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I think this is probably supposed to be super easy - both questions are worth a total of one mark I've just never seen most of this before.

y=XB+E where B is beta and E is the error term.

E~MVN(0, sigma^2*W) where W=diag(w1^2,..., wn^2)

a) Find the ordinary least squares estimate Beta(ols) b) Find the distribution of Beta(ols)

What does it mean for something to be MVN exactly? How does an ordinary least squares estimate differ from normal? What is this W matrix (does it even matter or is it just an arbitrary matrix?)? My Prof makes his notes up on the fly and I can't find any of this information anywhere.

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1 Answer 1

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  1. 'MVN' stands for multivariate normal, so E~MVN(0, sigma^2*W) means the distribution of E is a zero mean multivariate normal distribution with covariance matrix sigma^2*W.

  2. So long as E has zero mean, the ols gives the same form of estimate $$Beta(ols)=X^+y$$ where $X^+$ is the pseudoinverse of $X$, and $$Beta(ols) \sim MVN(X^+XB,\sigma^2X^+W{X^+}^T)$$

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