The model

Let yt be a scalar time series variable and let Xt be a N-dimensional multiple time series variable. Throughout, yt is taken to be the variable to be forecast while Xt is the vector time series variable that contain useful information for forecasting yt+ j – It is assumed that X( can be represented by the factor structure,

where Ft is the r x 1 common factor and et is the N x 1 idiosyncratic disturbance. The idiosyncratic disturbances are in general correlated across series and over time; specific assumptions used for the asymptotic analysis are given in section 3.

Our main objective is to estimate E(yt+ ^ | Xt). We model yt + ^ as,

enter (2.2); and that lags of yt do not enter (2.2). This first assumption is the key assumption that permits the dimension reduction necessary for handling very large Xt> The second assumption is not restrictive, because Ft in (2.2) can be reinterpreted as including lags without changing any essential argument. The third assumption also is not restrictive in the sense that yt+ j can be reinterpreted as a quasidifference so that lagged values of yt can be incorporated into the model.

where h is a diagonal N xN scaling matrix and r}t and ^ are, respectively, r x 1 and N xr stochastic disturbances. Specific assumptions about h are stated in section 3.

Depending on what further assumptions are made concerning the disturbances and the factor loading matrices, this model contains several important special cases. One is the static factor model in which the factor loadings are constant (so A^Aq), et is serially uncorrelated, Ft and {ejt} are mutually uncorrelated and are i.i.d.. If e-t and ejt are independent for i^j, the model is referred to as an exact static factor model. If the idiosyncratic disturbances are weakly correlated across series, the model is an approximate (static) factor model (cf. Chamberlain and Rothschild (1983) and Connor and Korajczyk (1986, 1993)).

Another important special case of (2. l)-(2.4) is the dynamic factor model without time variation, as has been studied by, among others, Geweke (1977), Sargent and Sims (1977), Engle and Watson (1981), Sargent (1989), Stock and Watson (1991), and Quah and Sargent (1993). In the standard dynamic factor model, dynamics are introduced in three ways: the factors are assumed to evolve according to a time series process; the idiosyncratic error terms are serially correlated; and the factors can enter with lags (or, in general, leads).