Exam 11: Regression With a Binary Dependent Variable

The population logit model of the binary dependent variable Y with a single regressor is $\operatorname { Pr } \left( Y = 1 \mid X _ { 1 } \right) = \frac { 1 } { 1 + e ^ { - \left( \beta _ { 0 } + \beta _ { 1 } X _ { 1 } \right) } }$ 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?

The logit model derives its name from

The binary dependent variable model is an example of a

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

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: Regression (1) White (2) Nonwhite Regression model Logit Logit Constant 1.459 -2.874 (0.685) (1.423) Age -0.275 0.084 (0.037) (0.068) age squared 0.00463 0.00021 (0.00044) (0.00081) Education -0.171 -0.127 (0.026) (0.038) farm status -0.687 -0.498 (0.173) (0.346) South 0.376 -0.520 (0.098) (0.180) expected family 0.0018 0.0011 earnings (0.00019) (0.00024) family composition 4.123 2.751 (0.294) (0.345) Pseudo- 0.266 0.189 Percent Correctly 82.0 83.4 Predicted 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. Variable (1) White mean (2) Nonwhite mean age 46.1 42.9 age squared 2,263.5 1,965.6 education 12.6 10.4 farm status 0.03 0.02 south 0.3 0.5 expected family earnings 2,336.4 1,507.3 family composition 0.2 0.3 (a)Interpret the results.Do the coefficients have the expected signs? Why do you think age was entered both in levels and in squares?

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: $\widehat { \beta } _ { \text {probit } } \approx 0.625 \times \widehat { \beta } _ { \text {Logit } } , \widehat { \beta } _ { \text {linear } } \approx 0.25 \times \widehat { \beta } _ { \text {Logit } }$ . 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?

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

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

(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.

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

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?

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

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?

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

(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): $L = - \frac { n } { 2 } \log \left( 2 \pi \sigma ^ { 2 } \right) - \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \mu _ { Y } \right) ^ { 2 }$ 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?

The linear probability model is

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

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?

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

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