10.6 FORECASTING WITH PANEL DATA

10.6 FORECASTING WITH PANEL DATA

Although our analysis has thus far focused on time-series forecasting, nothing prevents the forecasting methods from being used in a different context such as cross sections or panel data. The key point is that the predictor variables are predetermined either in the time dimension or in a cross-sectional sense. For example, we could build a forecasting model on a large cross-section of credit-card holders using data on household characteristics, past payment records, etc. Here the implicit time dimension is that we know whether a payment in the data subsequently turned out fraudulent.

In a predictive context, panel regressions may take the form

$$y_{it+1}=c+X_{it}^{\prime }\delta +u_{it+1},i=1,\dots ,N,t=1,\dots ,T,\text{\hspace{1em}(10.28)}$$

where i refers to individual units such as households, stocks, etc., while t refers to the time dimension. Stacking the data into an $(NT\times 1)$ vector

$\mathbf{y}=(y_{11},\dots ,y_{1T},y_{21},\dots ,y_{2T},\dots ,y_{N1},\dots ,y_{NT}{)}^{\prime }$ , and letting $Z=({\iota }_{NT}X),{\beta }^{\prime }=$ $(c^{\prime },{\delta }^{\prime })$ , with X being an $(NT\times K)$ matrix of the data, (10.28) can be written more compactly as

$$\begin{array}{r}y=Z\beta +u,\end{array}$$

where u is an $(NT\times 1)$ vector stacked in the same way as $y.$ . For the random effects model $u_{it}={\omega }_i+v_{it}$ , where ${\omega }_i\sim \mathrm{iid}(0,{\sigma }_{\omega }^2),v_{it}\sim \mathrm{iid}(0,{\sigma }_v^2),i=1,\dots ,N$ , with ${\omega }_i$ and $v_{it}$ being mutually independent across i and $t,$ as well as independent of $X_{it}{.}^6$ The vector of error terms can then be written as

$$u=(I_N\otimes {\iota }_T)\omega +v,$$

where $\omega =({\omega }_1,\dots ,{\omega }_N)$ and $\mathbf{v}=(v_{11},\dots ,v_{1T},v_{21},\dots ,v_{2T},\dots ,v_{N1},\dots ,v_{NT}{)}^{\prime }$ Moreover, the variance–covariance matrix for u simplifies to

$$\Sigma =E(u{u}^{\prime })={\sigma }_{\omega }^2(I_N\otimes ({\iota }_T{\iota }_T^{\prime }))+{\sigma }_v^2(I_N\otimes I_T).$$

Assuming a known covariance matrix , the best linear unbiased estimator of $\beta$ is obtained by generalized least squares, ${\hat{\beta }}_{\mathrm{GLS}}=(Z^{\prime }{\Sigma }^{-1}Z{)}^{-1}{Z}^{\prime }{\Sigma }^{-1}y;$ ; see Baltagi (2013). Moreover, as shown by Goldberger (1962), the best linear unbiased predictor of ${\gamma }_{iT+1}$ is

$$f_{iT+1}=Z_{it}^{\prime }{\hat{\beta }}_{\mathrm{GLS}}+{\gamma }_i^{\prime }{\Sigma }^{-1}{\hat{u}}_{\mathrm{GLS}},\text{\hspace{1em}(10.29)}$$

where ${\hat{u}}_{\mathrm{GLS}}=y-Z{\hat{\beta }}_{\mathrm{GLS}}$ , and ${\gamma }_i=E\left[u_{iT+1}u\right]$ captures any ability of the current vector of errors to predict future shocks to the i th variable in period $T+1$ . For the random effects model, $u_{iT+1}={\omega }_i+v_{iT+1}$ , and so ${\gamma }_i={\sigma }_{\omega }^2(l_i\otimes {\iota }_T)$ , where $l_i$ is an $(N\times 1)$ vector of $0\mathsf{s},$ except for a 1 in the i th place. See Baltagi (2013) for further details and discussion.

Two issues arise with the approach to forecasting in (10.29). First, this approach assumes that $\tilde{\gamma }$ is known, an assumption that is very unlikely to hold, and so introduces estimation errors when implemented in practice. Second, the forecast is best among unbiased estimators but, as we have seen throughout the book, there is often no reason to require forecasts to be unbiased. Even under MSE loss, bias can usually be traded off against forecast error variance in a way that reduces the expected loss.

Work by Baillie and Baltagi (1999) suggest that despite these deficiencies, what they label a “full-fledged GLS predictor” of $y_{i,T+1}$ that uses plug-in maximum likelihood estimates in place of unknown parameters, produces better forecasting performance compared with OLS or fixed effect predictors that ignore the random error structure in the data. The performance of the GLS estimator is particularly strong in the presence of large contributions from the random error, i.e., when ${\sigma }_{\omega }^2/(\overline{{\sigma }_{\omega }^2}+{\sigma }_v^2)$ is high.