Question

Problem 2

Let X1 , X2 , ⋯, XN be a random sample of size n from normal distribution with mean μ and variance σ2 .

a). Find the maximum likelihood estimator of σ2  2 . (2 points)

b). Find the asymptotic distribution of the maximum likelihood estimator of σ2  2

obtained in part (a).

Problem 1

Consider a k-variables linear regression model, i.e.,

Y = X 1β1 + X 2 β2 + ε,

Where, X1 is (N  k1 ) , X 2 is (N  k2 ) and k = k1 + k2 . As you may recall, adding columns to the X matrix (including additional regressors in the model) gives positive definite increase in R2. The adjusted R2 ( R 2 ) attempts to avoid this phenomenon of ever increase in R2. Show that the additional k2 number of variables (regressors) in this model increases R 2 if the calculated F-statistic in testing the joint statistical significance of coefficients of these additional

regressors (β2 ) is larger than one.

Answer 1

problem 2

(a)

$variance, v=v(\Sigma \frac {xi}{n})$

$=\frac {1}{n^2}\Sigma v(xi)$

$=\frac {n}{n^2} \sigma^2=\frac {\sigma^2}{n}$

$\therefore$ the likelihood estimator will be:

$\sigma^2= s^2\frac {n}{n-1}$

$=\frac {\Sigma (xi-x)^2}{n-1}$

$s^2$ is the constant estimator.

(b)

The sample median estimator of the median $X_n$ corresponding to p=0.5. $X_n$ is then a normal distribution with parameters $\mu$ and $\sigma2$.

Problem 2