Deck 11: Regression With a Binary Dependent Variable

Question 1

In the probit model $\operatorname { Pr } \left( Y = 1 \mid = \Phi \left( \beta _ { 0 } + \beta _ { 1 } X \right) , \Phi \right.$

A) is not defined for $\Phi ( 0 )$
B) is the standard normal cumulative distribution function.
C) is set to 1.96 .
D) can be computed from the standard normal density function.

Answer 1

is the standard normal cumulative distribution function.

Question 2

Nonlinear least squares

A)solves the minimization of the sum of squared predictive mistakes through sophisticated mathematical routines, essentially by trial and error methods.
B)should always be used when you have nonlinear equations.
C)gives you the same results as maximum likelihood estimation.
D)is another name for sophisticated least squares.

Answer 2

A

Question 3

The linear probability model is

A)the application of the multiple regression model with a continuous left-hand side variable and a binary variable as at least one of the regressors.
B)an example of probit estimation.
C)another word for logit estimation.
D)the application of the linear multiple regression model to a binary dependent variable.

Answer 3

D

Question 4

The binary dependent variable model is an example of a

A)regression model, which has as a regressor, among others, a binary variable.
B)model that cannot be estimated by OLS.
C)limited dependent variable model.
D)model where the left-hand variable is measured in base 2.

Question 5

The major flaw of the linear probability model is that

Question 6

The logit model derives its name from

A)the logarithmic model.
B)the probit model.
C)the logistic function.
D)the tobit model.

Question 7

In the probit model $\operatorname { Pr } \left( Y = 1 \mid X _ { 1 } , X _ { 2 } , \ldots , X _ { k } \right) = \Phi \left( \beta _ { 0 } + \beta _ { 1 } X _ { 1 } + \beta _ { x } X _ { 2 } + \ldots + \beta _ { k } X _ { k } \right)$

A) the $\beta \text { 's }$ do not have a simple interpretation.
B) the slopes tell you the effect of a unit increase in X on the probability of Y .
C) $\beta _ { 0 }$ cannot be negative since probabilities have to lie between 0 and 1 .
D) $\beta _ { 0 }$ is the probability of observing Y when all X 's are 0 .

Question 8

In the binary dependent variable model, a predicted value of 0.6 means that

A)the most likely value the dependent variable will take on is 60 percent.
B)given the values for the explanatory variables, there is a 60 percent probability that the dependent variable will equal one.
C)the model makes little sense, since the dependent variable can only be 0 or 1.
D)given the values for the explanatory variables, there is a 40 percent probability that the dependent variable will equal one.

Question 9

In the expression $\operatorname { Pr } ( \text { deny } = 1 \mid P / I \text { Ratio, black } ) = \Phi ( - 2.26 + 2.74 P / \text { ratio } + 0.71 \text { black } )$
the effect of increasing the $R ^ { 2 }$ ratio from 0.3 to 0.4 for a white person

A) is 0.274 percentage points.
B) is 6.1 percentage points.
C) should not be interpreted without knowledge of the regression
D) is 2.74 percentage points.

Question 10

The maximum likelihood estimation method produces, in general, all of the following desirable properties with the exception of

A)efficiency.
B)consistency.
C)normally distributed estimators in large samples.
D)unbiasedness in small samples.

Question 11

When having a choice of which estimator to use with a binary dependent variable, use

A)probit or logit depending on which method is easiest to use in the software package at hand.
B)probit for extreme values of X and the linear probability model for values in between.
C)OLS (linear probability model)since it is easier to interpret.
D)the estimation method which results in estimates closest to your prior expectations.

Question 12

In the expression $\operatorname { Pr } \left( Y = 1 \mid = \boldsymbol { \Phi } \left( \beta _ { 0 } + \beta _ { 1 } X \right) \right.$

A) $\left( \beta _ { 0 } + \beta _ { 1 } X \right)$ plays the role of z in the cumulative standard normal distribution function.
B) $\beta _ { 1 }$ cannot be negative since probabilities have to lie between 0 and 1 .
C) $\beta _ { 0 }$ cannot be negative since probabilities have to lie between 0 and 1 .
D) $\left( \beta _ { 0 } + \beta _ { 1 } X \right) > 0$ since probabilities have to lie between 0 and 1 .

Question 13

(Requires material from Section 11.3 - possibly skipped) For the measure of fit in your regression model with a binary dependent variable, you can meaningfully use the

A) regression $R ^ { 2 }$
B) size of the regression coefficients.
C) pseudo $R ^ { 2 }$
D) standard error of the regression.

Question 14

The probit model

A)is the same as the logit model.
B)always gives the same fit for the predicted values as the linear probability model for values between 0.1 and 0.9.
C)forces the predicted values to lie between 0 and 1.
D)should not be used since it is too complicated.

Question 15

The following tools from multiple regression analysis carry over in a meaningful manner to the linear probability model, with the exception of the

Question 16

The logit model can be estimated and yields consistent estimates if you are using

A)OLS estimation.
B)maximum likelihood estimation.
C)differences in means between those individuals with a dependent variable equal to one and those with a dependent variable equal to zero.
D)the linear probability model.

Question 17

$E \left( Y \mid X _ { 1 } , \ldots , X _ { k } \right) = \operatorname { Pr } \left( Y = 1 \mid X _ { 1 } , \ldots , X _ { k } \right)$
means that

A) for a binary variable model, the predicted value from the population regression is the probability that Y=1 , given X .
B) dividing Y by the X 's is the same as the probability of Y being the inverse of the sum of the X 's.
C) the exponential of Y is the same as the probability of Y happening.
D) you are pretty certain that Y takes on a value of 1 given the X 's.

Question 18

(Requires Appendix Material)The following are examples of limited dependent variables, with the exception of

A)binary dependent variable.
B)log-log specification.
C)truncated regression model.
D)discrete choice model.

Question 19

In the linear probability model, the interpretation of the slope coefficient is

A)the change in odds associated with a unit change in X, holding other regressors constant.
B)not all that meaningful since the dependent variable is either 0 or 1.
C)the change in probability that Y=1 associated with a unit change in X, holding others regressors constant.
D)the response in the dependent variable to a percentage change in the regressor.

Question 20

(Requires advanced material)Only one of the following models can be estimated by OLS :

Question 21

Sketch the regression line for the linear probability model with a single regressor.Indicate
for which values of the slope and intercept the predictions will be above one and below
zero.Can you rule out homoskedasticity in the error terms with certainty here?

Question 22

(Requires Appendix material and Calculus)The log of the likelihood function (L)for the
simple regression model with i.i.d.normal errors is as follows (note that taking the
logarithm of the likelihood function simplifies maximization.It is a monotonic
transformation of the likelihood function, meaning that this transformation does not affect
the choice of maximum): X
Derive the maximum likelihood estimator for the slope and intercept.What general
properties do these estimators have? Explain intuitively why the OLS estimator is
identical to the maximum likelihood estimator here.

Question 23

Consider the following probit regression

Question 24

You have a limited dependent variable (Y)and a single explanatory variable (X).You
estimate the relationship using the linear probability model, a probit regression, and a
logit regression.The results are as follows: (a)

Question 25

The following problems could be analyzed using probit and logit estimation with the exception of whether or not

A)a college student decides to study abroad for one semester.
B)being a female has an effect on earnings.
C)a college student will attend a certain college after being accepted.
D)applicants will default on a loan.

Question 26

A study analyzed the probability of Major League Baseball (MLB)players to "survive"
for another season, or, in other words, to play one more season.The researchers had a
sample of 4,728 hitters and 3,803 pitchers for the years 1901-1999.All explanatory
variables are standardized.The probit estimation yielded the results as shown in the
table: where the limited dependent variable takes on a value of one if the player had one more
season (a minimum of 50 at bats or 25 innings pitched), number of seasons played is
measured in years, performance is the batting average for hitters and the earned run
average for pitchers, and average performance refers to performance over the career.
16
(a)Interpret the two probit equations and calculate survival probabilities for hitters and
pitchers at the sample mean.Why are these so high?

Question 27

Consider the following logit regression:

Question 28

When estimating probit and logit models,

Question 29

(Requires Appendix material)Briefly describe the difference between the following
models: censored and truncated regression model, count data, ordered responses, and
discrete choice data.Try to be specific in terms of describing the data involved.

Question 30

(Requires advanced material)Maximum likelihood estimation yields the values of the coefficients that

A)minimize the sum of squared prediction errors.
B)maximize the likelihood function.
C)come from a probability distribution and hence have to be positive.
D)are typically larger than those from OLS estimation.

Question 31

Besides maximum likelihood estimation of the logit and probit model, your textbook
mentions that the model can also be estimated by nonlinear least squares.Construct the
sum of squared prediction mistakes and suggest how computer algorithms go about
finding the coefficient values that minimize the function.You may want to use an
analogy where you place yourself into a mountain range at night with a flashlight shining
at your feet.Your task is to find the lowest point in the valley.You have two choices to
make: the direction you are walking in and the step length.Describe how you will
proceed to find the bottom of the valley.Once you find the lowest point, is there any
guarantee that this is the lowest point of all valleys? What should you do to assure this?

Question 32

The Report of the Presidential Commission on the Space Shuttle Challenger Accident in
1986 shows a plot of the calculated joint temperature in Fahrenheit and the number of O-
rings that had some thermal distress.You collect the data for the seven flights for which
thermal distress was identified before the fatal flight and produce the accompanying plot. (a)Do you see any relationship between the temperature and the number of O-ring failures?
If you fitted a linear regression line through these seven observations, do you think the
slope would be positive or negative? Significantly different from zero? Do you see any
problems other than the sample size in your procedure?

Question 33

Your task is to model students' choice for taking an additional economics course after the
first principles course.Describe how to formulate a model based on data for a large
sample of students.Outline several estimation methods and their relative advantage over
other methods in tackling this problem.How would you go about interpreting the
resulting output? What summary statistics should be included?

Question 34

A study investigated the impact of house price appreciation on household mobility.The
underlying idea was that if a house were viewed as one part of the household's portfolio,
then changes in the value of the house, relative to other portfolio items, should result in
investment decisions altering the current portfolio.Using 5,162 observations, the logit
equation was estimated as shown in the table, where the limited dependent variable is one
if the household moved in 1978 and is zero if the household did not move:
14 where male, black, married78, and marriage change are binary variables.They indicate,
respectively, if the entity was a male-headed household, a black household, was married,
and whether a change in marital status occurred between 1977 and 1978.A7983 is the
appreciation rate for each house from 1979 to 1983 minus the SMSA-wide rate of
appreciation for the same time period, and PNRN is a predicted appreciation rate for the
unit minus the national average rate.
(a)Interpret the results.Comment on the statistical significance of the coefficients.Do the
slope coefficients lend themselves to easy interpretation?

Question 35

Earnings equations establish a relationship between an individual's earnings and its
determinants such as years of education, tenure with an employer, IQ of the individual,
professional choice, region within the country the individual is living in, etc.In addition,
binary variables are often added to test for "discrimination" against certain sub-groups of
the labor force such as blacks, females, etc.Compare this approach to the study in the
textbook, which also investigates evidence on discrimination.Explain the fundamental
differences in both approaches using equations and mathematical specifications whenever
possible.

Question 36

To measure the fit of the probit model, you should:

Question 37

The population logit model of the binary dependent variable Y with a single regressor is Logistic functions also play a role in econometrics when the dependent variable is not a
binary variable.For example, the demand for televisions sets per household may be a
function of income, but there is a saturation or satiation level per household, so that a
linear specification may not be appropriate.Given the regression model sketch the regression line.How would you go about estimating the coefficients?

Question 38

(Requires Appendix material and Calculus)The logarithm of the likelihood function (L)
for estimating the population mean and variance for an i.i.d.normal sample is as follows
(note that taking the logarithm of the likelihood function simplifies maximization.It is a
monotonic transformation of the likelihood function, meaning that this transformation
does not affect the choice of maximum):
Derive the maximum likelihood estimator for the mean and the variance.How do they
differ, if at all, from the OLS estimator? Given that the OLS estimators are unbiased,
what can you say about the maximum likelihood estimators here? Is the estimator for the
variance consistent?

Question 39

(Requires advanced material)Nonlinear least squares estimators in general are not

A)consistent.
B)normally distributed in large samples.
C)efficient.
D)used in econometrics.

Question 40

A study tried to find the determinants of the increase in the number of households headed
by a female.Using 1940 and 1960 historical census data, a logit model was estimated to
predict whether a woman is the head of a household (living on her own)or whether she is
living within another's household.The limited dependent variable takes on a value of one
if the female lives on her own and is zero if she shares housing.The results for 1960
using 6,051 observations on prime-age whites and 1,294 on nonwhites were as shown in
the table: where age is measured in years, education is years of schooling of the family head, farm
status is a binary variable taking the value of one if the family head lived on a farm, south
is a binary variable for living in a certain region of the country, expected family earnings
was generated from a separate OLS regression to predict earnings from a set of
regressors, and family composition refers to the number of family members under the age
of 18 divided by the total number in the family.
The mean values for the variables were as shown in the table. (a)Interpret the results.Do the coefficients have the expected signs? Why do you think age
was entered both in levels and in squares?