Assessing Correlation among Independent Variables

Normality Assumption Assumption : The Independent Variables Are Uncorrelated If the independent variables are perfectly correlated, then, technically speaking, regression analysis is impossible. Of course, in this case the solution is obvious; merely drop one of the perfectly correlated variables from the equation. In practical terms, then, the problem arises with variables that are highly but not perfectly correlated with each other. In this situation, the highly correlated independent variables fluctuate together, and it becomes difficult to obtain reliable estimates about the effect on the dependent variable of each independent variable by itself. Thus, when two or more independent variables are highly correlated, the coefficient estimates of these variables will be unreliable and the standard errors of the coefficients will be high. One rule of thumb advocates avoiding the simultaneous use of independent variables that have a correlation coefficient whose absolute value exceeds 0.8. Alternatively, Klein's Rule asserts that multicollinearity will not be a problem if the correlation between any two independent variables is less than the multiple correlation coefficient (R) of the regres-sion. Collinearity only affects coefficient estimates but not the explanatory power (R) of the regression. Therefore, collinearity problems may be detected by a regression equation with high R° but low t statistics for coefficient estimates or when the signs of one or more coefficients are opposite from those expected. Therefore, if the regression equation is to be used only for prediction of future values of the dependent variable, and not for making inferences about individual coefficient estimates, the presence of multicollinearity may not be a serious problem. Assumption : The Errors Are Normally Distributed This normality assumption is not required in order to obtain OLS estimates to fit a regression line through a set of observations. But our normal statistical procedures performing F tests and t tests, and constructing confidence intervals-are, strictly speaking, only valid if the error term can be approximated by a normal distribution. In practice, it would be unusual for nonnormality to create serious problems for the analyst. An Example We will ilustrate the diagnostic procedures by examining the following situation. CHATHAM BOX COMPANY The Chatham Box Company (CBC) manufactures cardboard boxes in a highly automated factory. Although the company cost accountant suspects that factory overhead is related to the number of machine-hours worked, no study has been undertaken to confirm whether his intuition is supported by evidence. The company cost accountant has assembled the following data and has provided the following background information: The accompanying tablesummarizes monthly operations in the 50 months since the plant opened. No major production line or equipment changes have occurred since we opened. The costs of the supply items included in overhead seem to be increasing by about 1% per month.