Question 1

The homoskedasticity-only F-statistic is

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 2

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 3

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 4

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 5

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   ? 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   = FU,and   F = Ω-1.Find F in the above case,and describe what effect the transformation has on the original data.

Accepted Answer

@#IMG-DLM& GLS= ( @#IMG-DLM& Ω-1X)-1( @#IMG-DLM& Ω-1Y).The key point f

Question 6

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 7

Your textbook derives the OLS estimator as   =   X)-1   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 in this case?

Accepted Answer

In order for a matrix to be invertible,i

Question 8

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 9

The OLS estimator for the multiple regression model in matrix form is

Accepted Answer

A) (X'X)-1X'Y 
B) X(X'X)-1X' - PX 
C) (X'X)-1X'U 
D) (XΩ-1X)-1XΩ-1Y 
A) (X'X)-1X'Y 
B) X(X'X)-1X' - PX 
C) (X'X)-1X'U 
D) (XΩ-1X)-1XΩ-1Y

Question 10

Let Y =   and X =   Find   X,   Y, (   X)-1 and finally (   X)-1   Y.

Accepted Answer

@#IMG-DLM& X = @#IMG-DLM& , @#IMG-DLM& Y

Question 11

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 12

- β

Accepted Answer

A) cannot be calculated since the population parameter is unknown. 
B) = (   X)-1
  U. 
C) = Y -   . 
D) = β + (   X)-1
  U 
A) cannot be calculated since the population parameter is unknown. 
B) = (   X)-1
  U. 
C) = Y -   . 
D) = β + (   X)-1
  U

Question 13

The multiple regression model can be written in matrix form as follows:

Accepted Answer

A) Y = Xβ. 
B) Y = X + U. 
C) Y = βX + U. 
D) Y = Xβ + U. 
A) Y = Xβ. 
B) Y = X + U. 
C) Y = βX + U. 
D) Y = Xβ + U.

Question 14

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 15

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 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 is singular. 
D) the conditional distribution of U given X is N(0n,In).

Question 16

Let PX = X(   X)-1   and MX = In - PX.Then MX MX =

Accepted Answer

A) X(   X)-1
  - PX. 
B)    
C) In. 
D) MX. 
A) X(   X)-1
  - PX. 
B)    
C) In. 
D) MX.

Question 17

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 18

Using the model Y = Xβ + U,and the extended least squares assumptions,derive the OLS estimator   .Discuss the conditions under which   X is invertible.

Accepted Answer

The derivation copies the relevant parts

Question 19

The linear multiple regression model can be represented in matrix notation as Y= Xβ + U,where X is of order n×(k+1).k represents the number of

Accepted Answer

A) regressors. 
B) observations. 
C) regressors excluding the "constant" regressor for the intercept. 
D) unknown regression coefficients. 
A) regressors. 
B) observations. 
C) regressors excluding the "constant" regressor for the intercept. 
D) unknown regression coefficients.

Question 20

Consider the following population regression function: Y = Xβ + U
where Y=   ,X=   ,β =   ,U=   Given the following information on population growth rates (Y)and education (X)for 86 countries   ,   ,   ,   ,   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 = @#IMG-DLM& = @#IMG-DLM& X"Y = = @#IMG-DLM& = @#IMG-DLM& (X'X)-

The homoskedasticity-only F-statistic is

The assumption that X has full column rank implies that

The OLS estimator

Your textbook derives the OLS estimator as = X)-1 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 in this case?

An estimator of β is said to be linear if

The OLS estimator for the multiple regression model in matrix form is

Let Y = and X = Find X, Y, ( X)-1 and finally ( X)-1 Y.

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

- β

The multiple regression model can be written in matrix form as follows:

The GLS estimator

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

Let PX = X( X)-1 and MX = In - PX.Then MX MX =

The formulation Rβ= r to test a hypotheses

Using the model Y = Xβ + U,and the extended least squares assumptions,derive the OLS estimator .Discuss the conditions under which X is invertible.

The linear multiple regression model can be represented in matrix notation as Y= Xβ + U,where X is of order n×(k+1).k represents the number of

Consider the following population regression function: Y = Xβ + U where Y= ,X= ,β = ,U= Given the following information on population growth rates (Y)and education (X)for 86 countries , , , , 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.

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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Exam 18: The Theory of Multiple Regression

The homoskedasticity-only F-statistic is

The assumption that X has full column rank implies that

The OLS estimator

Your textbook derives the OLS estimator as = X)-1 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 in this case?

An estimator of β is said to be linear if

The OLS estimator for the multiple regression model in matrix form is

Let Y = and X = Find X, Y, ( X)-1 and finally ( X)-1 Y.

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

- β

The multiple regression model can be written in matrix form as follows:

The GLS estimator

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

Let PX = X( X)-1 and MX = In - PX.Then MX MX =

The formulation Rβ= r to test a hypotheses

Using the model Y = Xβ + U,and the extended least squares assumptions,derive the OLS estimator .Discuss the conditions under which X is invertible.

The linear multiple regression model can be represented in matrix notation as Y= Xβ + U,where X is of order n×(k+1).k represents the number of

Consider the following population regression function: Y = Xβ + U where Y= ,X= ,β = ,U= Given the following information on population growth rates (Y)and education (X)for 86 countries , , , , 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.

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

Filters