Exam 11: Regression With a Binary Dependent Variable

The major flaw of the linear probability model is that a. the actuals can only be 0 and 1 , but the predicted are almost always different from that. b. the regression $R ^ { 2 }$ cannot be used as a measure of fit. c. people do not always make clear-cut decisions. d. the predicted values can lie above 1 and below 0 .

Consider the following probit regression $\operatorname { Pr } ( Y = 1 \mid X ) = \Phi ( 8.9 - 0.14 \times X )$ Calculate the change in probability for $X$ increasing by 10 for $X = 40$ and $X = 60$ . Why is there such a large difference in the change in probabilities?

(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

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

(Requires advanced material)Only one of the following models can be estimated by OLS : a. $Y = A K ^ { \alpha } L ^ { \beta } + u$ . b. $\operatorname { Pr } ( Y = 1 \mid X ) = \Phi \left( \beta _ { 0 } + \beta _ { 1 } X \right)$ . c. $\operatorname { Pr } ( Y = 1 \mid X ) = F \left( \beta _ { 0 } + \beta _ { 1 } X \right) = \frac { 1 } { 1 + e ^ { - \left( \beta _ { 0 } + \beta _ { 1 } X \right) } }$ . d. $Y = A K ^ { \alpha } L ^ { \beta } u$ .

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

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 Regression model Logit constant -3.323 (0.180) Male -0.567 (0.421) Black -0.954 (0.515) Married 78 0.054 (0.412) marriage 0.764 change (0.416) A7983 -0.257 (0.921) PNRN -4.545 (3.354) Pseudo- 0.016 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?

The probit model

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

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

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

When estimating probit and logit models, a. the $t$ -statistic should still be used for testing a single restriction. b. you cannot have binary variables as explanatory variables as well. c. $F$ -statistics should not be used, since the models are nonlinear. d. it is no longer true that the $\bar { R } ^ { 2 } < R ^ { 2 }$ .

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

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: Regression (1) Hitters (2) Pitchers Regression model probit probit constant 2.010 1.625 (0.030) (0.031) number of seasons -0.058 -0.031 played (0.004) (0.005) performance 0.794 0.677 (0.025) (0.026) average performance 0.022 0.100 (0.033) (0.036) 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?

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.

To measure the fit of the probit model, you should: a. use the regression $R ^ { 2 }$ . b. plot the predicted values and see how closely they match the actuals. c. use the log of the likelihood function and compare it to the value of the likelihood function. d. use the fraction correctly predicted or the pseudo $R ^ { 2 }$ .

Consider the following logit regression: $\operatorname { Pr } ( Y = 1 \mid X ) = F ( 15.3 - 0.24 \times X )$ Calculate the change in probability for $X$ increasing by 10 for $X = 40$ and $X = 60$ . Why is there such a large difference in the change in probabilities?

Nonlinear least squares

The following tools from multiple regression analysis carry over in a meaningful manner to the linear probability model, with the exception of the a. $F$ -statistic. b. significance test using the $t$ -statistic. c. $95 \%$ confidence interval using $\pm 1.96$ times the standard error. d. regression $R ^ { 2 }$ .

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?