Deck 18: The Theory of Multiple Regression

Question 1

The multiple regression model in matrix form $\boldsymbol { Y } = \boldsymbol { X } \boldsymbol { \beta } + \boldsymbol { U }$ can also be written as

A) $Y _ { i } = \boldsymbol { \beta } _ { 0 } + \boldsymbol { X } _ { i } ^ { \prime } \boldsymbol { \beta } + u _ { i } , i = 1 , \ldots , n$
B) $Y _ { i } = \boldsymbol { X } _ { i } ^ { \prime } \boldsymbol { \beta } _ { i } , i = 1 , \ldots , n$
C) $Y _ { i } = \boldsymbol { \beta } \boldsymbol { X } _ { i } ^ { \prime } + u _ { i } , i = 1 , \ldots , n$
D) $Y _ { i } = \boldsymbol { X } _ { i } ^ { \prime } \boldsymbol { \beta } + u _ { i } , i = 1 , \ldots , n$

Answer 1

$Y _ { i } = \boldsymbol { X } _ { i } ^ { \prime } \boldsymbol { \beta } + u _ { i } , i = 1 , \ldots , n$

Question 2

Minimization of $\sum _ { i = 1 } ^ { n } \left( Y _ { i } - b _ { 0 } - b _ { 1 } X _ { 1 i } - \cdots - b _ { k } X _ { k i } \right) ^ { 2 }$ results in

A) $X ^ { \prime } \boldsymbol { Y } = \boldsymbol { X } \hat { \boldsymbol { \beta } }$
B) $\boldsymbol { X } \hat { \boldsymbol { \beta } } = \mathbf { 0 } _ { k + 1 }$
C) $\boldsymbol { X } ^ { \prime } ( \boldsymbol { Y } - \boldsymbol { X } \hat { \boldsymbol { \beta } } ) = \mathbf { 0 } _ { k + 1 }$
D) $\boldsymbol { R } \boldsymbol { \beta } = \boldsymbol { r } \text {. }$

Answer 2

$\boldsymbol { X } ^ { \prime } ( \boldsymbol { Y } - \boldsymbol { X } \hat { \boldsymbol { \beta } } ) = \mathbf { 0 } _ { k + 1 }$

Question 3

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

Answer 3

D

Question 4

The GLS estimator is defined as

Question 5

The difference between the central limit theorems for a scalar and vector-valued random variables is

A)that n approaches infinity in the central limit theorem for scalars only.
B)the conditions on the variances.
C)that single random variables can have an expected value but vectors cannot.
D)the homoskedasticity assumption in the former but not the latter.

Question 6

The OLS estimator

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

The extended least squares assumptions in the multiple regression model include four assumptions from Chapter 6 $\left( u _ { i } \text { has conditional mean zero; } \left( \boldsymbol { X } _ { i } , Y _ { j } \right) , i = 1 , \ldots , n \right.$ are i.i.d. draws from their joint distribution; $\boldsymbol { X } _ { i } \text {and } u _ { i }$ have nonzero finite fourth moments; there is no perfect multicollinearity). In addition, there are two further assumptions, one of which is

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 8

The assumption that X has full column rank implies that

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 9

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

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 10

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

A) $\text { from } \left( \boldsymbol { X } ^ { \prime } \boldsymbol { X } \right) ^ { - 1 } \boldsymbol { X } ^ { \prime } \boldsymbol { U }$
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 11

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

Question 12

The formulation $\boldsymbol { R } \boldsymbol { \beta } = \boldsymbol { r }$ to test a 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 13

In the case when the errors are homoskedastic and normally distributed, conditional on X, then

Question 14

A joint hypothesis that is linear in the coefficients and imposes a number of restrictions can be written as

Question 15

Let there be q joint hypothesis to be tested. Then the dimension of r in the expression $R \beta = r$ is

A) $q \times 1$
B) $q \times ( k + 1 )$
C) $( k + 1 ) \times 1$
D) $q .$

Question 16

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

Question 17

The linear multiple regression model can be represented in matrix notation as $\boldsymbol { Y } = \boldsymbol { X } \boldsymbol { \beta } + \boldsymbol { U }$ where X is of order n x(k+1) . k represents the number of

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

Question 18

The Gauss-Markov theorem for multiple regression proves that

Question 19

One of the properties of the OLS estimator is

Question 20

$\hat { \beta } - \beta$

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

Question 21

Define the GLS estimator and discuss its properties when is known. Why is this estimator sometimes called infeasible GLS? What happens when is unknown? What would the matrix look like for the case of independent sampling with heteroskedastic errors, where
Since the inverse of the error variancecovariance matrix is needed to compute the GLS estimator, find https://d2lvgg3v3hfg70.cloudfront.net/TB34225555/. The textbook shows that the original model
will be transformed into where
Find in the above case, and describe what effect the transformation has on the original data.

Question 22

Write the following four restrictions in the form where the hypotheses are to be tested simultaneously.

Can you write the following restriction in the same format? Why not?

Question 23

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 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.

Question 24

q = 5 +3 p - 2 y
q = 10 - p + 10 y
p = 6 y

Question 25

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

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 26

Using the model and the extended least squares assumptions, derive the OLS estimator https://d2lvgg3v3hfg70.cloudfront.net/TB34225555/. Discuss the conditions under which is invertible.

Question 27

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 vector would look like in this case.

Question 28

Prove that under the extended least squares assumptions the OLS estimator is unbiased and that its variance-covariance matrix is

Question 29

Assume that the data looks as follows:

Question 30

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

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 31

The GLS estimator

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 32

In order for a matrix A to have an inverse, its determinant cannot be zero.Derive the
determinant of the following matrices:

Question 33

Your textbook derives the OLS estimator as 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′X in this case?

Question 34

Given the following matrices

Question 35

For the OLS estimator to exist, must be invertible. This is the case when has full rank. What is the rank of a matrix? What is the rank of the product of two matrices? Is it possible that could have rank n ? What would be the rank of in the case Explain intuitively why the OLS estimator does not exist in that situation.

Question 36

An estimator of $\beta$ is said to be linear if

A) it can be estimated by least squares.
B) it is a linear function of $Y _ { 1 } , \ldots , Y _ { n }$
C) there are homoskedasticity-only errors.
D) it is a linear function of $X _ { 1 } , \ldots , X _ { n }$

Question 37

Give several economic examples of how to test various joint linear hypotheses using matrix notation. Include specifications of where you test for (i) all coefficients other than the constant being zero, (ii) a subset of coefficients being zero, and (iii) equality of coefficients. Talk about the possible distributions involved in finding critical values for your hypotheses.

Question 38

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