This exercise deals with the loan data described in Exercise 6.12. The ‘‘Chow’’ test was introduced in Chapter 7.2.3 for testing the equality of coefficients in two regressions on subsets of observations. Here we ask a similar question concerning the parameters of the probit model for deliquency for the two subpopulations of borrowers who either have mortgage insurance (INSUR ¼ 1) or do not (INSUR ¼ 0).

(a) Estimate the probit model for DELINQUENT using all explanatory variables except INSUR and all observations. Call the value of the log-likelihood function evaluated at the maximum likelihood estimates ln LR.

(b) Re-estimate the model in

(a) using the sample observations for which INSUR

¼ 0. Call the value of the log-likelihood function evaluated at the maximum likelihood estimates ln L0.

(c) Re-estimate the model in

(b) using the sample observations for which INSUR

¼ 1. Call the value of the log-likelihood function evaluated at the maximum likelihood estimates ln L1.

(d) Compare the estimates from the models in (a),

(b) and (c). What major differences in coefficient signs, magnitudes, and significance do you observe?

(e) Re-estimate the model in (a), including each explanatory variable, INSUR, and interactions with INSUR. Compare the value of the log-likelihood function from the fully interacted model, call it ln LU, to ln L0þln L1. If you have done things correctly,thenlnLU shouldequallnL0þlnL1.Canyouexplainwhythismust be so?

(f) Carry out a likelihood ratio version of the Chow test by computing LR ¼ 2 ln ð Þ LU  ln LR . What is the appropriate critical value for a test at the 5% level of significance? What conclusion do you draw about the subgroups of individuals who do and do not have mortgage insurance? Do the two groups behave in the same way?