Question 1

For this question you may assume that linear combinations of normal variates are
themselves normally distributed.Let a, b, and c be non-zero constants.
(a) $$X \text { and } Y \text { are independently distributed as } N \left( a , \sigma ^ { 2 } \right) \text {. What is the distribution of } ( b X + c Y ) \text { ? }$$

Accepted Answer

The answer of For this question you may assume that...

Question 2

(Requires Appendix material) Your textbook considers various distributions such as the standard normal, $ t, \chi^{2} $, and $ F $ distribution, and relationships between them.
 (a) $$\text { Using statistical tables, give examples that the following relationship holds: } F _ { n _ { 1 } , \infty } = \frac { \chi _ { n _ { 1 } } ^ { 2 } } { n _ { 1 } } \text {. }$$

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The answer of (Requires Appendix material) Your textbook considers various...

Question 3

Consider the model $Y _ { i } = \beta _ { 1 } X _ { i } + u _ { i }$, where the $X _ { i }$ and the $u _ { i }$ are mutually independent i.i.d. random variables with finite fourth moment and $E \left( u _ { i } \right) = 0$. Let $\widehat { \beta } _ { 1 }$ denote the OLS estimator of $\beta _ { 1 }$. Show that
$$\sqrt { n } \left( \widehat { \beta } _ { 1 } - \beta _ { 1 } \right) = \frac { \frac { \sum _ { i = 1 } ^ { n } X _ { i } u _ { i } } { \sqrt { n } } } { \frac { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } } { n } } .$$

Accepted Answer

The answer of Consider the model \(Y _ { i...

Question 4

Feasible WLS does not rely on the following condition:
A)the conditional variance depends on a variable which does not have to appear in the regression function.
B)estimating the conditional variance function.
C)the key assumptions for OLS estimation have to apply when estimating the conditional variance function.
D)the conditional variance depends on a variable which appears in the regression function.

Accepted Answer

Feasible Weighted Least Squares (WLS) can be applied even when the conditional variance depends on a variable that does not appear in the regression function, making D incorrect. The method involves estimating the conditional variance function and assumes that the key OLS assumptions apply to this estimation process.

Question 5

Consider the model $Y _ { i } = \beta _ { 1 } X _ { i } + u _ { i }$, where $u _ { i } = c X _ { i } ^ { 2 } e _ { i }$ and all of the $X$ 's and $e$ 's are i.i.d. and distributed $N ( 0,1 )$. (a)Which of the Extended Least Squares Assumptions are satisfied here? Prove your
assertions.

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The answer of Consider the model \(Y _ { i...

Question 6

Your textbook states that an implication of the Gauss-Markov theorem is that the sample average, $\bar { Y }$, is the most efficient linear estimator of $E \left( Y _ { i } \right)$ when $Y _ { 1 } , \ldots , Y _ { n }$ are i.i.d. with $E \left( Y _ { i } \right) = \mu _ { Y }$ and $\operatorname { var } \left( Y _ { i } \right) = \sigma _ { Y } ^ { 2 }$. This follows from the regression model with no slope and the fact that the OLS estimator is BLUE.
Provide a proof by assuming a linear estimator in the $Y$ 's, $\tilde { \mu } = \sum _ { i = 1 } ^ { n } a _ { i } Y _ { i }$. (a)State the condition under which this estimator is unbiased.

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The answer of Your textbook states that an implication of...

Question 7

What does the Gauss-Markov theorem prove? Without giving mathematical details,
explain how the proof proceeds.What is its importance?

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The answer of What does the Gauss-Markov theorem prove? Without...

Question 8

The advantage of using heteroskedasticity-robust standard errors is that
A)they are easier to compute than the homoskedasticity-only standard errors.
B)they produce asymptotically valid inferences even if you do not know the form of the conditional variance function.
C)it makes the OLS estimator BLUE, even in the presence of heteroskedasticity.
D)they do not unnecessarily complicate matters, since in real-world applications, the functional form of the conditional variance can easily be found.

Accepted Answer

Heteroskedasticity-robust standard errors allow for consistent inference even if the form of the conditional variance is unknown, which is important in real-world applications where the functional form of the conditional variance may not be easily identifiable. Option A is incorrect, as heteroskedasticity-robust standard errors may actually be more computationally intensive than homoskedasticity-only standard errors. Option C is incorrect, as heteroskedasticity-robust standard errors do not necessarily make the OLS estimator BLUE (Best Linear Unbiased Estimator) in the presence of heteroskedasticity. Option D is incorrect, as the functional form of the conditional variance may not be easily found in real-world applications.

Question 9

The WLS estimator is called infeasible WLS estimator when
A)the memory required to compute it on your PC is insufficient.
B)the conditional variance function is not known.
C)the numbers used to compute the estimator get too large.
D)calculating the weights requires you to take a square root of a negative number.

Accepted Answer

The infeasible WLS estimator requires knowledge of the conditional variance function, which may not be known in practice. None of the other options listed are relevant to the infeasible WLS estimator.

Question 10

One of the earlier textbooks in econometrics, first published in 1971, compared
"estimation of a parameter to shooting at a target with a rifle.The bull's-eye can be taken
to represent the true value of the parameter, the rifle the estimator, and each shot a
particular estimate." Use this analogy to discuss small and large sample properties of
estimators.How do you think the author approached the n → ∞ condition? (Dependent
on your view of the world, feel free to substitute guns with bow and arrow, or missile.)

Accepted Answer

The answer of One of the earlier textbooks in econometrics,...

Question 11

Consider the simple regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$ where $X _ { \mathrm { i } } > 0$ for all $i$, and the conditional variance is $\operatorname { var } \left( u _ { i } \mid X _ { i } \right) = \theta X _ { i } ^ { 2 }$ where $\theta$ is a known constant with $\theta > 0$. (a) Write the weighted regression as $\tilde { Y } _ { i } = \beta _ { 0 } \tilde { X } _ { 0 i } + \beta _ { 1 } \tilde { X } _ { 1 i } + \tilde { u } _ { i }$. How would you construct $\tilde { Y } _ { i }$, $\tilde { X } _ { 0 i }$ and $\tilde { X } _ { 1 i } ?$

Accepted Answer

The answer of Consider the simple regression model \(Y _...

Question 12

In practice, the most difficult aspect of feasible WLS estimation is
A)knowing the functional form of the conditional variance.
B)applying the WLS rather than the OLS formula.
C)finding an econometric package that actually calculates WLS.
D)applying WLS when you have a log-log functional form.

Accepted Answer

Feasible WLS estimation involves weighting the observations based on their conditional variances. To do this, one must know the functional form of the conditional variance. This can be challenging as it is often not easily observable or known a priori.

Question 13

(Requires Appendix material)State and prove the Cauchy-Schwarz Inequality.

Accepted Answer

The answer of (Requires Appendix material)State and prove the Cauchy-Schwarz...

Question 14

(Requires Appendix Material)This question requires you to work with Chebychev's
Inequality.
(a)State Chebychev's Inequality.

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The answer of (Requires Appendix Material)This question requires you to...

Question 15

"One should never bother with WLS.Using OLS with robust standard errors gives
correct inference, at least asymptotically." True, false, or a bit of both? Explain carefully
what the quote means and evaluate it critically.

Accepted Answer

The answer of "One should never bother with WLS.Using OLS...

Question 16

"I am an applied econometrician and therefore should not have to deal with econometric
theory.There will be others who I leave that to.I am more interested in interpreting the
estimation results." Evaluate.

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The answer of "I am an applied econometrician and therefore...

Question 17

Discuss the properties of the OLS estimator when the regression errors are
homoskedastic and normally distributed.What can you say about the distribution of the
OLS estimator when these features are absent?

Accepted Answer

The answer of Discuss the properties of the OLS estimator...

Question 18

(Requires Appendix material) If the Gauss-Markov conditions hold, then OLS is BLUE. In addition, assume here that $X$ is nonrandom. Your textbook proves the Gauss-Markov theorem by using the simple regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$ and assuming a linear estimator $\tilde { \beta } _ { 1 } = \sum _ { i = 1 } ^ { n } a _ { i } Y _ { i }$. Substitution of the simple regression model into this expression then results in two conditions for the unbiasedness of the estimator:
$$\sum _ { i = 1 } ^ { n } a _ { i } = 0 \text { and } \sum _ { i = 1 } ^ { n } a _ { i } X _ { i } = 1 .$$
The variance of the estimator is $\operatorname { var } \left( \tilde { \beta } _ { 1 } \mid X _ { 1 } , \ldots , X _ { n } \right) = \sigma _ { u } ^ { 2 } \sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 }$.
Different from your textbook, use the Lagrangian method to minimize the variance subject to the two constraints. Show that the resulting weights correspond to the OLS weights. .

Accepted Answer

The answer of (Requires Appendix material) If the Gauss-Markov conditions...

Question 19

Estimation by WLS
A)although harder than OLS, will always produce a smaller variance.
B)does not mean that you should use homoskedasticity-only standard errors on the transformed equation.
C)requires quite a bit of knowledge about the conditional variance function.
D)makes it very hard to interpret the coefficients, since the data is now weighted and not any longer in its original form.

Accepted Answer

The answer of Estimation by WLS
A)although harder than OLS, will...

Quiz 17: The Theory of Linear Regression With One Regressor

Feasible WLS does not rely on the following condition:

Consider the model $Y _ { i } = \beta _ { 1 } X _ { i } + u _ { i }$ , where $u _ { i } = c X _ { i } ^ { 2 } e _ { i }$ and all of the $X$ 's and $e$ 's are i.i.d. and distributed $N ( 0,1 )$ . (a)Which of the Extended Least Squares Assumptions are satisfied here? Prove your assertions.

What does the Gauss-Markov theorem prove? Without giving mathematical details, explain how the proof proceeds.What is its importance?

The advantage of using heteroskedasticity-robust standard errors is that

The WLS estimator is called infeasible WLS estimator when

In practice, the most difficult aspect of feasible WLS estimation is

(Requires Appendix material)State and prove the Cauchy-Schwarz Inequality.

(Requires Appendix Material)This question requires you to work with Chebychev's Inequality. (a)State Chebychev's Inequality.

"One should never bother with WLS.Using OLS with robust standard errors gives correct inference, at least asymptotically." True, false, or a bit of both? Explain carefully what the quote means and evaluate it critically.

"I am an applied econometrician and therefore should not have to deal with econometric theory.There will be others who I leave that to.I am more interested in interpreting the estimation results." Evaluate.

Discuss the properties of the OLS estimator when the regression errors are homoskedastic and normally distributed.What can you say about the distribution of the OLS estimator when these features are absent?

Estimation by WLS