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Housing demand and disequilibrium
JOURNAL
OF URBAN
ECONOMICS
Housing
17, 42-51(1985)
Demand MARK
and Disequilibrium DYNARSKI
Department of Economics. tinioersig of Californra, Dad,
California 95616
Received September 13, 1982; revised May 2,1983 Researchers have recently begun relying heavily on samples of recent movers to estimate housing demand parameters. Estimates from mover, equilibrium, and disequilibrium samples are compared to determine whether significant differences exist. The results suggest that income elasticities are not affected by the sample distinctions, but demographic effects vary across the different subsamples. c, 1985 Academic
Press. Inc.
I. INTRODUCTION Researchers have recently begun to rely heavily on samples of recent movers to estimate housing demand parameters.’ The justification is that the financial and psychic costs of moving cause households to endure housing disequilibrium for long periods of time, thus violating the equilibrium assumption of cross-section demand estimation. Recent movers have overcome these costs to attain equilibrium. But recent mover samples may be nonrandomly generated from the equilibrium population. Estimates from such samples would thus be biased away from the true population parameters.’ The purpose of this paper is to compare estimates from mover, disequilibrium, and equilibrium samples, to determine whether significant differences exist. The information gained will be useful in modeling housing consumption and in estimating housing demand parameters. The equilibrium/disequilibrium distinction defines the appropriate population, according to demand theory. We test below to determine whether the distinction is substantive. The mover/nonmover distinction introduces a possible bias in sampling from the appropriate equilibrium population. The true equilibrium population is likely to contain both movers and non‘See Hanushek and Quigley [lo. 11. 121, Friedman and Weinberg [7], Mills and Sullivan [20], and Cronin [3] for examples using data from the Experimental Housing Allowance Program. Carliner [2] and Ihlanfeldt (141 use data from the Panel Study of Income Dynamics. The FHA data base used by Maisel, Bumham. and Austin [18]. Polinsky and Ellwood [23], and Rosen [24], should also be considered a mover subsample, as such data are collected only for recent home purchasers. ‘Mover selection bias is discussed generally by de Leuuw [4] and in the context of the Housing Allowance Program by Rosen [26]. Conference discussion of the Mills and Sullivan paper in the Brookings volume [l] suggests that the problem is widely recognized by researchers using the Housing Allowance data. 0094-1190/85 Copyright All n&s
$3.00
0 19R5 by Academic Press, Inc. of reproduction in any form resewed
42
HOUSING
DEMAND
AND
DISEQUILIBRIUM
43
movers: households that have recently moved are in equilibrium (by assumption), but households that have not moved are not necessarily in disequilibrium. A similar test is made of that distinction, to determine if selection bias is a problem.3 In Section II a housing demand function is estimated for a random sample drawn from the Panel Study of Income Dynamics. In Section III the same function is estimated and compared for various subsamples. Specific Panel data allow us to split the full sample into disequilibrium and mover subsamples. The comparison technique used is similar to a Chow test for the equality of coefficients between two samples. The results suggest that income elasticities are not much affected by the distinctions. Price elasticities are only slightly different for renters, and almost identical for homeowners. The paper concludes by comparing these results with existing housing demand studies. II. DATA BASE AND VARIABLE DEFINITIONS The data base is one year (1970) of the Panel Study of Income Dynamics [29]. The nonrandomly chosen sample of low income families (included for poverty research) was excluded, as were families whose household heads had recently changed. Following Rosen [25], families with new heads were excluded because of the difficulty in obtaining r&able permanent income estimates. The final sample contained 1866 homeowners and 1122 renters. A log-linear form was chosen for estimation.4 The equation used was
where E/P, is housing expenditure deflated by a nonhousing price index, Y/P, is permanent income (also in nonhousing units), PJP, is the relative price of housing, and X, is a set of n demographic shift variables. The disturbance is assumed normally distributed with zero mean and constant variance.5 Because the dependent variable is housing expenditure, the price ‘It can be readily demonstrated that if the probability of moving and house value are positively correlated, the income elasticity for the mover sample will be biased upward. See, generally, Hausman and Wise [13]. 4Translog runs had insignificant quadratic terms, and linear expenditure runs duplicated the log-linear results. The question of appropriate functional form is not relevant in this context. ‘A Park test for heteroscedasticity [g, Chap. 31, in which the log of the squared residuals from the estimated equation is regressed on income, found the income coefficient to be - .OOOB3 (S.E. .OOOOl) with an RZ of 009. The negative coefficient is puzzling, but the overall results suggest that the degree of heteroscedasticity is very low, hence efficiency losses are minimal. The statistical tests below require only that the degree of heteroscedasticity, if it exists, be the same for various subsamples.
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DYNARSKI
elasticity is found as the price coefficient minus one. It is assumed that house value is proportional to expenditure for housing services, and hence can be substituted for the dependent variable. With proportionality. only the constant is affected by this procedure. The nonhousing price deflator is explained below. Permanent income is defined as the average deflated taxable income for husband and wife for the years 1969-1975,6 plus 6% of home equity (implicit rent), plus transfer income.’ The taxable income plus implicit rent figure is used to compute the “permanent” tax liability from the 1970 tax tables [31]. No data were available for state and local income taxes. The use of actual future income is intended to capture households’ expectations of their future income; such expectations are likely to be an important element in the quantity decision. The use of lagged income only in the computation of permanent income, which has been the most popular method of computing permanent income, ignores the impact of expected future income increases (or decreases)on housing demand, probably overestimating demand for those nearing retirement and underestimating demand for first-time or young home buyers. The unit price of housing is considered here as three components: a metropolitan average, an intracity (distance) factor, and an individual tax factor for homeowners. The actual index used is p, = (1 - m16)(GP,)e-~”
where P, is the net price of housing, m is the individual’s marginal tax rate before housing related deductions, 8 is the proportion of housing costs untaxed, GP, is the BLS metropolitan housing index (for a lower income family of four-see [30]), h is the intracity price gradient (the rate at which unit housing prices decline with distance from the central city), and k is distance. 6 is set to zero for renters. Because the locations of sampled families were confidential, the actual price gradient for the individual’s metropolitan area could not be used. Instead, a fixed rate of 2% per mile was assumed, on the basis of Muth’s well-known study [21]. As distance was “The Panel included regional consumer price indices only for the years 1970-1972. For earlier and later years, the closest given value was inflated (or deflated) by the change in the aggregate CPI. This procedure ignores intercity relative price changes but such changes are likely to be small in so short a time span. ‘The addition of implicit rent to income introduces some degree of simultaneity. The problem can be alleviated by splitting the sample into income classes and substituting mean implicit rent for each agent in a given class, as in Rosen [24], but this technique is impractical except in very large samples. such as those used by Rosen and Feldstein and Taylor 16) in a different context.
HOUSING
DEMAND
AND
TABLE
45
DISEQUILIBRIUM 1
Demographic Variables Name
Definition
Mean
26
.25 .32 .28
Children DEP 1 DEP 2 DEP 3 SEX
One child under 17 Two children under 17 More than two children Female
.14 .16 .17 .24
RACE
Non-white (Black, Hispanic, other)
.27
AGE 1 AGE 2 AGE 3
NOW. Dummy variables are “1” if the condition column is true: zero otherwise.
in the second
given only in bracketed form, the midpoint of the bracket was entered, on the assumption that individuals are uniformly distributed within each bracket. The Polinsky-Ellwood unit price index [23], which works directly with land prices and hence has a gradient component, is superior to (2) because intracity variation is captured exactly, but the Panel data do not allow a better approximation.8 As noted by Polinsky [22], failure to account for intracity price variation biases the income elasticity downward. Computation of the tax factor is somewhat involved, and is relegated to the Appendix. The nonbousing price index was not among the data. But given the total budget index and the housing index, which were among the data, and the weight given to housing in the total budget index, the nonhousing index can be found by solving the identity B = aH + (1 - a)NH, where B is the total budget index, H is the housing index, and (Yis the housing weight, taken from [30]. If the housing index is divided by the total budget index to arrive at the relative housing price, an increase in the gross housing price index will be understated by the relative index, as both the numerator and the denominator will rise. The price elasticity would thus be biased upward (in absolute value). The demographic variables chosen were the age, sex, and race of the household head, plus the number of children under 17 years of age in the ‘The use of the BLS gross price index as a proxy for the central city housing price level is an approximation of unknown accuracy. Presumably the BLS low income index samples central city homes are more heavily than suburban homes, hence the two values may be quite close.
46
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household. Age is a proxy for credit constraints, which work against younger households, or for possible taste variation between younger and older households. Race and sex were dummy variables included to account for possible discrimination or taste variation. The number of young children was included on the hypothesis that large families tend to have large homes. For reference, Table 1 contains definitions and means for the demographic variables. OLS results for the full sample of homeowners are given in column (1) of Table 2. The income elasticity of .77 is very close to the .75 suggested by Polinsky [22], whereas the price elasticity of - .85 is higher than the - .67 found by Polinsky and Ellwood [23] and Rosen [24]. The crude gradient
TABLE -.__. Independent
(1) Full
W Y/f, )
,769 (28.4) ,150 (4.94) ,337 (4.16) ,361 (4.56) ,541 (6.71) -.049 (1.10) ,028 (.63) .033 (.752) .125 (3.00) -.203 (4.11) 2.39 .39 1866
Ln( p,/f,) AGE 1 AGE 2 AGE 3 DEP 1 DEP 2 DEP 3 SEX RACE CONST R’ N
2
Housing Demand Estimates-Homeowners ___Dependent: Ln( E/P,)
(2) Wdiseq .I64 (27.24) ,145 (4.56) ,305 (3.38) .295 (3.35) .454 (5.07) -.070 (1.47) ,014 (.290) .025 (.530) ,097 (2.19) -.193 (3.7) 2.51 .39 1664
(3) Eq mover/nonmover
(-.007) (.063) (.133) (1.14) (.007) (.031) (.267) (1.27) (.438)
wm (.114)
(-800) c.123) (.951) (.047) (.351) (.285) (2.10) (- .lOl) (.571) (-.237) 202
.813 (7.76) ,134 (1.31) ,099 (0.64) ,309 (1.98) ,438 (2.56) 046 (0.34) .262 (1.80) ,209 (1.46) -.258 (1.65) .015 (.095) 2.014 .41 1503
(-.047) (.438) (.015) (.141) (.280) (1.45) (.033) (0.17) (.060)
(.290) (-.126) (0.88) (-,278) (1.81) (p.205) (1.35) i.384) (2.36) (-,232) (1.42) (.459) 161
Notes. t-statistics are in parentheses below the coefficients. In columns (2) and (3). numbers to the right of the coefficients are shift coefficients for the subsample defined in the column heading, with r-statistics below. The sample in column (3) is the equilibrium subsample from column (2).
HOUSING
DEMAND
AND
47
DISEQUILIBRIUM
approximation may account for some of these differences: if a gradient of - 1.5% is used instead of - 2%, the income elasticity is unchanged but the price elasticity drops to - .76. The lower gradient is purely arbitrary, however, and the 2% figure is retained in later calculations. If only the gross price index is used the income elasticity is .81 and the price elasticity - 1.21. With the gradient included but taxes ignored, the income elasticity is .77 and the price elasticity is - .92. Leaving out the demographic variables, the income elasticity is .71 and the price elasticity is - .87. Evidently the income elasticity estimate is robust with respect to alternative price specifications, a fortunate event considering income’s importance as a determinant of housing demand and the empirical difficulty in computing an appropriate
TABLE
3
Housing Demand Estimates-Renters Dependent: Ln(RENT/P,) Independent
(1) Full
Ln(Y/h )
.501 (12.2) - .025 (.291) ,036 (S89) - ,015 (.239) ,078 (1.19) ,032 (.546) .057 (.825) - ,071 (1.W ,150 (3.07) - .Oll (.189) ,042 ,231 1122
W pH/po) AGE 1 AGE 2 AGE 3 DEP 1 DEP 2 DEP 3
RACE CONST R’ N
(2) Eq/diseq .477 (9.63) ,029 (0.26) - ,005 (0.05) - ,076 (0.94) .OOl (0.01) ,061 (0.80) .OSl (0.53) ,202 (3.34) ,201 (3.34) - ,016 (0.22) ,185 ,253 696
(.095) (1.03) (- ,146) (0.79) (.104) (0.83) (.142) (1.03) (.259) (1.74) ( - ,072) (0.59) ( -~ ,012) (0.09) (-,158) (1.54) ( -- ,158) (1.54) ( -~ 008) (0.06) (- -516) 426
~~
(3) Eq mover/nonmover -____ (.024) ,476 (5.53) (0.23) - ,272 (.423) (1.36) (1.77) ,026 ( - ,040) (0.23) (.023) - ,204 (.222) (1.14) (1.39) .061 ( - .022) (0.36) (0.10) ,271 (- ,326) (2.18) (2.07) ,187 ( - ,270) (1.33) (1.38) ,014 ( - ,196) (0.09) (1.01) .308 ( - S24) (2.66) (0.90) - .070 (.lOO) (0.48) (.590) .104 (- ,032) ,274 536 160
Notes. r-statistics are in parentheses below the coefficients. In columns (2) and (3), numbers to the right of the coefficients are shift coefficients for the subsample defined in the column heading, with r-statistics below. The sample in column (3) is the equilibrium subsample from column (2).
48
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housing price index. The results for the demographic variables indicate that expenditure increases with age and with number of children, though the standard errors for the latter coefficients are large. Female-headed households spend significantly more than male-headed households, whereas nonwhite households spend significantly less than white households. Rosen [25] and Lee and Trost [15] found similar results using the samedata base. OLS results for renters are given in column (1) of Table 3. The income elasticity is SO, smaller than that of homeowners, while the price elasticity is - .97, only slightly higher than the - .92 found for homeowners when taxes are ignored. The lower income elasticity for renters is a frequent empirical result [3, 14, 15, 191.Demographic variables appear to have little systematic effect on renter expenditure: the effects of age and children are nonmonotonic, and the race coefficient is very small and insignificant. Only the coefficient for female-headedhouseholds is significant. III. SUBSAMPLE ESTIMATES To compare the housing demand behavior of equilibrium and disequilibrium agents we must first decide how the subsamples are to be defined. One way of so doing is to use responses from agents themselves. Besides information about recent moves, the Panel also contains information about whether individuals are planning to move, and their reasons for wishing to move. Those who state that they are planning to move for housing consumption reasons are clearly in disequilibrium at their present consumption level.’ All others are assumedto be in equilibrium. Writing the equilibrium regression model as E(Y]X,) = X,p, and the disequilibrium model as E( Y(X2) = X,p,, we can test the joint null hypothesis pi = & by computing three-regressions and residual sums of squares corresponding to the equilibrium sample (RSS,), the disequilibrium sample (RSS,), and the combined sample (RSS,). The Chow test statistic is (RSS, - (RSS, + RSS,))/k (RSS, + RSS,)/(n, + n2 - 2k) ’ which under the null hypothesis has an F distribution with (k, n, + nz 2k) degreesof freedom, where k is the number of estimated parameters and 9The Panel classified planned moves into four categories: job-related (take another job; get closer to job), housing-related (more or less space; less rent; better neighborhood; better house), response to outside events (eviction; health reasons), and ambiguous (save money; neighbors moved away). Almost 60% of those planning to move were doing so for housingrelated reasons; see [29].
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(n,, n 2) are the numbers of observations in the equilibrium and disequilibrium samples.” A more revealing method of testing for coefficient differences is to run a regression on the full sample of the form y = xp, + X&M
+ E
(3)
where M is a dummy variable equal to one for those in the disequilibrium sample.” The separate null hypotheses &; = 0, i = 1,. . . , k-, are tested routinely using the reported t statistics. The advantage of this test over the Chow test is its ability to pinpoint the variables having different impacts in the two samples. In particular, elasticity differences can be separated from differences in the demographic coefficients. Turning first to homeowners, the Chow test statistic is 3.98, significant at the 99% confidence level.l2 Coefficients for the shift regression (3) are given in column (2) of Table 2. The column on the left lists the coefficients for the equilibrium sample, with t statistics below. The right column lists the shift coefficients for the disequilibrium, with t statistics for the shifts below. (The actual coefficient for the disequilibrium sample is found by adding the left and right columns.) Reasons for the positive Chow test are found by examining the t-statistics in the right column. Two of the shifts (AGE 3 and SEX) are significant of the 95% level; one other (AGE 2) is significant at about the 80% level. The income elasticities are very close, however, whereas the price elasticity is somewhat lower for the disequilibrium sample, though the difference is measured imprecisely. The significant shifts suggest that older families in disequilibrium are consuming more housing than their counterparts in equilibrium, perhaps because they have remained in their home after their children have left and are now consuming more housing than they need. That female households in disequilibrium should consume more housing than their equilibrium counterparts is harder to explain. For the mover/nonmover case, we eliminate disequilibrium observations from the sample and test for coefficient differences using the above techniques. Movers are defined as households whose place of residence had changed within the previous year. The Chow test statistic for the null hypothesis that mover and equilibrium nonmover coefficients are equal is 1.63, significant at the 90% level. The results from the combined regression with shift coefficients are given in column (3), Table 2. The coefficient for “See Maddala [17, p. 1981. Results of the Chow test are robust with heteroscedastic disturbances if at least one of the samples is large, which is the case here. On this point see Toyoda [28]. “Gujarati discusses regressions of this form [9, Sect. 13.61. Note that, unlike the Chow test, the disturbance is here assumed to have common variance in the two samples. “The critical f value with (11. m) df is 2.25 at the .Ol significance level.
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SEX is again significant at the 95% level, but the shift is opposite that of the disequilibrium coefficient. Young movers (AGE 1) consume less housing than young nonmovers. and movers with two or more children (DEP 2 and DEP 3) consume more housing than nonmovers in the same categories. Nonwhite movers also consume more housing than nonmovers. These results highlight the role of moving as an equilibrium adjustment mechanism: at various points in the life cycle moves are necessary to avoid overor undercrowding. Note, however, that the responsiveness of movers to income and price levels is very similar to that for nonmovers. Income is by far the most powerful predictor of housing consumption, and on that basis movers and nonmovers are alike. We turn now to renters. The Chow test for differences between equilibrium and disequilibrium coefficients is positive, with a test statistic of 2.59. The shift regression coefficients are in column (2) Table 3. None of the coefficients is significantly different from zero at the 95% confidence level (two-tailed tests). Surprisingly, shift coefficients for age groups show that all those in disequilibrium are consuming more than they wish. This seems likely for old households but less likely for young households, who typically would be living in too small a rental unit before moving to an owner-occupied unit. As with homeowners, however, the elasticity differences are trivial. For the mover/nonmover renter case, the Chow test statistic is 2.36. Shift coefficients are in column (3) Table 3. Movers with one child consume significantly more than nonmovers, with the effect diminishing for larger families. Middle-aged movers consume considerably less than nonmovers, but younger and older movers consume more. Movers are also more responsive to price than nonmovers, with a price elasticity of - 1.27 versus - .85 for nonmovers. Systematic deviations from the gross price index because of long-term rental discounts would bias the measured nonmover elasticity upward (away from zero); hence the true difference is probably larger than that found in column (3). The income elasticity difference is again small. Further refinement of the equilibrium/disequilibrium distinction is impossible with the data at hand. Considering the richness of the Panel Study relative to other economic data bases, it is unlikely that anything more than a mover/nonmover distinction is possible with other data. The above results indicate that income elasticity estimates are not much affected in any case, and for homeowners, price elasticity estimates are also invariant. Sample distinctions are inconsequential; only the rental price elasticity was significantly affected by the mover distinction. It is doubtful that this conclusion carries over to samples where those in disequilibrium are known to be all above (or all below) the equilibrium, as in the Housing Allowance Program. In such cases responsiveness of the nonmover sample will be lower than for movers, biasing elasticities downward. Our results suggest that the use of
HOUSING
DEMAND
AND
DISEQUILIBRIUM
51
mover subsamples allows unbiased estimation of the income elasticity, but the price elasticity may be too high in absolute value. Because demographic coefficients are of little policy value, it would not seem to matter much that they differ among samples. If the estimated equation is to be used for prediction, however, results could greatly differ according to which equation is used for the prediction. For example, mean house value for a female owner household is $18,086 for the equilibrium sample. If mover coefficients are used, mean value is only $11,994, about a 34% difference. Similarly, mean rent for female households using full sample coefficients is $1145 annually; using equilibrium coefficients (column (2)) the figure is $1206, only a 5% difference but nonetheless significant. Mean rent for mover families with one child is $1333; if equilibrium coefficients are used, mean rent is only $1071, a 20% difference. The joint effect of demographic coefficient differences may net to zero because of interactions, but to determine that here would take us too far afield. The focus to this point has been on the conditional mean demand function, with equal variance around the mean assumed. The notion of disequilibrium can be interpreted as a distance from the mean: random events push individuals away from their mean housing demand, some individuals being pushed farther from the mean than others. To test the hypothesis that the disequilibrium population has greater variance than the equilibrium population around an equal conditional mean, we use the mean square residuals from the separate equilibrium and disequilibrium regressions. Under the null hypothesis of equal variance, the ratio of the two is distributed F( n,, n2) with n, and n, equal to the degrees of freedom for the equilibrium and disequilibrium samples.The computed test statistic for homeowners is 1.16, significant at about the 93% level. For renters, the test value is 1.003, clearly insignificant. Evidently, the difference in variance between the two samples is not very pronounced. Moreover, the mean square residual from the mover subsample was larger than that of the disequilibrium subsample, contrary to the hypothesis that movers are in equilibrium and thus closer to the conditional mean.13 But until more is known about dynamic housing demand behavior, these results are difficult to interpret. A household may be very far from its equilibrium housing demand (found by solving first-order conditions), yet becauseof attachment to the area may claim to be in equilibrium, meaning it prefers not to move. Likewise, a household with no attachment may be close to its equilibrium demand level, yet claim to be in disequilibrium because the welfare loss “The mean square residual was .447 for mover homeowners and .376 for disequilibrium homeowners; the F ratio is significant at the 90% level. The renter values were ,285 and .242, also significant. That the ratios should be so close for homeowners and renters (1.189 versus 1.178) is probably only a curiosum.
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incurred by being off the demand curve exceeds total (pecuniary and psychic) moving costs. The topic is beyond the scope of this paper; a more complete discussion of housing demand behavior with residential attachment and transactions costs is presented in [5]. IV. COMPARISONS WITH EXISTING STUDIES A review of past work on mover versus nonmover income elasticities indicates that the mover elasticity is typically larger than the nonmover elasticity, but significance tests of the differences have reached opposite conclusions. I suggest below that the presence of price misspecifications in the articles surveyed may be a cause of the confusion. Using census tract median data for renters, De Leeuw [4] found an elasticity of 1.19 for movers versus .72 for nonmovers, for a regression of rent on income only. No statistical test of the difference is reported, though presumably such a large difference is significant. The lack of any housing price variable, however, biases the income elasticity estimates, and if income and price are correlated differently in mover and nonmover subsamples, an artificial difference in the estimated elasticities is introduced. On this basis, then, little can be said about mover/nonmover differences. Carliner [2], using Panel data, reports mover and nonmover income elasticities of .612 and .541 for owners, and .518 and .448 for renters. His specification included a permanent income term and a price index constructed as the ratio of the BLS housing and total budget indexes. A Chow test for coefficient differences between movers and nonmovers failed to reject the null hypothesis. The results are similar to those in Tables 2 and 3 above, though the price term did not correct for intracity and tax variation, and, as noted by Lee and Kong (161,is only a true relative price measure if the nonhousing index is in the denominator, rather than the total goods index. Polinsky and Ellwood [22] present further evidence that this form of price misspecification has only minor effects on the income elasticity estimate. Ihlanfeldt [14] found mover and nonmover elasticities of 1.03 and .68 for owners, and .45 and .38 for renters, using Panel data. Only the owner difference was significant, with a t statistic for the difference of 1.82. The price specification in this case was the BLS housing and nonhousing index entered separately, with no intracity or tax variation. To determine the sensitivity of the elasticity difference to the price specification, I excluded demographic variables from Eq. (1) above, and estimated mover and nonmover income elasticities of .839 and .687, with a t-statistic on the difference of 1.56. Reestimating the equation using separate BLS housing and nonhousing indexes (and ignoring other sources of price variation), the new elasticities were .848 and .691, with a r-statistic on the difference of 1.92. Evidently the change in the price specification is sufficient to lower the
53
HOUSING DEMAND AND DISEQUILIBRIUM TABLE 4 Effect of Disequilibrium on Cross-SectionalEstimates of Income and Price Elasticities of Housing Demand
Income elasticity, owners Income elasticity, renters Price elasticity, owners Price elasticity, renters
Households in equilibrium
Households in disequilibrium
Difference from equilibrium
,164 ,471 - .855 - ,971
.757 ,512 - .I22 - 1.117
- ,007 (- ,063) ,095 (1.03) ,133 (1.14) - ,146 (0.79)
Sources. Tables 2 and 3. A household is defined to be in disequilibrium if it states that it is planning a future move for housing consumption reasons. (See footnote 9.) Numbers in parentheses are r-statistics for the elasticity difference shown.
standard error of the difference, though the change in the difference itself is trivial (- .152 to - .157). The same exercise for the renter sample had a similar result on the standard error of the difference, suggesting that the price specification itself may give rise to the significant t-statistic for the owner mover/nonmover difference.14 There is little guidance in the econometrics literature on the use of tests for coefficient differences in the presence of a misspecitied variable. Omitting a relevant variable, for example, biases the remaining coefficients of a regression equation differently for different subsamples, as I have noted above in discussing De Leeuw’s results. If the correlations between included and excluded variables are different for different subsamplesin the probability limit, F tests for coefficient differences will clearly reject the null hypothesis with probability one, even if it is true for the correctly specified model. Whether the conditions for proper testing are satisfied in the models surveyed here is problematic.15 14For renters, the baseline results were mover and nonmover income elasticities of ,420 and ,395, with a r-statistic for the difference of 0.28. Using Ihlanfeldt’s price specification, the elasticities were .439 and ,394, with a r-statistic for the difference of 0.60. iSThe lack of price variation in some Experimental Housing Allowance Program data has allowed researchers to legitimately exclude price from the demand equation. Hanushek and Quigley [lo] report that income elasticities dropped 9% for the Phoenix control sample and 43% for the Pittsburgh control sample when computed over the full sample rather than over recent movers, but statistics on the differenceswere not reported. Mills and Sullivan [20] exclude from their sample households who did not respond to the housing allowance by moving or upgrading the structure as being in disequilibrium, but do not report result when such individuals were left in the sample. Because agents not responding to the allowances show zero elasticity, full sample estimates would necessarily be lower than equilibrium estimates, making these data unsuitable for a test of the mover/nonmover distinction.
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TABLE
5
Effect of Mover Selection on Cross-Sectional Estimates of Income and Price Elasticities of Housing Demand ____. __.Nonmovers Difference Recent movers Income elasticity, owners Income elasticity, renters Price elasticity, owners Price elasticity, renters
.813 .476 - .866 - 1.27
.766 ,500 - ,851 - ,849
- ,047 (0.44) ,024 (0.23) ,015 (0.14) -423 (1.77)
Sources. Tables 2 and 3. A household is defined to be a mover if it has changed locations within the previous year. Numbers in parentheses are r-statistics for the elasticity difference shown.
V. CONCLUSIONS In this paper a housing demand equation is estimated and used to test for coefficient differences between subsamples of equilibrium and disequilibrium agents, and between recent movers and nonmovers. The major finding is that elasticity differences are numerically small and statistically insignificant. For reference the elasticity estimates are compiled in Tables 4 and 5 for the equilibrium/disequilibrium and mover/nonmover comparisons. Demographic coefficient estimates differ significantly across subsamples, however, but because the signs of such coefficients are typically of greater importance than their magnitude, this finding is of lesserinterest. APPENDIX A full discussion of the tax correction for the price of housing is given here. The procedure follows Rosen [24] closely, with several variations noted. Tax subsidies to homeowners lower housing prices, with the effective subsidy rising as agents move into higher marginal tax brackets. Formally, if the individual’s marginal tax rate is m and tax provisions allow him to deduct or not claim 6% of his housing costs, the net price of housing is pH = (1 - m * 6)GP,
(A.11
where GP, is the gross housing price. To compute 8, tax-exempt costs of homeownership must be divided by total costs. Pretax homeownership costs are defined as r,el/+r,(l
-e)V+D+M+
T-
V*
where V is home market value, rc is the opportunity cost of capital, r, is
HOUSJNG DEMAND AND DISEQUILIBRIUM
55
the mortgage interest rate, e is the share of owner’s equity, D is depreciation, M is maintenance, T is property taxes, and I’* is expected capital gains. The first term is imputed rent, which is not reported as income. The second is mortgage interest, which, along with property taxes, can be deducted from income for those who itemize. Let Z be a dummy variable equal to one for itemizers. The tax-exempt percentage is then S=
r,eV+ Z(r,(l
- e)V+
r,eV+r,(l-e)V+T+D+M-V*’
T) (A.21
Equation (A.2) is similar to the 6 computed by Rosen [24, Eq. (3)] except for the itemizing indicator Z. In Rosen’s equation those who use the standard deduction receive no subsidy through the tax, whereas here such individuals receive a subsidy in the form of unreported implicit rent, a substantial amount for many homeowners. Expected capital gains are assumed to be zero, a good approximation for 1970. Maintenance and depreciation are fixed at 1.25 and 2.25%, from estimates in the literature (see [27]). The mortgage rate is set equal to the opportunity cost of capital at 6%. Net equity, mortgage debt outstanding, and property taxes were among the data. (No individual mortgage interest rate data were available.) Housing-related tax deductions allow many homeowners to itemize, though they might have taken the standard deduction otherwise. To avoid endogeneity of the price term to the greatest extent possible, Z is set to unity only for those homeowners who would have itemized before housing deductions are taken into account. The procedure is to first compute taxable income by subtracting from gross income the standard deduction and an exemption for each family member. Actual taxable income is found by inverting actual taxes paid through the tax table. If actual taxable income is less than computed taxable income, the individual itemized. For those individuals, 6% of mortgage debt outstanding plus property tax payments are added to actual taxable income, and the new level is compared with computed taxable income. If the new level remains below the computed level, the individual would have itemized before taking his housing deductions, and Z equals 1. A similar endogeneity problem exists for the marginal tax rate. Individuals with large housing deductions lower their marginal rate, thus raising their observed housing price and biasing the price elasticity toward zero. The marginal rate on the jrst dollar of housing deductions is required, and is found by adding implicit rent to actual taxable income for those taking the standard and finding the appropriate rate on that income level from the 1970 tax table. For itemizers implicit rent, mortgage interest, and property taxes were added to actual taxable income, from which the marginal rate is then found.
56
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ACKNOWLEDGMENTS I thank Bruce Hamilton, John Haltiwanger, Art Sullivan, Jay Helms, Leon Wegge, and an anonymous referee for helpful comments and advice.
REFERENCES 1. K. Bradbury and A. Downs, “Do Housing Allowances Work?’ Brookings, Washington, D.C. (1981). 2. G. Carliner, Income elasticity of housing demand, Rec. Econom. Statist.. 55. 528-532 (1973). 3. F. Cronin, Estimation of dynamic linear expenditure functions for housing, Rev. Econom. Sratisr., 64. 97-104 (1982). 4. F. De Leeuw, The demand for housing-a review of the cross-section evidence, Reu. Econom. Statist.. 53, I-10 (1971). 5. M. Dynarski, “Housing Choices with Residential Attachment.” University of California, Davis, Working Paper No. 193 (1982). 6. M. Feldstein and A. Taylor, The income tax and charitable contributions, Econometrica, 44, 1201-1222 (1976). 7. J. Friedman and D. Weinberg, The demand for rental housing: Evidence from the housing allowance demand experiment, J. Urban Econom., 9, 311-331 (1981). 8. S. Goldfeld and R. Quandt, “Nonlinear Methods in Econometrics.” North-Holland, London (1972). 9. D. Gujarati, “Basic Econometrics,” McGraw-Hill, New York (1981). 10. E. Hanushek and J. Quigley, The dynamics of the housing market: A stock-adjustment model of housing consumption. J. Urban Econom., 6, 90-111 (1979). 11. E. Hanushek and J. Quigley, An explicit model of intrametropolitan mobility, Land Econom., 54, 411-428 (1978). 12. E. Hanushek and J. Quigley, Consumption aspects, in “Do Housing Allowances Work?” (K. Bradbury and A. Downs, Eds.), Brookings, Washington, D.C. (1981). 13. J. Hausman and D. Wise, Social experimentation, truncated distributions, and efficient estimation, Econometrica. 45, 919-938 (1977). 14. K. Ihlanfeldt, An empirical investigation of alternative approaches to estimating the equilibrium demand for housing, J. Urban Econom., 9, 97-105 (1981). 15. L. F. Lee and R. Trost, Estimation of some limited dependent variable models with applications to housing demand, J. Econometrics, 8. 357-382 (1978). 16. T. H. Lee and C. Kong, Elasticities of housing demand, Sourhern Econom. J.. 44,298-305 (1977). 17. G. Maddala, “Econometrics,” McGraw-Hill, New York (1977). 18. S. Maisel. J. Burnham, and J. Austin, The demand for housing: A comment, Reo. Econom. Statisr., 53, 410-413 (1971). 19. S. Mayo, Theory and estimation in the economics of housing demand, J. Urban Econom., 10, 95-116 (1981). 20. E. Mills and A. Sullivan, Market effects, in “Do Housing Allowances Work?” (K. Bradbmy and A. Downs, Eds.), Brookings, Washington (1981). 21. R. Muth, Urban residential land and housing markets, in “Issues in Urban Economics,” (H. Perloff and L. Wingo, Eds.), Johns Hopkins, Baltimore (1968). 22. A. M. Polinsky, The demand for housing: A study in specification and grouping, Econometrica, 45. 447-461 (1977). 23. A. M. Polinsky and D. Ellwood, An empirical reconciliation of micro and grouped estimates of the demand for housing, Rev. Econom. Srarisr.. 61, 199-205 (1979). 24. H. Rosen, Owner-occupied housing and the federal income tax: Estimates and simulations, J. Urban Econom., 6, 247-266 (1979).
HOUSING DEMAND AND DISEQUILIBRIUM
57
25. H. Rosen, Housing decisions and the U.S. income tax: An econometric analysis, J. Public Econom., 11, l-25 (1979). 26. H. Rosen, “Housing Behavior and the Experimental Housing Allowance Program: What Have We Learned?,” NBER Working Paper No. 657 (1981). 27. .I. Shelton, The cost of renting versus owning a home. Land Econom., 44, 59-72 (1968). 28. T. Toyoda, Use of the Chow test under heteroscedasticity. Econometrica. 42, 601-608 (1974). 29. University of Michigan, “Panel Study of Income Dynamics.” 2 ~01s..Ann Arbor, Mich. (1972). 30. U.S. Bureau of Labor Statistics, “Three Budgets for an Urban Family of Four.” GPO. Washington, D.C. (1970). 31. U.S. Internal Revenue Service, “Statistics of Income,” GPO, Washington. D.C. (1971).
OF URBAN
ECONOMICS
Housing
17, 42-51(1985)
Demand MARK
and Disequilibrium DYNARSKI
Department of Economics. tinioersig of Californra, Dad,
California 95616
Received September 13, 1982; revised May 2,1983 Researchers have recently begun relying heavily on samples of recent movers to estimate housing demand parameters. Estimates from mover, equilibrium, and disequilibrium samples are compared to determine whether significant differences exist. The results suggest that income elasticities are not affected by the sample distinctions, but demographic effects vary across the different subsamples. c, 1985 Academic
Press. Inc.
I. INTRODUCTION Researchers have recently begun to rely heavily on samples of recent movers to estimate housing demand parameters.’ The justification is that the financial and psychic costs of moving cause households to endure housing disequilibrium for long periods of time, thus violating the equilibrium assumption of cross-section demand estimation. Recent movers have overcome these costs to attain equilibrium. But recent mover samples may be nonrandomly generated from the equilibrium population. Estimates from such samples would thus be biased away from the true population parameters.’ The purpose of this paper is to compare estimates from mover, disequilibrium, and equilibrium samples, to determine whether significant differences exist. The information gained will be useful in modeling housing consumption and in estimating housing demand parameters. The equilibrium/disequilibrium distinction defines the appropriate population, according to demand theory. We test below to determine whether the distinction is substantive. The mover/nonmover distinction introduces a possible bias in sampling from the appropriate equilibrium population. The true equilibrium population is likely to contain both movers and non‘See Hanushek and Quigley [lo. 11. 121, Friedman and Weinberg [7], Mills and Sullivan [20], and Cronin [3] for examples using data from the Experimental Housing Allowance Program. Carliner [2] and Ihlanfeldt (141 use data from the Panel Study of Income Dynamics. The FHA data base used by Maisel, Bumham. and Austin [18]. Polinsky and Ellwood [23], and Rosen [24], should also be considered a mover subsample, as such data are collected only for recent home purchasers. ‘Mover selection bias is discussed generally by de Leuuw [4] and in the context of the Housing Allowance Program by Rosen [26]. Conference discussion of the Mills and Sullivan paper in the Brookings volume [l] suggests that the problem is widely recognized by researchers using the Housing Allowance data. 0094-1190/85 Copyright All n&s
$3.00
0 19R5 by Academic Press, Inc. of reproduction in any form resewed
42
HOUSING
DEMAND
AND
DISEQUILIBRIUM
43
movers: households that have recently moved are in equilibrium (by assumption), but households that have not moved are not necessarily in disequilibrium. A similar test is made of that distinction, to determine if selection bias is a problem.3 In Section II a housing demand function is estimated for a random sample drawn from the Panel Study of Income Dynamics. In Section III the same function is estimated and compared for various subsamples. Specific Panel data allow us to split the full sample into disequilibrium and mover subsamples. The comparison technique used is similar to a Chow test for the equality of coefficients between two samples. The results suggest that income elasticities are not much affected by the distinctions. Price elasticities are only slightly different for renters, and almost identical for homeowners. The paper concludes by comparing these results with existing housing demand studies. II. DATA BASE AND VARIABLE DEFINITIONS The data base is one year (1970) of the Panel Study of Income Dynamics [29]. The nonrandomly chosen sample of low income families (included for poverty research) was excluded, as were families whose household heads had recently changed. Following Rosen [25], families with new heads were excluded because of the difficulty in obtaining r&able permanent income estimates. The final sample contained 1866 homeowners and 1122 renters. A log-linear form was chosen for estimation.4 The equation used was
where E/P, is housing expenditure deflated by a nonhousing price index, Y/P, is permanent income (also in nonhousing units), PJP, is the relative price of housing, and X, is a set of n demographic shift variables. The disturbance is assumed normally distributed with zero mean and constant variance.5 Because the dependent variable is housing expenditure, the price ‘It can be readily demonstrated that if the probability of moving and house value are positively correlated, the income elasticity for the mover sample will be biased upward. See, generally, Hausman and Wise [13]. 4Translog runs had insignificant quadratic terms, and linear expenditure runs duplicated the log-linear results. The question of appropriate functional form is not relevant in this context. ‘A Park test for heteroscedasticity [g, Chap. 31, in which the log of the squared residuals from the estimated equation is regressed on income, found the income coefficient to be - .OOOB3 (S.E. .OOOOl) with an RZ of 009. The negative coefficient is puzzling, but the overall results suggest that the degree of heteroscedasticity is very low, hence efficiency losses are minimal. The statistical tests below require only that the degree of heteroscedasticity, if it exists, be the same for various subsamples.
44
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elasticity is found as the price coefficient minus one. It is assumed that house value is proportional to expenditure for housing services, and hence can be substituted for the dependent variable. With proportionality. only the constant is affected by this procedure. The nonhousing price deflator is explained below. Permanent income is defined as the average deflated taxable income for husband and wife for the years 1969-1975,6 plus 6% of home equity (implicit rent), plus transfer income.’ The taxable income plus implicit rent figure is used to compute the “permanent” tax liability from the 1970 tax tables [31]. No data were available for state and local income taxes. The use of actual future income is intended to capture households’ expectations of their future income; such expectations are likely to be an important element in the quantity decision. The use of lagged income only in the computation of permanent income, which has been the most popular method of computing permanent income, ignores the impact of expected future income increases (or decreases)on housing demand, probably overestimating demand for those nearing retirement and underestimating demand for first-time or young home buyers. The unit price of housing is considered here as three components: a metropolitan average, an intracity (distance) factor, and an individual tax factor for homeowners. The actual index used is p, = (1 - m16)(GP,)e-~”
where P, is the net price of housing, m is the individual’s marginal tax rate before housing related deductions, 8 is the proportion of housing costs untaxed, GP, is the BLS metropolitan housing index (for a lower income family of four-see [30]), h is the intracity price gradient (the rate at which unit housing prices decline with distance from the central city), and k is distance. 6 is set to zero for renters. Because the locations of sampled families were confidential, the actual price gradient for the individual’s metropolitan area could not be used. Instead, a fixed rate of 2% per mile was assumed, on the basis of Muth’s well-known study [21]. As distance was “The Panel included regional consumer price indices only for the years 1970-1972. For earlier and later years, the closest given value was inflated (or deflated) by the change in the aggregate CPI. This procedure ignores intercity relative price changes but such changes are likely to be small in so short a time span. ‘The addition of implicit rent to income introduces some degree of simultaneity. The problem can be alleviated by splitting the sample into income classes and substituting mean implicit rent for each agent in a given class, as in Rosen [24], but this technique is impractical except in very large samples. such as those used by Rosen and Feldstein and Taylor 16) in a different context.
HOUSING
DEMAND
AND
TABLE
45
DISEQUILIBRIUM 1
Demographic Variables Name
Definition
Mean
26
.25 .32 .28
Children DEP 1 DEP 2 DEP 3 SEX
One child under 17 Two children under 17 More than two children Female
.14 .16 .17 .24
RACE
Non-white (Black, Hispanic, other)
.27
AGE 1 AGE 2 AGE 3
NOW. Dummy variables are “1” if the condition column is true: zero otherwise.
in the second
given only in bracketed form, the midpoint of the bracket was entered, on the assumption that individuals are uniformly distributed within each bracket. The Polinsky-Ellwood unit price index [23], which works directly with land prices and hence has a gradient component, is superior to (2) because intracity variation is captured exactly, but the Panel data do not allow a better approximation.8 As noted by Polinsky [22], failure to account for intracity price variation biases the income elasticity downward. Computation of the tax factor is somewhat involved, and is relegated to the Appendix. The nonbousing price index was not among the data. But given the total budget index and the housing index, which were among the data, and the weight given to housing in the total budget index, the nonhousing index can be found by solving the identity B = aH + (1 - a)NH, where B is the total budget index, H is the housing index, and (Yis the housing weight, taken from [30]. If the housing index is divided by the total budget index to arrive at the relative housing price, an increase in the gross housing price index will be understated by the relative index, as both the numerator and the denominator will rise. The price elasticity would thus be biased upward (in absolute value). The demographic variables chosen were the age, sex, and race of the household head, plus the number of children under 17 years of age in the ‘The use of the BLS gross price index as a proxy for the central city housing price level is an approximation of unknown accuracy. Presumably the BLS low income index samples central city homes are more heavily than suburban homes, hence the two values may be quite close.
46
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household. Age is a proxy for credit constraints, which work against younger households, or for possible taste variation between younger and older households. Race and sex were dummy variables included to account for possible discrimination or taste variation. The number of young children was included on the hypothesis that large families tend to have large homes. For reference, Table 1 contains definitions and means for the demographic variables. OLS results for the full sample of homeowners are given in column (1) of Table 2. The income elasticity of .77 is very close to the .75 suggested by Polinsky [22], whereas the price elasticity of - .85 is higher than the - .67 found by Polinsky and Ellwood [23] and Rosen [24]. The crude gradient
TABLE -.__. Independent
(1) Full
W Y/f, )
,769 (28.4) ,150 (4.94) ,337 (4.16) ,361 (4.56) ,541 (6.71) -.049 (1.10) ,028 (.63) .033 (.752) .125 (3.00) -.203 (4.11) 2.39 .39 1866
Ln( p,/f,) AGE 1 AGE 2 AGE 3 DEP 1 DEP 2 DEP 3 SEX RACE CONST R’ N
2
Housing Demand Estimates-Homeowners ___Dependent: Ln( E/P,)
(2) Wdiseq .I64 (27.24) ,145 (4.56) ,305 (3.38) .295 (3.35) .454 (5.07) -.070 (1.47) ,014 (.290) .025 (.530) ,097 (2.19) -.193 (3.7) 2.51 .39 1664
(3) Eq mover/nonmover
(-.007) (.063) (.133) (1.14) (.007) (.031) (.267) (1.27) (.438)
wm (.114)
(-800) c.123) (.951) (.047) (.351) (.285) (2.10) (- .lOl) (.571) (-.237) 202
.813 (7.76) ,134 (1.31) ,099 (0.64) ,309 (1.98) ,438 (2.56) 046 (0.34) .262 (1.80) ,209 (1.46) -.258 (1.65) .015 (.095) 2.014 .41 1503
(-.047) (.438) (.015) (.141) (.280) (1.45) (.033) (0.17) (.060)
(.290) (-.126) (0.88) (-,278) (1.81) (p.205) (1.35) i.384) (2.36) (-,232) (1.42) (.459) 161
Notes. t-statistics are in parentheses below the coefficients. In columns (2) and (3). numbers to the right of the coefficients are shift coefficients for the subsample defined in the column heading, with r-statistics below. The sample in column (3) is the equilibrium subsample from column (2).
HOUSING
DEMAND
AND
47
DISEQUILIBRIUM
approximation may account for some of these differences: if a gradient of - 1.5% is used instead of - 2%, the income elasticity is unchanged but the price elasticity drops to - .76. The lower gradient is purely arbitrary, however, and the 2% figure is retained in later calculations. If only the gross price index is used the income elasticity is .81 and the price elasticity - 1.21. With the gradient included but taxes ignored, the income elasticity is .77 and the price elasticity is - .92. Leaving out the demographic variables, the income elasticity is .71 and the price elasticity is - .87. Evidently the income elasticity estimate is robust with respect to alternative price specifications, a fortunate event considering income’s importance as a determinant of housing demand and the empirical difficulty in computing an appropriate
TABLE
3
Housing Demand Estimates-Renters Dependent: Ln(RENT/P,) Independent
(1) Full
Ln(Y/h )
.501 (12.2) - .025 (.291) ,036 (S89) - ,015 (.239) ,078 (1.19) ,032 (.546) .057 (.825) - ,071 (1.W ,150 (3.07) - .Oll (.189) ,042 ,231 1122
W pH/po) AGE 1 AGE 2 AGE 3 DEP 1 DEP 2 DEP 3
RACE CONST R’ N
(2) Eq/diseq .477 (9.63) ,029 (0.26) - ,005 (0.05) - ,076 (0.94) .OOl (0.01) ,061 (0.80) .OSl (0.53) ,202 (3.34) ,201 (3.34) - ,016 (0.22) ,185 ,253 696
(.095) (1.03) (- ,146) (0.79) (.104) (0.83) (.142) (1.03) (.259) (1.74) ( - ,072) (0.59) ( -~ ,012) (0.09) (-,158) (1.54) ( -- ,158) (1.54) ( -~ 008) (0.06) (- -516) 426
~~
(3) Eq mover/nonmover -____ (.024) ,476 (5.53) (0.23) - ,272 (.423) (1.36) (1.77) ,026 ( - ,040) (0.23) (.023) - ,204 (.222) (1.14) (1.39) .061 ( - .022) (0.36) (0.10) ,271 (- ,326) (2.18) (2.07) ,187 ( - ,270) (1.33) (1.38) ,014 ( - ,196) (0.09) (1.01) .308 ( - S24) (2.66) (0.90) - .070 (.lOO) (0.48) (.590) .104 (- ,032) ,274 536 160
Notes. r-statistics are in parentheses below the coefficients. In columns (2) and (3), numbers to the right of the coefficients are shift coefficients for the subsample defined in the column heading, with r-statistics below. The sample in column (3) is the equilibrium subsample from column (2).
48
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housing price index. The results for the demographic variables indicate that expenditure increases with age and with number of children, though the standard errors for the latter coefficients are large. Female-headed households spend significantly more than male-headed households, whereas nonwhite households spend significantly less than white households. Rosen [25] and Lee and Trost [15] found similar results using the samedata base. OLS results for renters are given in column (1) of Table 3. The income elasticity is SO, smaller than that of homeowners, while the price elasticity is - .97, only slightly higher than the - .92 found for homeowners when taxes are ignored. The lower income elasticity for renters is a frequent empirical result [3, 14, 15, 191.Demographic variables appear to have little systematic effect on renter expenditure: the effects of age and children are nonmonotonic, and the race coefficient is very small and insignificant. Only the coefficient for female-headedhouseholds is significant. III. SUBSAMPLE ESTIMATES To compare the housing demand behavior of equilibrium and disequilibrium agents we must first decide how the subsamples are to be defined. One way of so doing is to use responses from agents themselves. Besides information about recent moves, the Panel also contains information about whether individuals are planning to move, and their reasons for wishing to move. Those who state that they are planning to move for housing consumption reasons are clearly in disequilibrium at their present consumption level.’ All others are assumedto be in equilibrium. Writing the equilibrium regression model as E(Y]X,) = X,p, and the disequilibrium model as E( Y(X2) = X,p,, we can test the joint null hypothesis pi = & by computing three-regressions and residual sums of squares corresponding to the equilibrium sample (RSS,), the disequilibrium sample (RSS,), and the combined sample (RSS,). The Chow test statistic is (RSS, - (RSS, + RSS,))/k (RSS, + RSS,)/(n, + n2 - 2k) ’ which under the null hypothesis has an F distribution with (k, n, + nz 2k) degreesof freedom, where k is the number of estimated parameters and 9The Panel classified planned moves into four categories: job-related (take another job; get closer to job), housing-related (more or less space; less rent; better neighborhood; better house), response to outside events (eviction; health reasons), and ambiguous (save money; neighbors moved away). Almost 60% of those planning to move were doing so for housingrelated reasons; see [29].
HOUSING
DEMAND
AND
DISEQUILIBRIUM
49
(n,, n 2) are the numbers of observations in the equilibrium and disequilibrium samples.” A more revealing method of testing for coefficient differences is to run a regression on the full sample of the form y = xp, + X&M
+ E
(3)
where M is a dummy variable equal to one for those in the disequilibrium sample.” The separate null hypotheses &; = 0, i = 1,. . . , k-, are tested routinely using the reported t statistics. The advantage of this test over the Chow test is its ability to pinpoint the variables having different impacts in the two samples. In particular, elasticity differences can be separated from differences in the demographic coefficients. Turning first to homeowners, the Chow test statistic is 3.98, significant at the 99% confidence level.l2 Coefficients for the shift regression (3) are given in column (2) of Table 2. The column on the left lists the coefficients for the equilibrium sample, with t statistics below. The right column lists the shift coefficients for the disequilibrium, with t statistics for the shifts below. (The actual coefficient for the disequilibrium sample is found by adding the left and right columns.) Reasons for the positive Chow test are found by examining the t-statistics in the right column. Two of the shifts (AGE 3 and SEX) are significant of the 95% level; one other (AGE 2) is significant at about the 80% level. The income elasticities are very close, however, whereas the price elasticity is somewhat lower for the disequilibrium sample, though the difference is measured imprecisely. The significant shifts suggest that older families in disequilibrium are consuming more housing than their counterparts in equilibrium, perhaps because they have remained in their home after their children have left and are now consuming more housing than they need. That female households in disequilibrium should consume more housing than their equilibrium counterparts is harder to explain. For the mover/nonmover case, we eliminate disequilibrium observations from the sample and test for coefficient differences using the above techniques. Movers are defined as households whose place of residence had changed within the previous year. The Chow test statistic for the null hypothesis that mover and equilibrium nonmover coefficients are equal is 1.63, significant at the 90% level. The results from the combined regression with shift coefficients are given in column (3), Table 2. The coefficient for “See Maddala [17, p. 1981. Results of the Chow test are robust with heteroscedastic disturbances if at least one of the samples is large, which is the case here. On this point see Toyoda [28]. “Gujarati discusses regressions of this form [9, Sect. 13.61. Note that, unlike the Chow test, the disturbance is here assumed to have common variance in the two samples. “The critical f value with (11. m) df is 2.25 at the .Ol significance level.
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SEX is again significant at the 95% level, but the shift is opposite that of the disequilibrium coefficient. Young movers (AGE 1) consume less housing than young nonmovers. and movers with two or more children (DEP 2 and DEP 3) consume more housing than nonmovers in the same categories. Nonwhite movers also consume more housing than nonmovers. These results highlight the role of moving as an equilibrium adjustment mechanism: at various points in the life cycle moves are necessary to avoid overor undercrowding. Note, however, that the responsiveness of movers to income and price levels is very similar to that for nonmovers. Income is by far the most powerful predictor of housing consumption, and on that basis movers and nonmovers are alike. We turn now to renters. The Chow test for differences between equilibrium and disequilibrium coefficients is positive, with a test statistic of 2.59. The shift regression coefficients are in column (2) Table 3. None of the coefficients is significantly different from zero at the 95% confidence level (two-tailed tests). Surprisingly, shift coefficients for age groups show that all those in disequilibrium are consuming more than they wish. This seems likely for old households but less likely for young households, who typically would be living in too small a rental unit before moving to an owner-occupied unit. As with homeowners, however, the elasticity differences are trivial. For the mover/nonmover renter case, the Chow test statistic is 2.36. Shift coefficients are in column (3) Table 3. Movers with one child consume significantly more than nonmovers, with the effect diminishing for larger families. Middle-aged movers consume considerably less than nonmovers, but younger and older movers consume more. Movers are also more responsive to price than nonmovers, with a price elasticity of - 1.27 versus - .85 for nonmovers. Systematic deviations from the gross price index because of long-term rental discounts would bias the measured nonmover elasticity upward (away from zero); hence the true difference is probably larger than that found in column (3). The income elasticity difference is again small. Further refinement of the equilibrium/disequilibrium distinction is impossible with the data at hand. Considering the richness of the Panel Study relative to other economic data bases, it is unlikely that anything more than a mover/nonmover distinction is possible with other data. The above results indicate that income elasticity estimates are not much affected in any case, and for homeowners, price elasticity estimates are also invariant. Sample distinctions are inconsequential; only the rental price elasticity was significantly affected by the mover distinction. It is doubtful that this conclusion carries over to samples where those in disequilibrium are known to be all above (or all below) the equilibrium, as in the Housing Allowance Program. In such cases responsiveness of the nonmover sample will be lower than for movers, biasing elasticities downward. Our results suggest that the use of
HOUSING
DEMAND
AND
DISEQUILIBRIUM
51
mover subsamples allows unbiased estimation of the income elasticity, but the price elasticity may be too high in absolute value. Because demographic coefficients are of little policy value, it would not seem to matter much that they differ among samples. If the estimated equation is to be used for prediction, however, results could greatly differ according to which equation is used for the prediction. For example, mean house value for a female owner household is $18,086 for the equilibrium sample. If mover coefficients are used, mean value is only $11,994, about a 34% difference. Similarly, mean rent for female households using full sample coefficients is $1145 annually; using equilibrium coefficients (column (2)) the figure is $1206, only a 5% difference but nonetheless significant. Mean rent for mover families with one child is $1333; if equilibrium coefficients are used, mean rent is only $1071, a 20% difference. The joint effect of demographic coefficient differences may net to zero because of interactions, but to determine that here would take us too far afield. The focus to this point has been on the conditional mean demand function, with equal variance around the mean assumed. The notion of disequilibrium can be interpreted as a distance from the mean: random events push individuals away from their mean housing demand, some individuals being pushed farther from the mean than others. To test the hypothesis that the disequilibrium population has greater variance than the equilibrium population around an equal conditional mean, we use the mean square residuals from the separate equilibrium and disequilibrium regressions. Under the null hypothesis of equal variance, the ratio of the two is distributed F( n,, n2) with n, and n, equal to the degrees of freedom for the equilibrium and disequilibrium samples.The computed test statistic for homeowners is 1.16, significant at about the 93% level. For renters, the test value is 1.003, clearly insignificant. Evidently, the difference in variance between the two samples is not very pronounced. Moreover, the mean square residual from the mover subsample was larger than that of the disequilibrium subsample, contrary to the hypothesis that movers are in equilibrium and thus closer to the conditional mean.13 But until more is known about dynamic housing demand behavior, these results are difficult to interpret. A household may be very far from its equilibrium housing demand (found by solving first-order conditions), yet becauseof attachment to the area may claim to be in equilibrium, meaning it prefers not to move. Likewise, a household with no attachment may be close to its equilibrium demand level, yet claim to be in disequilibrium because the welfare loss “The mean square residual was .447 for mover homeowners and .376 for disequilibrium homeowners; the F ratio is significant at the 90% level. The renter values were ,285 and .242, also significant. That the ratios should be so close for homeowners and renters (1.189 versus 1.178) is probably only a curiosum.
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incurred by being off the demand curve exceeds total (pecuniary and psychic) moving costs. The topic is beyond the scope of this paper; a more complete discussion of housing demand behavior with residential attachment and transactions costs is presented in [5]. IV. COMPARISONS WITH EXISTING STUDIES A review of past work on mover versus nonmover income elasticities indicates that the mover elasticity is typically larger than the nonmover elasticity, but significance tests of the differences have reached opposite conclusions. I suggest below that the presence of price misspecifications in the articles surveyed may be a cause of the confusion. Using census tract median data for renters, De Leeuw [4] found an elasticity of 1.19 for movers versus .72 for nonmovers, for a regression of rent on income only. No statistical test of the difference is reported, though presumably such a large difference is significant. The lack of any housing price variable, however, biases the income elasticity estimates, and if income and price are correlated differently in mover and nonmover subsamples, an artificial difference in the estimated elasticities is introduced. On this basis, then, little can be said about mover/nonmover differences. Carliner [2], using Panel data, reports mover and nonmover income elasticities of .612 and .541 for owners, and .518 and .448 for renters. His specification included a permanent income term and a price index constructed as the ratio of the BLS housing and total budget indexes. A Chow test for coefficient differences between movers and nonmovers failed to reject the null hypothesis. The results are similar to those in Tables 2 and 3 above, though the price term did not correct for intracity and tax variation, and, as noted by Lee and Kong (161,is only a true relative price measure if the nonhousing index is in the denominator, rather than the total goods index. Polinsky and Ellwood [22] present further evidence that this form of price misspecification has only minor effects on the income elasticity estimate. Ihlanfeldt [14] found mover and nonmover elasticities of 1.03 and .68 for owners, and .45 and .38 for renters, using Panel data. Only the owner difference was significant, with a t statistic for the difference of 1.82. The price specification in this case was the BLS housing and nonhousing index entered separately, with no intracity or tax variation. To determine the sensitivity of the elasticity difference to the price specification, I excluded demographic variables from Eq. (1) above, and estimated mover and nonmover income elasticities of .839 and .687, with a t-statistic on the difference of 1.56. Reestimating the equation using separate BLS housing and nonhousing indexes (and ignoring other sources of price variation), the new elasticities were .848 and .691, with a r-statistic on the difference of 1.92. Evidently the change in the price specification is sufficient to lower the
53
HOUSING DEMAND AND DISEQUILIBRIUM TABLE 4 Effect of Disequilibrium on Cross-SectionalEstimates of Income and Price Elasticities of Housing Demand
Income elasticity, owners Income elasticity, renters Price elasticity, owners Price elasticity, renters
Households in equilibrium
Households in disequilibrium
Difference from equilibrium
,164 ,471 - .855 - ,971
.757 ,512 - .I22 - 1.117
- ,007 (- ,063) ,095 (1.03) ,133 (1.14) - ,146 (0.79)
Sources. Tables 2 and 3. A household is defined to be in disequilibrium if it states that it is planning a future move for housing consumption reasons. (See footnote 9.) Numbers in parentheses are r-statistics for the elasticity difference shown.
standard error of the difference, though the change in the difference itself is trivial (- .152 to - .157). The same exercise for the renter sample had a similar result on the standard error of the difference, suggesting that the price specification itself may give rise to the significant t-statistic for the owner mover/nonmover difference.14 There is little guidance in the econometrics literature on the use of tests for coefficient differences in the presence of a misspecitied variable. Omitting a relevant variable, for example, biases the remaining coefficients of a regression equation differently for different subsamples, as I have noted above in discussing De Leeuw’s results. If the correlations between included and excluded variables are different for different subsamplesin the probability limit, F tests for coefficient differences will clearly reject the null hypothesis with probability one, even if it is true for the correctly specified model. Whether the conditions for proper testing are satisfied in the models surveyed here is problematic.15 14For renters, the baseline results were mover and nonmover income elasticities of ,420 and ,395, with a r-statistic for the difference of 0.28. Using Ihlanfeldt’s price specification, the elasticities were .439 and ,394, with a r-statistic for the difference of 0.60. iSThe lack of price variation in some Experimental Housing Allowance Program data has allowed researchers to legitimately exclude price from the demand equation. Hanushek and Quigley [lo] report that income elasticities dropped 9% for the Phoenix control sample and 43% for the Pittsburgh control sample when computed over the full sample rather than over recent movers, but statistics on the differenceswere not reported. Mills and Sullivan [20] exclude from their sample households who did not respond to the housing allowance by moving or upgrading the structure as being in disequilibrium, but do not report result when such individuals were left in the sample. Because agents not responding to the allowances show zero elasticity, full sample estimates would necessarily be lower than equilibrium estimates, making these data unsuitable for a test of the mover/nonmover distinction.
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TABLE
5
Effect of Mover Selection on Cross-Sectional Estimates of Income and Price Elasticities of Housing Demand ____. __.Nonmovers Difference Recent movers Income elasticity, owners Income elasticity, renters Price elasticity, owners Price elasticity, renters
.813 .476 - .866 - 1.27
.766 ,500 - ,851 - ,849
- ,047 (0.44) ,024 (0.23) ,015 (0.14) -423 (1.77)
Sources. Tables 2 and 3. A household is defined to be a mover if it has changed locations within the previous year. Numbers in parentheses are r-statistics for the elasticity difference shown.
V. CONCLUSIONS In this paper a housing demand equation is estimated and used to test for coefficient differences between subsamples of equilibrium and disequilibrium agents, and between recent movers and nonmovers. The major finding is that elasticity differences are numerically small and statistically insignificant. For reference the elasticity estimates are compiled in Tables 4 and 5 for the equilibrium/disequilibrium and mover/nonmover comparisons. Demographic coefficient estimates differ significantly across subsamples, however, but because the signs of such coefficients are typically of greater importance than their magnitude, this finding is of lesserinterest. APPENDIX A full discussion of the tax correction for the price of housing is given here. The procedure follows Rosen [24] closely, with several variations noted. Tax subsidies to homeowners lower housing prices, with the effective subsidy rising as agents move into higher marginal tax brackets. Formally, if the individual’s marginal tax rate is m and tax provisions allow him to deduct or not claim 6% of his housing costs, the net price of housing is pH = (1 - m * 6)GP,
(A.11
where GP, is the gross housing price. To compute 8, tax-exempt costs of homeownership must be divided by total costs. Pretax homeownership costs are defined as r,el/+r,(l
-e)V+D+M+
T-
V*
where V is home market value, rc is the opportunity cost of capital, r, is
HOUSJNG DEMAND AND DISEQUILIBRIUM
55
the mortgage interest rate, e is the share of owner’s equity, D is depreciation, M is maintenance, T is property taxes, and I’* is expected capital gains. The first term is imputed rent, which is not reported as income. The second is mortgage interest, which, along with property taxes, can be deducted from income for those who itemize. Let Z be a dummy variable equal to one for itemizers. The tax-exempt percentage is then S=
r,eV+ Z(r,(l
- e)V+
r,eV+r,(l-e)V+T+D+M-V*’
T) (A.21
Equation (A.2) is similar to the 6 computed by Rosen [24, Eq. (3)] except for the itemizing indicator Z. In Rosen’s equation those who use the standard deduction receive no subsidy through the tax, whereas here such individuals receive a subsidy in the form of unreported implicit rent, a substantial amount for many homeowners. Expected capital gains are assumed to be zero, a good approximation for 1970. Maintenance and depreciation are fixed at 1.25 and 2.25%, from estimates in the literature (see [27]). The mortgage rate is set equal to the opportunity cost of capital at 6%. Net equity, mortgage debt outstanding, and property taxes were among the data. (No individual mortgage interest rate data were available.) Housing-related tax deductions allow many homeowners to itemize, though they might have taken the standard deduction otherwise. To avoid endogeneity of the price term to the greatest extent possible, Z is set to unity only for those homeowners who would have itemized before housing deductions are taken into account. The procedure is to first compute taxable income by subtracting from gross income the standard deduction and an exemption for each family member. Actual taxable income is found by inverting actual taxes paid through the tax table. If actual taxable income is less than computed taxable income, the individual itemized. For those individuals, 6% of mortgage debt outstanding plus property tax payments are added to actual taxable income, and the new level is compared with computed taxable income. If the new level remains below the computed level, the individual would have itemized before taking his housing deductions, and Z equals 1. A similar endogeneity problem exists for the marginal tax rate. Individuals with large housing deductions lower their marginal rate, thus raising their observed housing price and biasing the price elasticity toward zero. The marginal rate on the jrst dollar of housing deductions is required, and is found by adding implicit rent to actual taxable income for those taking the standard and finding the appropriate rate on that income level from the 1970 tax table. For itemizers implicit rent, mortgage interest, and property taxes were added to actual taxable income, from which the marginal rate is then found.
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ACKNOWLEDGMENTS I thank Bruce Hamilton, John Haltiwanger, Art Sullivan, Jay Helms, Leon Wegge, and an anonymous referee for helpful comments and advice.
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25. H. Rosen, Housing decisions and the U.S. income tax: An econometric analysis, J. Public Econom., 11, l-25 (1979). 26. H. Rosen, “Housing Behavior and the Experimental Housing Allowance Program: What Have We Learned?,” NBER Working Paper No. 657 (1981). 27. .I. Shelton, The cost of renting versus owning a home. Land Econom., 44, 59-72 (1968). 28. T. Toyoda, Use of the Chow test under heteroscedasticity. Econometrica. 42, 601-608 (1974). 29. University of Michigan, “Panel Study of Income Dynamics.” 2 ~01s..Ann Arbor, Mich. (1972). 30. U.S. Bureau of Labor Statistics, “Three Budgets for an Urban Family of Four.” GPO. Washington, D.C. (1970). 31. U.S. Internal Revenue Service, “Statistics of Income,” GPO, Washington. D.C. (1971).