Deck 17: The Theory of Linear Regression With One Regressor

Question 1

It is possible for an estimator of $\mu _ { Y }$ to be inconsistent while

A) converging in probability to $\mu _ { Y }$
B) $S _ { n } \stackrel { p } { \rightarrow } \mu _ { Y }$
C) unbiased.
D) $\operatorname { Pr } \left[ \left| S _ { n } - \mu _ { Y } \right| \geq \delta \right] \rightarrow 0$

Answer 1

unbiased.

Question 2

When the errors are heteroskedastic, then

A)WLS is efficient in large samples, if the functional form of the heteroskedasticity is known.
B)OLS is biased.
C)OLS is still efficient as along as there is no serial correlation in the error terms.
D)weighted least squares is efficient.

Answer 2

A

Question 3

If the errors are heteroskedastic, then

A)the OLS estimator is still BLUE as long as the regressors are nonrandom.
B)the usual formula cannot be used for the OLS estimator.
C)your model becomes overidentified.
D)the OLS estimator is not BLUE.

Answer 3

D

Question 4

The Gauss-Markov Theorem proves that

Question 5

Asymptotic distribution theory is

A)not practically relevant, because we never have an infinite number of observations.
B)only of theoretical interest.
C)of interest because it tells you what the distribution approximately looks like in small samples.
D)the distribution of statistics when the sample size is very large.

Question 6

Besides the Central Limit Theorem, the other cornerstone of asymptotic distribution theory is the

A)normal distribution.
B)OLS estimator.
C)Law of Large Numbers.
D)Slutsky's theorem.

Question 7

$E \left( \frac { 1 } { n - 2 } \sum _ { i = 1 } ^ { n } \hat { u } _ { i } ^ { 2 } \right)$

A) is the expected value of the homoskedasticity only standard errors.
B) $= \sigma _ { u } ^ { 2 }$
C) exists only asymptotically.
D) $= \sigma _ { u } ^ { 2 } / ( \mathrm { n } - 2 )$

Question 8

If, in addition to the least squares assumptions made in the previous chapter on the simple regression model, the errors are homoskedastic, then the OLS estimator is

A)identical to the TSLS estimator.
B)BLUE.
C)inconsistent.
D)different from the OLS estimator in the presence of heteroskedasticity.

Question 9

The extended least squares assumptions are of interest, because

A)they will often hold in practice.
B)if they hold, then OLS is consistent.
C)they allow you to study additional theoretical properties of OLS.
D)if they hold, we can no longer calculate confidence intervals.

Question 10

Finite-sample distributions of the OLS estimator and t-statistics are complicated, unless

Question 11

Question 12

.

Question 13

The link between the variance of $\bar { Y }$ and the probability that $\bar { Y }$ is within $\pm \delta \text { of } \mu _ { Y }$ is provided by

A) Slutsky's theorem.
B) the Central Limit Theorem.
C) the Law of Large Numbers.
D) Chebychev's inequality.

Question 14

You need to adjust $S _ { \hat { u } } ^ { 2 }$ by the degrees of freedom to ensure that $s _ { \hat { u } } ^ { 2 }$ is

A) an unbiased estimator of $\sigma _ { u } ^ { 2 }$
B) a consistent estimator of $\sigma _ { u } ^ { 2 }$
C) efficient in small samples.
D) F -distributed.

Question 15

An implication of $\sqrt { n } \left( \hat { \beta } _ { 1 } - \beta _ { 1 } \right) \stackrel { d } { \rightarrow } N \left( 0 , \frac { \operatorname { var } \left( v _ { i } \right) } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } } \right)$ is that

A) $\hat { \beta } _ { 1 } \text { is unbiased. }$
B) $\hat { \beta } _ { 1 } \text { is consistent }$
C) OLS is BLUE.
D) there is heteroskedasticity in the errors.

Question 16

All of the following are good reasons for an applied econometrician to learn some econometric theory, with the exception of

Question 17

Under the five extended least squares assumptions, the homoskedasticity-only t- distribution in this chapter

Question 18

The class of linear conditionally unbiased estimators consists of

Question 19

Slutsky's theorem combines the Law of Large Numbers

A)with continuous functions.
B)and the normal distribution.
C)and the Central Limit Theorem.
D)with conditions for the unbiasedness of an estimator.

Question 20

The following is not part of the extended least squares assumptions for regression with a single regressor:

Question 21

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)

Question 22

(a)

Question 23

Question 24

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.

Question 25

(a)Which of the Extended Least Squares Assumptions are satisfied here? Prove your
assertions.

Question 26

(a)State the condition under which this estimator is unbiased.

Question 27

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

Question 28

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.

Question 29

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.

Question 30

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.)

Question 31

(a)

Question 32

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.

Question 33

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

Question 34

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

Question 35

"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.

Question 36

"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.

Question 37

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?

Question 38

.

Question 39

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.