Identifying Omitted Variables

Plotting Deflated Overhead Correlated error terms can occasionally be fixed by taking first dif ferences between successive observations, subtracting each observation from the preceding observation. This procedure is a mechanical solution that does not provide any insight into the underlying structural cause of the dependency. Recall that the error term in the regression analysis incorporates the cumulative effects of all the variables omitted from the model. Mechanical approaches, such as computing first differences, do not identify the source and nature of these omitted variables. We should attempt to identify the omitted variables or the cause of the time dependency still remaining in the data. In this case, we can recall the cost accountant's statement about the price increases in the overhead costs. Perhaps if we deflate the data, to eliminate the estimated monthly price increases, we will get a clearer picture of the underlying relationship between machine-hours and over-head. We divide each of the reported overhead costs by (1.01)* -1, where n is the observation number, to reflect the estimate of 1% per month inflation in the data. After this transformation, the overhead data are expressed in terms of month 1 dollars. The estimated regression line for the deflated data is Deflated overhead = 194,667 + 10.73 machine-hours The R' of the regression is 0.78, a considerable improvement over the 0.10 obtained from the initial regression. We now have a much better fit to the data. The t statistic of the machine-hours coefficient equals 13.0 and is distributed with 48 degrees of freedom. This value is statistically significant at any confidence level one might ever want to use. We have overwhelming evidence of a statistically significant relationship between machine-hours and deflated overhead The plot of deflated overhead versus machine-hours (Exhibit 5-5) appears to be approximately linear. Also, a plot (not shown) of the error terms versus time and of the independent and dependent variables does not reveal any systematic patterns that would suggest the expected value of the error term is different from zero. Plots of the error terms (not shown) against a time variable, machine-hours, and the predicted value of overhead also seem to have a constant width, suggesting that the variance of the error term is not a function of the magnitude of any of these variables. The Durbin-Watson statistic for the regression equals 1.61. With 50 observations and at 95% confidence, the cutoff points for the Durbin- Watson tests are du = 1.59 and d, = 1.50. We therefore cannot reject the null hypothesis that successive error terms are independent. It is worth noting that adjusting the data for the 1% monthly inflation not only significantly improved the accuracy of the model, but also resolved the issue of strong autocorrelation in the original data. The steady price increase caused early observations to be below the predicted values and the later values to be above predicted values. Thus, if the error term from one observation was negative (positive), it was very likely that error terms from neighboring observations would also be negative (positive). The ability to predict something about nearby error terms violates the independence assumption and is the source of the measured autocorrelation in the original data.