Assumption of Independent Error Terms

Model Misspecification Errors Assumption : The Variance of the Error Term Is Constant The ordinary least-squares procedure weights each observation equally. This equally weighted procedure is optimal if the underlying precision or variance is the same for every observation. In other words, the OLS procedure requires that the variance of the error term not be a function of (1) time,(2) the size of any of the independent variables, or (3) the size of the predicted value of the dependent variable. If these three conditions are met, the condition called homoscedasticity exists, and the variance of the distribution of the error term is constant for every observation; the variance is independent of time and the magnitude of all the variables in the model. When the data do not have these properties, we describe the data as being heteroscedastic. In this case, the OLS estimates will no longer be the best (minimum variance) estimates that we can produce from the given set of data. Departures from homoscedasticity can be detected by plotting the data and observing whether error terms seem to vary systematically over time or for large values of the independent variable(s). In practice, the homoscedasticity assumption is frequently violated; the most common violation occurs when a higher variance exists for large observations of the independent or dependent variables. Attempts to correct this condition, however, rarely make a large change in the estimated parameters, so that a casual benefit-cost analysis suggests that we not devote much more space here discussing this particular problem. The main problem created by heteroscedasticity is that the standard errors from the regression equation may be inaccurate. Assumption : The Error Terms Are Independent When error terms are correlated with each other, we have less precision in our estimates than the statistical procedures would lead us to believe. In effect, we are getting less information than assumed (a smaller effective sample size) because the same information influences more than one observation. In statistical terms, the standard errors produced by the regression will be misestimated, thus invalidating normal cosfadard eatements. For ex anie pasted, seria corelation causes he downwardly biasestimated by the persion seria underestimated. The doer or cobied standard chors is pon to be confidence intervals, ioner far are tient estimate or tors will ions com the regression equation, timates too narrow, aus ing are analyst to assign more precision to estimates than is property warranted. A number of tests can be used to detect correlain armong the ermor terms. Themes commonly used test is the Durbie-ae on statistic, which provide measure of association between successive values of the error perises musin- Waison statistic that is computed from the regression data is compared against two tabulated values, de arees othat depend on the desired confidence level of the test and theration freedom of the data which is determined by the number of observations and the number of independent variables in the regression). If fee Durbin-Watson statistic is greater than d, then we do not reject the null hypothesis that there is no positive correlation between successive error terms. If the Durbin-Watson statistic is less than d, then we accept the alternative hypothesis that there is a positive correlation between successive error terms. If the Durbin-Watson statistic lies between d, and d, the test is inconclusive and we can neither accept nor reject the null hypothesis. To test for negative correlation between successive error terms, we must subtract the computed Durbin-Watson statistic from 4. We then follow the same procedure to test for negative correlation as the procedure used to test for positive correlation. Roughly speaking, a Durbin-Watson statistic around 2 suggests no evidence of correlation of the successive error terms. A Durbin-Watson statistic less than 1.5 is usually evidence of positive correlation of the successive error terms, and a value greater than 2.5 is usually evidence of negative correlation. The Durbin-Watson test is confined to testing for correlation between successive error terms (this is called first-order serial correlation): if we let r be the correlation between successive error terms, we can show that the Durbin-Watson statistic approximately equals 2 * [1 - r]). The regression assumption is stronger than that tested by the Durbin-Watson statistic. The regression assumption requires no correlation of any kind (of any order) between the error terms. For this reason, some regression packages allow the analyst to compute higherorder correlations of the errors. For example, if a quarterly or seasonal effect exists but is not included in the model, one might find that a fourth-order correlation of the residuals exists in monthly data. That is, every fourth error term is correlated.) The most common cause of correlated error terms is a model mis-specification error. That is, the model is not a good representation of the actual relationship. The following are common misspecification errors: 1. The omission of an important variable, such as a seasonal effect, that affects the dependent variable. In this case, the omitted variable should be identified and added to the model. 2. The relationship is nonlinear perhaps because of inflationary trends. In this case, the data should be transformed (deflated), or an explicit nonlinear regression should be used. 3. A shift in the production process has occurred, such as a change in the production equipment that is used but has not been incorporated into the model. In this case, the shift should be incorporated into the model, or the regression should be confined to the period since the shift.