Question 1

Assume that the variance depends on a third variable,Wi,which does not appear in the regression function,and that var(ui|Xi,Wi)= θ0+θ1  .One way to estimate θ0 and θ1consistently is to regress
A)  i on
 using OLS
B)  i on
 using OLS
C)  on
 using OLS
D)  on
 using OLS

Accepted Answer

The answer of Assume that the variance depends on a...

Question 2

If the functional form of the conditional variance function is incorrect,then
A)the standard errors computed by WLS regression routines are invalid
B)the OLS estimator is biased
C)instrumental variable techniques have to be used
D)the regression R2 can no longer be computed

Accepted Answer

If the functional form of the conditional variance is incorrect, the weights used in Weighted Least Squares (WLS) regression will be incorrect, leading to invalid standard errors.

Question 3

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 4

(Requires Appendix material)If X and Y are jointly normally distributed and are uncorrelated,
A)then their product is chi-square distributed with n-2 degrees of freedom
B)then they are independently distributed
C)then their ratio is t-distributed
D)none of the above is true

Accepted Answer

If X and Y are jointly normally distributed and uncorrelated, then their covariance is zero. From the properties of jointly normal distribution, this implies that X and Y are independent. Therefore, their product is also independent, and it is not chi-square distributed with any degrees of freedom. Similarly, their ratio is also independent, and it is not t-distributed.

Question 5

Homoskedasticity means that
A)var(ui|Xi)= 
B)var(Xi)= 
C)var(ui|Xi)= 
D)var(  i|Xi)=

Accepted Answer

The answer of Homoskedasticity means that
A)var(ui|Xi)= 
B)var(Xi)= 
C)var(ui|Xi)= 
D)var( ...

Question 6

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,first...

Question 7

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

Knowing the functional form of the conditional variance is the most difficult aspect of feasible WLS estimation because it requires prior knowledge or assumptions about the variance structure, which may not be straightforward or easy to identify in practice. This can involve making educated guesses, running diagnostics, or conducting additional tests to determine the appropriate variance specification, which can be time-consuming and require a lot of data. Conversely, applying WLS rather than OLS or finding an econometric package that calculates WLS is relatively straightforward and can be achieved with minimal effort or expertise. Applying WLS with a log-log functional form may also present some challenges, but this is typically less difficult than determining the functional form of the conditional variance.

Question 8

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

Accepted Answer

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

Question 9

If the variance of u is quadratic in X,then it can be expressed as
A)var(ui|Xi)= 
B)var(ui|Xi)= θ0 + θ1 
C)var(ui|Xi)= θ0 + θ1 
D)var(ui|Xi)=

Accepted Answer

The answer of If the variance of u is quadratic...

Question 10

In practice,you may want to use the OLS estimator instead of the WLS because
A)heteroskedasticity is seldom a realistic problem
B)OLS is easier to calculate
C)heteroskedasticity robust standard errors can be calculated
D)the functional form of the conditional variance function is rarely known

Accepted Answer

The answer of In practice,you may want to use the...

Question 11

"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.

Accepted Answer

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

Question 12

Suppose that the conditional variance is var(ui|Xi)= λh(Xi)where λ is a constant and h is a known function.The WLS estimator is
A)the same as the OLS estimator since the function is known
B)can only be calculated if you have at least 100 observations
C)the estimator obtained by first dividing the dependent variable and regressor by the square root of h and then regressing this modified dependent variable on the modified regressor using OLS
D)the estimator obtained by first dividing the dependent variable and regressor by h and then regressing this modified dependent variable on the modified regressor using OLS

Accepted Answer

The WLS (weighted least squares) estimator in this scenario is obtained by weighting the observations based on the inverse of the conditional variance, which in this case is 1/h(Xi). Therefore, we can modify the dependent variable and regressor by dividing them by the square root of h(Xi), yielding the transformed model: y* = y / sqrt(h(Xi)) and X* = X / sqrt(h(Xi)). We can then regress y* on X* using OLS to obtain the WLS estimator.

Question 13

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

Using heteroskedasticity-robust standard errors produces asymptotically valid inferences even if you do not know the form of the conditional variance function. This is important because in real-world applications, the functional form of the conditional variance can be difficult or impossible to find. Option A is incorrect because heteroskedasticity-robust standard errors are not necessarily easier to compute than homoskedasticity-only standard errors. Option C is incorrect because while it is true that heteroskedasticity-robust standard errors can help reduce bias in the OLS estimator, they do not necessarily make it BLUE (best linear unbiased estimator). Option D is incorrect because the functional form of the conditional variance can be difficult to find in many real-world applications.

Question 14

In order to use the t-statistic for hypothesis testing and constructing a 95% confidence interval as  1.96 standard errors,the following three assumptions have to hold:
A)the conditional mean of ui,given Xi is zero; (Xi,Yi),i = 1,2,…,n are i.i.d.draws from their joint distribution;Xi and ui have four moments
B)the conditional mean of ui,given Xi is zero; (Xi,Yi),i = 1,2,…,n are i.i.d.draws from their joint distribution;homoskedasticity
C)the conditional mean of ui,given Xi is zero; (Xi,Yi),i = 1,2,…,n are i.i.d.draws from their joint distribution;the conditional distribution of ui given Xi is normal
D)none of the above

Accepted Answer

The answer of In order to use the t-statistic for...

Question 15

Consider the model Yi = β1Xi + ui,where ui = c  ei 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.
(b)Would an OLS estimator of β1 be efficient here?
(c)How would you estimate β1 by WLS?

Accepted Answer

The answer of Consider the model Yi = β1Xi +...

Question 16

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 17

"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 18

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 WLS estimator is called infeasible when the conditional variance function (also known as the weight function) is not known. This makes it impossible to calculate the optimal weights for the regression analysis.

Question 19

The large-sample distribution of  1 is
A)  (
 1-β1)
 N(0
 where νi= (Xi-μx)ui
B)  (
 1-β1)
 N(0
 where νi= ui
C)  (
 1-β1)
 N(0
 where νi= Xiui
D)  (
 1-β1)
 N(0

Accepted Answer

The answer of The large-sample distribution of  1 is
A)...

Question 20

Assume that var(ui|Xi)= θ0+θ1  .One way to estimate θ0 and θ1 consistently is to regress
A)  i on
 using OLS
B)  on
 using OLS
C)  on
 iusing OLS
D)  on
 using OLS but suppressing the constant ("restricted least squares")

Accepted Answer

The answer of Assume that var(ui|Xi)= θ0+θ1  .One way...

Quiz 17: The Theory of Linear Regression With One Regressor

Assume that the variance depends on a third variable,Wi,which does not appear in the regression function,and that var(ui|Xi,Wi) = θ0+θ1 .One way to estimate θ0 and θ1consistently is to regress

If the functional form of the conditional variance function is incorrect,then

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?

(Requires Appendix material) If X and Y are jointly normally distributed and are uncorrelated,

Homoskedasticity means that

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

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

If the variance of u is quadratic in X,then it can be expressed as

In practice,you may want to use the OLS estimator instead of the WLS because

"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.

Suppose that the conditional variance is var(ui|Xi) = λh(Xi) where λ is a constant and h is a known function.The WLS estimator is

The advantage of using heteroskedasticity-robust standard errors is that

In order to use the t-statistic for hypothesis testing and constructing a 95% confidence interval as 1.96 standard errors,the following three assumptions have to hold:

Consider the model Yi = β1Xi + ui,where ui = c ei 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. (b)Would an OLS estimator of β1 be efficient here? (c)How would you estimate β1 by WLS?

Feasible WLS does not rely on the following condition:

"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.

The WLS estimator is called infeasible WLS estimator when

The large-sample distribution of 1 is

Assume that var(ui|Xi) = θ0+θ1 .One way to estimate θ0 and θ1 consistently is to regress