Unobserved Heterogeneity in Models of Competing Mortgage.docx

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1、1 Unobserved Heterogeneity in Models of Competing Mortgage Termination Risks John M. Clapp,* Yongheng Deng* and Xudong An* Abstract This paper extends unobserved heterogeneity to the multinomial logit model (MNL) framework in the context of mortgages terminated by refinance, move, or default. It tes

2、ts for the importance of unobserved heterogeneity when borrower characteristics such as income, age and credit score are included to capture lender-observed heterogeneity. It does this by comparing the proportional hazard model (PHM) to MNL with and without mass-point estimates of unobserved heterog

3、eneous groups of borrowers. The mass point mixed hazard model (MMH) yields larger and more significant coefficients for several important variables in the move model, whereas the MNL model without unobserved heterogeneity performs well with the refinance estimates. The MMH clearly dominates the alte

4、rnative models in-sample and out-of-sample. However, it is sometimes difficult to obtain convergence for the models estimated jointly with mass points. JEL classification: G21; C25; C41; C52; D12 * School of Business Administration, University of Connecticut, Storrs CT, 06269-1041 or john.clappbusin

5、ess.uconn.edu. *School of Policy, Planning and Development, University of Southern California, Los Angeles CA, 90089- 0626 or ydengusc.edu. * School of Policy, Planning and Development, University of Southern California, Los Angeles CA, 90089- 0626 or xudonganusc.edu. 2 Introduction Recent researche

6、s on mortgage borrowers behavior have proposed several models for the competing risks of mortgage termination by refinancing, moving and/or default (Deng, Quigley and Van Order 2000, Clapp et al. 2001, Deng and Quigley 2001).1 Clapp et al. (2001) present evidence that a multinomial logit model (MNL)

7、 with restructured event history data is an attractive alternative to duration models such as the proportional hazard model (PHM). The MNL allows direct competition among the choices: the probabilities of termination risks, and the probability of continuing to pay, must sum to one. Thus, an increase

8、 in one termination probability must be offset by a decline in probability for one or more of the alternatives. On the other hand, the MNL cannot allow correlations among the termination risks through unobservable variables, as implied by the independence from irrelevant alternatives (IIA) assumptio

9、n.2 In addition, the MNL requires the i.i.d. assumption for a given agent observed over time3 following standard practice, we stack the observations of historical events for each agent into our likelihood function. This logic also requires complicated formulation of variables measuring duration depe

10、ndency. By way of contrast, the hazard function in a proportional hazard model (PHM) is constructed in a path dependent framework: i.e., the hazard rate of termination is conditioned on the subject surviving up to time t-1. Therefore, any event between t and t-1 is not an i.i.d. event. The full maxi

11、mum likelihood estimation approach also allows researchers to estimate a PHM with correlated competing risks.4 Although the multinomial logit model (MNL) and proportional hazard model (PHM) differ in the above-mentioned perspectives, they are both widely used in the literature of mortgage terminatio

12、n risks and demonstrated to be effective in the studies. Since Green and Shoven (1986) 3 first introduced the proportional hazard model (PHM) to analyze mortgage termination by refinance, there have been several major developments to improve the application of PHM to mortgage termination analysis. R

13、ecent applications include more sophisticated and realistic modeling frameworks. For example, Schwartz and Torous (1989) developed a contingent claim framework for valuation of GNMA mortgage-backed securities through the integration of an empirical PHM to estimate the aggregate GNMA mortgage pools p

14、repayment experience. Stanton (1995) extends the Schwartz and Torous (1989) model by allowing transaction cost of prepayment in the modeling of mortgage pools rational prepayment behavior. The application of logit models to mortgage termination issues is well established. Mattey and Wallace (2001),

15、Ambrose and Capone (1998), Berkovec et al. (1998), Archer, et al. (1996), Quigley and Van Order (1995), Philips et al. (1995), and Cunningham and Capone (1990) have used binomial logit or MNL models. The PHM is established in the literature, but to a lesser extent (See, Ambrose and Sanders 2003, Pav

16、lov 2001, Bennett et al. 2001, Ambrose and Capone 2000, Vandell, et al. 1993, Schwatz and Torous 1989, and Green and Shoven 1986).5 Deng, Quigley and Van Order (2000) address competing risks of mortgage termination in a proportional hazard framework that allows correlated competing risks and account

17、s for the unobserved heterogeneity as discrete mass points. Their approach models individual mortgage borrowers as coming from two or more distinct groups with unobserved characteristics. The model cannot directly observe which group each individual belongs to, but it can estimate the discrete proba

18、bility distribution that each type influences the hazard function. The technique assumes a discrete number of groups; the researcher obtains maximum likelihood estimators of the mass-point distribution, i.e., the idiosyncratic risk as well as the probability associated with 4 such risk from each gro

19、up.6 Moreover, estimated mass-point parameters shift the baseline hazard function, allowing for a different hazard function for each unobserved group. This idea is potentially important to mortgage lenders because borrower characteristics are observed only at the time of loan application. Any unobse

20、rved changes in borrower characteristics may have a large impact on default or prepayment rates. This is particularly relevant to the move decision, where changes in employment or family status are likely to play an important role. Therefore, a statistical method for modeling unobserved borrower cha

21、racteristics may improve the power to predict mortgage terminations by move, refinance or default. This paper develops a mass-point mixed multinomial logit model (MML) that accounts for unobserved heterogeneity. Our extension of unobserved heterogeneity to the MNL model is motivated by the advantage

22、s mentioned above, and by the extensive use of MNL in the literature. Previous literature shows that the mass-point mixed technique adds significantly to the proportional hazard model (PHM), so it is worth testing for its contribution to the MNL model. We want to test for improvements in model effic

23、iency and predictive power associated with accounting for unobserved heterogeneity. Part of our agenda is to develop and implement a framework for cross-model-validation of mortgage termination risks. This allows us to judge any improvement in predictive accuracy that might be associated with adding

24、 unobserved subgroups to any model of mortgage terminations. Finally, we compare proportional hazard model (PHM) and MNL in terms of estimated 5 coefficients, statistical significance and out-of-sample predictive ability. Such comparisons allow judgment about the qualitative differences among the mo

25、dels. The predictive test is a particularly demanding standard for unobserved heterogeneity, where the number of unobserved groups, their location parameter, and their frequency are difficult to estimate from the micro loan history data. Our extension of the mass point mixed framework to the MNL mod

26、el can be positioned in the literature as follows: Model No Heterogeneity Unobserved Heterogeneity Proportional Hazard Model (PHM) with Competing Risks Han-Hausman (1990) MMH model of DQVO (2000) Multinomial logit model (MNL) with Event History Clapp et al. (2001) MML model developed here The compar

27、ison of these four models will test for economically significant (i.e., important) differences and for out-of-sample predictive ability. The remainder of this paper is organized as follows: The next section summarizes option theory as applied to mortgage termination and it develops observable variab

28、les that intervene in the termination decision; section 3 discusses empirical methods of model estimation, the role of unobserved heterogeneity, out-of sample prediction and cross-model validation; section 4 describes the data; section 5 discusses results and section 6 presents conclusions. 6 Observ

29、able Variables and Mortgage Termination Decisions Each month the borrower must decide whether to make the next regularly scheduled payment, refinance the mortgage, move and prepay the mortgage or default. This section summarizes observable variables associated with the borrowers decision and provide

30、s an overview of theory for each choice.7 The relevant variables can be classified as personal characteristics (income, age, etc.), loan characteristics (loan amount, loan-to-value ratio and note rate), financial market conditions (the stochastic path of market interest rates) and housing market con

31、ditions. Hendershott and Van Order (1987) and Kau and Keenan (1995) showed that the right to refinance the mortgage provides the borrower a call option on the mortgage debt with a strike price equal to the unpaid mortgage balance. Viewing the problem narrowly as the decision to exercise a call optio

32、n or not, the relationship between the market value of the loan and the unpaid mortgage balance is the primary determinant in the choice to refinance. When the default option is added, house prices and interest rates become the two observable variables of primary interest. The option-theoretic appro

33、ach does not address the move decision.8 The economic theory on household mobility points to a strong role for borrower characteristics in the move choice.9 Clearly, a choice model with the move alternative needs more than the financial options related variables. Observable Variables Explaining the

34、Refinance Decision Options pricing theory applied to mortgage refinancing implies that the borrower should exercise the option to call the debt whenever the market value of the mortgage exceeds the current 7 balance by enough to cover the costs of refinancing. Transaction costs are treated as a cons

35、tant increase in the strike price of the call option. However, borrowers do not exercise the option to refinance as ruthlessly as do owners of other financial options (See, Green and LaCour-Little 1999, Deng, Quigley and Van Order 2000). This has led some researchers, such as Stanton (1995) and Gree

36、n and LaCour-Little (1999), to treat transaction costs as varying across borrowers. However, in both studies, even implausibly high levels of transaction costs could not fully explain the observed prepayment behavior. Nevertheless, transactions costs suggest some observable variables that can be inc

37、luded in models of refinancing: For example, the larger the loan balance, then the greater the dollar amount of benefits from refinancing. This increases the probability of refinancing because fixed transactions costs (e.g., the time costs of refinancing) are more likely to be covered. Since transac

38、tion costs alone seem insufficient to explain the under-exercise of the prepayment option, a number of researchers have incorporated the effects of institutional constraints on a borrowers ability to refinance. For example, Archer, Ling, and McGill (1996) used American Housing Survey data from 1985

39、and 1987 to examine the influence of post-origination income and collateral constraints on prepayment behavior. They found higher annual payment-to- income and loan-to-value ratios were negatively related to prepayments, after controlling for call option values. Deng, Quigley and Van Order (1996) fo

40、und the importance of trigger events, such as unemployment and divorce, in affecting mortgage borrowers prepayment behavior. Caplin, Freeman, and Tracy (1997) found that regional recessions depressed prepayment rates by as much as 50% in states with declining property markets. Mattey and Wallace (20

41、01) and 8 Downing, Stanton and Wallace (2001) found evidence that differences in house-price dynamics across regions are an important source of heterogeneity between mortgage pool performance. Bennett, et al. (2001) found strong evidence that poor credit history as well as high current LTV significa

42、ntly reduced the probability of refinancing. These empirical findings are intuitive, for if collateral value declines below the loan balance, additional cash will be required to refinance. Similarly, a borrower whose income or financial position deteriorates may be unable to refinance due to payment

43、-to-income or credit quality constraints. In addition, making the right refinancing decision requires ongoing monitoring of market conditions and ready access to lenders. To the extent that particular demographic groups (e.g., minorities) have more limited access to information or lenders, we would

44、expect that group to have higher transactions costs of refinancing (Deng and Gabriel 2004). To summarize, the probability of refinancing is an increasing function of the market value of the loan, borrower income and the loan balance. It is a negative function of the current LTV, the probability of n

45、egative equity, the local unemployment rate, minority status and a low credit score dummy. Observable Variables Explaining the Move Decision Household mobility is a mechanism whereby households adjust their housing consumption to changes in circumstances (Rossi 1955). Theory says that a households d

46、ecision to move is based on housing “dissatisfaction”, household characteristics and exogenous circumstances (e.g., job or family composition changes). The dissatisfaction that ultimately results in a move is the direct 9 result of “changes in the needs of a household, changes in the social and phys

47、ical amenities offered by a particular location, or a change in the standards used to evaluate these factors” (Speare 1974, p. 175). Green and Shoven (1986) and Quigley (1987) documented a significant “lock-in” effect arising from below market rate financing. They found that homeowners with low mort

48、gage rates (relative to current market rates) delayed moving. We extend this reasoning to an in-the-money refinancing option, i.e., the borrower has a high mortgage rate relative to current market rates. In this case, the borrower has an added incentive to move to a new house, since the move effecti

49、vely refinances the mortgage as well as dealing with housing dissatisfaction. Thus, we expect the market value of the mortgage10 and the loan balance to be positively related to the move decision; the reasons are the same as for the expected positive signs in the refinancing equation. Turning to borrower characteristics explaining the move decision, the age of the head of household has consistently been shown to have a strong, significant negative effect on household mobility (See, e.g., Quigl