By suitable redefinition of the factors and the idiosyncratic disturbances, the dynamic factor model can be rewritten in the form (2.1) with At constant. To see this, let Zt denote anxl vector of time series variables which are assumed to satisfy the dynamic factor model,
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for i= 1.. ,n, where ft is a vector of factors and L is the lag operator. In the econometric literature using dynamic factor models, {^} and i = l,…,n are taken to be mutually independent. Let a-(L) have order q and let g-(L) be a finite order lag polynomial with roots outside the unit circle. (Typically normality of these disturbances is further assumed to motivate using the Kalman filter to compute the maximum likelihood estimates of the factors.) The model is completed by making an additional assumption specifying the stochastic process followed by the factors, such as a Gaussian vector autoregression, where the factors are distributed independently of {v-t}.


There are at least two ways to rewrite this model in the form (2.1) with time invariant parameters. The first is to let Xt = Zt, Ft = (f[, f[_j,…f| )’, Л = (ckq … a^), and et = vr With these definitions, (2.5) and (2.6) are equivalent to (2.1), where the idiosyncratic errors et are serially correlated, Ft has dimension r=dim(ft) x(q +1), and At=A. In this representation, the factors Ft are dynamically singular in the sense that the spectral density matrix of Ft has rank dim(ft).
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where N=dim(x[)=np and r=dim(Fj) = (p+q)xdim(ft). As in the first representation, the factors f| are dynamically singular. Operationally, these two representation suggest different estimation strategies, the first by extracting dynamic factors using contemporaneous values of Xt only, the second by using lags of Xt as well. A potential advantage of the second representation is that additional indicators (lagged values of Xt) are introduced for the estimation of ft, which could improve finite sample performance.