Question 1

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 produce valid inferences even if the form of the conditional variance function is unknown, which is different from homoskedasticity-only standard errors. This is a significant advantage in applied econometrics because the conditional variance function is often unknown. Option A is false since computing heteroskedastic-robust standard errors is more complicated than computing homoskedasticity-only standard errors. Option C is incorrect since heteroskedastic-robust standard errors do not make the OLS estimator BLUE, but rather more efficient. Option D is false because in real-world applications, the functional form of the conditional variance is often unknown, and finding it can be challenging.

Question 2

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...

Question 3

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 becomes infeasible when the conditional variance function is unknown, as it is not possible to calculate the weights needed for the estimator.

Question 4

Assume that var(u_i|X_i)= ?₀+?₁ $x _ { i } ^ { 2 }$ One way to estimate ?₀ and ?₁ consistently is to regress A) $\hat {\mathrm { u }}$ _i on $x _ { i } ^ { 2 }$ using OLS B) $\hat { u } _ { i } ^ { 2 }$ on $x _ { i } ^ { 2 }$ using OLS C) $\hat { u } _ { i } ^ { 2 }$ on $\sqrt { x }$ _iusing OLS D) $\hat { u } _ { i } ^ { 2 }$ on $x _ { i } ^ { 2 }$ using OLS but suppressing the constant ("restricted least squares")

Accepted Answer

To estimate the variances $\sigma^2_{0}$ and $\sigma^2_{1}$ consistently, one should regress the squared residuals ($\hat{u}_i^2$) on the squared independent variable ($x_i^2$) using Ordinary Least Squares (OLS). This approach directly relates the variance of the residuals to the squared value of the independent variable, which is the essence of heteroskedasticity modeling.

Question 5

If the variance of u is quadratic in X, then it can be expressed as A)var(u_i|X_i)= $\theta _ { 0 } ^ { 2 }$ B)var(u_i|X_i)= ?₀+ ?₁ $x _ { i } ^ { 1 / 2 }$ C)var(u_i|X_i)= ?₀+ ?₁ $x _ { i } ^ { 2 }$ D)var(u_i|X_i)= $\sigma _ { \mathrm { u } } ^ { 2 }$

Accepted Answer

The correct choice is C because it expresses the variance of $u$ as a quadratic function of $X$, indicated by the term $x_i^2$, which aligns with the definition of variance being quadratic in $X$. Choices A and D suggest constant variances, and B suggests a variance that increases with the square root of $X$, not quadratically.

Question 6

Assume that the variance depends on a third variable, Wi, which does not appear in the regression function, and that var(u_i|X_i,W_i)= ?₀+?₁ $\frac { 1 } { W _ { i } }$ One way to estimate ?₀ and ?₁consistently is to regress A) $\hat {\mathrm { u }}$ _i on $w _ { i } ^ { 2 }$ using OLS B) $\hat {\mathrm { u }}$ _i on $\frac { 1 } { W _ { i } }$ using OLS C) $\hat { u } _ { i } ^ { 2 }$ on $\frac { X _ { i } } { W _ { i } }$ using OLS D) $\hat { u } _ { i } ^ { 2 }$ on $\frac { 1 } { W _ { i } }$ using OLS

Accepted Answer

Given the variance depends on $Wi$ as $var(u|X,W) = \alpha_0 + \alpha_1 \frac{1}{W_i}$, to estimate $\alpha_0$ and $\alpha_1$ consistently, one should regress the squared residuals ($\hat{u}_i^2$) on the inverse of $W_i$ ($\frac{1}{W_i}$) using Ordinary Least Squares (OLS). This approach directly addresses the heteroscedasticity issue by modeling the variance as a function of $W_i$.

Question 7

(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 are uncorrelated, it does not necessarily mean that their product is chi-square distributed or that their ratio is t-distributed. However, it does imply that they are independent, since the covariance between X and Y is 0. Therefore, choice B is the correct answer.

Question 8

The large-sample distribution of $\hat { \beta }$ ₁is A) $\sqrt { n }$ ( $\hat { \beta }$ ₁-?₁) $\begin{array} { l } \mathrm { d } \ \rightarrow \end{array}$ N(0 $\frac { \operatorname { var } \left( v _ { i } \right) } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } }$ where ?_i= (X_i-?_x)u_i B) $\sqrt { n }$ ( $\hat { \beta }$ ₁-?₁) $\begin{array} { l } \mathrm { d } \ \rightarrow \end{array}$ N(0 $\frac { \operatorname { var } \left( v _ { i } \right) } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } }$ where ?_i= u_i C) $\sqrt { n }$ ( $\hat { \beta }$ ₁-?₁) $\begin{array} { l } \mathrm { d } \ \rightarrow \end{array}$ N(0 $\frac { \operatorname { var } \left( v _ { i } \right) } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } }$ where ?_i= X_iu_i D) $\sqrt { n }$ ( $\hat { \beta }$ ₁-?₁) $\begin{array} { l } \mathrm { d } \ \rightarrow \end{array}$ N(0 $\frac { \sigma _ { u } ^ { 2 } } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } }$

Accepted Answer

The large-sample distribution of $\hat { \beta }$ ₁is $\sqrt { n }$ ( $\hat { \beta }$ ₁-?₁) $\begin{array} { l } \mathrm { d } \\rightarrow\end{array}$ N(0 $\frac { \operatorname { var } \left( v _ { i } \right) } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } }$ where ?_i= (X_i-?_x)u_i because the error term is assumed to be independent of the explanatory variable.

Question 9

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 10

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 function is incorrect, then the standard errors computed by WLS regression routines are invalid. This is because WLS relies on accurate estimates of the variance function to appropriately weight the observations in the regression. If the variance function is misspecified, then the weighting will be incorrect, leading to incorrect standard errors.

Question 11

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 12

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

Accepted Answer

The first assumption is the conditional mean of u_i, given X_i is zero. This is the assumption of linearity. The second assumption is that (X_i,Y_i), i = 1,2, …, n are i.i.d. draws from their joint distribution. This is the assumption of independence. The third assumption is that X_i and u_i have four moments. This is the assumption of finite variance.

Question 13

"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...

Question 14

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

In order to implement feasible WLS, we need to know the conditional variance function. This can be difficult, particularly if the true functional form is unknown and needs to be estimated. Choosing the wrong functional form can also lead to biased estimates.

Question 15

Homoskedasticity means that A)var(u_i|X_i)= $\sigma _ { \mathrm { ui } } ^ { 2 }$ B)var(X_i)= $\sigma _ { \mathrm { u } } ^ { 2 }$ C)var(u_i|X_i)= $\sigma _ { \mathrm { u } } ^ { 2 }$ D)var( $\hat {\mathrm { u }}$ _i|X_i)= $\sigma _ { \mathrm { ui } } ^ { 2 }$

Accepted Answer

Homoskedasticity refers to the assumption that the variance of the error terms in a regression model is constant across all levels of the independent variable. Choice C correctly defines homoskedasticity as the variance of the error term $u$ given the independent variables $X$ being constant ($\sigma _ { \mathrm { u } } ^ { 2 }$).

Question 16

Consider the model Y_i = β₁X_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. (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 Y_i = β₁X_i +...

Question 17

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 18

Suppose that the conditional variance is var(u_i|X_i)= λh(X_i)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 estimator in this case is 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. This is because the variance of the error term is proportional to h, and hence, by dividing by the square root of h, we can obtain a modified regression where the errors have constant variance. The choice A is incorrect as the fact that the function is known does not imply that the estimator will be the same as OLS. The choice B is incorrect as the number of observations is not related to the calculation of the estimator. The choice D is incorrect as dividing by h would increase the variance of the errors.

Question 19

"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 20

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...

Quiz 17: The Theory of Linear Regression With One Regressor

The advantage of using heteroskedasticity-robust standard errors is that

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

The WLS estimator is called infeasible WLS estimator when

Assume that var(u_i|X_i) = ?₀+?₁ $x _ { i } ^ { 2 }$ One way to estimate ?₀ and ?₁ consistently is to regress

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

Assume that the variance depends on a third variable, Wi, which does not appear in the regression function, and that var(u_i|X_i,W_i) = ?₀+?₁ $\frac { 1 } { W _ { i } }$ One way to estimate ?₀ and ?₁consistently is to regress

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

The large-sample distribution of $\hat { \beta }$ ₁is

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

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

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

"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.

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

Homoskedasticity means that

Feasible WLS does not rely on the following condition:

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

"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?

Quiz 17: The Theory of Linear Regression With One Regressor

The advantage of using heteroskedasticity-robust standard errors is that

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

The WLS estimator is called infeasible WLS estimator when

Assume that var(ui|Xi) = ?0+?1 xi2x _ { i } ^ { 2 }xi2​ One way to estimate ?0 and ?1 consistently is to regress

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

Assume that the variance depends on a third variable, Wi, which does not appear in the regression function, and that var(ui|Xi,Wi) = ?0+?1 1Wi\frac { 1 } { W _ { i } }Wi​1​ One way to estimate ?0 and ?1consistently is to regress

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

The large-sample distribution of β^\hat { \beta }β^​ 1 is

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

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

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

"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.

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

Homoskedasticity means that

Feasible WLS does not rely on the following condition:

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

"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?

Assume that var(u_i|X_i) = ?₀+?₁ $x _ { i } ^ { 2 }$ One way to estimate ?₀ and ?₁ consistently is to regress

Assume that the variance depends on a third variable, Wi, which does not appear in the regression function, and that var(u_i|X_i,W_i) = ?₀+?₁ $\frac { 1 } { W _ { i } }$ One way to estimate ?₀ and ?₁consistently is to regress

The large-sample distribution of $\hat { \beta }$ ₁is

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

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