Suppose we are interested in the linear model yi = 0 + 1x1i + 2x2i + ei . Also suppose the columns x1 and x2 of the design matrix for this model have mean 0 and length 1. (That is, x 0 1x1 = 1 and x 0 2x2 = 1. This is a very particular situation that is unlikely to happen in practice; it just makes our arithmetic easier for a moment.). Then if r is the correlation between x1 and x2, we have the following: X0X = n 0 0 0 1 r 0 r 1 and X0X 1 = 1 0 0 0 1/(1 r 2 ) r/(1 r 2 ) 0 r/(1 r 2 ) 1/(1 r 2 ) (a) In our setup where the predictors have mean 0 and length 1, explain why SXX = 1. Use that to show that the VIF formula on page 203 matches 2 (X0X) 1 (above). (b) Determine what values of r will make the variance of 1 and 2 large. Explain why, using what you know about the variance of the vector(Page 203 attached)

6.5 Case Study: Effect of Wine Critics' Ratings on Prices of Bordeaux Wines 203 6.4.1 Multicollinearity and Variance Inflation Factors First, consider a multiple regression model with two predictors Y = Bot Bix, + Bzxz te Let r, denote the correlation between x, and x, and S, denote the standard devia- tion of x.. Then it can be shown that 02 Var(B;) j =1,2 1-12 (n-1)S Notice how the variance of B; gets larger as the absolute value of / increases. Thus, correlation amongst the predictors increases the variance of the estimated regression coefficients. For example, when r? =0.99 the variance of B; is 1 - 12 1-0.992 = 30.25 times larger than it would be if ? = 0. The term 1- 12 is called a variance inflation factor (VIF). Next consider the general multiple regression model Y = Bot Bix, + Bzx2 + ... + Box, te Let R, denote the value of R obtained from the regression of x, on the other x's (i.e., the amount of variability explained by this regression). Then it can be shown that 02 Var(B, ) = X 1-R. (n-1)$2 j = 1,.... P The term 1/(1- R.) is called the jth variance inflation factor (VIF). The variance inflation factors for the bridge construction example are as follows: log (DArea) ] log (CCost) log (Dwgs) log (Length) log (Spans) 7. 164619 8 . 483522 3. 408900 8 . 014174 3. 878397