TAILIEUCHUNG - Handbook of Economic Forecasting part 72

Handbook of Economic Forecasting part 72. 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 | 684 E. Ghysels et al. where as in the previous section S represents the periodicity of the data while here Pj is the order of the autoregressive component for season j p max p1 . pS Dj sn s is again a seasonal dummy that is equal to 1 in season j and zero otherwise and Sn s iid 0 ff2 . The PAR model of 44 45 requires a total of 3S j 1 Pj parameters to be estimated. This basic model can be extended by including periodic moving average terms Tiao and Grupe 1980 Lutkepohl 1991 . Note that this process is nonstationary in the sense that the variances and covariances are time-varying within the year. However considered as a vector process over the S seasons stationarity implies that these intra-year variances and covariances remain constant over years n 0 1 2 . It is this vector stationarity concept that is appropriate for PAR processes. Substituting from 45 into 44 the model for season s is s L ySn s Ps L s Ts Sn S Sn s 46 where jL 1- 1jL----------- Pj jLPj. Alternatively following Boswijk and Franses 1996 the model for season s can be represented as p-1 1 asL ySn s 8s Ms Sn s flks 1 as-kL ySn s-k 8Sn s 47 k 1 where aS-Sm as for s 1 . S m 1 2 . and jL is a pj - 1 -order polynomial in L. Although the parameterization of 47 is useful it should also be appreciated that the factorization of s L implied in 47 is not in general unique del Barrio Castro and Osborn 2004 . Nevertheless this parameterization is useful when the unit root properties of ySn s are isolated in 1 - aSL . In particular the process is said to be periodically integrated if n s 1 48 s 1 with the stochastic part of 1 - asL ySn s being stationary. In this case 48 serves to identify the parameters of 47 and the model is referred to as a periodic integrated autoregressive PIAR model. To distinguish periodic integration from conventional nonperiodic integration we require that not all as 1 in 48 . An important consequence of periodic integration is that such series cannot be decomposed into distinct seasonal .

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