Consider the model y, =yi + ej where for X = {z1,..., IN}, and for all i, E (e,X) = c # 0. and Var (eX) =of. Denote bythe OLS estimator for Y.Which of the following is correct?None of the above.O For all i, E (7 + e|X) = y 7 +cofConditional on X. the variance of the OLS estimator, 7 of y is Var (§|X)O The OLS estimator for y is biased.O E (y|X) = c

From the given information, yi=γxi+ei X=x1, x2, ...., XN Eei|X=c≠0 Varei|X=σ2 Here, the ols…

Answered: Consider the model y, , + e, where for…

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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Question

Consider the model y, =yi + ej where for X = {z1,..., IN}, and for all i, E (e,X) = c # 0. and Var (eX) =of. Denote by
the OLS estimator for Y.
Which of the following is correct?
None of the above.
O For all i, E (7 + e|X) = y 7 +c
of
Conditional on X. the variance of the OLS estimator, 7 of y is Var (§|X)
O The OLS estimator for y is biased.
O E (y|X) = c

Expert Solution

Step 1

From the given information,

$y_{i} = γ x_{i} + e_{i}$

$X = (x 1, x 2, . . . ., X_{N})$

$E (e_{i} | X) = c \neq 0$

$V a r (e_{i} | X) = σ^{2}$

Here, the ols estimator is unbiased estimator.

That is, $E (\hat{γ}) = γ$

Hence, the option "the OLS estimator for $γ$ is biased" is incorrect.

Consider,

$E (\bar{y} | X) = E (\hat{γ} \bar{x} + e_{i} | X) = E (\hat{γ} \bar{x} | X) + E (e_{i} | X) = \hat{γ} E (\bar{x} | X) + c = \hat{γ} \bar{x} + c$

Hence, the option $E (\hat{γ} \bar{x} + e_{i} | X) = \hat{γ} \bar{x} + c$ is correct.

And $E (\bar{y} | X) = c$ is incorrect.

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Consider the model y, , + e, where for X= {r1,…..,¤N}, and for all i, E (e|X) = c # 0. and Var (eX) = the OLS estimator for Y. Which of the following is correct? O None of the above. O For all i. E (9+ ei|X) = y E + c