These representations exploit the fact that aj(L) has finite order. If it has infinite order then these static representations have infinitely many factors. Parametric time domain dynamic factor models explored to date in the literature assume finite order c^(L) (cf. Sargent (1989),

Stock and Watson (1991), and Quah and Sargent (1993)). Whether this is a problem is an empirical issue.

It is convenient at this point to introduce some additional notation. Let X*t denote the observation on variable i at time t and let the T x 1 vector X- = (X- j, X^, ■ • • ’ • Also let F-t be the observation on the ith factor at time t, let the Txr matrix F=(Fj, F2,…,F^)’, let Pp = F(F’F)’!F\ and let At = (\lt, \2t,…,XNt)\ where \tis rXl vector ^actor loadings on variable i at time t. Let F° denote the true value of F. Let denote the i’th row of Throughout, с and d denote generic finite positive constants. For any matrix M, its (i,j) element is M*j and its norm is ||M || = {tr(M’M)}i//2. For a real symmetic matrix M, mineval(M) and maxeval(M) denote its minimum and maximum eigenvalue.

Finally, to address the problem of missing data in an unbalanced panel, let /-t be a nonrandom indicator function, where I^= 1 if the i^1 variable is observed at date t, and /jt=0 otherwise.

Estimation

There are three challenges in the estimation of the factors. First, the number of parameters is large; with N—500 and k = 15, for example, the initial factor loading matrix Aq has 7500 elements. Second, if both Ft and At are treated as stochastic, then the model is a bilinear form in random variables. Third, in our application we must handle an unbalanced panel, since different macroeconomic time series are available over different periods of time.

The standard method of estimation of dynamic factor models is by maximum likelihood using the Kalman filter. Application of the Kalman filter to dynamic factor models can be justified by assuming that the factor loading matrices are constant and by making suitable parametric assumptions on the disturbances (mutual independence, Gaussianity, and a parametric serial correlation structure); then the Gaussian likelihood can be evaluated using the Kalman filter and the likelihood can be maximized accordingly. This has been implemented in low dimensional systems (e.g. Engle and Watson (1981), Sargent (1989), Stock and Watson (1991)) and in higher dimensional systems (N=60) where the maximization is done using a modification of this appproach based on the EM algorithm (Quah and Sargent (1993)). Although the Kalman filter is easily modified for missing data, nonlinear filters are needed to compute the likelihood when Ft and At are both random. Moreover, likelihood maximization when N is very large does not seem promising from a computational perspective.