Question

4. Consider the following model: Let X = (1,X1,X2, Xs)". Suppose and that X is not perfectly colinear. Suppose further that ElY]oo and Exoo for 1 j S 3. Using a large sample of i.i.d. observations from (XX1.x2, Xs), you estimate this equation using Ols. Let β-(%, 33) de- note the resulting estimate of β = (Ah-…β3). (a) Let θ-2A-β3. Is θη-2A-A3 a consistent estimate of θ? Is it an unbiased estimate of 0? (b) Express Var(0,] in terms of VarlB], VarlBal, and CovBB (c) How would you test the null hypothesis H 01 versus the alternative K : θ 1 at the 5% significance level? In particular, write down your test statistic, your critical value, and rule you would use to determine whether or not to reject the null hypothesis (d) What is the p-value for your test?

Answer 1

a) Yes it a consistent estimate.

As we have estimated the parameter using OLS. So when we increase the size of sample then the estimated value converges to the   true value hence it is consistent.

Yes it is an unbiased estimate.

an estimater is unbiased when  

E(ei)

therefore,

$E(\Theta^{_{1}^{}} )=E(2\beta _{1}^{'}-\beta _{3}^{'}) =2E(\beta _{1}^{'})-\beta ^{_{3}^{'}} =2\beta _{1}-\beta _{3}$

As estimate of beta are unbaised

thus,

E(ei)

b)

Varier) Var(29-Be)

As

Var(x + y) = Var(z) Var(y) + 2Cor(2, y)

Thus,

$Var(2\beta ^{_{1}^{'}}-\beta ^{_{3}^{'}})=4Var(\beta ^{_{1}^{'}})+Var(\beta ^{_{3}^{'}})-2*2Cov(\beta ^{_{1}^{'}},\beta ^{_{3}^{'}})$

c) We will use the Z score to test he statistics.For $\alpha$ equal to percent we will take the value of $z=\pm 1.96$ to find the confidence interval

We will reject the null hypohesis if where a is the Z score i.e the ciical value else we will accept the null hyothesis.

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