Question 6 Discrimination laws in the USA require that two people, identical but

Question

Question 6 Discrimination laws in the USA require that two people, identical but for their race, should be equally likely to have a mortgage application accepted. In order to investigate whether there is any discrimination by race in the lending market, the Federal Reserve Bank of Boston collected data on 2,380 mortgage applications filed in Boston in 1990 Table 4 provides descriptions of the key variables and some mortgage denial regressions are presented in Table 5 In these data 28% of black applicants were denied a mortgage while only 9% of white applicants were denied. Discuss whether or not this constitutes evidence of racial discrimination in lending practices Consider the linear probability model (LPM) results in column (1) of Table 5. Suppose, two applicants, one white and one black, apply for a mortgage. They have the same values for all regressors other than race. How much more likely is the black applicant to be denied a mortgage? Provide a complete interpretation of this result in terms of both economic and statistical significance. Explain any differences between this result and that obtained in (i) Your econometrically naïve friend comments on the large differences in the estimated coefficient on the black dummy variable across the three sets of regression results and how strong the evidence is in favour of discrimination that emerges from the logit estimates. Explain what is wrong with this interpretation and provide an appropriate comparison of the LPM and logit estimates of the impact of race on the probability of being denied a mortgage Consider the probit estimates in column (3) of Table 5. Provide a complete interpretation of the estimated coefficient on the insurance dummy and the impact of this variable on the probability of mortgage denial (i) (ii) (iii) (iv) Table 4: Variables included in regression models of mortgage decisions Variable Definition Deny PI Sample mean 0.12 0.33 1 if mortgage application was denied, 0 otherwise Ratio of total monthly debt repayments to total monthly income Consumer credit score index that varies from a minimumm of 1 if no slow payments or delinquencies to a maximum of 6 if have delinquent credit history with payments 90 days overdue Mortgage credit score index that varies from a minimum| of 1 if no late mortgage payments to a maximum of 4 if more than two late mortgage payments 1 if any public record of credit problems (eg bankruptcy), 0 otherwise CScore MScore 1.7 0.07 0.02 0.14 Insurance1 if applicant applied for mortgage insurance and was Black Self denied, 0 otherwise 1 if applicant is black, 0 otherwise 1 if applicant self-employed, 0 otherwise 0.12

Explanation / Answer

ANSWER B

The linear probability models are modeled as such: E (Y | X) = Prob ( Y = 1 | X)

= A + B1*X1 + B2*X2 + B3*X3 + B4*X4 + …

where, Y = application denied (=1, if yes & = 0, if not)

and X = are the individual independent regressors (or predictors)

Let’s say X1 stands for a flag, which indicates race. So it takes value = 1, for Black and = 0, if white

The question says every other regressor i.e. X2, X3 … are the same for both Blacks and Whites and hence the only variable that is different is X1

From the table we see that the coefficient B1 in LPM model for Black (i.e. X1) is 0.084. This implies that everything else remaining the same, whenever an applicant is Black (i.e. X1), the probability Prob ( Y = 1 | X) i.e. probability of denial goes up by 0.084. 0.084 times more likely!

The asymptotic standard error for the coefficient B1 is 0.023, which means the t-test which measures whether or not the coefficient i.e. B1 = 0 or not can be computed by:

Null: Estimated B1 = Hypothesized B1 (=0)

Alternate: Estimated B1 Hypothesized B1 (=0)

t-value = (0.084 – 0) / SE = 0.084 / 0.023 = 3.65

From the t-table, at 95% significance and degrees of freedom (2379), gives a t-value which can be assumed to be from the same population = 1.960. This is the extreme t-value which can be due to ‘chance’ i.e. if Null was true.

Since our estimated t-value, 3.65 > 1.96, we can safely assume that NULL is rejected and Alternate is true i.e. Estimated B1 0.

ANSWER C

We saw the LPM in previous question.

The logit probability models are modeled as such: E (Y | X) = Prob (Y = 1 | X)

= 1 / (1 + e-Z) where,

Z = A + B1*X1 + B2*X2 + B3*X3 + B4*X4 + …

where, Y = application denied (=1, if yes & = 0, if not)

and X = are the individual independent regressors (or predictors)

The LPM provides a direct impact on Prob (Y = 1 | X), whereas in Logit models they are different.

Prob (Y = 1 | X) = 1 / (1 + e-Z)

Prob (Y = 1 | X) = eZ / (eZ + 1)

1/ Prob (Y = 1 | X) = (eZ + 1) / eZ

1/ Prob (Y = 1 | X) = 1 + e-Z

[1/ Prob (Y = 1 | X)] – 1 = e-Z

[1 - Prob (Y = 1 | X)] / Prob (Y = 1 | X) = e-Z

[Prob (Y = 0 | X)] / Prob (Y = 1 | X) = e-Z

Prob (Y = 1 | X) / [Prob (Y = 0 | X)] = eZ

Natural log i.e. LN

LN(Prob (Y = 1 | X) / Prob (Y = 0 | X)) = eZ

In english, the above statement implies log of odds of denial are directly proportional to Z (where, Z stated above is = Z = A + B1*X1 + B2*X2 + B3*X3 + B4*X4 + …).

Odds are nothing but probability of two classes in a ratio i.e. Prob (Y = 1 | X) / Prob (Y = 0 | X). In english, this means odds of denial (against acceptance).

Hence from this equation, value of B1 = 0.688, implies everything else remaining the same, the log of odds goes up by 0.688.

With little more math…

odds à goes up by e0.688

odds à goes up by ~ 1.99, for convenience sake, let’s assume 2

The difference between odds and probability explained here: Say odds is equal to c

Odds = Prob (Y = 1 | X) / Prob (Y = 0 | X) = c

          Prob (Y = 1 | X) / (1 - Prob (Y = 1 | X)) = 2

          Prob (Y = 1 | X) = c * (1 - Prob (Y = 1 | X))

           Prob (Y = 1 | X) = c – c*Prob (Y = 1 | X))

           (1+c) * Prob (Y = 1 | X) = c

           Prob (Y = 1 | X) = c / (1+c)

Say, for Whites odds are cw & Blacks it is cb

Prob (Y = 1 | X1 = 1 (Black) | everything else same) = cb / (1+ cb)

Prob (Y = 1 | X1 = 0 (Black) | everything else same) = cw / (1+ cw)

So, when we say odds go up by 2 it means….

cb / (1+ cb) = 2 + cw / (1+ cw)

So, there’s a huge difference between Prob (Y = 1 | X) coefficient interpretation in LPM vs. LOGIT models.

Total Applications 2380 Mean of Blacks 0.14 Number of Blacks 333.2 333 Rounded Number of Whites 2047 Rounded Mean of Denials 0.12 Number of Denials 285.6 286 Rounded Blacks Whites # Samples 333 2047 Mean Denial 0.28 0.09 # Denials (rounded) 93 184 ANSWER A The difference in means i.e. 0.28 and 0.09 alone does not qualify as evidence that the evaluation of applications for denial between blacks and whites are from the same population i.e. no racial discriminination. We need to perform t-tests to understand the difference in means better. Only if the difference is statistically significant, we can say that they come from different populations i.e. there's evidence of racial discrimination in how applications are processed between blacks and whites

Related Questions

Navigate

• Browse (All)
• Browse Q
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.