## Modelling mortgage demand under credit rationing

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Most (pooled) cross-section research has used a simple discrete choice model such as a probit or logit specification to model the mortgage/housing choices of constrained and non-constrained households. Also, most of the research has been concerned with the impact of mortgage underwriting criterion on the probability of home ownership. A comparatively neglected consideration is the nature of the interdependence of the discrete and continuous choices. That is the decision to take out mortgage debt (that is enter or renew owner occupation) together with the decision on the size of debt or gearing. Theoretical work on mortgage demand typically takes the tenure choice decision as given. This issue can be usefully explored via the so called double hurdle model (Cragg 1971; Leece 1995a, 2000b). The discussion of this model also provides a framework for a consideration of approaches to estimation.

If a household makes an equilibrium decision then any choices involving zero debt (including renting rather than owning) will also be equilibrium choices. Also, the influences upon the discrete and continuous choice will be the same. This can be modelled using a Tobit. The Tobit model is based upon a latent unobservable demand for mortgage debt M* generated by the index M* = bxj + ui, where Xj is a set of independent variables, bi are parameter estimates and ui is an error term. The observable mortgage demand M equals M* if M* > 0. That is, the underlying demand in terms of the size of mortgage is only observed when M* > 0. The expected value of a mortgage is the expected mean value conditional upon M* > 0, multiplied by the probability of observing a positive value (described as a non-limit observation). This can be taken as the probability of entering owner occupation financed by a mortgage. Thus the general form of a mortgage demand equation (a Tobit) that contains zero values is given by expression (6.1).

E[M] = 0* Probability[M = 0] + E[M|M* > 0]* Probability[M* > 0] (6.1)

In the case of Tobit estimation zero values are not discarded and aspects of both discrete and continuous choice are retained in the estimation. This further assumes that any household with non-zero mortgage demand will always obtain a loan. This is a restrictive assumption when applied to mortgage markets because some zero observations could represent households which had a positive demand for M but were precluded from obtaining one, that is they were rationed. Households might also be rationed in the size of loan available and decide not to enter owner occupation (Ortalo-Magne & Rady 1998, 1999, 2002). This relates to some degree to the discussion of estimation of mortgage demand functions in Chapter 2. Zero values occurred where households had paid off their mortgage debt, but the household owned their home. It may be more valid in that case to assume that zero observations are equilibrium choices, compared to, say, those arising out of credit rationing. The general form of a truncated regression used to estimate mortgage demand where credit rationing may be present is given by equation (6.2). Note that zero values are discarded and that there is a correction for the truncation of the observed distribution of choices ali.

In summary, most analysis of mortgage demand under credit rationing has been concerned with discrete choice models that estimate the likelihood of home ownership. It could be argued that the estimation of mortgage demand under credit rationing is of equal interest. This raises the question of how best to estimate such demand. The model suggested here is a double hurdle model. The double hurdle model estimates a probit for the discrete choice and a truncated regression on the non-zero observations. This allows for the possibility that not all mortgage demands will be met and that different parameter estimates and even variables apply to the discrete and continuos choices. Though there are other approaches to modelling the selectivity issues involved with credit rationing, the discussion highlights the possibility of complex relationships between the discrete and continuous choices. These can be modelled in other ways, and at times selectivity problems are just not evident (Leece 1995b). There are other forms of the double hurdle model (e.g. with correlated error terms, or Box & Cox specifications), mainly applied to discrete and continuos choices in markets other than that for mortgage debt (see Burton et al. 1994, 2000; De Sarbo & Choi 1999). There is considerable scope for further applications of these models in this area of research.

An example of estimation of mortgage demand under mortgage credit rationing

This section presents an example of a mortgage demand equation estimated using a basic double hurdle model (Leece 1995a). The estimation involves UK data using a pooled cross-section/time series sample. The study period spans pre- and post-financial deregulation. It is generally considered that disequilibrium credit rationing was significantly reduced, or disappeared, during the early 1980s (Meen 1990). The results reported here represent the only cross-section study of mortgage credit rationing for the United Kingdom covering the period of financial deregulation. The purpose of the exercise is to indicate changes in household behaviour during this time frame. The exercise also offers a basic example of the double hurdle methodology. The focus of the discussion is a truncated regression.

The truncated regression (see Table 6.1) is estimated on a sample of mortgage holders taken from the Family Expenditure Survey (1986). These estimates were made along with the estimation of a probit equation on mortgage and non-mortgage holders, implicitly a tenure choice decision. The choice to take out mortgage debt is the first hurdle, and the size of mortgage demanded is the second. The discussion focuses upon the second of these two hurdles. The research involved a number of sources of possible bias. The real mortgage balance, used as a dependent variable, was an estimate, thus introducing the possibility of measurement error.4 Exact identification of who was, and who was not, rationed in the first hurdle (tenure choice) was also not possible. These difficulties all emerged from the limitations of UK household level data covering this time frame.

Despite data problems the estimated mortgage demand equation offers some insight into the effects of financial deregulation in the UK. A number of interactions were modelled to detect changes pre- and post-financial deregulation. For example, a dummy variable for pre-1980 observations was interacted with the nominal and the real gross mortgage

 Variable Coefficient t-value Constant 262669.00 1.114 North -2370.67 -1.217 Yorkshire -2537.12 -1.640 Northwest -1278.95 -0.900 East Midlands -1423.83 -0.899 West Midlands 566.28 0.359 East Anglia 2502.97 1.119 London 9072.14 3.666 South East 9957.32 6.008 South West 2905.28 1.815 Pre-1980 mortgage (Yes = 1) 10927.00 0.036 Total household expenditure 8147.34 1.010 Age of head of household (HOH) -750.70 -7.725 Gross HOH real income 79.93 3.481 HOH manual worker (Yes = 1) -1158.40 -0.513 HOH married (Yes = 1) 1804.25 0.681 Pre-1980 mortage x gross HOH real income -52.39 -2.645 HOH male (Yes = 1) -2370.67 -1.217 Number of rooms in the property 3384.39 3.290 HOH manual worker x gross HOH real income 2.13 0.390 Child present age 5 and under 18 years -1543.27 -1.668 Nominal gross mortgage interest rate at time of origination 2298.80 2.330 Pre-1980 mortgage x nominal gross mortgage interest rate at -7205.30 -4.860 time of origination Real mortgage interest rate at time of origination 2153.26 2.185 Pre-1980 mortgage x real mortgage interest rate at time of -1998.82 -2.106 origination Expected relative house price inflation -646.83 -1.296 Loan-to-value ratio for first time buyers -71154.40 -1.358 Pre-1980 mortgage x loan-to-value ratio for first time buyers 13655.90 0.196

### Notes:

1. Total household expenditure is an instrument estimated from an expenditure equation.
2. The residual from the household expenditure equation is included as a test of exogeneity.

interest rate. The signs and coefficients on the interaction variables indicate that real interest rates had more sizeable effects post-financial deregulation. This might indicate a lessening of credit constraints, though the net of tax nominal mortgage interest rate had an unexpected positive sign. The income of the head of household was also statistically significant in the truncated regression, though this had not been the case in the probit. A key feature of the double hurdle model is the possibility that some different variables effect the discrete and the continuous choice.

Though real interest rate effects were stronger post-1980 in the UK the nominal mortgage interest rate was still a relevant specification in any post-1980 mortgage demand equation. Thus the overall conclusion of the study was that some households continued to be rationed post-financial deregulation. This was also the case for the discrete choice modelled by the probit equation. A log likelihood ratio test of the explanatory power of the double hurdle model compared to a Tobit estimated on this data offered a more powerful explanation of variation in mortgage demand. Thus the null hypothesis of perfect credit markets with no rationing was rejected. However, there was some easing of credit rationing post-1980, though the exact form of the rationing in either period was not identified.

The double hurdle model raised questions about the interdependence of discrete and continuous choices when analysing mortgage demand. Further analysis of these results can be found in Leece (1995a). The methodology facilitated a testing of the likelihood, if not the exact extent, of mortgage credit rationing. The empirical study also indicated the impact of deregulation on household behaviour in the UK mortgage market. However, mortgage markets have changed even more since the mid-1980s. Generally, financial deregulation has been accompanied by the increased securitisation of mortgage debt, particularly in the US. These factors should have contributed to the significant lessening, or removal, of anything other than equilibrium credit rationing based upon default risk. However, temporary or dynamic credit rationing might be evident if mortgage market adjustment is sluggish.

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### Responses

• Hannele
How to estimate discrete choice tobit model?
8 years ago