Equity, Vacancy, and Time to Sale in Real Estate.

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1 Title: Author: Address: Equity, Vacancy, and Time to Sale in Real Estate. Thomas W. Zuehlke Department of Economics Florida State University Tallahassee, Florida U.S.A. Phone: Fax: Date: August 2001 Abstract: JEL: The estimates presented in this paper suggest that the effect of minimum down payment requirements on expected market duration is not confined to declining markets. Statistical evidence of a threshold effect for loan-tovalue ratios is found in a sample of single family homes from a period of relatively stable housing prices. The value of occupancy status as a measure of opportunity cost is also confirmed. Finally, the estimates suggest that the assumption of a common duration elasticity is a potential specification error. The hypothesis of a common duration elasticity is strongly rejected for all models considered. C41, R31 1

2 Equity, Vacancy, and Time to Sale in Real Estate Genesove and Mayer (1997) examine the relationship between an owner s equity position and time to sale in the Boston condominium market during a period of declining market prices. They find a positive relationship between expected time to sale and a seller s loan to value ratio. They hypothesize that minimum down payment requirements on the purchase of their next home force owners with relatively small equity positions to be more selective when considering purchase offers for their current home; they set higher reservation prices, have longer expected market duration, and higher expected selling prices. If the value of the seller s next purchase is constrained by the minimum down payment requirement, then every extra dollar received from the sale of the existing home will give... more than a dollar s worth of additional housing value. They also hypothesize a threshold for this effect, since the minimum down payment requirement is not likely to be binding for sellers with sufficiently high equity positions. Zuehlke (1987) examines time to sale in a market for single family housing. Vacant houses are found to exhibit a lower baseline hazard rate (for probability of sale), but a higher duration elasticity than occupied houses. 1 This suggests that owners of vacant housing have a stronger incentive to adopt a strategy of diminishing reservation prices. One incentive is provided by decumulation of assets among sellers with decreasing absolute risk aversion. As search continues, wealth is reduced by the costs of search, and the seller is less likely to engage in the uncertainty of further search when confronted with an offer. Ceteris paribus, the owner of a vacant house faces higher costs of search because of the forgone return to equity, and consequently, has a stronger incentive to adopt a strategy of diminishing reservation prices. 2

3 Another such incentive occurs if vacant houses are more likely to deteriorate as a result of vandalism or neglect. The seller might act to offset potential losses by adopting a sequence of diminishing reservation prices. This paper considers both equity position and occupancy status as determinants of time to sale. It also tests the breadth of the Genesove and Mayer hypothesis, by considering the impact of minimum down payment requirements during a period of stable market prices. From a theoretical perspective, minimum down payment requirements are expected to constrain sellers with small equity positions, even when housing prices are stable. It may be more difficult to measure this effect statistically, however, absent the equity losses that occur with declining market prices. Weibull hazard models are estimated in both Genesove and Mayer, and Zuehlke, although a different parameter set is adopted in each. Their methodology is generalized to control for heterogeneity and to allow observation specific duration elasticities. Data The data used in this paper consists of 276 observations of single family homes listed with the Multiple Listing Service of Tallahassee. The information provided for each house includes market duration, the asking price, a fairly comprehensive set of housing attributes, the occupancy status of the house, the mortgage balance including any second mortgages, whether or not the house had sold, and the selling price conditional on sale. The sample mean of market duration (over both completed and incomplete duration) is 134 days. In this sample, 38 percent of the houses were vacant, and 47 percent of the houses were sold. 3

4 The loan to value ratio is measured as the ratio of the mortgage balance, including any second mortgages, to the initial asking price. Asking price is used because selling price is available only for the sold subsample. While selling price is generally a preferred measure of value, there is a very strong correlation between selling price and asking price. A simple least squares regression of selling price on asking price for the sold subsample, found selling prices to be approximately 494 dollars plus 95 percent of asking prices. The coefficient of determination for this regression is percent. The results reported below are virtually unchanged if the predicted selling price from this regression is used in place of asking price when constructing the loan to value ratio. 2 The sample mean of the loan-to-value ratio is 46 percent. The sample period chosen by Genesove and Mayer, Spring 1990 through Fall1992, was characterized by a 30 percent decline in average selling prices. Because of the resulting losses in equity, approximately 40 percent of owners were financially constrained, in the sense of having loan to value ratios in excess of 80 percent. Relative to a stable market, there is a disproportionately large number of financially constrained owners. The sample period considered in this paper, Winter 1982, was one of stable average selling prices. In this sample, only 6.25 percent of the owners had loan to value ratios exceeding 80 percent. Weibull Hazard Models The Weibull hazard model, in its simplest form, assumes that the log of the hazard function is linear in the log of duration. That is, ln[h(t)] = " + $ ln(t) where h(t) denotes the hazard function and t denotes the current value of duration. The 4

5 parameter " is the baseline (t=1) hazard rate, and the parameter $ is the duration elasticity. The parameter " is often made a function of a vector of observable regressors, X, and a heterogeneity component, v. Heterogeneity reflects the effect of unmeasured individual-specific, but timeinvariant, regressors. Letting " = v + X*, the model becomes ln[h(t)] = v + X* + $ ln(t) This is a proportional hazard model. Variation in the elements of X results in proportional variation in the baseline hazard rate, leaving the duration elasticity unchanged. The coefficients in * give the marginal impact of the regressors on the log of the baseline hazard rate. 3 The relationship between a given specification of the hazard function and the resulting distribution of duration times is necessary in order to specify the likelihood function. 4 The distribution function for completed duration, conditional upon v is v+ Xδ β+ 1 Ftv ( ) = 1 exp( e t ) The distribution function of the standard Weibull model, which does not control for heterogeneity, is obtained under the restriction v=0. Absent heterogeneity, there are no unusual estimation problems. Estimation when v is unobserved was originally considered by Kiefer and Wolfowitz (1956), who showed that the parameters can be estimated consistently by the method of Maximum Likelihood, as long as v can be represented by a common distribution function, not necessarily of known parametric form. Estimation proceeds by maximizing the unconditional likelihood function, which corresponds to the expectation of the conditional likelihood over the distribution of v. The unconditional distribution function for completed duration is: v+ Xδ β+ 1 Ft ( ) = 1 E[exp( e t )] v 5

6 There are no special estimation problems if the distribution of v is of known parametric form. For example, if e v has Gamma distribution with mean one and variance, then the unconditional distribution function for completed duration is 2 X 1 Ft () = 1 ( 1+ σ e t ) δ β+ σ 2 The distribution function of the standard Weibull model is obtained as the limit of the Weibull- Gamma model as F 2 approaches zero. The likelihood function for this specification can be maximized with conventional gradient methods. An alternative method of controlling for heterogeneity is to estimate the distribution of v with non-parametric methods. Laird (1978) shows that, for cases of practical interest, the nonparametric estimator is a discrete distribution with a finite, but unknown, number of support points. In this case, the expectation over v reduces to a simple weighted sum, and the distribution function is m v + Xδ + j β 1 Ft () = 1 [exp( e t )] p j= 1 where m is the unknown number of support points v j each with probability p j. The Weibull-NP model reduces to the standard Weibull model is there is a single support point with unit j probability. 5 Lindsey (1981) recommends estimation of the support points and their probabilities with the EM algorithm, and use of the Gâteaux derivative to verify that the resulting solution is a global MLE. The parameters of the hazard function, * and $, are estimated with conventional gradient methods. While most applications assume a common duration elasticity for all observations, it is possible to make the duration elasticity a function of regressors. This is the approach taken in Zuehlke (1987), where a separate duration elasticity is estimated for vacant and occupied 6

7 housing. This generalization can be accommodated in the model above, by replacing $ with X2. The coefficients in 2 give the marginal impact of a regressor on the duration elasticity. Distinct sets of regressors can be specified for the baseline parameter and duration elasticity by the imposition of exclusion restrictions on * and 2. The sample used in this paper includes both sold and unsold houses. Completed duration, T i, is observed for those houses that sell, while censored or incomplete duration, t i, is observed for houses that remain unsold. For the unsold subsample, the only information available is that completed duration must exceed current duration: T i > t i. The log-likelihood function for this type of sample is: n ln L( δ, θ ) = 1 { Ji ln[ f ( Ti)] + ( 1 Ji) ln[ 1 F( ti)]} i= where J i is a binary variable that indicates a house that is sold, and the density function for completed duration, f( ), is obtained from the known distribution function by differentiation. Results The models estimated in Genesove and Mayer make the baseline hazard rate a function of regressors, while imposing a common duration elasticity (across observations). Table 1 reports ML estimates of hazard models with a similar structure. Each model has a common duration elasticity, and a baseline hazard rate that is a function of the regressors LAP, VAC, LTV, and LTV-SPLINE. 6 The regressor LAP is the log of asking price, VAC is a binary variable that equals one when the house is vacant, and LTV is the loan-to-value ratio time 100. The regressor LTV-SPLINE is constructed as (LTV-80) times a binary variable this indicates LTV>80. Genesove and Mayer use this spline to capture the threshold effect of the minimum 7

8 down payment constraint. It allows the marginal effect of an increase in the loan-to-value ratio on the baseline hazard rate to differ above and below 80 percent. The estimates in the first column are the standard Weibull model with no correction for heterogeneity. The estimates in the second column are the Weibull model with a Gamma distribution for heterogeneity. The estimates in the third column are the Weibull model with non-parametric estimation of the distribution for heterogeneity. The estimated duration elasticities in the Weibull and Weibull-Gamma models are positive (0.12 and 0.19), but are not statistically different than zero at conventional levels. The estimated duration elasticity for the Weibull-NP model is positive (0.46) and statistically significant. These results are not surprising given that the presence of unmeasured heterogeneity is known to impart a downward bias in the estimated duration elasticity. The Weibull hazard model assumes a linear relationship between the log of the hazard function and the log of duration. Since VAC is a binary variable, its inclusion in the vector of baseline regressors serves to shift the intercept. It allows a different baseline hazard rate for vacant and occupied houses. With all three Weibull specifications, the coefficient of VAC is small and statistically insignificant. Genesove and Mayer hypothesize that minimum down payment requirements will be binding only for sellers with sufficiently low equity stakes, or sufficiently high loan-to-value ratios. Among these sellers, higher loan-to-value ratios are expected to result in longer expected duration through a lower baseline hazard rate. This is the motivation for including both LTV and LTV-SPLINE as regressors. Under this scenario, one would expect the coefficient of LTV to be zero and the coefficient of LTV-SPLINE to be negative. The results reported in Table 1 8

9 provide, at best, only weak evidence in support of the Genesove and Mayer hypothesis. For all three specifications, the estimated coefficient of LTV is positive (ranging from to 0.011) and statistically significant, while the estimated coefficient of LTV-SPLINE is negative (ranging from to ), but statistically insignificant. Taken together, there is some evidence of a threshold at a loan-to-value ratio of 80 percent, but it is LTV ratios below 80 percent that are statistically significant. The estimated coefficient of LAP is negative (ranging from to -0.71) for all three specifications. These estimates are significantly different than zero at conventional levels, although in the case of the Weibull-Gamma model, only at the 10 percent level. These estimates give the intuitively pleasing result, that houses with higher asking prices have lower hazard probabilities and longer expected market duration. The heterogeneity distributions are of only tangential interest. The maximizing value of F 2 for the Weibull-Gamma model is small and statistically insignificant. The maximizing value of m, the number of support points in the Weibull-NP model, is four. Lindsey (1981) shows that at the global MLE: 1) the Gâteaux derivative is non-positive at all points, and 2), the support points are local maximum corresponding to zero values of the Gâteaux derivative. The plot of the Gâteaux derivative in Figure 1 shows that these conditions are satisfied for the estimates reported in Table 1. The numerical values of the support points (v j ) and their corresponding probabilities (p j ) are superimposed on the figure. The estimates reported in Table 2 relax the assumption of a common duration elasticity. Both the duration elasticity and the baseline hazard rate are functions of the regressors LAP, VAC, LTV, and LTV-SPLINE. The format of the table is similar to that of Table 1, where the 9

10 columns correspond to estimates of the Weibull, Weibull-Gamma, and Weibull-NP models, respectively. The estimates reported in Table 2 indicate that vacant houses have a lower baseline hazard probability, but a higher duration elasticity than occupied houses. This is consistent with the hypothesis that owners of vacant housing are more likely to adopt a strategy of diminishing reservation prices. For all three specifications, the duration coefficient of VAC (ranging from 0.57 to 0.91) is significantly positive, while the baseline coefficient of VAC (ranging from to -4.61) is significantly negative. Using the Weibull-NP estimates, the hazard probability of a vacant house is lower than that of an occupied house for the first 159 days of market duration, and is higher thereafter. When the assumption of a common duration elasticity is relaxed, the coefficient estimates of LTV and LTV-SPLINE are more consistent with the expectations of Genesove and Mayer. Variation in LTV ratios below 80 percent, as measured by the coefficient of LTV, do not have a statistically significant effect on the estimated hazard rate. For all three specifications, the estimated duration coefficient of LTV is negative (ranging from to ) and statistically insignificant, while the baseline coefficient of LTV is positive (ranging from to 0.022) and statistically insignificant. The coefficients of LTV-SPLINE capture the differential impact of variation in LTV rates above 80 percent. The duration coefficients of LTV-SPLINE are positive (ranging from to 0.289), and are significant at approximately the 10 percent level for both the Weibull- Gamma and Weibull-NP models. The baseline coefficients of LTV-SPLINE are negative (ranging from to ), and as with the duration coefficients, are significant at 10

11 approximately the 10 percent level for both the Weibull-Gamma and Weibull-NP models. This pattern is consistent with the threshold effect predicted by Genesove and Mayer. It also suggests that sellers with high LTV ratios might be adopting a diminishing reservation price strategy. When the minimum down payment constraint is binding, sellers might be more selective during the initial stages of search, attempting to secure a relatively high offer, then becoming progressively less selective if the market demonstrates that such an offer is not forthcoming. The baseline coefficient of LAP is negative (ranging from-1.46 to -1.80), but is significant at conventional levels only with the Weibull-NP model. The duration coefficient of LAP is positive (ranging from 0.22 to 0.26) and again only significant with the Weibull-NP model. Perhaps the best interpretation of these results is that owners adopting a strategy of diminishing reservation prices are more likely to set a higher initial asking price. The estimated heterogeneity distributions are qualitatively similar to those reported for the models with common duration elasticity. The maximizing value of F 2 in the Weibull- Gamma model is again small and statistically insignificant, and the maximizing number of support points in the Weibull-NP model is four. The plot of the Gâteaux derivative in Figure 2 shows that the conditions for a global MLE are satisfied. Finally, a comparison of Tables 1 and 2 reveals that the coefficient estimates are more sensitive to the specification of the hazard function than to the choice of heterogeneity distribution. A likelihood ratio test of the assumption of a common duration elasticity is provided in Table 2. This test statistic is asymptotically distributed as chi-square with 4 degrees of freedom. The reported test statistics (ranging from to 20.37) exceed the one percent critical value of 13.3 for all three models. The assumption of a common duration elasticity is strongly rejected. 11

12 Conclusion The estimates presented in this paper suggest that the effect of minimum down payment requirements on expected market duration is not confined to declining markets. Statistical evidence of a threshold effect for loan-to-value ratios is found in a sample of single family homes from a period of relatively stable housing prices. The value of occupancy status as a measure of opportunity cost is also confirmed. Finally, the estimates suggest that the assumption of a common duration elasticity is a potential specification error. The hypothesis of a common duration elasticity is strongly rejected for all models considered. 12

13 TABLE 1: ML Estimates of Hazard Models (Asymptotic t ratios in parentheses) Weibull Weibull-Gamma Weibull-NP Log-Likelihood Duration Elasticity $ ( ) ( ) ( ) Baseline Coefficients (*) Intercept ( ) ( ) VAC ( ) ( ) ( ) LTV ( ) ( ) ( ) LTV-SPLINE ( ) ( ) ( ) LAP ( ) ( ) ( ) Heterogeneity Parameter F ( ) 13

14 Gâteaux Derivative of the Weibull-NP Model in Table 1 1. Support points correspond to local maximum of the Gâteaux derivative. 2. Corresponding support probabilities shown in parentheses. 14

15 TABLE 2: ML Estimates of Hazard Models (Asymptotic t ratios in parentheses) Weibull Weibull-Gamma Weibull-NP Log-Likelihood LR test (X2 constant) Duration Coefficients (2) Intercept ( ) ( ) ( ) VAC ( ) ( ) ( ) LTV ( ) ( ) ( ) LTV-SPLINE ( ) ( ) ( ) LAP ( ) ( ) ( ) Baseline Coefficients (*) Intercept ( ) ( ) VAC ( ) ( ) ( ) LTV ( ) ( ) ( ) LTV-SPLINE ( ) ( ) ( ) LAP ( ) ( ) ( ) Heterogeneity Parameter F ( ) 15

16 Gâteaux Derivative of the Weibull-NP Model in Table 2 1. Support points correspond to local maximum of the Gâteaux derivative. 2. Corresponding support probabilities shown in parentheses. 16

17 References Genesove, David, and Christopher J. Mayer, Equity and Time to Sale in the Real Estate Market, American Economic Review 87, June 1997, pp Kiefer, J. and Wolfowitz, J. "Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Nuisance Parameters," Annals of Mathematical Statistics 27, 1956, pp Laird, Nan. "Nonparametric Maximum Likelihood Estimation of a Mixing Distribution," Journal of American Statistical Association, 73, 1978, pp Lancaster, Tony, Econometric Methods for the Duration of Unemployment, Econometrica 47, July 1979, pp Lindsay, Bruce, "Properties of the Maximum Likelihood Estimator of a Mixing Distributionn," in C. Taille et al. (eds.), Statistical Distributions in Scientific Work, Vol. 5, 1981, pp Zuehlke, Thomas W., Duration Dependence in the Housing Market, Review of Economics and Statistics 69, November 1987, pp

18 Endnotes 1. Genesove and Mayer (1997) consider differences in the decisions of owner-occupants and investors, but do not consider differences between the decisions of owners of vacant and occupied properties. 2. Genesove and Mayer (1997) use assessed value when construction the loan to value ration. They found their results to be robust to the use of either asking price or selling price in place of assessed value. 3. Genesove and Mayer (1997) solve the fundamental Weibull relationship for ln(t) in terms of X. Their coefficient estimates correspond to - */$, and their standard error of the regression to 1/$. The specification adopted in this paper facilitates the use of methods that correct for unobserved heterogeneity. 4. Details may be found in Lancaster (1979). 5. In this model, one must impose either the condition * 1 =0 (and interpret the elements of v as different intercepts) or the condition : v =0 (and interpret the elements of v as increments the the intercept * 1 ). We will impose the former condition. 6. The set of housing attributes available in this data set were jointly insignificant, and their inclusion as regressors did not change the nature of the results. This is not surprising. In simple bidding models, housing attributes are correlated with probability of sale only if the marginal valuations of buyers and sellers differ. See Zuehlke(1987). 18