10.1 FORECASTING WITH FACTOR MODELS

10.1 FORECASTING WITH FACTOR MODELS

When presented with the information embedded in a large set of potentially relevant predictor variables, $x_{it},i=1,\dots ,N$ , the simplest forecasting approach would be to consider a linear model of the form

$$y_{t+1}=\alpha +\sum _{i=1}^N{\beta }_i(L)x_{it}+\gamma (L)y_t+{\epsilon }_{yt+1},\text{\hspace{1em}(10.2)}$$

where $\begin{array}{r}{\beta }_i(L)={\sum }_{j=0}^{P_x}{\beta }_{ij}{L}^j\end{array}$ is a lag polynomial of order ${\varphi }_x$ which contains the regression coefficients for the ith variable, and $\begin{array}{r}\gamma (L)={\sum }_{j=0}^{p_y}{\gamma }_j{L}^j\end{array}$ is a lag polynomial of order $Py$ . While we assume a one-period forecast horizon, the model can easily accommodate arbitrary forecast horizons h by projecting $y_{t+h}$ on variables observed at time t. Ignoring potential serial correlation in ${\epsilon }_y,$ this model can be estimated by OLS, assuming that the total number of mean parameters, $N_{\beta }=N\times p_x+p_y+1$ , is small relative to the length of the time series, $T$ . Often, however, $N_{\beta }>T$ and so conventional estimation methods are not feasible. Instead it is common practice to assume that, while a few $(n_{x^{\ast }})$ key predictor variables, $x^{\ast }$ , directly affect $y,$ most of the x-variables affect $\mathbf{y}$ only through a set of $q$ common components, $F_{;}$ , where $q$ is much smaller than N.

Focusing our attention on linear models, this suggests using a model of the form

$$y_{t+1}=\alpha +{\beta }_f(L{)}^{\prime }{F}_t+\gamma (L)y_t+{\varphi }^{\prime }(L)x_t^{\ast }+{\epsilon }_{yt+1},\text{\hspace{1em}(10.3)}$$

where $F_t=(F_{1t},\dots ,F_{qt}{)}^{\prime }$ is a set of $q$ common factors extracted from the list of N original predictors, $x_t$ . Here ${\beta }_f(L),\gamma (L)$ , and $\varphi (L)$ are vector lag polynomials of order $p_f,p_y$ , and ${\nabla }^{p_{x^{\ast }}}$ , respectively, so that this model has $p_f\times q+p_y+p_{x^{\ast }}\times$ $n_{x^{\ast }}+1$ parameters. Economic theory could justify inclusion of certain $x^{\ast }$ -variables— e.g., the Phillips curve suggests including current and lagged unemployment rates in a model for the inflation rate. If idiosyncratic variation in these variables matters to $y_{:}$ , they should be included since the common factors suppress this information. Typically it is assumed that the identity of such variables is known ex ante since otherwise the search for such variables could quickly become infeasible.

To see how this matters in practice, suppose that all lag polynomials contain two lags $(p_y=p_x=p_f=p_{x^{\ast }}=2)$ and $N=200$ , while $q=4$ and $n_{x^{\ast }}=2$ . Then the forecasting model in (10.2) requires fitting 403 parameters, while the model in (10.3) requires estimating only 15 parameters. Of course the representation in (10.3) offers parsimony only when the number of factors, $q:$ , and sources of “direct” influence, $n_{x^{\ast }}$ represent a significant reduction from the N original variables $x_t$

Forecasters are interested in the common factors only to the extent that they help improve predictions of the target variable, $y_{t+1}$ . If the factors were known, forecasting with (10.3) would simply amount to choosing the right number of lags for $F,x^{\ast }$ and $\mathbf{y}$ and estimating their coefficients. Provided that the variables have all been transformed so that they are stationary, OLS estimation and model selection criteria such as BIC could be used to select the lag orders.

In practice, the common factors are usually unknown and so must be extracted from the data. Forecasting with common factor models can therefore be thought of as a two-step process. In the first step, estimates of the common factors are extracted. In the second step, the factors, along with past values of the predicted variable and possibly other relevant variables are used to select and estimate a forecasting model.

To operationalize the forecast, suppose that a set of factor estimates, ${\hat{\mathbf{F}}}_t$ , have been extracted. These are then used in (10.3) along with current and past values of $\mathbf{y}$ and $x^{\ast }$ to select and estimate a forecasting model. The actual forecasts thus take the form

$${\hat{f}}_{t+1\mid t}=\hat{\alpha }+\sum _{j=1}^{p_f}{\hat{\beta }}_j^{\prime }{\hat{F}}_{t+1-j}+\sum _{j=1}^{p_y}{\hat{\gamma }}_j{y}_{t+1-j}+\sum _{j=1}^{p_{x^{\ast }}}{\hat{\varphi }}_j^{\prime }{x}_{t+1-j}^{\ast },\text{\hspace{1em}(10.4)}$$

where $\{\hat{\alpha },{\hat{\beta }}_j,{\hat{\gamma }}_j,{\hat{\varphi }}_j\}$ are obtained from a regression of the predicted variable on a constant and lagged values of the extracted factors, the predicted variable, and the $x^{\ast }$ -variables.