RIDGE REGRESSION

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where RJ is the coefficient of determination when Xj is regressed on the other explanatory variables. If Xj is exactly collinear with the other variables, that is, if Xj is an exact linear function of them, then RJ = 1 and VIFj = 00. If Xj is completely uncorrelated with the other variables, then RJ = 0 and VIFj = 1. For the cement data, the VIFs for XI' X 2 , X 3 , and X 4 are 38.5, 254.4, 46.9, and 282.5. There is a collinearity problem for a variable Xj if RJ is close to 1, that is, if VIFj is large. The computer package Minitab prints a warning when VIF; is larger than 100. There is a direct connection between the variance inflation factor of Xj and the standard deviation of the least-squares estimate of its regression coefficient:

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where ~j = ~LSj and Sj is the standard deviation of the observed values of X j Hence the size of SD(~) is determined by three factors: a factor ...jVIFj due to the relationship of Xj with the other explanatory variables, a factor (T /Sj depending on the variation of the random errors relative to the variation of the measurements of X j , and a factor 1/ ~ depending on the sample size. Inaccuracy of ~j can be due to collinearity of Xj with other explanatory variables or to large random errors or to a small sample size. The phrase "variance inflation factor" comes from the fact that Var(~) is VIFj times larger than what it would be if Xj were uncorrelated with the other explanatory variables.

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Matrix Formulation of the Problem of Collinearity. In terms of the matrix Z of explanatory variables, collinearity means that some column of Z is approximately a linear combination of the other columns. This implies that the matrix Z'Z, which must be inverted to calculate the least-squares estimate of 1', is nearly singular. Inverting a singular matrix is like taking the reciprocal of the number 0; it is not a valid operation. Inverting a nearly singular matrix is similar to taking the reciprocal of a very small number; some of the entries in the inverse matrix are likely to be very large. The variance of the least-squares estimate YLSj is equal to (T2 times the jth diagonal entry in (Z'Z)-I. Therefore near-singularity of Z'Z is likely to be associated with large variances for some of the least-squares estimates. This indicates how collinearity leads to inaccuracy of regression estimates. Besides the statistical problem of large variances for the estimates, collinearity also poses a computational problem. It is difficult to achieve numerical accuracy in computing the inverse of a nearly singular matrix.

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NOTES

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Further Justification of Ridge Estimation. By viewing the problem of collinearity as the problem of the near-singularity of Z'Z, we are led to the method of ridge regression, which modifies Z'Z so that it is farther from singularity. The matrix Z'Z is modified so that it is closer to what it would be for data in which there is no coIIinearity, that is, data in which all the explanatory variables are uncorrelated with one another. We now need to know what Z'Z looks like for such data. The matrix Z'Z is n - 1 times the sample correlation matrix of the explanatory variables. To verify this, note that the (j, k) entry of Z'Z is LiZijZik' Since Zij = (x ij - x)/Sj' where Xj and Sj are the sample mean and standard deviation of the observed values of variable X j , the (j, k) entry of Z'Z is L/X ij - XjXX ik - xk)/(SjSk)' Recall that Sjk = L/Xij - XjXX ik xk)/(n - 1) is the sample covariance of Xj and X k . So the (j, k) entry is (n - l)Sjk/(SjSk)' By definition, Sjd(SjSk) is the sample correlation between Xj and X k . In the most favorable case, in which all the explanatory variables are uncorrelated, the sample correlation matrix is simply the identity matrix I, and so Z'Z = (n - 0/. When there is collinearity, we can move Z'Z closer to the most favorable case by adding a multiple of 1 to Z'Z, that is, by replacing Z'Z by Z'Z + kl. This leads to (8.4).

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NOTES 8.3a. The formula for -}\s is obtained from (3.2). Direct application of (3.2) gives us -}\s = L(Zi - ZXYi - Y)/L(Zi - Z)2. Since z = 0, this becomes -}\s = LZi(Yi - Y)j LZi2 Moreover, LZi(Yi - Y) = LZiYi because LZiY = YLZi = y(nz) = O. 8.3b. To verify (8.2), let g = (y) and note that Yar(y) = [(y - gf]. Now MSE(y) = [(y - y)2] = [ y - g) + (g - y 2] = [(y _ 0 2 +

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(g -

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+ 2(y - gxg - y)] = [(y - 0 2] + (g Yar(y) + [ (y) - yF because (y - 0 = (y)

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y)2 -

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8.3c. Formula (8.3) follows from (8.2). By (8.2), MSE(e-}\s) = Yar(e-}\s) + [ (e-}\s) - yF, which equals e 2 Yar(YLS) + [e (-}\s) - yF = e 2 u + [ey - y F = e 2 u + (e - 1)2y2. The derivative with respect to e is 2eu + 2(e - l)y2 = 2[c(u + y2) - y2], which is 0 when e(u + y2) = y2. 8.3d. It is difficult to figure out the exact mean squared error of the ridge estimate y but we can estimate it by computer simulations. For example, let us simulate a model similar to the process that generated the height data. Consider the model Yi ~ 170 + 5z i + e i for i = 1,2, ... ,32, where the z/s

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