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  ( X) -1. B

Question 2

Your textbook shows that the following matrix (Mx = In - Px)is a symmetric idempotent matrix.Consider a different Matrix A,which is defined as follows: A = I -  ιι' and ι =  a.Show what the elements of A look like.
b.Show that A is a symmetric idempotent matrix
c.Show that Aι = 0.
d.Show that A  =  ,where  is the vector of OLS residuals from a multiple regression.

Accepted Answer

a.A =  -   =  b.A' =  ' =  = A
A×A =(I -  ιι')×(I -  ιι')= (I -  ιι' -  ιι' +  ιι' ιι')
But ιι' ιι' =  ×  =  ,and   =   =  ιι'
This means that the last two terms in the above equation cancel each other,and therefore A×A = A,that is,A is idempotent.
c.Aι = (I -  ιι')ι = ι -  ιι' ι = 0 since ιι' = n
d.A  = (I -  ιι')  =  -  ιι'  =  since ι'  = 0

Question 3

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

Accepted Answer

A)  Yi = β0 + X  β + ui,i = 1,…,n.
B)  Yi = X  βi,i = 1,…,n.
C)  Yi = βX  + ui,i = 1,…,n.
D)  Yi = X  β + ui,i = 1,…,n. D

Question 4

In Chapter 10 of your textbook,panel data estimation was introduced.Panel data consist of observations on the same n entities at two or more time periods T.For two variables,you have
(Xit,Yit),i = 1,... ,n and t = 1,... ,T
where n could be the U.S.states.The example in Chapter 10 used annual data from 1982 to 1988 for the fatality rate and beer taxes.Estimation by OLS,in essence,involved "stacking" the data.
(a)What would the variance-covariance matrix of the errors look like in this case if you allowed for homoskedasticity-only standard errors? What is its order? Use an example of a linear regression with one regressor of 4 U.S.states and 3 time periods.
(b)Does it make sense that errors in New Hampshire,say,are uncorrelated with errors in Massachusetts during the same time period ("contemporaneously")? Give examples why this correlation might not be zero.
(c)If this correlation was known,could you find an estimator which was more efficient than OLS?

Accepted Answer

The answer of In Chapter 10 of your textbook,panel data...

Question 5

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 = -  in the same format? Why not?

Accepted Answer

The answer of Write the following four restrictions in the...

Question 6

The GLS estimator is defined as

Accepted Answer

A)  (  Ω-1X) -1 ( Ω-1Y) .
B)  (  X) -1 Y.
C)    Y.
D)  (  X) -1 U.

Question 7

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  ) = Ω(X) ,where Ω(X) is n × n matrix-valued that can depend on X.
D)  E(U  ) = 0n.

Question 8

The heteroskedasticity-robust estimator of  is obtained

Accepted Answer

A)  from (  X) -1 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 9

One of the properties of the OLS estimator is

Accepted Answer

A)  X  = 0k+1.
B)  that the coefficient vector  has full rank.
C)    (Y - X ) = 0k+1.
D)  (  X) -1= Y

Question 10

Prove that under the extended least squares assumptions the OLS estimator  is unbiased and that its variance-covariance matrix is  (X'X)-1.

Accepted Answer

The answer of Prove that under the extended least squares...

Question 11

Consider the multiple regression model from Chapter 5,where k = 2 and the assumptions of the multiple regression model hold.
(a)Show what the X matrix and the β vector would look like in this case.
(b)Having collected data for 104 countries of the world from the Penn World Tables,you want to estimate the effect of the population growth rate (X1i)and the saving rate (X2i)(average investment share of GDP from 1980 to 1990)on GDP per worker (relative to the U.S. )in 1990.What are your expected signs for the regression coefficient? What is the order of the (X'X)here?
(c)You are asked to find the OLS estimator for the intercept and slope in this model using the
formula  = (X'X)-1 X'Y.Since you are more comfortable in inverting a 2×2 matrix (the inverse of a 2×2 matrix is,  =   )
you decide to write the multiple regression model in deviations from mean form.Show what the X matrix,the (X'X)matrix,and the X'Y matrix would look like now.
(Hint: use small letters to indicate deviations from mean,i.e. ,zi = Zi -  and note that
Yi =  0 +  1X1i +  2X2i +  i  =  0 +  1  1 +  2  2.
Subtracting the second equation from the first,you get
yi =  1x1i +  2x2i +  i)
(d)Show that the slope for the population growth rate is given by  1 =  (e)The various sums needed to calculate the OLS estimates are given below:  = 8.3103;  = .0122;  = 0.6422  = -0.2304;  = 1.5676;  = -0.0520
Find the numerical values for the effect of population growth and the saving rate on per capita income and interpret these.
(f)Indicate how you would find the intercept in the above case.Is this coefficient of interest in the interpretation of the determinants of per capita income? If not,then why estimate it?

Accepted Answer

The answer of Consider the multiple regression model from Chapter...

Question 12

The TSLS estimator is

Accepted Answer

A)  (X'X) -1 X'Y
B)  (X'Z(Z'Z) -1 Z'X) -1 X'Z(Z'Z) -1 Z' Y
C)  (XΩ-1X) -1(XΩ-1Y) 
D)  (X'Pz) -1PzY

Question 13

A =  ,B =  ,and C =  show that  =  +  and  =   .

Accepted Answer

The answer of A =  ,B =  ,and...

Question 14

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)  the conditional distribution of ui given Xi is normal.
D)  invertibility of the matrix of regressors.

Question 15

For the OLS estimator  = (  X)-1  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 answer of For the OLS estimator  = (...

Question 16

The Gauss-Markov theorem for multiple regression states that the OLS estimator

Accepted Answer

A)  has the smallest variance possible for any linear estimator.
B)  is BLUE if the Gauss-Markov conditions for multiple regression hold.
C)  is identical to the maximum likelihood estimator.
D)  is the most commonly used estimator.

Question 17

Minimization of  results in

Accepted Answer

A)    Y = X .
B)  X  = 0k+1.
C)    (Y - X ) = 0k+1.
D)  Rβ = r.

Question 18

Write down,in general,the variance-covariance matrix for the multiple regression error term U.Using the assumptions cov(ui,uj|XiXj)= 0 and var(ui|Xi)=  .Show that the variance-covariance matrix can be written as  In.

Accepted Answer

The answer of Write down,in general,the variance-covariance matrix for the...

Question 19

Write the following three linear equations in matrix format Ax = b,where x is a 3×1 vector containing q,p,and y,A is a 3×3 matrix of coefficients,and b is a 3×1 vector of constants.
q = 5 +3 p - 2 y
q = 10 - p + 10 y
p = 6 y

Accepted Answer

The answer of Write the following three linear equations in...

Question 20

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.

Exam 18: The Theory of Multiple Regression

The Gauss-Markov theorem for multiple regression proves that

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

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 = - in the same format? Why not?

The GLS estimator is defined as

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

The heteroskedasticity-robust estimator of is obtained

One of the properties of the OLS estimator is

Prove that under the extended least squares assumptions the OLS estimator is unbiased and that its variance-covariance matrix is (X'X)-1.

The TSLS estimator is

A = ,B = ,and C = show that = + and = .

The Gauss-Markov theorem for multiple regression states that the OLS estimator

Minimization of results in

Write down,in general,the variance-covariance matrix for the multiple regression error term U.Using the assumptions cov(ui,uj|XiXj)= 0 and var(ui|Xi)= .Show that the variance-covariance matrix can be written as In.

Write the following three linear equations in matrix format Ax = b,where x is a 3×1 vector containing q,p,and y,A is a 3×3 matrix of coefficients,and b is a 3×1 vector of constants. q = 5 +3 p - 2 y q = 10 - p + 10 y p = 6 y

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