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Suppose that you have simple two variable model:

Y=b0+b1X1+e

The least squares estimator for the slope coefficient, b1 can be obtained with

b1=cov(X1,Y)/var(X1)

the intercept term can be calculated from the means of X1 and Y

b0=mean(Y)-b1*mean(X1)

In a larger model, Y=b0+b1X1+b2X2+e

the estimator for b1 can be found with

b1=(cov(X1,Y)var(X2)-cov(X2,Y)cov(X1,X2))/(var(X1)var(X2)-cov(X1,X2)2)

to find b2, simply swap the X1 and X2 terms in the above to get

b2=(cov(X2,Y)var(X1)-cov(X1,Y)cov(X1,X2))/(var(X1)var(X2)-cov(X1,X2)2)

Find the intercept with

b0=mean(Y)-b1*mean(X1)-b2*mean(X2)

Beyond two regressors, it just gets ugly.

Q: How do you use a variance-covariance matrix to obtain least squares estimates?

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