Due on Thursday November 17, 2022 (by 11:59 PM EST)

Note: No credit will be given if you report only the final answers without showing formulas and                calculations when appropriate. This applies to both theoretical and empirical questions. For the               empirical questions, make sure to submit the R scripts and output on Latte. No credit will be given if the R output is missing.

Problem 1

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, expectedfamily earnings was generated from a separate OLS regression to predict earnings from a set of regressors, andfamily         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?

(b) Calculate the difference in the predicted probability between whites and nonwhites at the   sample mean values of the explanatory variables. Why do you think the study did not combine the observations and allowed for a nonwhite binary variable to enter?

(c) What would be the effect on the probability of a nonwhite woman living on her own, if        education andfamily composition were changed from their current mean to the mean of whites, while all other variables were left unchanged at the nonwhite mean values?

Problem 2

The logit regression in Chapter 11 of your textbook reads:

 = F(-4.13 + 5.37 P/Iratio + 1.27 black)

(a) Using a spreadsheet program such as Excel, plot the following logistic regression function

with a single X, i =                                   where  0 = -4. 13,  1 = 5.37, 2 = 1.27.

Enter values for X1 in the first column starting from 0 and then increment these by 0.1 until you

reach 2.0. Let X2 be 0 at first. Then enter the logistic function formula in the next column. Next allow X2 to be 1 and calculate the new values for the logistic function in the third column.         Finally produce the predicted probabilities for both blacks and whites, connecting the predicted values with a line.

(b) Using the same spreadsheet calculations, list how the probability increases for blacks and for whites as the P/I ratio increases from 0.5 to 0.6.

(c) What is the difference in the rejection probability between blacks and whites for a P/I ratio of

0.5 and for 0.9? Why is the difference smaller for the higher value here?

(d) Table 11.2 on page 368 of your textbook lists a logit regression (column 2) with further  explanatory variables. Given that you can only produce simple plots in two dimensions, how would you proceed in (a) above if there were more than a single explanatory variable?

Problem 3

You have a sample size of size ! = 1 with data $%  = 2, (%  = 1. You are interested in the value of ) in the regression * = +) + - (Note that there is no intercept).

a.   Plot the sum of squared residuals ($%  − 0(%)2  as a function of 0 .

b.   Show that the least squares estimated of ) is equal to 2.

c.   Using 345678   = 1, plot the Ridge penalty term 345678 02  as a function of 0 .

d.   Using 345678  = 1, plot the Ridge penalized sum of squared residuals, ($%  − 0(%)2  + 34(2)5678 02 .

e.   Find the value of )945678 .

f.   Using 345678  = 0.5, repeat parts (c) and (d). Also, find the new value of )945678 .

g.   Using 345678  = 3, repeat parts (c) and (d). Also, find the new value of )945678 .

h.   Use the graphs that you produced in parts (a)-(d) for the various values of 345678  to explain why a large value of 345678  results in more shrinkage of the OLS estimates.