3. Professor Heterosa considers the simple regression model without intercept: $y_i = \beta X_i + u_i$ (i = 1, 2, ...n) where $\beta$ is a scalar. In addition to the above four standard assumptions, let us assume that the error term is a random pick from a normal distribution. She has collected the data from two countries: $n_1$ observations from Italy, and $n_2$ observations from Spain. She suspects there exists a country difference in error variances such that the error variance for Italy's $n_1$ observations is $\sigma_1^2$, and the error variance for Spain's $n_2$ observations is $\sigma_2^2$ where $n_1 + n_2 = n$. Thus, $var(u_i) = \sigma_1^2$ for i = 1, 2, ..., $n_1$ and $var(u_i) = \sigma_2^2$ for i = $n_1$+1, $n_1$+2, ..., n. She wants to test $H_0: \sigma_1^2 = \sigma_2^2$ against $H_1 : \sigma_1^2 \ne \sigma_2^2$.
(a) If Professor Heterosa applies White's test for the hypotheses, she must change the hypotheses. Explain how to change the hypotheses and how to estimate the parameters in the new hypotheses. Is White's test a good idea in this case? [10]
(b) If Professor Heterosa applies Breusch-Pagan LM test for the hypotheses, she must change the hypotheses. Explain how to change the hypotheses and how to estimate the parameters in the new hypotheses. [10]
(c) Now, all the tests have shown that the error variances are different between Italy and Spain, i.e. $\sigma_1^2 \ne \sigma_2^2$. How would you estimate $\beta$? [10]
4. Consider the regression model $y = X\beta + u$ where X is an (n×k) matrix and y is an (n×1) vector. Assume that the error terms have independent standard normal distribution, i.e. $u \sim N(0, I)$. Let us consider a restriction $H_0: R\beta = q$ where R is an (rxk) matrix.
(a) Derive the restricted maximum likelihood estimator of $\beta$, $\beta^{MLE}$, which maximizes the likelihood function subject to $R\beta = q$. Note that the likelihood function, L, is as follows:
$L = \prod_{i=1}^n f(u_i) = (2\pi)^{-\frac{n}{2}} e^{-\frac{u'u}{2}}$
(b) The RLS estimator, $\beta^*$, is $\beta^* = \hat{\beta} - (X'X)^{-1}R'[R(X'X)^{-1}R']^{-1}(R\hat{\beta} - q)$ where $\hat{\beta} = (X'X)^{-1}X'y$. Cramer-Rao Lower Bound (CRLB) for the variance of any estimator of a parameter $\theta$ is the inverse of (Fisher's) Information Matrix, I($\theta$), where $I(\theta) = -E[\frac{\partial^2 log(L)}{\partial \theta \partial \theta'}]$. Show that $\beta^*$ is fully efficient. If necessary, use the Partitioned Inverse Rule. [10]
Partitioned Inverse Rule. [10]
For the general 2 × 2 partitioned matrix, one form of the partitioned inverse is
$\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}^{-1} = \begin{bmatrix} F_1(I + A_{12}F_2A_{21}F_1) & -A_{11}A_{12}F_2 \\ -F_2A_{21}A_{11} & F_2 \end{bmatrix}$
where
$F_2 = (A_{22} - A_{21}A_{11}^{-1}A_{12})^{-1}$
The upper left block could also be written as
$F_1 = (A_{11} - A_{12}A_{22}^{-1}A_{21})^{-1}$
