Question 1

The Gauss-Markov theorem for multiple regression proves that

Accepted Answer

A) MX is an idempotent matrix. 
B) the OLS estimator is BLUE. 
C) the OLS residuals and predicted values are orthogonal. 
D) the variance-covariance matrix of the OLS estimator is $\sigma _ { u } ^ { 2 }$ ( $X ^ { \prime }$ X)-1. 
A) MX is an idempotent matrix. 
B) the OLS estimator is BLUE. 
C) the OLS residuals and predicted values are orthogonal. 
D) the variance-covariance matrix of the OLS estimator is $\sigma _ { u } ^ { 2 }$ ( $X ^ { \prime }$ X)-1.

Question 2

The presence of correlated error terms creates problems for inference based on OLS. These can be overcome by

Accepted Answer

A) using HAC standard errors. 
B) using heteroskedasticity-robust standard errors. 
C) reordering the observations until the correlation disappears. 
D) using homoskedasticity-only standard errors. 
A) using HAC standard errors. 
B) using heteroskedasticity-robust standard errors. 
C) reordering the observations until the correlation disappears. 
D) using homoskedasticity-only standard errors.

Question 3

Write the following four restrictions in the form Rβ = r, where the hypotheses are to be tested simultaneously.
β3 = 2β5,
β1 + β2 = 1,
β4 = 0,
β2 = -β6.
Can you write the following restriction β2 = - $\frac { \beta _ { 3 } } { \beta _ { 1 } }$ in the same format? Why not?

Accepted Answer

@#IMG-DLM&_TB5979_11_TB5979_11_TB5979_11_TB5979_11

Question 4

The GLS assumptions include all of the following, with the exception of

Accepted Answer

A) the Xi are fixed in repeated samples. 
B) Xi and ui have nonzero finite fourth moments. 
C) E(U $U'$ $|X$ )= ?(X), where ?(X)is n × n matrix-valued that can depend on X. 
D) E(U $|X$ )= 0n. 
A) the Xi are fixed in repeated samples. 
B) Xi and ui have nonzero finite fourth moments. 
C) E(U $U'$ $|X$ )= ?(X), where ?(X)is n × n matrix-valued that can depend on X. 
D) E(U $|X$ )= 0n.

Question 5

You have obtained data on test scores and student-teacher ratios in region A and region B of your state. Region B, on average, has lower student-teacher ratios than region A. You decide to run the following regression.
Yi = β0+ β1X1i + β2X2i + β3X3i+ui
where X1 is the class size in region A, X2  is the difference between the class size between region A and B, and X3  is the class size in region B. Your regression package shows a message indicating that it cannot estimate the above equation. What is the problem here and how can it be fixed? Explain the problem in terms of the rank of the X matrix.

Accepted Answer

There is perfect multicollinearity here,

Question 6

The homoskedasticity-only F-statistic is

Accepted Answer

A)  $\frac{(R \hat{\beta}-\mathrm{r})^{\prime}\left[R^{\prime}\left(\mathrm{X}^{\prime} \mathrm{X}\right)^{-1} R\right]^{-1}(R \hat{\beta}-r) / q}{\mathrm{~s}_{\mathrm{u}}^{2}}$ 
B)  $\frac{(R \hat{\beta}-\mathrm{r})^{\prime}\left[R^{\prime}\left(X^{\prime} X\right)^{-1} R\right]^{-1}(R \hat{\beta}-r)}{\mathrm{s}_{\mathrm{u}}^{2}}$ 
C)  $\frac{(R \hat{\beta}-\mathrm{r})^{\prime}\left[R^{\prime} \hat{\Sigma}_{\beta} R\right]^{-1}(R \hat{\beta}-r)}{\mathrm{q}}$ 
D)  $\frac{\hat{\mathrm{U}} \mathrm{P} Z \hat{\mathrm{U}}}{\hat{\mathrm{U}^{\prime} \mathrm{M}_{\mathrm{Z}} \mathrm{U}}}$ 
A)  $\frac{(R \hat{\beta}-\mathrm{r})^{\prime}\left[R^{\prime}\left(\mathrm{X}^{\prime} \mathrm{X}\right)^{-1} R\right]^{-1}(R \hat{\beta}-r) / q}{\mathrm{~s}_{\mathrm{u}}^{2}}$ 
B)  $\frac{(R \hat{\beta}-\mathrm{r})^{\prime}\left[R^{\prime}\left(X^{\prime} X\right)^{-1} R\right]^{-1}(R \hat{\beta}-r)}{\mathrm{s}_{\mathrm{u}}^{2}}$ 
C)  $\frac{(R \hat{\beta}-\mathrm{r})^{\prime}\left[R^{\prime} \hat{\Sigma}_{\beta} R\right]^{-1}(R \hat{\beta}-r)}{\mathrm{q}}$ 
D)  $\frac{\hat{\mathrm{U}} \mathrm{P} Z \hat{\mathrm{U}}}{\hat{\mathrm{U}^{\prime} \mathrm{M}_{\mathrm{Z}} \mathrm{U}}}$

Question 7

Consider the following population regression function: Y = X? + U
where Y= $\left[ \begin{array} { l } 
Y _ { 1 } \
Y _ { 2 } \
\ldots \
Y _ { n }
\end{array} \right]$ , X= $\left[ \begin{array} { l l } 
1 & X _ { 1 } \
1 & X _ { 2 } \
\ldots & \ldots \
1 & X _ { n }
\end{array} \right]$ , ? = $\left[ \begin{array} { l } 
\beta _ { 0 } \
\beta _ { 1 }
\end{array} \right]$ , U= $\left[ \begin{array} { c } 
u_1 \
u_2 \
\ldots \
u_\text{n}
\end{array} \right]$ Given the following information on population growth rates (Y)and education (X)for 86 countries $\sum _ { i = 1 } ^ { n } Y _ { i } = 1.594$ , $\sum _ { i = 1 } ^ { n } x _ { i } = 449.6$ , $\sum _ { i = 1 } ^ { n } Y _ { i } ^ { 2 } = 0.03982$ , $\sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } = 3,022.76$ , $\sum _ { i = 1 } ^ { n } x _ { i } Y _ { i } = 6.4697$ a)find X'X, X'Y, (X'X)-1 and finally (X'X)-1 X'Y.
b)Interpret the slope, and if necessary, the intercept.

Accepted Answer

a)X'X =
b)c) @#IMG-DLM&_TB5979_11_TB5979_11_TB5979

Question 8

Define the GLS estimator and discuss its properties when Ω is known. Why is this estimator sometimes called infeasible GLS? What happens when Ω is unknown? What would the Ω matrix look like for the case of independent sampling with heteroskedastic errors, where var(ui | Xi)= ch(Xi)= σ2
  $x _ { 1 i } ^ { 2 }$ ? Since the inverse of the error variance-covariance matrix is needed to compute the GLS estimator, find Ω-1. The textbook shows that the original model Y = Xβ + U will be transformed into $\widetilde { Y } = \widetilde { X } \beta + \widetilde { U } , \text { where } \widetilde { Y } = F Y , \widetilde { X } = F X \text {, and } \widetilde { U }$ = FU, and $F$ F = Ω-1. Find F in the above case, and describe what effect the transformation has on the original data.

Accepted Answer

@#IMG-DLM&_TB5979_11_TB5979_11_TB5979_11_TB5979_11

Question 9

Let PX = X( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ and MX = In - PX. Then MX MX =

Accepted Answer

A) X( $X ^ { \prime }$ X)-1   $X ^ { \prime }$  - PX. 
B)  $M _ { X } ^ { 2 }$ 
C) In. 
D) MX. 
A) X( $X ^ { \prime }$ X)-1   $X ^ { \prime }$  - PX. 
B)  $M _ { X } ^ { 2 }$ 
C) In. 
D) MX.

Question 10

The GLS estimator is defined as

Accepted Answer

A) ( $X ^ { \prime }$ ?-1X)-1 ( $X ^ { \prime }$ ?-1Y). 
B) ( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ Y. 
C)  $A ^ { \prime }$ Y. 
D) ( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ U. 
A) ( $X ^ { \prime }$ ?-1X)-1 ( $X ^ { \prime }$ ?-1Y). 
B) ( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ Y. 
C)  $A ^ { \prime }$ Y. 
D) ( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ U.

The Gauss-Markov theorem for multiple regression proves that

The presence of correlated error terms creates problems for inference based on OLS. These can be overcome by

Write the following four restrictions in the form Rβ = r, where the hypotheses are to be tested simultaneously. β₃ = 2β₅, β₁ + β₂ = 1, β₄ = 0, β₂ = -β₆. Can you write the following restriction β₂ = - $\frac { \beta _ { 3 } } { \beta _ { 1 } }$ in the same format? Why not?

The GLS assumptions include all of the following, with the exception of

The homoskedasticity-only F-statistic is

Let P_X = X( $X ^ { \prime }$ X)^-¹ $X ^ { \prime }$ and M_X = I_n - P_X. Then M_XM_X =

The GLS estimator is defined as

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Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

Linear Regression With Multiple Regressors

Hypothesis Tests and Confidence Intervals in Multiple Regression

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Experiments and Quasi-Experiments

Introduction to Time Series Regression and Forecasting

Estimation of Dynamic Causal Effects

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The Theory of Linear Regression With One Regressor

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Exam 18: The Theory of Multiple Regression

The Gauss-Markov theorem for multiple regression proves that

The presence of correlated error terms creates problems for inference based on OLS. These can be overcome by

The GLS assumptions include all of the following, with the exception of

The homoskedasticity-only F-statistic is

Let PX = X( X′X ^ { \prime }X′ X)-1 X′X ^ { \prime }X′ and MX = In - PX. Then MX MX =

The GLS estimator is defined as

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 Linear Regression With One Regressor

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Let P_X = X( $X ^ { \prime }$ X)^-¹ $X ^ { \prime }$ and M_X = I_n - P_X. Then M_XM_X =