are mutually independent Wiener processes and aK P and kc are parameters the latter being the frequency of the cycle. The characteristic roots of the matrix containing p and kc are log p ikc so the condition for t to be a stationary process is p 1. Seasonal The continuous time seasonal model is the sum of a suitable number of trigonometric components Yj t generated by processes of the form 134 with p equal to unity and kc set equal to the appropriate seasonal frequency kj for j 1 . s 2 . Ch. 7 Forecasting with Unobserved Components Time Series Models 385 . Stock variables The discrete state space form for a stock variable generated by a continuous time process consists of the transition equation 129 together with the measurement equation yr Za tT st ZaT eT t 1 . T 135 where er is a .
