Exam 11: Regression With a Binary Dependent Variable

Consider the following probit regression Pr(Y = 1 | X)= Φ(8.9 - 0.14 × 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 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): L = - $\frac { n } { 2 }$ log(2πσ²)- $\frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \mu _ { Y } \right) ^ { 2 }$ 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?

(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.

The estimated logit regression in your textbook is $\widehat{\operatorname { Pr } ( \text { deny } = 1 \mid \text { P/Iratio,black } )}$ = F(-4.13 + 5.37 P/Iratio + 1.27 black) Using a spreadsheet program, such as Excel, generate a table with predicted probabilities for both whites and blacks using P/I Ratiovalues between 0 and 1 and increments of 0.05.

(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): L = - $\frac { n } { 2 }$ log(2π)- $\frac { n } { 2 }$ log σ² - $\frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \beta _ { 0 } - \beta _ { 1 } X _ { i } \right) ^ { 2 }$ 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.

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?

The estimated logit regression in your textbook is $\widehat{\operatorname { Pr } ( \text { deny } = 1 \mid \text { P/Iratio,black } )}$ = F(-4.13 + 5.37 P/Iratio + 1.27 black) Is there a meaningful interpretation to the slope for the P/I Ratio? Calculate the increase of a rejection probability for both blacks and whites as the P/I Ratio increases from 0.1 to 0.2. Repeat the exercise for an increase from 0.65 to 0.75. Why is the increase in the probability higher for blacks at the smaller value of the P/I Ratio but higher for whites at the larger P/I Ratio?

The population logit model of the binary dependent variable Y with a single regressor is Pr(Y=1 | X₁)= $\frac { 1 } { 1 + e ^ { - \left( \beta _ { 0 } + \beta _ { 1 } X _ { 1 } \right) } }$ 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 Y_i = $\frac { \beta _ { 0 } } { 1 + \beta _ { 1 } e ^ { - \beta _ { 2 } X _ { i } } }$ + u_i, sketch the regression line. How would you go about estimating the coefficients?

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: =2.858-0.037\timesX (0.007) Pr (Y=1\midX)=F(15.297-0.236\timesX) Pr (Y=1\midX)=\Phi(8.900-0.137\timesX) (0.058) (a)Although you cannot compare the coefficients directly, you are told that "it can be shown" that certain relationships between the coefficients of these models hold approximately. These are for the slope: $\hat \beta _ { \text {probit } }$ ? 0.625 × $\hat \beta \text { Logit }$ , $\hat \beta \text { linear }$ ? 0.25 × $\hat \beta \text { Logit }$ Take the logit result above as a base and calculate the slope coefficients for the linear probability model and the probit regression. Are these values close? (b)For the intercept, the same conversion holds for the logit-to-probit transformation. However, for the linear probability model, there is a different conversion: $\hat \beta _ { 0 , \text { linear } }$ ? 0.25 × $\hat \beta 0 , \text { Logit }$ + 0.5 Using the logit regression as the base, calculate a few changes in X (temperature in degrees of Fahrenheit)to see how good the approximations are.

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.