Question

1. Consider the model y = X\beta + u, where y is a $T \times 1$ vector of observations on a dependent variable, X is a $T \times k$ full-rank matrix of observations on exogenous variables, $\beta$ is a $k \times 1$ vector of unknown parameters, and u is a $T \times 1$ vector of unobserved normally distributed disturbances, with $E(u) = 0$,and with $E(uu') = \sigma^2I$. (a) [20%] Derive the ordinary least squares estimator of $\beta$, $\hat{\beta}$, and show it is unbiased. (b) [20%] Derive the variance-covariance matrix for $\hat{\beta}$. (c) [20%] Let $\hat{u} = y - X\hat{\beta}$, show $\hat{u} = My = Mu$, where $M = (I - X(X'X)^{-1}X')$ (d) [20%] What is $E(\hat{u}\hat{u}')$? Using this derive an unbiased estimator of $\sigma^2$. (e) [20%] How would you estimate the standard error of the $i$th element of $\beta$, $\beta_i$?

          1. Consider the model
y = X\beta + u,
where y is a $T \times 1$ vector of observations on a dependent variable, X is
a $T \times k$ full-rank matrix of observations on exogenous variables, $\beta$ is a
$k \times 1$ vector of unknown parameters, and u is a $T \times 1$ vector of unobserved
normally distributed disturbances, with $E(u) = 0$,and with $E(uu') = \sigma^2I$.
(a) [20%] Derive the ordinary least squares estimator of $\beta$, $\hat{\beta}$, and show
it is unbiased.
(b) [20%] Derive the variance-covariance matrix for $\hat{\beta}$.
(c) [20%] Let $\hat{u} = y - X\hat{\beta}$, show $\hat{u} = My = Mu$, where $M = (I - X(X'X)^{-1}X')$
(d) [20%] What is $E(\hat{u}\hat{u}')$? Using this derive an unbiased estimator of
$\sigma^2$.
(e) [20%] How would you estimate the standard error of the $i$th element
of $\beta$, $\beta_i$?

$1. Consider the model y = Xβ+ u, where y is a T × 1 vector of observations on a dependent variable, X is a T × k full-rank matrix of observations on exogenous variables, β is a k × 1 vector of unknown parameters, and u is a T × 1 vector of unobserved normally distributed disturbances, with E(u) = 0,and with E(uu') = σ^2I. (a) [20%] Derive the ordinary least squares estimator of $\beta$, $\hat{\beta}$, and show it is unbiased. (b) [20%] Derive the variance-covariance matrix for $\hat{\beta}$. (c) [20%] Let $\hat{u} = y - X\hat{\beta}$, show $\hat{u} = My = Mu$, where $M = (I - X(X'X)^{-1}X')$ (d) [20%] What is $E(\hat{u}\hat{u}')$? Using this derive an unbiased estimator of σ^2. (e) [20%] How would you estimate the standard error of the $i$th element of β, ?$

Added by Kathy G.

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