2 strictly speaking multicollinearity refers to the

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2 Strictly speaking, multicollinearity refers to the existence of more than one exact linear relationship, and collinearity refers to the existence of a single linear relationship. But this distinction is rarely maintained in practice, and multicollinearity refers to both cases.
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 4 Why do we obtain the result shown in (4)? Recall the meaning of 1 ˆ : It gives the rate of change in the average value of t Y as 1 t X changes by a unit, holding 2 t X constant. But if 2 t X and 1 t X are perfectly collinear, there is no way 2 t X can be kept constant: As 1 t X changes, so does 2 t X by the factor 0.9. What would happen if the relationship is as follows? 2 1 (0.9) t t t X X u (6) where t u is a stochastic error term . In this case, you can verify it by simply adding to 2 t X some random numbers (such as 1, 0, 4 and -1), the coefficient of correlation r 12 would not be unity, but it would take a very close value to unity. This situation is known as the less than perfect multicollinearity. If multicollinearity is less than perfect, as in (6), the regression coefficients, although determinate, possess large standard errors (in relation to the coefficients themselves), which means the coefficients cannot be estimated with great precision or accuracy. In passing, note that multicollinearity, as we have defined it, refers only to linear relationships among the X variables. It does not rule out nonlinear relationships among them.
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 5 Large variances due to Multicollinearity Note that, 2 1 2 2 2 12 ˆ var( ) 1 t t x r (7) and 2 2 2 2 2 12 ˆ var( ) 1 t x r (8) Note also that,
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 6 2 12 1 2 2 2 2 12 1 2 ˆ ˆ cov( , ) 1 t t r r x x   (9) It is apparent from (7) and (8) that as 12 r tends toward 1, that is, as collinearity increases, the variances of the two estimators increase and in the limit when 12 r = 1, they are infinite. It is equally clear from (9) that as 12 r increases toward 1, the covariance of the two estimators also increases in absolute value. 2. Consequences of Multicollinearity In cases of near or high multicollinearity, one is likely to encounter the following consequences: 1. Although BLUE, the OLS estimators have large variances and covariances, making precise estimation difficult. 2. Because of consequence 1, the confidence intervals tend to be much wider, leading to the acceptance of the “zero null hypothesis” (i.e., the true population coefficient is zero) more readily. Hence, the probability of accepting a false hypothesis (i.e., type II error ) increases.
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