Question 1

The probability of an event A or B (Pr(A or B))to occur equals

Accepted Answer

A) Pr(
A) × Pr(
B) . 
B) Pr(
A) + Pr(
B) if A and B are mutually exclusive. 
C)  $\frac { \operatorname { Pr } ( A ) } { \operatorname { Pr } ( B ) }$ 
D) Pr(
A) + Pr(
B) even if A and B are not mutually exclusive. 
A) Pr(
A) × Pr(
B) . 
B) Pr(
A) + Pr(
B) if A and B are mutually exclusive. 
C)  $\frac { \operatorname { Pr } ( A ) } { \operatorname { Pr } ( B ) }$ 
D) Pr(
A) + Pr(
B) even if A and B are not mutually exclusive.

Question 2

In econometrics, we typically do not rely on exact or finite sample distributions because

Accepted Answer

A) we have approximately an infinite number of observations (think of re-sampling). 
B) variables typically are normally distributed. 
C) the covariances of Yi, Yj are typically not zero. 
D) asymptotic distributions can be counted on to provide good approximations to the exact sampling distribution (given the number of observations available in most cases). 
A) we have approximately an infinite number of observations (think of re-sampling). 
B) variables typically are normally distributed. 
C) the covariances of Yi, Yj are typically not zero. 
D) asymptotic distributions can be counted on to provide good approximations to the exact sampling distribution (given the number of observations available in most cases).

Question 3

The mean of the sample average $\bar { Y }$ , $E ( \bar { Y } )$ , is

Accepted Answer

A)  $\frac { 1 } { n } \mu _ { \gamma }$ 
B)  $\mu_y$ 
C)  $\frac { \mu _ { y } } { \sqrt { n } }$ 
D)  $\frac { \sigma _ { y } } { \mu _ { y } }$ for n > 30. 
A)  $\frac { 1 } { n } \mu _ { \gamma }$ 
B)  $\mu_y$ 
C)  $\frac { \mu _ { y } } { \sqrt { n } }$ 
D)  $\frac { \sigma _ { y } } { \mu _ { y } }$ for n > 30.

Question 4

Let Y be a random variable. Then var(Y)equals

Accepted Answer

A)  $\sqrt { \left. E \left[ Y - \mu _ { y } \right) ^ { 2 } \right] }$ 
B)  $E \left[ \left| \left( Y - \mu _ { y } \right) \right| \right]$ 
C)  $E \left[ \left( Y - \mu _ { y } \right) ^ { 2 } \right]$ 
D)  $E [ ( Y - \mu ) ]$ 
A)  $\sqrt { \left. E \left[ Y - \mu _ { y } \right) ^ { 2 } \right] }$ 
B)  $E \left[ \left| \left( Y - \mu _ { y } \right) \right| \right]$ 
C)  $E \left[ \left( Y - \mu _ { y } \right) ^ { 2 } \right]$ 
D)  $E [ ( Y - \mu ) ]$

Question 5

The probability of an outcome

Accepted Answer

A) is the number of times that the outcome occurs in the long run. 
B) equals M × N, where M is the number of occurrences and N is the population size. 
C) is the proportion of times that the outcome occurs in the long run. 
D) equals the sample mean divided by the sample standard deviation. 
A) is the number of times that the outcome occurs in the long run. 
B) equals M × N, where M is the number of occurrences and N is the population size. 
C) is the proportion of times that the outcome occurs in the long run. 
D) equals the sample mean divided by the sample standard deviation.

Question 6

The expectations augmented Phillips curve postulates
?p = ? - f (u - $\bar { u }$ ),
where ?p is the actual inflation rate, ? is the expected inflation rate, and u is the unemployment rate, with "-" indicating equilibrium (the NAIRU - Non-Accelerating Inflation Rate of Unemployment). Under the assumption of static expectations (? = ?p-1), i.e., that you expect this period's inflation rate to hold for the next period ("the sun shines today, it will shine tomorrow"), then the prediction is that inflation will accelerate if the unemployment rate is below its equilibrium level. The accompanying table below displays information on accelerating annual inflation and unemployment rate differences from the equilibrium rate (cyclical unemployment), where the latter is approximated by a five-year moving average. You think of this data as a population which you want to describe, rather than a sample from which you want to infer behavior of a larger population. The data is collected from United States quarterly data for the period 1964:1 to 1995:4.
Joint Distribution of Accelerating Inflation and Cyclical Unemployment, 1964:1-1995:4
$\begin{array}{|c|c|c|c|}
\hline & \begin{array}{c}
(u-\bar{u})>0 \
(Y=0)
\end{array} & \begin{array}{c}
(u-\bar{u}) \geq 0 \
(Y=1)
\end{array} & \text { Total } \
\hline \begin{array}{c}
\Delta p-\Delta p-1>0 \
(X=0)
\end{array} & 0.156 & 0.383 & 0.539 \
\hline \begin{array}{c}
\Delta p-\Delta p-1 \leq 0 \
(X=1)
\end{array} & 0.297 & 0.164 & 0.461 \
\hline \text { Total } & 0.453 & 0.547 & 1.00 \
\hline
\end{array}$
(a)Compute E(Y)and E(X), and interpret both numbers.
(b)Calculate E(Y $\mid X$ = 1)and E(Y $\mid X$ = 0). If there was independence between cyclical unemployment and acceleration in the inflation rate, what would you expect the relationship between the two expected values to be? Given that the two means are different, is this sufficient to assume that the two variables are independent?
(c)What is the probability of inflation to increase if there is positive cyclical unemployment? Negative cyclical unemployment?
(d)You randomly select one of the 59 quarters when there was positive cyclical unemployment ((u - $\bar { u }$ )> 0). What is the probability there was decelerating inflation during that quarter?

Accepted Answer

(a)E(Y)= 0.547 . 54.7 percent of the qua

Question 7

Show that the correlation coefficient between Y and X is unaffected if you use a linear transformation in both variables. That is, show that corr(X,Y)= corr(X*, Y*), where X* = a + bX and Y* = c + dY, and where a, b, c, and d are arbitrary non-zero constants.

Accepted Answer

corr(X^*, Y^*)

Question 8

The conditional distribution of Y given X = x, Pr(Y = y $\mid X$ =x), is

Accepted Answer

A)  $\frac { \operatorname { Pr } ( Y = y ) } { \operatorname { Pr } ( X = x ) }$ 
B)  $\sum _ { i = 1 } ^ {l } \operatorname { Pr } \left( X = x _ { i } , Y = y \right)$ 
C)  $\frac { \operatorname { Pr } ( X = x , Y = y ) } { \operatorname { Pr } ( Y = y ) }$ 
D)  $\frac { \operatorname { Pr } ( X = x , Y = y ) } { \operatorname { Pr } ( X = x ) }$ 
A)  $\frac { \operatorname { Pr } ( Y = y ) } { \operatorname { Pr } ( X = x ) }$ 
B)  $\sum _ { i = 1 } ^ {l } \operatorname { Pr } \left( X = x _ { i } , Y = y \right)$ 
C)  $\frac { \operatorname { Pr } ( X = x , Y = y ) } { \operatorname { Pr } ( Y = y ) }$ 
D)  $\frac { \operatorname { Pr } ( X = x , Y = y ) } { \operatorname { Pr } ( X = x ) }$

Question 9

The kurtosis of a distribution is defined as follows:

Accepted Answer

A)  $\frac { E \left[ \left( Y - \mu _ { \gamma } \right) ^ { 4 } \right] } { \sigma _ { \gamma } ^ { 4 } }$ 
B)  $\frac { E \left[ \left( Y ^ { 4 } - \mu _ { y } ^ { 4 } \right) \right] } { \sigma _ { y } ^ { 2 } }$ 
C)  $\frac { \text { skewness } } { \operatorname { vat } ( Y ) }$ 
D) E[(Y - ${ } ^ { \mu } \gamma$ 4) 
A)  $\frac { E \left[ \left( Y - \mu _ { \gamma } \right) ^ { 4 } \right] } { \sigma _ { \gamma } ^ { 4 } }$ 
B)  $\frac { E \left[ \left( Y ^ { 4 } - \mu _ { y } ^ { 4 } \right) \right] } { \sigma _ { y } ^ { 2 } }$ 
C)  $\frac { \text { skewness } } { \operatorname { vat } ( Y ) }$ 
D) E[(Y - ${ } ^ { \mu } \gamma$ 4)

Question 10

To infer the political tendencies of the students at your college/university, you sample 150 of them. Only one of the following is a simple random sample: You

Accepted Answer

A) make sure that the proportion of minorities are the same in your sample as in the entire student body. 
B) call every fiftieth person in the student directory at 9 a.m. If the person does not answer the phone, you pick the next name listed, and so on. 
C) go to the main dining hall on campus and interview students randomly there. 
D) have your statistical package generate 150 random numbers in the range from 1 to the total number of students in your academic institution, and then choose the corresponding names in the student telephone directory. 
A) make sure that the proportion of minorities are the same in your sample as in the entire student body. 
B) call every fiftieth person in the student directory at 9 a.m. If the person does not answer the phone, you pick the next name listed, and so on. 
C) go to the main dining hall on campus and interview students randomly there. 
D) have your statistical package generate 150 random numbers in the range from 1 to the total number of students in your academic institution, and then choose the corresponding names in the student telephone directory.

Question 11

The systolic blood pressure of females in their 20s is normally distributed with a mean of 120 with a standard deviation of 9. What is the probability of finding a female with a blood pressure of less than 100? More than 135? Between 105 and 123? You visit the women's soccer team on campus, and find that the average blood pressure of the 25 members is 114. Is it likely that this group of women came from the same population?

Accepted Answer

Pr(Y<100)= 0.0131; Pr(Y>135)= 0.04

Question 12

A few years ago the news magazine The Economist listed some of the stranger explanations used in the past to predict presidential election outcomes. These included whether or not the hemlines of women's skirts went up or down, stock market performances, baseball World Series wins by an American League team, etc. Thinking about this problem more seriously, you decide to analyze whether or not the presidential candidate for a certain party did better if his party controlled the house. Accordingly you collect data for the last 34 presidential elections. You think of this data as comprising a population which you want to describe, rather than a sample from which you want to infer behavior of a larger population. You generate the accompanying table:

Joint Distribution of Presidential Party Affiliation and Party Control
of House of Representatives, 1860-1996 $$\begin{array} { | c | c | c | c | } 
\hline & \begin{array} { c } 
\text { Democratic Control } \
\text { of House } ( Y = 0 )
\end{array} & \begin{array} { c } 
\text { Republican Control } \
\text { of House } ( Y = 1 )
\end{array} & \text { Total } \
\hline \begin{array} { c } 
\text { Democratic } \
\text { President } ( X = 0 )
\end{array} & 0.412 & 0.030 & 0.441 \
\hline \begin{array} { c } 
\text { Republican } \
\text { President } ( X = 1 )
\end{array} & 0.176 & 0.382 & 0.559 \
\hline \text { Total } & 0.588 & 0.412 & 1.00 \
\hline
\end{array}$$ (a)Interpret one of the joint probabilities and one of the marginal probabilities.
(b)Compute E(X). How does this differ from E(X $\mid Y$ = 0)? Explain.
(c)If you picked one of the Republican presidents at random, what is the probability that during his term the Democrats had control of the House?
(d)What would the joint distribution look like under independence? Check your results by calculating the two conditional distributions and compare these to the marginal distribution.

Accepted Answer

(a)38.2 percent of the presidents were R

Question 13

Think of an example involving five possible quantitative outcomes of a discrete random variable and attach a probability to each one of these outcomes. Display the outcomes, probability distribution, and cumulative probability distribution in a table. Sketch both the probability distribution and the cumulative probability distribution.

Accepted Answer

Answers will vary by student.

Question 14

From the Stock and Watson (http://www.pearsonhighered.com/stock_watson)website the chapter 8 CPS data set (ch8_cps.xls)into a spreadsheet program such as Excel. For the exercise, use the first 500 observations only. Using data for average hourly earnings only (ahe), describe the earnings distribution. Use summary statistics, such as the mean, median, variance, and skewness. Produce a frequency distribution ("histogram")using reasonable earnings class sizes.

Accepted Answer

\[\begin{array} { | l | l | } 
\hline \t

Question 15

Consider the following linear transformation of a random variable y = $\frac { x - \mu _ { x } } { \sigma _ { x } }$ where ?x is the mean of x and ?x is the standard deviation. Then the expected value and the standard deviation of Y are given as

Accepted Answer

A) 0 and 1 
B) 1 and 1 
C) Cannot be computed because Y is not a linear function of X 
D)  $\frac { \mu } { \sigma _ { x } }$ and ?x 
A) 0 and 1 
B) 1 and 1 
C) Cannot be computed because Y is not a linear function of X 
D)  $\frac { \mu } { \sigma _ { x } }$ and ?x

Question 16

Two random variables X and Y are independently distributed if all of the following conditions hold, with the exception of

Accepted Answer

A) Pr(Y = y $\mid X$ = x)= Pr(Y = y). 
B) knowing the value of one of the variables provides no information about the other. 
C) if the conditional distribution of Y given X equals the marginal distribution of Y. 
D) E(Y)= E[E(Y $\mid X$ )]. 
A) Pr(Y = y $\mid X$ = x)= Pr(Y = y). 
B) knowing the value of one of the variables provides no information about the other. 
C) if the conditional distribution of Y given X equals the marginal distribution of Y. 
D) E(Y)= E[E(Y $\mid X$ )].

Question 17

The central limit theorem states that

Accepted Answer

A) the sampling distribution of $\frac { \bar { Y } - \mu _ {y} } { \sigma \bar { Y } }$ is approximately normal. 
B)  $\bar { Y } \xrightarrow{p} \mu _ { \gamma }$ 
C) the probability that $\bar { Y }$ is in the range $\mu_y$ ± c becomes arbitrarily close to one as n increases for any constant c > 0. 
D) the t distribution converges to the F distribution for approximately n > 30. 
A) the sampling distribution of $\frac { \bar { Y } - \mu _ {y} } { \sigma \bar { Y } }$ is approximately normal. 
B)  $\bar { Y } \xrightarrow{p} \mu _ { \gamma }$ 
C) the probability that $\bar { Y }$ is in the range $\mu_y$ ± c becomes arbitrarily close to one as n increases for any constant c > 0. 
D) the t distribution converges to the F distribution for approximately n > 30.

Question 18

The mean and variance of a Bernoille random variable are given as

Accepted Answer

A) cannot be calculated 
B) np and np(1-p) 
C) p and $\sqrt { p ( 1 - p ) }$ 
D) p and (1- p) 
A) cannot be calculated 
B) np and np(1-p) 
C) p and $\sqrt { p ( 1 - p ) }$ 
D) p and (1- p)

Question 19

The conditional expectation of Y given X, E(Y $X = x )$ , is calculated as follows:

Accepted Answer

A)  $\sum _ { i= 1 } ^ { k } Y _ { i } \operatorname { Pr } \left( X = x _ {i} \mid Y = y \right)$ 
B) E $[ E ( Y \mid X ) ] ]$ 
C)  $\sum _ { i= 1 } ^ { k } y _ { l } \operatorname { Pr } \left( Y = y _ {i} \mid X = x \right)$ 
D)  $\sum _ { i = 1 } ^ { k } E \left( Y \mid X = x _ { i } \right) \operatorname { Pr } \left( X = x _ { i } \right)$ 
A)  $\sum _ { i= 1 } ^ { k } Y _ { i } \operatorname { Pr } \left( X = x _ {i} \mid Y = y \right)$ 
B) E $[ E ( Y \mid X ) ] ]$ 
C)  $\sum _ { i= 1 } ^ { k } y _ { l } \operatorname { Pr } \left( Y = y _ {i} \mid X = x \right)$ 
D)  $\sum _ { i = 1 } ^ { k } E \left( Y \mid X = x _ { i } \right) \operatorname { Pr } \left( X = x _ { i } \right)$

Question 20

The central limit theorem

Accepted Answer

A) states conditions under which a variable involving the sum of Y1,..., Yn i.i.d. variables becomes the standard normal distribution. 
B) postulates that the sample mean $\bar { Y }$ is a consistent estimator of the population mean $\mu_y$ 
C) only holds in the presence of the law of large numbers. 
D) states conditions under which a variable involving the sum of Y1,..., Yn i.i.d. variables becomes the Student t distribution. 
A) states conditions under which a variable involving the sum of Y1,..., Yn i.i.d. variables becomes the standard normal distribution. 
B) postulates that the sample mean $\bar { Y }$ is a consistent estimator of the population mean $\mu_y$ 
C) only holds in the presence of the law of large numbers. 
D) states conditions under which a variable involving the sum of Y1,..., Yn i.i.d. variables becomes the Student t distribution.

The probability of an event A or B (Pr(A or B))to occur equals

In econometrics, we typically do not rely on exact or finite sample distributions because

The mean of the sample average $\bar { Y }$ , $E ( \bar { Y } )$ , is

Let Y be a random variable. Then var(Y)equals

The probability of an outcome

Show that the correlation coefficient between Y and X is unaffected if you use a linear transformation in both variables. That is, show that corr(X,Y)= corr(X^, Y^), where X^* = a + bX and Y^* = c + dY, and where a, b, c, and d are arbitrary non-zero constants.

The conditional distribution of Y given X = x, Pr(Y = y $\mid X$ =x), is

The kurtosis of a distribution is defined as follows:

To infer the political tendencies of the students at your college/university, you sample 150 of them. Only one of the following is a simple random sample: You

Consider the following linear transformation of a random variable y = $\frac { x - \mu _ { x } } { \sigma _ { x } }$ where ?_x is the mean of x and ?_x is the standard deviation. Then the expected value and the standard deviation of Y are given as

Two random variables X and Y are independently distributed if all of the following conditions hold, with the exception of

The central limit theorem states that

The mean and variance of a Bernoille random variable are given as

The conditional expectation of Y given X, E(Y $X = x )$ , is calculated as follows:

The central limit theorem

Economic Questions and Data

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

The Theory of Multiple Regression

Filters

Exam 2: Review of Probability

The probability of an event A or B (Pr(A or B))to occur equals

In econometrics, we typically do not rely on exact or finite sample distributions because

The mean of the sample average Yˉ\bar { Y }Yˉ , E(Yˉ)E ( \bar { Y } )E(Yˉ) , is

Let Y be a random variable. Then var(Y)equals

The probability of an outcome

Show that the correlation coefficient between Y and X is unaffected if you use a linear transformation in both variables. That is, show that corr(X,Y)= corr(X*, Y*), where X* = a + bX and Y* = c + dY, and where a, b, c, and d are arbitrary non-zero constants.

The conditional distribution of Y given X = x, Pr(Y = y ∣X\mid X∣X =x), is

The kurtosis of a distribution is defined as follows:

To infer the political tendencies of the students at your college/university, you sample 150 of them. Only one of the following is a simple random sample: You

Consider the following linear transformation of a random variable y = x−μxσx\frac { x - \mu _ { x } } { \sigma _ { x } }σx​x−μx​​ where ?x is the mean of x and ?x is the standard deviation. Then the expected value and the standard deviation of Y are given as

Two random variables X and Y are independently distributed if all of the following conditions hold, with the exception of

The central limit theorem states that

The mean and variance of a Bernoille random variable are given as

The conditional expectation of Y given X, E(Y X=x)X = x )X=x) , is calculated as follows:

The central limit theorem

Economic Questions and Data

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

The Theory of Multiple Regression

Filters

The mean of the sample average $\bar { Y }$ , $E ( \bar { Y } )$ , is

Show that the correlation coefficient between Y and X is unaffected if you use a linear transformation in both variables. That is, show that corr(X,Y)= corr(X^, Y^), where X^* = a + bX and Y^* = c + dY, and where a, b, c, and d are arbitrary non-zero constants.

The conditional distribution of Y given X = x, Pr(Y = y $\mid X$ =x), is

Consider the following linear transformation of a random variable y = $\frac { x - \mu _ { x } } { \sigma _ { x } }$ where ?_x is the mean of x and ?_x is the standard deviation. Then the expected value and the standard deviation of Y are given as

The conditional expectation of Y given X, E(Y $X = x )$ , is calculated as follows: