10.4 PRACTICAL ISSUES ARISING WITH FACTOR MODELS

10.4 PRACTICAL ISSUES ARISING WITH FACTOR MODELS

Several practical issues arise in the use of common factor models in economic forecasting. This section discusses identification of the factors, parameter instability, missing observations, factor-augmented VARs, and partial least squares methods.

10.4.1 Identification and Economic Interpretation of Factors

The unobserved common factors are fundamentally unidentified. For practical purposes, it is commonly found to be empirically important to standardize the underlying x-variables prior to the analysis, e.g., by subtracting the mean and dividing by the standard deviation of each series. However, without further identifying assumptions, it makes little sense to inquire into the sign and magnitude of the coefficients of ${\lambda }_i(L)$ and ${\lambda }_y(L)$ in the dynamic factor model (10.5), (10.8), or the static factor loadings - in (10.12). To see this, rewrite the contribution from the factors in the static model (10.12) as $\Lambda F_t^p=\Lambda R{R}^{-1}{F}_t^p$ for any nonsingular $r\times r^{\cdot }$ “rotation” matrix, R. Hence, data generated from a model with loading matrix - and factors $F_t^P$ will be identical to data from a rotated model with loading matrix $\tilde{\Lambda }=\Lambda R$ and factors ${\tilde{F}}_t^p=R^{-1}{F}_t^p$ If interest lies purely in mechanically computing a forecast, this is not a problem, of course, since all forecasts are identical under nonsingular rotations of the factors.

A forecast user may nevertheless be skeptical about a purely statistical “black-box” approach that does not allow economic interpretation of the factors. To facilitate interpretation of the factors, one can project the extracted factors on economic variables identified in advance. For example, the first principal component may be strongly correlated with nominal interest rates while the second principal component is strongly correlated with real output growth. Alternatively, one can first group economic variables into clusters such as interest rate or term structure variables, real growth variables, wages, and prices and extract common factors from each of these groups which are in turn used to predict the variable of interest. This would result in “interest rate” and “output” factors that are easier to interpret, although there is no guarantee that such factors closely mimic the factors obtained from a “global” factor extraction method that does not rely on bins.

10.4.2 Instability

Instability in the processes driving the observed variables can be an important issue. Estimates of the dynamic factors reflect the covariance structure embedded in ${\Sigma }_{XX}$ and also depend on the dynamic processes driving ${\epsilon }_{xt}$ . If these are not stable over time, this can lead to inconsistent estimates of the factors used in the forecasting model. A limited amount of instability is permitted, however, provided that this is sufficiently idiosyncratic across the individual series so that it washes out in the cross section and can be diversified away in the aggregate.

Specifically, Stock and Watson (2002a) allow for a stochastic drift in the factor loadings of the form

$$\begin{array}{l}{x}_{it}={\lambda }_{it}^{\prime }{F}_t+{\epsilon }_{xit},\\ \\ {\lambda }_{it}={\lambda }_{it-1}+g_{iT}{\epsilon }_{\lambda it},\end{array}\text{\hspace{1em}(10.25)}$$

where ${\epsilon }_{\lambda it}$ is an $(r\times 1)$ vector of random variables, while $g_{iT}$ is a scalar. The random walk evolution in the factor loadings captures the tendency of many economic relations to gradually change over time. Under this type of temporal instability, we can rewrite the model in (10.25) as

$$\begin{array}{l}{x}_{it}={\lambda }_{i0}{F}_t+a_{it},\\ a_{it}={\epsilon }_{xit}+({\lambda }_{it}-{\lambda }_{i0})F_t={\epsilon }_{xit}+g_{iT}\sum _{s=1}^t{\epsilon }_{\lambda is}^{\prime }{F}_t.\end{array}\text{\hspace{1em}(10.26)}$$

Provided that the time-series variation in the parameter values is “small,” sufficiently idiosyncratic in a cross-sectional sense, and $g_{iT}$ is independent of $F_t,{\epsilon }_{xjt}$ , and ${\epsilon }_{\lambda jt}$ for all $i,j,$ and t, Stock and Watson show that the underlying factors can still be consistently estimated from the cross section of X-variables and the forecast based on estimated factors will converge to the forecast given the true factors.

In subsequent work, Stock and Watson (2009) show that when the factor structure is subject to a single large break, full-sample principal component estimates still span the space of pre- and post-break factors, the only difference being that the preand post-break factor estimates use different linear combinations of the full-sample principal components.