Nearly collinear regressors: Construct the geometric vector representation of a regression with two explanatory variables in mean deviation form, by( = B 1 x( 1 þ B 2 x( 2, distinguishing between two...

Nearly collinear regressors: Construct the geometric vector representation of a regression with two explanatory variables in mean deviation form, by( = B₁x( 1 þ B₂x( 2, distinguishing between two cases: (a) X₁
and X= are highly correlated, so that the angle separating the x( 1 and x( 2 vectors is small, and (b) X₁
and X₂
are uncorrelated, so that the x( 1 and x( 2 vectors are orthogonal. By examining the regressor plane, show that slight changes in the position of the by( vector (due, e.g., to sampling fluctuations) can cause dramatic changes in the regression coefficients B₁
and B₂
in case (a) but not in case (b). The problem of collinearity is discussed further in Chapter 13.