IV_A_MULTICOL_2011

IV_A_MULTICOL_2011 - IV A 1 James B. McDonald Brigham Young...

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IV A 1 James B. McDonald Brigham Young University 10//11/2011 IV. Miscellaneous Topics A. Multicollinearity 1. Introduction The least squares estimator of β in the model y = Xβ + ε is defined by ˆ = (X'X) -1 X'y. As long as the columns of the X matrix are independent, (X'X) -1 exists and ˆ can be evaluated. If any one column of X can be expressed as a linear combination of the remaining columns, X'X = 0 and (X'X) -1 is not defined. Consider the matrix k 1 1 1 2 1 k 2 1 2 2 2 k k 1 k 2 kX Cor( , ) Cor( , ) ... Cor( ) X X X X X X Cor( , ) Cor( , ) ... Cor( ) X X X X X X Cor(X) = Cor( , ) Cor( , ) Cor( ) X X X X X 12 1k 21 2k k1 k2 1 ... 1 ... = 1 where ρ ij = correlation (X i ,X j ). Recall that 0 Cor(X) 1.
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IV A 2 One "polar" case is that in which the "independent" or exogenous variables are orthogonal or uncorrelated with each other, i.e., Cor(X) = I; hence, Cor(X) I = 1. Another polar case is the situation in which one exogenous variable can be written as a linear combination of the remaining exogenous variables, e.g., x t2 x t3 Sales Revenue t = β 1 + β 2 (Sales of right ski boots) + β 3 (Sales of left ski boots) + ε t . In this case, 23 32 1 Cor( , ) 1 1 XX Cor(X) = Cor( , ) 1 1 and Cor(X) = 0. While the extreme case of Cor(X) = 0 is not very common, frequent instances in which Cor(X) is small may arise in which some rather "strange" results may occur. We will define multicollinearity to exist whenever Cor(X) < 1. Cor(X) = 0 is referred to as exact multicollinearity. Multicollinearity is not necessarily bad, but it may make it difficult to accurately estimate the impact of individual variables on the expected value of the dependent variable. The question of interest is generally not whether we have multicollinearity, but what is the "degree" of multicollinearity, what are the associated consequences, and what can be done about it? While multicollinearity can contribute to imprecise estimates, it is not the only cause or explanation of imprecise estimation. In summary, the impact of multicollinearity is that if two or more independent variables move together , then it can be difficult to obtain precise estimates of the effects of the individual variables, β i = y t )/ X ti , which assumes the other X’s are fixed.
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IV A 3 2. A special case of two explanatory variables. In order to illustrate some of the consequences of multicollinearity, consider the following model: (1) y t = β 1 + β 2 x t2 + β 3 x t3 + ε t t = 1,2, . . ., n it can be shown (details are in the appendix) that (2) i 22 2 ˆ i i 23 23 1 = Var( ) Var( )(1- ) (1- ) X X n n where 23 2 3 Correlation (X ,X ). (3) s - ˆ = t ˆ i i ˆ i i The confidence intervals for β i are given by (4) . ) - )(1 x Var( n s t ˆ = s t ˆ 2 23 ti 2 2 / 1 2 / i ˆ 2 / i i Equation (2) can be used to illustrate the point made on page 3 about multicollinearity only being one of several factors which may impact estimator precision. From (2) we note that (other things being equal) increasing the sample size (n), increasing the variance of the variable whose coefficient is being estimated (X i ), reducing σ 2 , or reducing the square of the correlation between the independent variables will increase the precision of our estimators, i.e., reduce the variance of the estimator. A graphical analysis may be helpful.
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IV_A_MULTICOL_2011 - IV A 1 James B. McDonald Brigham Young...

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