## Measurement issues in specifying mortgage termination models

One difficulty in testing option theoretic models is that the theoretical variables cannot always be measured directly. For example, the net value of the option to default or prepay which can inhibit mortgage termination, even when the option is 'in the money'. In the case of default, the value of the put option will depend upon the stochastic behaviour of particular house prices. Measuring the extent to which an option is 'in the money' (its intrinsic value) can also be problematic. Given that the precise valuation of options is difficult, research generally adopts a probabilistic approach, arguing that the embedded options are more likely to be exercised the further they are 'in the money'.

Several measures of the intrinsic value of the call option have been used in the prepayment literature. For example, the spread between, or ratio of, the coupon rate and the prevailing mortgage interest rate (LaCour Little 1999; Peristiani et al. 1997; Pavlov 2001). Richard & Roll (1989) measure intrinsic value by a present value annuity ratio, or the ratio of the present value of remaining payments on the mortgage at the coupon rate and the new interest rate (see Archer et al. 1997; Deng et al. 1996, 2000; Bennett et al. 1998, 2001).3 However, for new mortgages they suggest that the ratio of the coupon and the market interest rate is a good proxy for the ratio of balances.4

Measures of the intrinsic value of the call option are imperfect, they do not account for transaction costs and there is no correction for the expected holding period of the mortgage. Peristiani et al. (1997) make a number of corrections for variations in holding period, none of which has a marked effect on their empirical findings. There is also a measurement issue for loan level analysis in assigning the value of the expected refinancing rate for those who do not refinance. For example, the points coupon trade off can produce interest rate variations. The literature has adopted a number of approaches to attributing the forgone rate for non-refinancers. For example, Bennett et al. (1998) use the average Freddie Mac commitment rate on a 30-year, fixed rate mortgage for the month that a loan was closed.

Some prepayment studies have distinguished between the intrinsic value of the option, which reflects the non-callable bond component of the mortgage (see expression (9.1)), and the value of the embedded option (Giliberto & Thibodeau 1989; Caplin et al. 1997a,b; Bennett et al. 1998, 2000, 2001). The net value of the call option will be effected by interest rate volatility. Giliberto & Thibodeau used the annual variance in the monthly FHLBB contract rate to measure the net value of the call option to prepay. Caplin et al. (1997a, b) used a GARCH measure of conditional variance. Bennett et al. (2000) measure interest rate volatility with the implied volatility from options on 10-year US Treasury-note futures contracts. The extent to which the call option has to be 'in the money' to trigger prepayment is of interest in itself. The impact of volatility on the value of an option is greatest when it is 'near the money'. The sensitivity of the option value to changes in volatility is known as the vega. Bennett et al. estimated this vega using it to infer the prepayment thresholds on a sample of mortgages.

To measure the extent to which the put option is 'in the money' requires an indication of the homeowner's equity in their property. This in turn requires a measure of the loan-to-value ratio. Moreover, this loan-to-value ratio is ideally measured subsequent to the origination of the mortgage, that is exposte. This exposte measure is not as readily available as data at the point of origination and is often estimated (see Foster & Van Order 1985; Quigley & Van Order 1990; Cunningham & Capone 1990; Capone & Cunningham 1992). This is also the case with other variables, such as income, and clearly these estimates add the potential for measurement error. Some studies have been fortunate enough to have data that does not require that these values be estimated (Archer et al. 1997).

The loan-to-value ratio is in fact a rather complex variable to interpret. Measured at origination it is more likely to reflect credit market constraints. Measured at this point of time it may also reflect information asymmetry between borrower and lender, say regarding property-specific house price volatility (see Deng et al. 2000). The loan-to-value ratio may also proxy personal characteristics such as attitudes to risk. There is the possibility that the loan-to-value ratio should be treated as simultaneously determined with the default decision (see Brueckner 1994b, 1994c). So the point of time to which the measured loan-to-value ratio relates is important. There is also a need to be precise regarding when a household defaults. This is often taken as mortgages subject to foreclosure, but a property might be sold during this period and thus avoid default (Phillips et al. 1996). Ambrose & Buttimer (2000) recognise this issue and follow Kau et al. (1992) by defining default as the lender's act of taking title to the property. Non-payment is termed 'delinquency'.

As with all econometric studies, default and prepayment research may also be subject to omitted variable bias. Mattey & Wallace (2001) note the effect on estimates based on mortgage pools of neglecting varying rates of house price inflation. A form of omitted variable bias is selectivity bias. For example, in the case of default studies lenders favour lending to borrowers who do not have an above average risk of default, thus any sample of borrowers is a particularly favourable selection. Ross (2000) has tentatively shown the merit of controlling for selection bias arising out of the approval process, and demonstrated how different databases, one for approval and one for defaulters and non-defaulters, can be used to correct for this bias. This requires an assumption that the underlying approval process is the same for both samples and that variables can be found that influence approval but not default (that is the approval equation can be identified). This is a potentially important source of selection bias.

More research into the multiple sources of selection bias in default and prepayment studies can be expected as more and better data becomes available. Pavlov (2001) clearly demonstrates the bias inherent in any estimation of a prepayment model that does not allow for the competing risk of default (moving). Deng et al. (2000) indicates the importance of unobserved heterogeneity for mortgage termination estimation. Competing risk and unobserved heterogeneity among borrowers are issues dealt with in the most recent application of econometric analysis. Thus, there are a range of omitted variable problems to be allowed some of which are related to the choice of estimation technique.

### Econometric estimation

Several estimation techniques have been used in the mortgage prepayment and default literature, with the choice of approach depending upon the key research question(s), the nature of the data set and the current state of the art. The discussion in this section does assume some broad familiarity with methods of statistical estimation, though intuitive explanations are presented where possible. The main focus of the exposition is on the modelling of prepayment and default as competing risk.5

A large number of studies have used variations on the Cox proportional hazards model (Green & Shoven 1986; Follain & Ondrich 1997; Deng et al. 2000; Pavlov 2001). Others have utilised either a single logit or probit equation (Archer et al. 1997; Green & LaCour Little 1999), or have estimated a multinomial logit model with separate equations for a number of choices (Zorn 1989; Cunningham & Capone 1990; Clapp et al. 2001). The Cox proportional hazards model and the multinomial logit specification have both been used to model prepayment and default behaviour as competing risks (Deng et al. 1996, 2000; Clapp et al. 2001; Ambrose & Buttimer 2000; Calhoun & Deng 2002). Models of competing risk represent the most significant recent development in the estimation of mortgage termination equations. The approach was first adopted by economists to study the variety of mutually exclusive ways that individuals could exit unemployment (Narendranathan & Stewart 1993; Mealli & Pudney 1996; McCall 1996). This has proved to be an appropriate choice for modelling the competing nature of prepayment and default (Deng et al. 1996, 2000; Ambrose & Capone 2000).

The competing risk models recognise that durations can terminate (observations exit) for several competing reasons. The essence of the modelling approach is that each risk factor (that is reason for exit) has its own hazard

(risk) function. Of course, when one event leads to termination it precludes termination due to the other risk factor, consequently observations are censored. For mortgage terminations prepayment risk is censored when default occurs, while for default the censoring occurs when prepayment is the reason for mortgage termination. This is equivalent to treating each hazard as if it has a latent duration that is not always observed. Thus the competing risk approach aggregates the separate hazard functions and corrects for the censoring effects. Applying this approach has demonstrated the inefficiency of parameter estimates in prepayment (default) models when default (prepayment) is not controlled for (see Pavlov 2001).

The general form of the competing risk model can be illustrated by reference to the appropriate log likelihood function. Equation (10.1) follows Deng et al. (2000) and illustrates the log likelihood to be maximised. Here we see the log likelihood as a sum of competing risks for individual i, beginning with the risk of prepayment (p), followed by the risk of default (d), and the risk due to censored observations such as mortgages which continue beyond the observation period (c). These choices are reflected in the subscripts in equation (10.1). Each argument (p, d, c) is an indicator variable taking the value of 1 when the appropriate exit or act of censoring occurs. The terms Fj(Ki) are the unconditional probabilities of termination due to a particular cause j. The essence of this approach is the partitioning and summation of the different hazard (risk) rates.

log L = dpi log (Fp(Ki)) + ddi log (Fd(Ki)) + dui log (Fu(K)) + dCi log (FC(K))

Estimation can be based upon the usual partial likelihood approach. Excluding alternative means of terminating a mortgage when looking at prepayment or default will result in inconsistent coefficient estimates (Pavlov 2001). Equally important, the chosen independent variables might have different effects on the individual hazards, or it might be that different variables should be included in the specifications. There are, in fact, a number of different algorithms and forms of the competing risk model. Deng et al. (2000) estimated a form which allowed for both the simultaneous determination of the two hazard rates and the presence of unobserved heterogeneity and time varying variables.6 The results of the various estimations are discussed below.

An alternative econometric approach to modelling competing risks is to estimate the various exit routes using a series of logit models in the form of a multinomial logit (Clapp et al. 2001; Calhoun & Deng 2002). That is to model a selection of discrete choices. The proportional hazards model does have the advantage that the probabilities of the various forms of mortgage termination are restricted to sum to unity, and so any one termination is at the expense of the other. This appealing restriction is not true of the multinomial logit model. However, the multinomial logit does overcome some of the restrictive assumptions of other survival models. For example, survival models assume that the hazard functions for the competing risks are independent. There is also the assumption that the covariates have a constant proportionate effect on a hazard rate. However, the multinomial logit model does assume that the choices (competing risks) are independent (see Clapp et al. 2001). So each estimation technique has its own set of restrictions to note when evaluating competing risk.

The theoretical and econometric modelling of mortgage termination behaviour is undoubtedly complex. In principle one might argue that prepayment and default should be considered in the context of a system of equations which at the very least would include specifications for household mobility and the demand for mortgage finance. Elmer & Seelig (1999) note the need to link 'individual financial characteristics such as borrowing, savings, and insolvency, to house prices, home equity, and other option related variables'.7 One might also recall the modelling of Buist & Yang (2000) and their incorporation of the labour market into the theoretical and empirical analysis. Inevitably studies of mortgage termination have been beset by data limitations, but the econometric modelling has developed significantly in its explicit recognition of the links between the various types of mortgage termination, and the existence of unobserved heterogeneity among borrowers.

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