Question 1

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

Accepted Answer

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)  $\hat { u }$ = PXY where PX = X( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ 
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)  $\hat { u }$ = PXY where PX = X( $X ^ { \prime }$ X)-1   $X ^ { \prime }$ C

Question 2

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 A

Question 3

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

(a)Under the extended least least squares assumptions, E(UU' | X)=

\sigma _ { u } ^ { 2 }

I_n.
In the above example of 4 U.S. states and 3 time periods, the identity matrix will be of order 12×12, or (nT)× (nT)in general. Specifically

\left( \begin{array} { c c c c } \sigma _ { u } ^ { 2 } & 0 & \cdots & 0 \\0 & \sigma _ { u } ^ { 2 } & \cdots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & \sigma _ { u } ^ { 2 }\end{array} \right)

(b)It is reasonable to assume that a shock to an adjacent state would have an effect on its neighboring state, particularly when the shock affects the larger of the two such as the case in Massachusetts. Other examples may be Texas and Arkansas, Michigan and Indiana, California and Arizona, New York and New Jersey, etc. A negative oil price shock, which affects the demand for automobiles produced in Michigan, will have repercussions for suppliers located not only in Michigan, but also elsewhere.
(c)In case of a known variance-covariance matrix of the error terms, the GLS estimator

\hat { \beta }

^GLS = (

X ^ { \prime }

?^-¹^X)^-¹(

X ^ { \prime }

?^-¹^Y)could be used. The variance-covariance matrix would be of the form

\Omega = \left( \begin{array} { c c c c c c c c c c c c } \sigma _ { 11 } & 0 & 0 & \sigma _ { 12 } & 0 & 0 & \sigma _ { 13 } & 0 & 0 & \sigma _ { 14 } & 0 & 0 \\0 & \sigma _ { 11 } & 0 & 0 & \sigma _ { 12 } & 0 & 0 & \sigma _ { 13 } & 0 & 0 & \sigma _ { 14 } & 0 \\0 & 0 & \sigma _ { 11 } & 0 & 0 & \sigma _ { 12 } & 0 & 0 & \sigma _ { 13 } & 0 & 0 & \sigma _ { 14 } \\\sigma _ { 12 } & 0 & 0 & \sigma _ { 22 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 24 } & 0 & 0 \\0 & \sigma _ { 12 } & 0 & 0 & \sigma _ { 22 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 24 } & 0 \\0 & 0 & \sigma _ { 12 } & 0 & 0 & \sigma _ { 22 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 24 } \\\sigma _ { 13 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 33 } & 0 & 0 & \sigma _ { 34 } & 0 & 0 \\0 & \sigma _ { 13 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 33 } & 0 & 0 & \sigma _ { 34 } & 0 \\0 & 0 & \sigma _ { 13 } & 0 & 0 & \sigma _ { 23 } & 0 & 0 & \sigma _ { 33 } & 0 & 0 & \sigma _ { 34 } \\\sigma _ { 14 } & 0 & 0 & \sigma _ { 24 } & 0 & 0 & \sigma _ { 34 } & 0 & 0 & \sigma _ { 44 } & 0 & 0 \\0 & \sigma _ { 14 } & 0 & 0 & \sigma _ { 24 } & 0 & 0 & \sigma _ { 34 } & 0 & 0 & \sigma _ { 44 } & 0 \\0 & 0 & \sigma _ { 14 } & 0 & 0 & \sigma _ { 24 } & 0 & 0 & \sigma _ { 34 } & 0 & 0 & \sigma _ { 44 }\end{array} \right)

(There is a subtle issue here for the case of a feasible GLS estimator, where the variances and covariances have to be estimated. It can be shown, in that case, that the GLS estimator does not exist unless n ? T, which is not the case for most panels. It is easier to see that the variance-covariance matrix is singular for n>T if the data is stacked by time period.)

Question 4

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

Accepted Answer

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

Question 5

Using the model Y = Xβ + U, and the extended least squares assumptions, derive the OLS estimator $\hat {\beta}$ Discuss the conditions under which $X ^ { \prime }$ X is invertible.

Accepted Answer

The derivation copies the relevant parts

Question 6

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 7

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. 
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 8

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 $\hat{\beta}$ = (X'X)-1 X'Y. Since you are more comfortable in inverting a 2×2 matrix (the inverse of a 2×2 matrix is, $\left( \begin{array} { l l l l l } 
a&b\c&d
\end{array} \right)^{-1}$ = $\frac { 1 } { a d - b c }$ $\left( \begin{array} { c c } 
d & - b \
- c & a
\end{array} \right)$ )
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 - $\bar { Z }$ and note that
Yi = $\hat { \beta }$ 0 + $\hat { \beta }$ 1X1i + $\hat { \beta }$ 2X2i + $\hat {u}$ i $\bar { Y }$ = $\hat { \beta }$ 0 + $\hat { \beta }$ 1
  $\bar { X }$ 1 + $\hat { \beta }$ 2
  $\bar { X }$ 2.
Subtracting the second equation from the first, you get
yi = $\hat { \beta }$ 1x1i + $\hat { \beta }$ 2x2i + $\hat {u}$ i)
(d)Show that the slope for the population growth rate is given by $\hat { \beta }$ 1 = $\frac { \sum_{i=1}^{n} y_{i} x 1 i \sum_{i=1}^{n} x_{2 i}^{2}-\sum_{i=1}^{n} y_{i} x_{2 i}\sum_{i=1}^{n} x_{1 i} x_{2 i}}{\sum_{i=1}^{n} x_{1 i}^{2}\sum_{i=1}^{n} x 2_{2 i}^{2}-(\left.\sum_{i=1}^{n} x_{1 i} x_{2 i}\right)^{2}
}$ (e)The various sums needed to calculate the OLS estimates are given below: $\sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 }$ = 8.3103; $\sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 }$ = .0122; $\sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 }$ = 0.6422 $\sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i }$ = -0.2304; $\sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i }$ = 1.5676; $\sum _ { i = 1 } ^ { n } x _ { 1 \mathrm { i } } x _ { 2 i }$ = -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

(a)
X = @#IMG-DLM&_TB5979_11_TB5979_11_TB5979_11_T

Question 9

Consider the following symmetric and idempotent Matrix A: A = I - $\frac { 1 } { n }$ ??' and ? = $\left[ \begin{array} { c } 
1 \
1 \
\ldots \
1
\end{array} \right]$ 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?

Accepted Answer

a. Note that @#IMG-DLM&_TB5979_11_TB5979_11_TB5979

Question 10

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

Accepted Answer

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

Question 11

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

A = @#IMG-DLM&_TB5979_11_TB5979_11_TB5979_11_TB597

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 
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

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. 
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 14

Assume that the data looks as follows:
Y = $ \left( \begin{array} { l } 
Y _ { 1 } \
Y _ { 2 } \
O \
Y _ { n }
\end{array}\right)$ , U = $\left( \begin{array} { c } 
u _ { 1 } \
u _ { 2 } \
\mathrm { O } \
u _ { n }
\end{array} \right)$ , X = $\left( \begin{array} { l } 
X _ { 11 } \
X _ { 12 } \
\mathrm { O } \
X _ { 1 n }
\end{array} \right)$ , and β = (β1)
Using the formula for the OLS estimator $\hat { \beta }$ = ( $X ^ { \prime }$ X)-1
  $X ^ { \prime }$ Y, derive the formula for $\hat { \beta }$ 1, the only slope in this "regression through the origin."

Accepted Answer

In this case, @#IMG-DLM&_TB5979_11_TB5979_11_TB597

Question 15

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.

Accepted Answer

Answers will vary by student. Many restr

Question 16

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 - $\frac { 1 } { n }$ ιι' and ι = $\left[ \begin{array} { c } 
1 \
1 \
\ldots \
1
\end{array} \right]$ 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 $\hat { \mathrm { U } }$ = $\hat { \mathrm { U } }$ , where $\hat { \mathrm { U } }$  is the vector of OLS residuals from a multiple regression.

Accepted Answer

a. A = @#IMG-DLM&_TB5979_11_TB5979_11_TB5979_11_TB

Question 17

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)= $\sigma _ { \mathrm { u } } ^ { 2 }$ Show that the variance-covariance matrix can be written as $\sigma _ { \mathrm { u } } ^ { 2 }$ In.

Accepted Answer

(var-cov)( @#IMG-DLM&_TB5979_11_TB5979_11_TB5979_1

Question 18

A = $\left( \begin{array} { l l } 
a _ { 11 } & a _ { 12 } \
a _ { 21 } & a _ { 22 }
\end{array} \right)$ , B = $\left( \begin{array} { l l } 
b _ { 11 } & b _ { 12 } \
b _ { 21 } & b _ { 22 }
\end{array} \right)$ , and C = $\left( \begin{array} { l l l } 
c _ { 11 } & c _ { 12 } & c _ { 13 } \
c _ { 21 } & c _ { 22 } & c _ { 23 }
\end{array} \right)$ show that $( A + B ) ^ { \prime }$ = $A ^ { \prime }$ + $B ^ { \prime }$ and $( A C ) ^ { \prime }$ = $C ^ { \prime }$ $A ^ { \prime }$

Accepted Answer

The answer of A = \(\left( \begin{array} { l l...

Question 19

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

Accepted Answer

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. 
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 20

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.

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

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

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

Using the model Y = Xβ + U, and the extended least squares assumptions, derive the OLS estimator $\hat {\beta}$ Discuss the conditions under which $X ^ { \prime }$ 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

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

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

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 TSLS estimator is

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)= $\sigma _ { \mathrm { u } } ^ { 2 }$ Show that the variance-covariance matrix can be written as $\sigma _ { \mathrm { u } } ^ { 2 }$ I_n.

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

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

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

Exam 18: The Theory of Multiple Regression

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

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

Minimization of ∑i=1n(Yi−b0−b1X1i−…−bkXki)2\sum _ { i = 1 } ^ { n } \left( Y _ { i } - b _ { 0 } - b _ { 1 } X _ { 1 i } - \ldots - b _ { k } X _ { k i } \right) ^ { 2 }∑i=1n​(Yi​−b0​−b1​X1i​−…−bk​Xki​)2 results in

Using the model Y = Xβ + U, and the extended least squares assumptions, derive the OLS estimator β^\hat {\beta}β^​ Discuss the conditions under which X′X ^ { \prime }X′ 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

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

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

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 TSLS estimator is

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

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

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

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

Using the model Y = Xβ + U, and the extended least squares assumptions, derive the OLS estimator $\hat {\beta}$ Discuss the conditions under which $X ^ { \prime }$ X is invertible.