10.2.1 Maximum Likelihood Estimation (Small N)

10.2.1 Maximum Likelihood Estimation (Small N)

When the number of variables, N, is small, a three-step parametric estimation approach can be followed to obtain an exact dynamic factor model. The first step uses the Kalman filter to estimate the parameters of a Gaussian likelihood function for $\{y_{t+1},x_t\}$ . In the second step, the Kalman filter and smoother is used to extract estimates of the unknown factor realizations, $\{F_t{\}}_{t=1}^T$ . In a third and final step, the forecast is computed by projecting the dependent variable on the factor estimates and any observable variables, ${\mathbf{x}}_t^{\ast }$ , deemed relevant. The approach involves nonlinear optimization which in practice limits the dimension of the problem.

Stock and Watson (2006) discuss conditions under which a well-identified Gaussian likelihood function exists for data-generating processes represented by equations (10.5)–(10.9). First, it is required that the idiosyncratic components can be represented by finite-order AR processes ${\delta }_i(L){\epsilon }_{xit}=v_{xit}$ and ${\delta }_y(L){\epsilon }_{yt}=v_{yt}$ as in equations (10.6) and (10.9). Second, it is required that the factor dynamics can be captured through a finite-order AR process, $\Gamma (L)F_t=v_{ft}$ as in (10.7), where the innovations $(v_{x1t},\dots ,v_{xNt},v_{yt},v_{f1t},\dots ,v_{frt})$ are mutually uncorrelated, i.i.d., and normally distributed. Finally, ${\lambda }_i(L)$ and ${\lambda }_y(L)$ in (10.5) and (10.8) should be finiteorder lag polynomials.

Under these assumptions, the Kalman filter can be used to compute the likelihood function for the dynamic system comprised of (10.5), (10.7), and (10.11). The resulting likelihood function can be maximized by means of the EM algorithm to obtain maximum likelihood estimates while the Kalman smoother can be used to provide estimates of the factors, ${\hat{\mathbf{F}}}_t$ . With these in place, a forecast can be computed from an OLS regression using data up to time $T:$ :

$$y_{T+1}=\beta (L){\hat{F}}_T+\gamma (L)y_T+{\epsilon }_{T+1}.\text{\hspace{1em}(10.14)}$$

Estimates of $\beta (L)$ and $\gamma (L)$ will be consistent because errors in the factor estimates, $F_T-{\hat{F}}_T$ , are uncorrelated with current and past values of $\hat{F}$ and y.

In an interesting application of the Kalman filter, Aruoba, Diebold, and Scotti (2009) use a single factor approach to extract a daily summary measure of the state of the business cycle based on data measured at weekly, monthly, and quarterly frequencies. This application makes use of the ability of the Kalman filter to allow the underlying state variable to evolve at a different frequency than the observables. Although Aruoba, Diebold, and Scotti do not use the extracted factor for prediction, clearly their factor could be used for such purposes. We further discuss this approach in chapter 21.