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

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= (Xi-?x)ui 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= ui 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= Xiui D) $\sqrt { n }$ ( $\hat { \beta }$ 1-?1) $\begin{array} { l } \mathrm { d } \\ \rightarrow \end{array}$ N(0 $\frac { \sigma _ { u } ^ { 2 } } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } }$ 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= (Xi-?x)ui 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= ui 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= Xiui D) $\sqrt { n }$ ( $\hat { \beta }$ 1-?1) $\begin{array} { l } \mathrm { d } \\ \rightarrow \end{array}$ N(0 $\frac { \sigma _ { u } ^ { 2 } } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } }$ A

Assume that var(ui|Xi)= ?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") 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")

Exam 17: The Theory of Linear Regression With One Regressor

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(Y_i)when Y₁,..., Y_n are i.i.d. with E(Y_i)= μ_Y and var(Y_i)= $\sigma \stackrel { 2 } { Y }$ 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. (b)Derive the variance of this estimator. (c)Minimize this variance subject to the constraint (condition)derived in (a)and show that the sample mean is BLUE.

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Question 1

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(a)E( $\tilde { \mu }$ )= $E \left( \sum _ { i = 1 } ^ { n } a _ { i } Y _ { i } \right) = \sum _ { i = 1 } ^ { n } a _ { i } E \left( Y _ { i } \right)$ = μ_Y
$\sum _ { i = 1 } ^ { n } a _ { i }$ Hence for this to be an unbiased estimator, the following condition must hold: $\sum _ { i = 1 } ^ { n } a _ { i } = 1$ (b)var( $\tilde { \mu }$ )= E( $\tilde { \mu }$ - E( $\tilde { \mu }$ ))² = $E \left( \sum _ { i = 1 } ^ { n } a _ { i } Y _ { i } - \mu _ { y } \right) ^ { 2 } = \left( \sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 } E \left( Y _ { i } - \mu _ { y } \right) ^ { 2 } \right.$ = $\sigma \stackrel { 2 } { Y }$ $\sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 }$ (c)Define the Lagrangian L = $\sigma \stackrel { 2 } { Y }$ $\sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 } - \lambda \left( \sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 } - 1 \right)$ , where λ is the Lagrange multiplier. To obtain the first order conditions, minimize L with respect to the n weights and the Lagrange multiplier, and solve the resulting (n+1)equations in the (n+1)unknowns. $\frac { \partial } { \partial a _ { i } }$ L = 0 = 2 $\sigma \stackrel { 2 } { Y }$ a_i - λ; i = 1,..., n $\frac { \partial } { \partial \lambda }$ L = 0 = $\sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 } - 1$ Summing the first equation 2 $\sigma \stackrel { 2 } { Y }$ $\sum _ { i = 1 } ^ { n } a _ { i }$ = nλ and bringing in the second equation subsequently, results in λ = $\frac { 2 \sigma ^ { 2 }_Y } { n }$ Substituting this result into the first equation then gives
2 $\sigma \stackrel { 2 } { Y }$ a_i= $\frac { 2 \sigma ^ { 2 }_Y } { n }$ ; i = 1,..., or a_i = $\frac { 1 } { n }$ ; i = 1,..., n. Since these are also the OLS weights, then OLS is BLUE.

If the errors are heteroskedastic, then

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Question 2

Correct Answer:

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D

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

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If the errors are heteroskedastic, then

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

"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 advantage of using heteroskedasticity-robust standard errors is that

The extended least squares assumptions are of interest, because

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

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

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

Feasible WLS does not rely on the following condition:

The following is not one of the Gauss-Markov conditions:

The class of linear conditionally unbiased estimators consists of

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

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

Asymptotic distribution theory is

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

"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.

Economic Questions and Data

Review of Probability

Review of Statistics

Linear Regression With One Regressor

Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

Linear Regression With Multiple Regressors

Hypothesis Tests and Confidence Intervals in Multiple Regression

Nonlinear Regression Functions

Assessing Studies Based on Multiple Regression

Regression With Panel Data

Regression With a Binary Dependent Variable

Instrumental Variables Regression

Experiments and Quasi-Experiments

Introduction to Time Series Regression and Forecasting

Estimation of Dynamic Causal Effects

Additional Topics in Time Series Regression

The Theory of Multiple Regression

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Exam 17: The Theory of Linear Regression With One Regressor

If the errors are heteroskedastic, then

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

"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 advantage of using heteroskedasticity-robust standard errors is that

The extended least squares assumptions are of interest, because

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

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

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

Feasible WLS does not rely on the following condition:

The following is not one of the Gauss-Markov conditions:

The class of linear conditionally unbiased estimators consists of

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

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

Asymptotic distribution theory is

It is possible for an estimator of μγ{ } ^ { \mu } \gammaμγ to be inconsistent while

"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.

Economic Questions and Data

Review of Probability

Review of Statistics

Linear Regression With One Regressor

Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

Linear Regression With Multiple Regressors

Hypothesis Tests and Confidence Intervals in Multiple Regression

Nonlinear Regression Functions

Assessing Studies Based on Multiple Regression

Regression With Panel Data

Regression With a Binary Dependent Variable

Instrumental Variables Regression

Experiments and Quasi-Experiments

Introduction to Time Series Regression and Forecasting

Estimation of Dynamic Causal Effects

Additional Topics in Time Series Regression

The Theory of Multiple Regression

Filters

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

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

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