The PANEL Procedure

This example continues to use the Christenson Associates airline data, which measures costs, prices of inputs, and utilization rates for six airlines over the time span 1970–1984. There are six cross sections and fifteen time observations. Here, you examine the different estimates that are generated from the one-way random-effects and two-way random-effects models, by using four different methods to estimate the variance components: Fuller and Battese, Wansbeek and Kapteyn, Wallace and Hussain, and Nerlove.

The data for this example are created by the PROC PANEL statements shown in Example 27.2. The following PROC PANEL statements generate the estimates:

proc panel data=airline;
   id I T;
   model "One-Way, FB"    lC = lQ lPF lF / ranone vcomp=fb;
   model "One-Way, WK"    lC = lQ lPF lF / ranone vcomp=wk;
   model "One-Way, WH"    lC = lQ lPF lF / ranone vcomp=wh;
   model "One-Way, NL"    lC = lQ lPF lF / ranone vcomp=nl;
   model "Two-Way, FB"    lC = lQ lPF lF / rantwo vcomp=fb;
   model "Two-Way, WK"    lC = lQ lPF lF / rantwo vcomp=wk;
   model "Two-Way, WH"    lC = lQ lPF lF / rantwo vcomp=wh;
   model "Two-Way, NL"    lC = lQ lPF lF / rantwo vcomp=nl;
   model "Pooled"         lC = lQ lPF lF / pooled;
   model "Between Groups" lC = lQ lPF lF / btwng;
   model "Between Times"  lC = lQ lPF lF / btwnt;
   compare / pstat(estimate) mstat(varcs varts varerr);
run;

The parameter estimates and variance components for all models and estimators are reported in Output 27.4.1 and Output 27.4.2. Both tables are created by the COMPARE statement.

In the random-effects model, individual constant terms are viewed as randomly distributed across cross-sectional units. They are not viewed as parametric shifts of the regression function, as they are in the fixed-effects model. This is appropriate when the sampled cross-sectional units are drawn from a large population. In this example, the six airlines are clearly a sample of all the airlines in the industry and not an exhaustive, or nearly exhaustive, list.

There are four techniques for computing the variance components in the one-way random-effects model. The method by Fuller and Battese (1974) (FB) uses a "fitting of constants" method to estimate them, the Wansbeek and Kapteyn (1989) (WK) method uses the true disturbances, the Wallace and Hussain (1969) (WH) method uses ordinary least squares residuals, and the Nerlove (1971) (NL) method uses one-way fixed-effects regression.

Looking at the estimates of the variance components for cross section and error in Output 27.4.2, you see that equal variance components for error for FB and WK are equal, whereas they are nearly equal for WH and NL.

All four techniques produce different variance components for cross sections. These estimates are then used to estimate the values of the parameters in Output 27.4.1. All the parameters appear to have similar and equally plausible estimates. Both the indices for output in revenue passenger miles (lQ) and fuel price (lPF) have small, positive effects on total costs, which you would expect. The load factor (LF) has a somewhat larger and negative effect on total costs, suggesting that as utilization increases, costs decrease.

Comparing the four two-way estimators, you find that the variance components for error produced by the FB and WK methods are equal, as they are in the one-way model. However, in this case, the WH and NL methods produce variance estimates that are dissimilar. The estimates of the variance component for cross sections are all different, but in a close range. The same cannot be said for the variance component for time series. As varied as each of the variance estimates might be, they produce parameter estimates that are similar and plausible. As in the one-way effects model, the indices for output (lQ) and fuel price (lPF) are small and positive. The load factor (LF) estimates are all negative and somewhat smaller in magnitude than the estimates that are produced in the one-way model. During the time the data were collected, the Civil Aeronautics Board dissolved, so it is possible that the time dummy variables are proxies for this dissolution. The dissolution would lead to the decay of time effects and an imprecise estimation of the effects of the load factors, even though the estimates are statistically significant.

The pooled estimates give you something to compare the random-effects estimates against. You see that signs and magnitudes of output and fuel price are similar but that the magnitude of the load factor coefficient is somewhat larger under pooling. Because the model appears to have both cross-sectional and time series effects, the pooled model should not be used.

Finally, you examine the between estimators. For the between-groups estimate, you are looking at each airline’s data averaged across time. You see in Output 27.4.1 that the between-groups parameter estimates are radically different from all other parameter estimates. This difference could indicate that the time component is not being appropriately handled with this technique. For the between-times estimate, you are looking at the average across all airlines in each time period. In this case, the parameter estimates are of the same sign and closer in magnitude to the previously computed estimates. Both the output and load factor effects appear to have more bearing on total costs.