Deck 18: The Theory of Multiple Regression

Question 1

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

A) $X ^ { \prime }$ Y = X $\hat \beta$
B)X $\hat \beta$ = 0_k+1.
C) $X ^ { \prime }$ (Y - X $\hat \beta$ )= 0_k+1.
D)R? = r.

Answer 1

$X ^ { \prime }$ (Y - X $\hat \beta$ )= 0_k+1.

Question 2

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

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.

Answer 2

by replacing the population moments in its definition by sample moments.

Question 3

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

A)the X_i are fixed in repeated samples.
B)X_i and u_i 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 $|X$ )= 0_n.

Answer 3

the X_i are fixed in repeated samples.

Question 4

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

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

Question 5

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

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

Question 6

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 7

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

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

Question 8

The GLS estimator is defined as

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.

Question 9

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

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

Question 10

The Gauss-Markov theorem for multiple regression 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 11

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

A)( $X ^ { \prime }$ X)^-1 $X ^ { \prime }$ Y.
B)R? = r.
C) $\hat \beta$ - ?.
D)R?= 0.

Question 12

The formulation Rβ= 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

One of the properties of the OLS estimator is

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

Question 14

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 15

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

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

Question 16

The Gauss-Markov theorem for multiple regression proves that

A)M_X 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 17

The extended least squares assumptions in the multiple regression model include four assumptions from Chapter 6 (ui has conditional mean zero; (X_i,Y_i), i = 1,…, n are i.i.d. draws from their joint distribution; X_i 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 18

$\hat \beta$ - ?

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 19

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 20

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

A)X( $X ^ { \prime }$ X)^-1 $X ^ { \prime }$ - P_X.
B) $M _ { X } ^ { 2 }$
C)I_n.
D)M_X.

Question 21

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 (X_1i)and the saving rate (X_2i)(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., z_i = Z_i - and note that
Y_i = ₀ + ₁X_1i + ₂X_2i + _i = ₀ + _1
1 + _2
2.
Subtracting the second equation from the first, you get
y_i = ₁x_1i + ₂x_2i + _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?

Question 22

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

Question 23

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

Question 24

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

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

Question 25

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

A) $\hat \beta$ is distributed N(?, $\sum _ { \hat { \beta } \mid X }$ ),where $\sum _ { \hat { \beta } \mid X }$ = $\sigma _ { u } ^ { 2 }$ I_(k+1).
B) $\hat \beta$ is distributed N(?, $\sum _ { \hat { \beta } }$ ), where $\sum _ { \hat { \beta } }$ = $\sum _ { \sqrt { n } \left( \hat { \boldsymbol { \beta } } _ { - } \boldsymbol { \beta } \right) }$ /n = $Q _ { X } ^ { - 1 }$ $\sum _ { V }$ $Q _ { X } ^ { - 1 }$ /n.
C) $\hat \beta$ is distributed N(?, $\sum _ { \hat { \beta } \mid X }$ ),where $\sum _ { \hat { \beta } \mid X }$ = $\sigma _ { u } ^ { 2 }$ ( $X ^ { \prime }$ X)^-1.
D) $\hat { u }$ = P_XY where P_X = X( $X ^ { \prime }$ X)^-1 $X ^ { \prime }$

Question 26

The homoskedasticity-only F-statistic is

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 27

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(u_i X_i)= ch(X_i)= 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.

Question 28

The extended least squares assumptions in the multiple regression model include four assumptions from Chapter 6 (u_i has conditional mean zero; (X_i,Y_i), i = 1,…, n are i.i.d. draws from their joint distribution; X_i 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)the conditional distribution of u_i given X_i is normal.
D)invertibility of the matrix of regressors.

Question 29

Give several economic examples of how to test various joint linear hypotheses using matrix notation. Include specifications of Rβ = r 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 30

An estimator of β is said to be linear if

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

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

Assume that the data looks as follows:
Y = , U = , X = , and β = (β₁)
Using the formula for the OLS estimator = ( X)^-1
Y, derive the formula for ₁, the only slope in this "regression through the origin."

Question 33

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
(X_it, Y_it), 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?

Question 34

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

Question 35

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 36

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 37

The TSLS estimator is

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'P_z)^-1P_zY

Question 38

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?

Question 39

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

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

Question 40

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

Question 41

Consider the following symmetric and idempotent Matrix A: A = I - ??' and ? = a. Show that by postmultiplying this matrix by the vector Y (the LHS variable of the OLS regression), you convert all observations of Y in deviations from the mean.
b. Derive the expression Y'AY. What is the order of this expression? Under what other name have you encountered this expression before?

Question 42

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

Question 43

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.
Y_i = β₀+ β₁X_1i + β₂X_2i + β₃X_3i+u_i
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.

Question 44

Your textbook shows that the following matrix (M_x = I_n - P_x)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.

Question 45

Consider the following population regression function: Y = X? + U
where Y= , X= , ? = , 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.

Question 46

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?

Question 47

In order for a matrix A to have an inverse, its determinant cannot be zero. Derive the determinant of the following matrices:
A = B = X'X where X = (1 10)

Question 48

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.

Question 49

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

Question 50

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 I_n.