Question 1

The formulation Rβ= r to test a hypotheses

Accepted Answer

A) allows for restrictions involving both multiple regression coefficients and single regression coefficients. 
B) is F-distributed in large samples. 
C) allows only for restrictions involving multiple regression coefficients. 
D) allows for testing linear as well as nonlinear hypotheses. 
A) allows for restrictions involving both multiple regression coefficients and single regression coefficients. 
B) is F-distributed in large samples. 
C) allows only for restrictions involving multiple regression coefficients. 
D) allows for testing linear as well as nonlinear hypotheses.

Question 2

The multiple regression model in matrix form Y = X? + U can also be written as

Accepted Answer

A) Yi = ?0 + X $\begin{array} { l } 
' \
i
\end{array}$ ? + ui, i = 1,…, n. 
B) Yi = X $\begin{array} { l } 
' \
i
\end{array}$ ?i, i = 1,…, n. 
C) Yi = ?X $\begin{array} { l } 
' \
i
\end{array}$ + ui, i = 1,…, n. 
D) Yi = X $\begin{array} { l } 
' \
i
\end{array}$ ? + ui, i = 1,…, n. 
A) Yi = ?0 + X $\begin{array} { l } 
' \
i
\end{array}$ ? + ui, i = 1,…, n. 
B) Yi = X $\begin{array} { l } 
' \
i
\end{array}$ ?i, i = 1,…, n. 
C) Yi = ?X $\begin{array} { l } 
' \
i
\end{array}$ + ui, i = 1,…, n. 
D) Yi = X $\begin{array} { l } 
' \
i
\end{array}$ ? + ui, i = 1,…, n.

Question 3

One implication of the extended least squares assumptions in the multiple regression model is that

Accepted Answer

A) feasible GLS should be used for estimation. 
B) E(U|X)= In. 
C)  $X ^ { \prime }$ X is singular. 
D) the conditional distribution of U given X is N(0n, In). 
A) feasible GLS should be used for estimation. 
B) E(U|X)= In. 
C)  $X ^ { \prime }$ X is singular. 
D) the conditional distribution of U given X is N(0n, In).

Question 4

An estimator of β is said to be linear if

Accepted Answer

A) it can be estimated by least squares. 
B) it is a linear function of Y1,…, Yn . 
C) there are homoskedasticity-only errors. 
D) it is a linear function of X1,…, Xn . 
A) it can be estimated by least squares. 
B) it is a linear function of Y1,…, Yn . 
C) there are homoskedasticity-only errors. 
D) it is a linear function of X1,…, Xn .

Question 5

One of the properties of the OLS estimator is

Accepted Answer

A) X $\hat \beta$ = 0k+1. 
B) that the coefficient vector $\hat \beta$ has full rank. 
C)  $X ^ { \prime }$ (Y - X $\hat \beta$ )= 0k+1. 
D) ( $X ^ { \prime }$ X)-1= $X ^ { \prime }$ Y 
A) X $\hat \beta$ = 0k+1. 
B) that the coefficient vector $\hat \beta$ has full rank. 
C)  $X ^ { \prime }$ (Y - X $\hat \beta$ )= 0k+1. 
D) ( $X ^ { \prime }$ X)-1= $X ^ { \prime }$ Y

Question 6

The leading example of sampling schemes in econometrics that do not result in independent observations is

Accepted Answer

A) cross-sectional data. 
B) experimental data. 
C) the Current Population Survey. 
D) when the data are sampled over time for the same entity. 
A) cross-sectional data. 
B) experimental data. 
C) the Current Population Survey. 
D) when the data are sampled over time for the same entity.

Question 7

Let Y = $\left( \begin{array} { l } 
- 2 \
3 \
10 \
2 \
2
\end{array} \right)$ and X = $\left( \begin{array} { l l } 
1 & 0 \
1 & 1 \
1 & 3 \
1 & - 1 \
1 & 2
\end{array} \right)$ Find $X ^ { \prime }$ X, $X ^ { \prime }$ Y, ( $X ^ { \prime }$ X)-1 and finally ( $X ^ { \prime }$ X)-1
  $X ^ { \prime }$ Y.

Accepted Answer

@#IMG-DLM&_TB5979_11_TB5979_11_TB5979_11_TB5979_11

Question 8

To prove that the OLS estimator is BLUE requires the following assumption

Accepted Answer

A) (Xi,Yi)i = 1, …, n are i.i.d. draws from their joint distribution 
B) Xi and ui have nonzero finite fourth moments 
C) the conditional distribution of ui given Xi is normal 
D) none of the above 
A) (Xi,Yi)i = 1, …, n are i.i.d. draws from their joint distribution 
B) Xi and ui have nonzero finite fourth moments 
C) the conditional distribution of ui given Xi is normal 
D) none of the above

Question 9

Write an essay on the difference between the OLS estimator and the GLS estimator.

Accepted Answer

Answers will vary by student, but some o

Question 10

The extended least squares assumptions in the multiple regression model include four assumptions from Chapter 6 (ui has conditional mean zero; (Xi,Yi), i = 1,…, n are i.i.d. draws from their joint distribution; Xi and ui have nonzero finite fourth moments; there is no perfect multicollinearity). In addition, there are two further assumptions, one of which is

Accepted Answer

A) heteroskedasticity of the error term. 
B) serial correlation of the error term. 
C) homoskedasticity of the error term. 
D) invertibility of the matrix of regressors. 
A) heteroskedasticity of the error term. 
B) serial correlation of the error term. 
C) homoskedasticity of the error term. 
D) invertibility of the matrix of regressors.

Question 11

Your textbook derives the OLS estimator as $\hat { \beta }$ = $\left( X ^ { \prime } \right.$ X)-1
  $X ^ { \prime }$ Y.
Show that the estimator does not exist if there are fewer observations than the number of explanatory variables, including the constant. What is the rank of $X ^ { \prime }$ X in this case?

Accepted Answer

In order for a matrix to be invertible,

Question 12

The OLS estimator

Accepted Answer

A) has the multivariate normal asymptotic distribution in large samples. 
B) is t-distributed. 
C) has the multivariate normal distribution regardless of the sample size. 
D) is F-distributed. 
A) has the multivariate normal asymptotic distribution in large samples. 
B) is t-distributed. 
C) has the multivariate normal distribution regardless of the sample size. 
D) is F-distributed.

Question 13

For the OLS estimator $\hat { \beta}$ = ( $X ^ { \prime }$ X)-1
  $X ^ { \prime }$ Y to exist, X'X must be invertible. This is the case when X has full rank. What is the rank of a matrix? What is the rank of the product of two matrices? Is it possible that X could have rank n? What would be the rank of X'X in the case n<(k+1)? Explain intuitively why the OLS estimator does not exist in that situation.

Accepted Answer

The rank of a matrix is the maximum numb

Question 14

In order for a matrix A to have an inverse, its determinant cannot be zero. Derive the determinant of the following matrices:
A = $\left( \begin{array} { r r } 
3 & 6 \
- 2 & 1
\end{array} \right)$ B = $\left( \begin{array} { r r r } 
1 & - 1 & 2 \
1 & 0 & 3 \
4 & 0 & 2
\end{array} \right)$ X'X where X = (1 10)

Accepted Answer

det (A)=15

Question 15

The GLS estimator

Accepted Answer

A) is always the more efficient estimator when compared to OLS. 
B) is the OLS estimator of the coefficients in a transformed model, where the errors of the transformed model satisfy the Gauss-Markov conditions. 
C) cannot handle binary variables, since some of the transformations require division by one of the regressors. 
D) produces identical estimates for the coefficients, but different standard errors. 
A) is always the more efficient estimator when compared to OLS. 
B) is the OLS estimator of the coefficients in a transformed model, where the errors of the transformed model satisfy the Gauss-Markov conditions. 
C) cannot handle binary variables, since some of the transformations require division by one of the regressors. 
D) produces identical estimates for the coefficients, but different standard errors.

Question 16

The assumption that X has full column rank implies that

Accepted Answer

A) the number of observations equals the number of regressors. 
B) binary variables are absent from the list of regressors. 
C) there is no perfect multicollinearity. 
D) none of the regressors appear in natural logarithm form. 
A) the number of observations equals the number of regressors. 
B) binary variables are absent from the list of regressors. 
C) there is no perfect multicollinearity. 
D) none of the regressors appear in natural logarithm form.

Question 17

$\hat \beta$ - ?

Accepted Answer

A) cannot be calculated since the population parameter is unknown. 
B) = ( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ U. 
C) = Y - $\hat { Y }$ 
D) = ? + ( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ U 
A) cannot be calculated since the population parameter is unknown. 
B) = ( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ U. 
C) = Y - $\hat { Y }$ 
D) = ? + ( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ U

Question 18

Prove that under the extended least squares assumptions the OLS estimator $\hat { \beta}$ is unbiased and that its variance-covariance matrix is $\sigma _ { u } ^ { 2 }$ (X'X)-1.

Accepted Answer

Start the proof by relating the OLS esti

Question 19

The heteroskedasticity-robust estimator of $\sum \sqrt { n ( \hat { \beta } - \beta ) }$ is obtained

Accepted Answer

A) from ( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ U. 
B) by replacing the population moments in its definition by the identity matrix. 
C) from feasible GLS estimation. 
D) by replacing the population moments in its definition by sample moments. 
A) from ( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ U. 
B) by replacing the population moments in its definition by the identity matrix. 
C) from feasible GLS estimation. 
D) by replacing the population moments in its definition by sample moments.

Question 20

Let there be q joint hypothesis to be tested. Then the dimension of r in the expression Rβ = r is

Accepted Answer

A) q × 1. 
B) q × (k+1). 
C) (k+1)× 1. 
D) q. 
A) q × 1. 
B) q × (k+1). 
C) (k+1)× 1. 
D) q.

The formulation Rβ= r to test a hypotheses

The multiple regression model in matrix form Y = X? + U can also be written as

One implication of the extended least squares assumptions in the multiple regression model is that

An estimator of β is said to be linear if

One of the properties of the OLS estimator is

The leading example of sampling schemes in econometrics that do not result in independent observations is

To prove that the OLS estimator is BLUE requires the following assumption

Write an essay on the difference between the OLS estimator and the GLS estimator.

The OLS estimator

The GLS estimator

The assumption that X has full column rank implies that

$\hat \beta$ - ?

Prove that under the extended least squares assumptions the OLS estimator $\hat { \beta}$ is unbiased and that its variance-covariance matrix is $\sigma _ { u } ^ { 2 }$ (X'X)^-¹.

The heteroskedasticity-robust estimator of $\sum \sqrt { n ( \hat { \beta } - \beta ) }$ is obtained

Let there be q joint hypothesis to be tested. Then the dimension of r in the expression Rβ = r is

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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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Regression With Panel Data

Regression With a Binary Dependent Variable

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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 formulation Rβ= r to test a hypotheses

The multiple regression model in matrix form Y = X? + U can also be written as

One implication of the extended least squares assumptions in the multiple regression model is that

An estimator of β is said to be linear if

One of the properties of the OLS estimator is

The leading example of sampling schemes in econometrics that do not result in independent observations is

To prove that the OLS estimator is BLUE requires the following assumption

Write an essay on the difference between the OLS estimator and the GLS estimator.

The OLS estimator

The GLS estimator

The assumption that X has full column rank implies that

β^\hat \betaβ^​ - ?

Prove that under the extended least squares assumptions the OLS estimator β^\hat { \beta}β^​ is unbiased and that its variance-covariance matrix is σu2\sigma _ { u } ^ { 2 }σu2​ (X'X)-1.

The heteroskedasticity-robust estimator of ∑n(β^−β)\sum \sqrt { n ( \hat { \beta } - \beta ) }∑n(β^​−β)​ is obtained

Let there be q joint hypothesis to be tested. Then the dimension of r in the expression Rβ = r is

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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$\hat \beta$ - ?

Prove that under the extended least squares assumptions the OLS estimator $\hat { \beta}$ is unbiased and that its variance-covariance matrix is $\sigma _ { u } ^ { 2 }$ (X'X)^-¹.

The heteroskedasticity-robust estimator of $\sum \sqrt { n ( \hat { \beta } - \beta ) }$ is obtained