Question

What is the value of ^B_1 in the limit "rx1, x2 > 1"?

Last two sentences are state the question. I also stated it above

Answer 1

Let r₁₂ be the correlation coefficient between X₁ & X₂. In the extreme collienearity, however the data is standardized the the correlation must tend to 1 from the definition of correlation coefficient. But in case of linear regression it introduces multicollinearity which has been described briefly in the next paragraph and its problem in the model.

As we know $\beta$₁ gives the rate of change in the average value of Y as X₁ changes by a unit holding X₂ constant. But if X₁ and X₂ are perfectly collinear, there is no way X₂ can be kept fixed. That is in extreme collinearity as X₁ changes, so does X₂ by a constant factor.

Now, for the above regression model by the method of least squares the variances and covariances of the estimates are as follows:

Var($\hat{}\beta$₁₎ = $\sigma$²/$\sum$x₁²(1 - r₁₂²)

Var($\hat{}\beta$₂₎ = $\sigma$²/$\sum$x₂²(1 - r₁₂²)

Cov($\hat{}\beta$₁,$\hat{}\beta$₂) = - r₁₂$\sigma$² / (1 - r_12²) $\sqrt{}$($\sum$x₁²$\sum$x₂²)

It is apparent from the above expressions of variances and covariances between the two regression coefficients, the variances of the two estimators increase and in the limit when r₁₂=1 , they are infinite. It is equally clear that as r₁₂ increases toward 1, the covariance of the two estimators also increases in absolute value.

As the least squares estimates are unbiased in the extreme collinearity the value of $\hat{}\beta$₁ will not change but its variance will be very large. This can be seen from the above expression.