Diagnostic Tests for Regression

Higher Order Correlation Specification Tests A1. Linearity One test for linearity plots the dependent variable against the independent variable(s) to determine whether the relationship is approximately linear. This test should usually be performed before any regression is run. Exhibit 5-1 has a great deal of dispersion (which explains why the R' is low), but no violent departures from linearity are evident. A2. Expected Value of the Error Term Is Zero A plot of the error terms from the regression versus a time variable (see Exhibit 5-2) is upward sloping, and the correlation between the error terms and the time variable is 0.94. An important explanatory variable that is correlated with time has probably been omitted from the model. For example, the level of production may have been increasing and the relationship between cost and production levels may not be linear (more specifically, costs increase at a more rapid rate than the increase in the level of production). Alternatively, the cost data could be trending upward because of inflation. A3. The Variance of the Error Terms Is Constant The plots of the error terms from this regression against a time variable, the independent variable, and the predicted value of the dependent variable appear to have a constant width. The width of the plot of the error terms does not seem to vary as a function of the size of any of these variables. We conclude that no evidence of heteroscedasticity exists. A4. Error Terms Are Independent The Durbin-Watson statistic from the regression is 0.33. From the table of Durbin-Watson cutoff points for 95% confidence, one independent variable, and 50 observations, we find du = 1.59 and d, = 1.50. Since 0.33 is less than 1.50, we reject the null hypothesis of uncorrelated errors and conclude that successive error terms are positively correlated. Recall that the Durbin-Watson test relates to only one aspect of correlation between error terms. There may be higher-order correlations among the error terms. When we plot the error terms against time, no evidence appears of a cyclical patter of residuals, such as a sine wave, that would suggest higher-order correlations. A formal test of higher-order correlation can be undertaken by lagging the residuals and computing higher-order correlations. A5. Independent Variables Are Not Correlated This assumption is irrelevant to the study because the regression only employed one independent variable.