For multiple linear regression model Y = Xβ + e, assume X is non-random.

Assume Gauss-Markov assumptions hold. Show that variance-covariance matrix of b, the least square estimates of β is σ²(X'X)^-1.

Cov(b) = Cov[(X'X)^-1 * X'Y]

= (X'X)^-1 * X'Cov(Y)X(X'X)^-1

= ... I know the rest of the steps

My questions are:

1) notation for variance-covariance matrix of b is both Var(b) or Cov(b), correct?

2) I don't understand how I go from Cov[(X'X)^-1 * X'Y] to (X'X)^-1 * X'Cov(Y)X(X'X)^-1. Can anyone clarify? Thanks

Here's a sketch of what you trying to get at:

Bhat = (X`X)^-1 X`y

Substituting y=XB +e in this expression gives

Bhat = (X`X)^-1 X`(XB + e)

= (X`X)^-1 X`XB + (X`X)^-1 X`e

= B + (X`X)^1 X`e

Thus,

Bhat – B = (X`X)^-1 X`e

Now by definition:

var-cov(Bhat) = E[Bhat – B)(Bhat – B)`]

= E{[ (X`X)^-1 X`e][X`X)^-1 X`e]`}

=E[ (X`X)^-1 X`ee`X(X`X)^-1 ]

where the last step is made by the fact that (AB)`=B`A`.

As such given your assumptions:

var-cov(Bhat) = (X`X)^-1 X` E[(ee`)X(X`X)^-1

=(X`X)^-1 X Sigma^2 I X(X`X)^-1

=Sigma^2(X`X)^-1

Note: I did this quickly - but it should give you the answer you're looking for.