The model is:

yi = β1 + β2x1i + β3x2i + ui                   (1)

At each step, state any additional assumption you need to use:

(1.1) Derive the OLS estimators without using vectors/matrix notations.

(1.2) Show that OLS estimator is unbiased.

(1.3) Assume that,

Discuss the behaviour of this estimator  as sample size increases to a very large number n → ∞.

(1.4) Comment on the previous part. In particular, can you think of a case where takes the form. above, and what would be the main purpose of such regression?

Question 2

Consider the regression model:

y = Xβ0 + u

where y is T ×1, X is T ×k and rank(X) = k, β0 is the k×1 parameter vector, and u ∼ N(0T , σ02IT) where σ0 2 is unknown but a positive constant.

(2.1) Using this result, propose a decision rule to test:

H0 : Rβ0 = r

HA : Rβ0 ≠ r

where R and r are respectively a q×k matrix and a q×1 vector of constants. Define the test-statistic associated with this hypothesis testing in terms of R, r, etc. What would constitute a Type I error in this context and what is the probability of a Type I error associated with your decision rule?

(2.2) Define the p-value of the test in previous part.

Question 3

The model is:

yi = β0 + β1X1,i + β2X2,i + ui

for i = 1, . . . , N and we wish to test the null hypothesis: H0 : β1 = β2 = 0.

(3.1) What is the alternative hypothesis? Re-write the regression model, and the null hypothesis in terms of notations used in the lecture (R, r, etc.), indicating the size of each variable.

Using the null hypothesis, what are the numerical values for elements in R, r, etc.

(3.2) What is the test statistic and its distribution when the variance of the error term is unknown?

(3.3) Represent elements in (X′X)−1 = {cjk}. What is [R(X′X)−1R′]−1 in terms of cjk elements?

(3.4) What is the test-statistic in terms of cjk elements?

(3.5) Suppose the test conclusion is to reject the null, comment on this conclusion.

(3.6) Suppose the test conclusion is to fail-to-reject the null, comment on this conclusion.

Question 4

Consider the probability density function, f(x; θ) = λe−λx. Find the MLE of λ and its variance (assuming that the sample is i.i.d.).

Question 5

Consider a simple linear regression model with non-stochastic regressors and i = 1, . . . , n:

yi = α + βxi + ui                       (2)

ui ∼ i.i.d N (0, σ2 )                    (3)

(5.1) Define the ML estimator for α and β.

(5.2) Clearly stating any assumption you need, derive the ML estimators for α and β.

(5.3) Is this estimator BLUE?