# Least squares estimator

I have many quite hard questions for least squares estimator Suppose we have a vector of $n$ observation $Y$ which has the distribution $N_n(X\beta,\sigma^2I)$, where $X$ is an $n \times p$ matrix of known values, which has full column rank $p$, and $\beta$ is a $p \times 1$ vector of unknown parameters. The least square estimator of $\beta$ is $$\hat{\beta}=(X'X)^{-1}X'Y$$

a. Determine the distribution of $\hat{\beta}$.

b. Let $\hat{Y}=X\hat{\beta}$. Determine the distribution of $\hat{Y}$.

c. Let $\hat{e}=Y-\hat{Y}$. Determine the distribution of $\hat{e}$.

d. By writing the random vertor $(\hat{Y}',\hat{e}')$ as a linear function of $Y$, show that the random vectors $\hat{Y}$ and $\hat{e}$ are independent.

e. Show that $\hat{\beta}$ solves the least square problem, that is $$||Y-X\hat{\beta}||^2=\min_{b\in \mathbb{R}^p}||Y-X\beta||^2$$.

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If this is homework you should add the tag. Also you should show your attempts at the various problems. Concerning the first problems, have a look at this: en.wikipedia.org/wiki/… – Stefan Hansen Oct 28 '12 at 9:33