Multicollinearity in Regression Analysis: Comment

Abstract

In recent paper this REVIEW,' Farrar and Glauber (hereafter FG) revisit problem of regression analysis. Viewing problem of as both facet and symptom of poor experimental design, 2 FG propose a three-stage hierarchy of increasing detailed tests presence, location, and pattern, 3 of multicollinearity. The first this series of three tests, on which other two are conditional, is desigped provide useful first measure of presence and severity of multicollinearity 4 sample on hand. Bartlett's well-known statistic testing joint distribution of sample correlations under assumption of vanishing parent correlations between variables is used by FG detecting multicollinearity. Bartlett shows that (natural) logarithm of intercorrelation determinant computed from sample drawn from multivariate, ortho-normal distribution, multiplied by factor k, is approximately distributed as Chi Square with v 1/2 n (n 1) degrees of freedom, where k = -[N 1 1/6 (2n + 5)], N is sample size and n is number of variables considered. If investigator concludes from first stage that exists and that it is severe enough warrant some action, FG propose regress consecutively each explanatory variable on remaining ones. The rp'silting F statistics will test for dependence of particular variables on other members 5 of set of explanatory variables. Finally, patterns of interdependence among independent variables are examined by testing significa-nce of partial correlations of every pair of explanatory variables, all other variables held constant. The main pillar of this three-level test is, of course, Bartlett's test which is properly used making inferences,6 under null hypothesis that all population correlations are zero. Since FG claim, however, that they are not interested drawing inferences from sample population (inferences from sample population . . . are possible . . . however, little importance is attached properties of population from which set of data has been drawn. Attention focuses largely, if not entirely, on sample itself 7), their use of Chi-Square statistic is questionable. Moreover, it is neither practical nor necessary assume orthogonality between parent economic variables, 4f one wishes make such inferences. Here we come heart of problem of multicollinearity. One may agree with FG that it is preferable think of in terms of [its] severity rather than its existence or nonexistence. 8 If one agrees with this approach, natural way proceed is indeed to define terms of departures from hypothesized statistical condition. 9 But what is this hypothesized condition? For FG this condition is the requirement that explanatory variables be truly independent of one another. 10 However, there is no such requirement least-squares solution. On contrary, least squares solu* I share with D. C. Farrar and R. R. Glauber my indebtedness Professor John R. Meyer who introduced us problem, and I am grateful his comments on an earlier draft. I am also grateful Professors J. Johnston, N. Wallace, D. Farrar, and R. Glauber valuable discussions. I am particularly thankful D. Farrar who did not spare his efforts order dig out old forgotten data, which enabled me recompute his regression equations. 1D. C. Farrar and R. R. Glauber, Multicollinearity Regression Analysis: The Problem Revisited, this REvIEw, XLIX (Feb. 1967). 2Ibid., p. 93. 3 Ibid., p. 104. ' Ibid., p. 101. BIbid., p. 104. Bartlett has originally developed this statistic order test number of meaningful components that can be extracted from set of variables. concise statement is given by Bartlett: A Note on Multiplying Factors Various x2 Approximations, Journal of Royal Statistical Society (B), XVI, no. 2 (1954), pp. 296-298. 7D. C. Farrar and R. R. Glauber, op. cit., 100. 8Ibid., p. 106. 9 Ibid., p. 92. 10Ibid., pp. 92 and 100.