The Demand for Owner-Occupied Housing: Implications from Intertemporal Analysis

JOURNAL OF HOUSING ECONOMICS ARTICLE NO.

7, 49–68 (1998)

HE980224

The Demand for Owner-Occupied Housing: Implications from Intertemporal Analysis Donald R. Haurin Departments of Economics and Finance, Ohio State University, 1010 Derby Hall, 154 North Oval Mall, Columbus, Ohio 43210 E-mail: [email protected]

and Eui-Chul Chung Department of Social Development, The Seoul Development Institute, San4-5, Yejang-dong, Jung-ku, Seoul 100-250, Korea E-mail: [email protected] Received January 13, 1998

A household’s consumption of housing depends on current and expected future values of influential variables. However, the cost variable used in most studies of housing demand is static. In contrast, we formulate a cost measure that incorporates both expected future changes in its components and the transaction costs associated with home ownership. Measuring annualized transaction costs requires us to estimate a household’s expected length of stay at the time of purchase. We use a parametric hazard rate model for this estimation. By projecting future values of the other components of user cost, we complete the measurement of a multiperiod version. Results from estimating housing demands show that our multiperiod transaction-adjusted user cost performs better than the static form and that inclusion of transaction costs is the most important adjustment.  1998 Academic Press

Studies of the demand for owner-occupied housing based on an intertemporal optimization framework have recently drawn considerable interest in the housing literature. Goodman (1995) and Goodman and Wassmer (1992) observe that households plan for future events; thus, their tenure choice and housing demand may differ from the outcomes predicted in a static model. Incorporating the desire to relocate and the presence of imperfect capital markets and borrowing constraints (Artle and Varaiya 1978, Zorn 1989) further complicates intertemporal models. A major finding, which has been well documented in previous theoretical and empirical studies, is that the expected length of stay is one of the critical factors influencing housing demand. Several authors have examined the 49 1051-1377/98 $25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.

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effect of the length of stay on housing decisions. Haurin and Lee (1989) used the predicted length of stay as an explanatory variable when estimating a structural model of the demand for owner-occupied housing. They found evidence that the choices of housing quantity, loan-to-value ratio, and planned length of stay are simultaneously determined. They also found a significant and positive effect of increases in the length of stay on the quantity of housing demanded. Further, they argued that omission of the length-of-stay variable from the structural housing demand equation results in a downward bias of the estimate of the owners’ income elasticity of demand. Henderson and Ioannides (1989) developed a joint tenure choice and residence spell model. In a structural equation for housing expenditures, they found that the expected length of stay was negative and significant for renters. Zorn (1988) estimated the expected duration of occupancy needed to calculate the flow cost of moving. He found that increased moving cost encourages households to stay in their current residences. Chambers and Simonson (1989), Goodman (1990), Hendershott and Shilling (1982), and Rosenthal (1988) studied the effects of transaction costs associated with home owning. These authors note that owners’ transaction costs may be large because they include real estate brokers’ fees, search costs for a dwelling and mortgage, and possibly the psychic costs of committing to ownership and the associated risks. Hendershott and Shilling’s basic assumption is that transaction costs are proportional to house value and they are amortized over the length of stay. In their measure of user costs, as the planned length of stay rises, user costs fall because annualized transaction costs fall. Rosenthal also found that owners facing higher transaction costs tend to increase the length of stay because transaction costs are amortized over a longer period. Chambers and Simonson hypothesized that tenure choice and length of stay are determined simultaneously and found supportive empirical evidence. We begin with a household’s two-period intertemporal utility maximization process. Our intuitive argument is that a household’s demand for owner-occupied housing is based on the best information currently available about the future paths of income, cost of owning, and household demographic characteristics. A household should expect some of the factors influencing the cost of owning to change over time; examples include the marginal tax rate, nominal interest rate, and rate of house price appreciation. Because it is costly to change housing consumption without moving, at the initial time of a move a rational household will account for expected future changes during the planned length of stay in a new dwelling. Our paper differs from previous studies in that the model explicitly incorporates the expected length of stay in the formulation of the user cost of owning. We develop a more appropriate measure of the owner cost variable, one that integrates the standard user cost of owning, transaction

OWNER-OCCUPIED HOUSING DEMAND

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costs, and the planned length of stay.1 Using data from the 1988 Panel Study of Income Dynamics, we estimate the demand for owner-occupied housing. Results from using our new cost measure are compared with results from using a static cost of owning. We find that the estimated price elasticity is greater when we use the new measure.2 This finding implies that estimates of housing demand price elasticities derived from a conventional measure may be biased downward. The first section of the paper presents a two-period model that explicitly introduces the length of stay into housing consumption and tenure choice. We also reformulate the user cost of owning so that it is based on a household’s intertemporal budget constraint. Sections 2 and 3 describe the empirical estimation procedure and our data set. The fourth section presents the length of stay and housing demand estimates. Price and income elasticities of demand for housing are calculated and compared. The final section presents our conclusions.

1. A THEORETICAL MODEL OF THE DEMAND FOR OWNER-OCCUPIED HOUSING

This section develops an intertemporal model of a household’s housing decisions under the assumption of perfect foresight. We divide a household’s planning horizon (T ) into two segments. At the beginning of the first period, it chooses a dwelling unit (H1) and determines how long to stay (D) in the chosen dwelling unit. It then chooses whether to move to another housing unit (H2). If it relocates, it incurs transaction costs and remains in H2 until the end of the planning horizon. The size of transaction costs depends on whether the household owns or rents. A household’s intertemporal utility function is U5

E

D

0

U1 (H1 , xt) e2rt dt 1

E

T

D

U2 (H2 , xt) e2rt dt,

(1)

where Uj , j 5 1, 2, is continuous, twice differentiable, and increasing in Hj 1 Harrington (1989) developed a different form of dynamic optimization model. He argues that there is not a single price elasticity of demand for housing; rather, the full intertemporal substitution effect must be estimated. However, he does not consider the transactions cost associated with owning a house; thus, he omits consideration of the impact of the expected length of stay. His formal model allows the quantity of housing to continuously adjust while we argue that housing quantity will effectively remain constant during the period of a stay in a residence. 2 Mayo (1981) and Harmon (1988) review studies of the price elasticity of demand for housing.

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and xt , with UHj . 0, Uxt . 0. The quantity of housing services is Hj , the intertemporally varying quantity of the composite good is xt , and r is a household’s rate of time preference. We assume that a household has no incentive to leave a bequest at the end of the planning horizon. The household selects Hj , xt , and D to maximize lifetime utility. We assume that a household faces only an intertemporal budget constraint. It has an income profile, yt , and after-tax income at time t is denoted as ydt . The price of the composite good is constant over the planning horizon and is normalized to 1. The per unit housing cost of renting is C tR . The per unit cost of owning (C tO) employs the concept of user cost and is represented as C tO 5 Pt h(1 2 tt)(it 1 tpt ) 1 r 1 e 2 gt j,

(2)

where Pt is the real value of a standardized (constant-quality) house at time t, tt is the marginal (or tenure choice) tax rate at time t, it is the riskfree interest rate at time t, tpt is the property tax rate at time t, r is a risk premium, e is the rate of depreciation and maintenance expenditures on housing, and gt is the rate of nominal house price appreciation.3 If a household moves to another dwelling unit at the end of a period, it incurs transaction costs. These costs depend on the chosen tenure status because owners incur greater costs than do renters. We assume that a renter’s transaction costs are a fixed amount. An owner’s transaction costs are variable because the selling cost depends on house value, this amount varying over time. We assume that transaction costs are amortized over the length of stay in the chosen dwelling unit. Expressing the transaction costs in terms of flow cost for period j yields4 TCj 5

wj bPHj e2rDj 1 (1 2 wj) me2rDj

E

Dj

Dj21

e2rt dt

,

(3)

3 Hendershott and Slemrod (1983) show that different tax rates are relevant for the tenurechoice and quantity-demanded decisions. They point out that what matters for tenure choice is the average tax saving per dollar of expense due to being an owner rather than a renter. The tax saving due to a marginal dollar of owner-occupied housing related expenses is relevant to the decision about quantity demanded. Depending on which tax rate is applied in Eq. (2), appropriate adjustments should be made to construct a correct budget constraint. 4 Haurin and Gill (1997) find that it is important to include transaction costs in the user cost measure when explaining home ownership. However, their empirical tests indicate that alternative specifications of transaction costs work equally well when comparing a measure with only a variable component proportional to house value with one having variable and fixed components.

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OWNER-OCCUPIED HOUSING DEMAND

where j 5 1 or 2, D0 5 0, D2 5 T, and bPHj is the owner’s transaction cost at rate b on house value PHj . A renter’s fixed transaction cost is m and r is the real after-tax interest rate. We specify a household’s tenure status in period j so that wj 5 1 if it owns and wj 5 0 if it rents. The partial derivative of TCj with respect to Dj is negative, meaning that transaction costs for period j decrease as the length of stay increases, being amortized over a longer period. A household’s intertemporal budget constraint is

E y e dt 1 E A*H e 5 E hx 1 C ( y , H ) H j e T

W0 1

0

D

0

D

R 2rt dt

t

1

0

O t

t

1

1

E w A*H e dt 1 E hx 1

2rt

2rt

dt 1

T

D

2

2

T

D

t

2rt

dt (4)

[w2 C tO ( yt , H2) 1 (1 2 w2)C tR] H2je2rt dt 1 bPH1 e2rD 1 w2 bPH2 e2rT 1 (1 2 w2) me2rT, where W0 is a household’s initial wealth and yRdt is a household’s after-tax income at time t if it rents. An owner’s after-tax income differs from a renter’s because of the federal income tax deductibility of mortgage interest and property tax payments. Also, implicit income earned from housing equity is not taxable. The terms labeled A* in (4) make the appropriate adjustments.5 Equation (4)’s specification is based on the assumption that the household is an owner in the first segment, but is free to own or rent in the second period, this assumption being consistent with our empirical work. By maximizing lifetime utility (1) subject to the lifetime budget constraint (4), a household chooses the optimal quantity of housing consumed for each period, H *j , the optimal expenditure on the residual composite good, x*t , and the optimal length of stay in the dwelling unit, D*, as well as the tenure state in the second period. From (4) and the first-order conditions, we find that the appropriate form of the user cost of owning for the first period is the transactionadjusted weighted average cost, C 1*( yt , H *1 , D*), defined as6 Total owner cost is C tO H1 5 Pt H1 h(1 2 tt)(it 1 tpt ) 1 r 1 e 2 gt j. Because the tenure choice tax rate is the average tax savings per dollar of housing, if a household itemizes then tt (it 1 tpt )Pt H1 5 T R 2 T O, where T R is the total tax liability if the household rents and T O is the tax liability if the household owns. Therefore, the total owner cost can be reexpressed as C tO H1 5 Pt H1 hit 1 tpt 1 r 1 e 2 gtj 2 (T R 2 T O). That is, the total tax savings resulting from being an owner is fully captured if the tenure choice tax rate is used for tt . However, because the marginal tax rate is not equal to the tenure choice tax rate (it is generally greater), only part of tax savings is included in the owner cost expression. The adjustment term, A*, corrects for this difference. See Chung (1994) for further details. 6 The first order conditions are derived in Chung (1994). 5

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E

D*

C 1*( yt , H *1 , D*) 5

0

C tO ( yt , H 1*) e2rt dt 1 bPe2rD*

E

D*

0

e2rt dt

,

(5)

where the second term in the numerator represents the present value of the transaction costs per unit of housing, evaluated at the time of a move. The higher are transaction costs, the greater the user cost. Therefore, the cost of owning that determines a household’s housing demand for the first period not only includes the expected path of the standard owner cost due to changes in income and its other components, but it also is influenced by the optimal length of stay, D*. The demand for owner-occupied housing is specified as a function of a household’s lifetime after-tax income ( yd), the transaction-adjusted weighted average cost of owning, and a vector of the taste shifters such as demographic characteristics, Z: H1 5 f ( yd , C 1*, Z ).

(6)

2. EMPIRICAL SPECIFICATION

The calculation of the weighted average cost of owning requires information about a household’s planned length of stay in a dwelling unit. We estimate the planned length of stay of owners using a parametric hazard rate model developed by Peterson (1986), this being capable of capturing the effects of time-varying covariates. The length of stay is divided into k exhaustive, nonoverlapping intervals such that t0 , t1 , ...... , tk21 , tk (t0 5 0 and tk 5 ti), where ti is the observed length of stay of household i, evaluated at the time when a household moves or when censoring occurs. Covariates X are assumed to remain constant within each of the k intervals, but may change from one interval to next. Let h(t u Xk) be the hazard function from tk21 to tk . Using the relationship between the hazard function and the survival function, the survival function for duration tk is7 S (tk u Xk) 5 pj51 prob(T $ tj u T $ tj21). k

(7)

7 Let S(t) be the survival function corresponding to h(t). Because h(t) 5 2d ln S(t)/dt, we know that S(t) 5 exp(2eh(t) dt). Therefore, the survival probability that T is greater than t or equal to tk is prob(T $ tj u T $ tj21) 5 exp[2ejj21 h(t u xj) dt].

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OWNER-OCCUPIED HOUSING DEMAND

The density at tk is f (tk u Xk) 5 h(tk) S(tk). Therefore the log-likelihood for household i is ln Li 5 gi ln f (tk u Xk) 1 ln S(tk) 5 gi ln h(tk u Xk) 2

OE k

j51

tj

tj21

h(t u Xj) dt, (8)

where gi 5 0 if an observation is right-censored and 1 if not. One of the appealing characteristics of this method is that the hazard rate in each time interval can be specified as a function of explanatory variables for that interval, thereby capturing effects of time-varying covariates. We assume a Weibull survival distribution, S(t) 5 exp[2(l(t)t) p]; thus, the hazard function is h(t) 5 l(t)p(l(t)t) p21. Here, l(t) 5 exp(2d 9Xit), where d and p are vectors of parameters to be estimated, and Xit is a vector of explanatory variables at time t. Under the Weibull assumption, the hazard rate changes monotonically over time, with p indicating whether the change is positive or negative. If p is greater than 1, the hazard rate rises over time. Next, a log transformation of ti enables explanatory variables to enter the estimation linearly so that ln ti 5 d 9Xit 1 vwi ,

(9)

where v 5 1/p and wi is an error variable that has an extreme value distribution. In the estimation, ln ti is specified as p

p

p

p

ln ti 5 f ( ydit , W it , Lit , Z it , ysdit , Lsit , Z sit , Z0) 1 vwi , p

(10) p

where ydit is the predicted real after-tax permanent income, W it is predicted p p real wealth, Lit is the predicted mortgage lock-in effect, and Z it is a vector of predicted measures of time-varying demographic characteristics such as marriage and the number of children under age 18. Z0 is a vector of timeinvariant demographic variables. We assume that the planned length of stay determines the actual length of stay, but it is also impacted by post-planning shocks. Measures of shocks are incorporated in an ad hoc way and are represented as a vector of variables with superscript s. Shocks are defined as the difference between realizations and expectations. Z sit is a vector of demographic shocks, including those to the number of children and marriage. Lsit is the difference between the actual value of the amount of lock-in and the predicted value, and ysdit is transitory real after-tax income. In a hazard rate model with time-varying covariates, the calculation of the expected duration is not simple. If all variables are assumed to be timeinvariant, exp(d 9Xi) is the expected duration of stay of a household. In the

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case where Xi varies over time, the survival probability in (7) must be used. We express it as k

S(tk u Xk) 5 p prob(T $ tj u T $ tj21) j51

5 prob(T $ 1 u T $ 0) ? prob(T $ 2 u T $ 1) ......

(11)

prob(T $ tk u T $ tk21). Thus, the probability that a household’s length of stay is greater than or equal to T is the product of conditional probabilities up to k. We assume that if the survival probability becomes less than 0.5, then a household moves from the initial dwelling. The expectation of the length of stay is calculated based only on the parameters of time-invariant variables and predicted time-varying variables; we set all shock measures to zero. This measure of expected length of stay is used to calculate the weighted average owner cost variable. We test log-linear and linear forms of the housing demand equation using a linear translation of demographic variables.8 Specifying y i as the error term and c and c as parameter vectors, f (Hi) 5 c0 1 c1 ydi 1 c2C iO 1 Zi c 1 yi .

(12)

Housing quantity is the real value of the house that a household purchased when it moved, divided by a real house price index.9

3. DATA AND MEASUREMENT OF VARIABLES

The data are drawn from the 20th wave of the Panel Study of Income Dynamics (PSID) (1984, 1989). The restriction of the sample to only households with the same head for 1973–1988 is necessary because we first 8 Goodman (1990) allowed demographic variables to interact with price and income variables in his housing demand equation. However, his estimates in the more complex specification differ little from those derived using the standard form. 9 Because we estimate the housing demand equation using only a subsample of households who moved and owned, it is possible our results are subject to sample selection bias because the households who have chosen the other alternatives are excluded from our sample. We considered this problem and included a sample selection correction variable obtained from estimating a multinomial logit model in the first stage, similar to Zorn (1989). However, the coefficients of the selection correction variable were not significant in any specification of the housing demand estimation. Therefore, we conclude there is no evidence of sample selection bias.

OWNER-OCCUPIED HOUSING DEMAND

57

observe households in 1973 and follow their housing choices through 1988 (N 5 2,739).10 House price information is available for only 23 MSAs (N 5 821).11 We select households who choose to be owners in the 1974– 1979 period (N 5 117) and follow them until 1988, recording their housing consumption and whether they relocated again. 3.1. Measurement of Variables for the Length of Stay Estimation A household’s length of stay is the observed length if they moved before 1988; if not, the stay is right-censored. Yearly total household income includes head’s and spouse’s labor, capital, and transfer income. Also, imputed rent is included for home owners. Conversion to real values uses MSA-specific deflators.12 Following Henderson and Ioannides (1987), real pre-tax permanent income is obtained by estimating a fixed effect model that captures the effects of unobserved household characteristics. We estimate yit 5 a1 5 Zit a2 1 ui 1 eit ,

(13)

where yit is the real pre-tax income of household i at time t, Zit is a vector of household demographic characteristics at time t, and ui is a fixed effect accounting for unobserved household characteristics.13 The stochastic errors, ui and eit are assumed to have zero means and eit is serially and crosssectionally uncorrelated with constant variance. Included among explanatory variables in (13) are head’s and spouse’s age, education, and their squares, number of children under 18 years of age residing with the household, health of the head, and number of years holding the same job and its square. Variables that are significant at the 5% level are head age and 10

The marital status of the head is allowed to change over time. The PSID reports household locations by state and county. To obtain locations by MSAs, Area Components of Standard Metropolitan Statistical Areas from the County and City Data Book (U.S. Bureau of Census, 1977) defined as of June 1977 are used to integrate counties into MSAs. The MSAs included in the sample are Atlanta, Baltimore, Boston, Buffalo, Chicago, Cincinnati, Cleveland, Dallas, Denver, Detroit, Houston, Kansas City, Los Angeles, Milwaukee, Minneapolis, New York, Philadelphia, Pittsburgh, San Diego, San Francisco, Seattle, St. Louis, and Washington, D.C. 12 For the 1974–1981 period, the deflator is the total budget net of total housing cost obtained from the U.S. Bureau of Labor Statistics Urban Family Budgets, Intermediate Budget Series for a Four-Person Family (1975–1981). After 1981, the index for all items less shelter is from the CPI Detailed Reports Consumer Price Index for All Urban Consumers (1982–1988). 13 This calculation allows the value of permanent income to change over time, following the method of Goodman and Kawai (1982). We implicitly assume that the length of stay p decision is affected by the time variation of ydit . In contrast, our measure of income used in the housing demand equation is lifetime income. 11

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its square, spouse age, spouse education and its square, number of children, and job tenure and its square. Predicted real pre-tax income is used to compute real after-tax permanent p income ( ydit ). After conversion to nominal pre-tax income using MSAspecific deflators, taxable income is calculated by subtracting deductions and personal exemptions from the predicted pre-tax income. Each household chooses the maximum deduction, comparing the standard deduction to itemization. Amounts of personal exemptions and standard deductions are reported in each year’s Federal tax table. In calculating itemized deductions, non-housing-related deductions including state income tax deductions are assumed to equal 6% of a household’s predicted pre-tax income.14 Applying each year’s tax table to each year’s taxable portion of the predicted permament pre-tax income yields the predicted tax and the after-tax permanent income. Both are divided by MSA-specific deflators to convert to real values. The shock to income is measured as the absolute value of real transitory after-tax income, ysdit , defined as the absolute value of the difference between actual and predicted after-tax permanent income. p We predict a household’s real wealth (W it) using a method similar to that used for estimating pre-tax income. It is endogenous because the level of wealth is a result of a household’s choice of housing and other goods consumption. Wealth is calculated as the sum of a household’s real housing equity if it is a home owner and its real capital income divided by that year’s zero coupon bond interest rate. These rates are obtained from the U.S. Treasury term structure data reported in McCulloch and Kwon (1993). Explanatory variables in the regression include college education indicators for the head and spouse and those variables used in the pre-tax income estimation. The measure of mortgage lock-in depends not only on the difference between the current mortgage rate and the original mortgage rate but also on the remaining mortgage principal. Denote the current mortgage rate as it , the original mortgage rate is i0 , and the remaining mortgage principal as Mt , this value observed in the sample. The measure of the lock-in effect is assumed to be (it 2 i0) Mt if it . i0 , but if interest rates fall, then the lock-in is zero.15 To obtain i0 we identify the time that a household moved and bought a house during 1974–1979, and i0 is assumed to be the zero coupon rate at that time. We include two lock-in variables in the estimation 14 When calculating the expected value of future personal exemptions, rather than the actual number of children, we use the predicted number of children. This is obtained by regressing the actual number of children against the characteristics of the household head and allowing for household-specific effects. The fixed effect assures that the predicted value nearly equals the actual value in the present, but time varying variables such as age will change the predicted future number of dependents over time. 15 The impact of declining interest rates is captured by the user cost variable.

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OWNER-OCCUPIED HOUSING DEMAND p

of length of stay. One is the predicted lock-in amount (Lit), this calculated by multiplying an owning household’s remaining mortgage principal in each year by (it 2 i0).16 The measure of shock in mortgage lock-in (Lsit) is the difference between the actual lock-in measure at time t and the amount of lock-in predicted at time 0 to exist at time t.17 The PSID reports the number of children under age 18 in a household. To calculate a measure of the unexpected change in the number of children, we predict the number of children using a fixed-effect model similar to that used for the pre-tax income estimation. The independent variables include age of head, age squared of head, education of head, education squared of head, health status of head, job tenure, job tenure squared, and whether college educated. All variables are significant at the 5% level except the education variables. The absolute value of the difference between the actual number of children under age 18 and the predicted value is used as a measure of the unexpected change. Actual marital status of head is reported in the PSID. The expected marital status of head is estimated using a probit model that yields a time path of the probability of marriage. Along with the head’s explanatory variables used in the pre-tax income estimation, we include head’s sex, race, and the real rent in the MSA. Rent is included because prior research found that household formation depends on the level of real rents in a respondent’s locality (Haurin et al. 1993).18 All variables are significant at the 1% level except the square of job tenure, college education, and age of the head, these being marginally significant. Marriage outcomes are correctly predicted in 85% of the cases. We include two measures of unexpected marital shocks. For households married at time t, we define S-Married as 1-prob(married)t . For single heads at time t, S-Divorce is u0-prob(married)tu. In both cases, positive values occur when marital status at time t differs from that predicted in the initial period. The other variables included in the estimation of length of stay are timeinvariant variables and time varying variables assumed to be known by a household. The time-invariant variables include whether the head is male or African-American and the year when a household moved into the 16

The predicted future spot rate is calculated from the data of McCulloch and Kwon (1993). The possibility of refinancing is allowed and is signaled by the movements of a household’s remaining mortgage principal. If a household does not move, but the remaining mortgage principal increases by 30% in a year, then we assume refinancing occurs. This relatively large increase is selected as the threshold to adjust for possible measurement errors in the mortgage balance data. 18 We assume real rents vary across geographic locations, but are constant over time. We also assume that, for a given location, real rents are invariant across both individuals and the level of housing consumption. The derivation of rents used data from Thibodeau (1989) is described by Chung (1994). 17

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house.19 The time-varying variables assumed to be known are the age, education, and employment status of the head and the region where the household lives. 3.2. Measurement of Variables for the Housing Demand Estimation We assume that the quantity of housing demanded is a function of lifetime household wealth. We follow Zorn (1988) and Henderson and Ioannides (1987) and calculate the value in an annuity whose present value equals a household’s lifetime wealth:

F O W0 1

yd 5

T

t50

O T

t50

ydt /(1 1 rit)t

G

,

(14)

1/(1 1 rit)t

where rit is the real after-tax interest rate. Initial real wealth (W0) is the value of wealth in the year of purchase, this derived as described in Section 3.1.20 The length of the planning horizon (T ) is set such that a household head is assumed to live until age 85.21 The real after-tax interest rate (rit) varies among households, p

rit 5 (1 2 t it) it 1 r 2 fgt ,

(15)

p

where t it is individual i’s predicted permanent marginal tax rate at time t,22 it is the zero coupon rate at time t, r is the risk premium, and fgt is the expected inflation rate at time t. Following Follain et al. (1987), we set r to 0.03. The expected inflation rate is calculated assuming that households expect the real interest rate to remain constant in the future. This assump19 We include dummy variables for the years 1974 to 1978, omitting 1979. These variables are included to pick up otherwise unmeasured effects of general economic conditions at the time a household moves in. 20 Because the move-in year of households who chose to move and own houses is different from 1974 to 1979, six different streams of zero coupon rates are used depending on when a household moved in. To obtain a series of future expected zero coupon rates, term-to-maturity data in the years households moved and owned are used and each future year’s spot rate is calculated. 21 When a family’s real after-tax permanent income is extrapolated for the post-1987 period, 1988 information about the independent variables is used. We assume that the marital status does not change after 1987 and the head does not change job positions until age 65 when he or she retires. The number of children under 18 years of age in the family unit changes over time. 22 We use the predicted (permanent) marginal tax rate, rather than the actual marginal tax rate, to be consistent with our use of permanent income.

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tion implies that the future expected inflation rates can be calculated as the future spot rate less the current real interest rate. 3.3. Constant-Quality House Price and the Quantity of Housing Using data for a 10-yr (1974–1983) period from the Annual Housing Survey (AHS), Thibodeau (1989) measured nominal housing prices for 60 metropolitan areas. He used a hedonic equation technique that regressed the price of housing on a set of housing characteristics. Tenure-specific house prices were measured by predicting the market rent or market value for constant-quality housing. However, because residential dwellings were surveyed in the AHS only at three- or four-year intervals in each metropolitan area, among the 23 MSAs used in our sample, fewer than 10 MSAs are included in each year for the 1974–1979 period. To interpolate between the years reported in Thibodeau’s index, we use the annual rate of change in real new house prices of 12 geographic regions for 1964–1982 provided by Diamond (1984). Diamond’s estimation used weights and coefficients generated from the hedonic equation used by the Bureau of the Census in the computation of the C-27 series, ‘‘Price Index of One-Family Houses Sold.’’ The quantity of housing (H ) is calculated by converting the reported nominal house value that a household purchased when it moved to real terms using MSA-specific deflators and then dividing real house values by the above constant-quality real house price index. 3.4. Weighted Average User Cost of Owning The weighted average user cost of owning in (2) differs among households because of the variation in Federal income tax deductions of mortgage interest and property tax payments. The property tax rate, tpt , is the reported property tax payment divided by house value. We assume that tpt does not vary over time. The rate of depreciation and maintenance expenditures on housing, e, is assumed to equal 0.02. For the rate of house price appreciation at time t, gt , we assume that there are no real capital gains or losses on housing and that inflation rates are the same across MSAs. Therefore, the expected inflation rate at time t is used for gt . Studies of the size of transaction costs associated with home ownership are few. Chambers and Simonson (1989) argue that 6 to 10% of house value is reasonable. Cunningham and Hendershott (1984) argue for a higher value, up to 12%. Malatesta and Hess (1986), using a sample of 100 transactions, find transaction costs are 12%. We take a conservative approach and assume that b is 0.06. After obtaining the expected length of stay (D) from the hazard rate estimation, we obtain the transaction-adjusted weighted average cost of owning from a discrete version of Eq. (5).

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In our estimation of housing demand, various other forms of the owner cost variable are tested including (1) the sum of the user cost at the time of purchase (a constant over time) and the annualized value of transaction costs, C tO50 1 [bPe2rD* /e0D* e2rt dt], (2) the transaction-adjusted weighted average cost less the annualized value of transaction costs, e0D* C tO e2rt dt/e0D* e2rt dt, and (3) the user cost of owning at the time of purchase, C tO50 . These variations include the typical static measure of user costs (3), a forward-looking measure that excludes transaction costs (2), and an easily implemented static measure that includes transaction costs. 3.5. Other Variables Other variables included in the housing demand estimation are household demographic charactertistics such as the age of a household head, the number of children under age 18, whether the head is married, male, or African-American, and the level of education, all of which are measured at the time when a household moved and bought a house.23 Dummy variables indicating the year of purchase are included to capture any differences in housing demand related to intertemporal variations in macro variables. 4. ESTIMATION RESULTS

Table 1 reports the results of the hazard rate estimation of the length of stay. About half of the sample completed their stay by 1988. Among the variables included in the estimation, sex and age of the head are significant. A household with a male head tends to move sooner than one with a female head and older heads stay longer. The probability of marriage is significant and positively impacts the length of stay. The lock-in variable has the anticipated positive coefficient, but it is not statistically significant. The only significant shock variable is the number of children; however, the marital status shock measures are near significance. The results indicate that positive or negative deviation from of the actual from the expected number of children reduces the length of stay. An unexpected change in marital status has differing effects depending on whether the event is a divorce or marriage. If the head divorces unexpectedly, he or she tends to move earlier, while if a marriage occurs unexpectedly, the household tends to stay longer. A likelihood ratio test indicates that the coefficients of the five measures of shocks are significantly different from zero at the 1% level.24 All of the year-of-move dummy variables are positive and the ones 23

Education level is categorized into eight levels ranging from highest grade completed of 0–5 to an advanced/professional degree. 24 The test statistic is 48.38 and the x 2 value with five degrees of freedom is 15.09 at the 1% level.

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TABLE 1 Length of Stay Hazard Model Estimation Variables

Coefficients a

Constant Male head African American Head age Head education Permanent income (3100) Wealth (3100) Predicted children Predicted lock-in (3100) Prob. of marriage Head employed S-permanent income (3100) S-lock-in (31000) S-children S-married S-divorced North Central West South Year 5 74 Year 5 75 Year 5 76 Year 5 77 Year 5 78 v Log-likelihood

21.71 (21.5) 24.17 (22.5) 0.33 (0.8) 0.06 (4.3) 0.11 (1.0) 20.17 (21.1) 20.02 (20.6) 20.09 (21.0) 0.25 (1.5) 6.28 (2.8) 0.10 (0.2) 0.18 (0.5) 0.01 (0.4) 20.38 (22.2) 3.06 (1.8) 20.62 (21.8) 0.10 (0.4) 20.03 (20.1) 0.23 (0.8) 0.82 (3.3) 0.52 (1.7) 0.18 (0.6) 0.81 (2.3) 0.85 (3.2) 0.47 (5.5) 2180.06

Means

0.85 0.44 41.86 5.42 177.47 161.94 1.52 62.60 0.62 0.88 12.25 72.90 0.68 0.18 0.10 0.38 0.16 0.30 0.19 0.22 0.14 0.14 0.18

Values in parentheses are t-statistics. N 5 1011 (117 households). a

for 1974, 1977, and 1978 are significant at the 5% level; thus, households that became owners in these years remained in place longer than those becoming owners in 1979, ceteris paribus. The coefficient of y, this being the inverse of p in the survival function, is significant and less than unity, implying that the hazard rate of moving increases monotonically over time.25 The expected length of stay of a household is calculated using observed values of the time-invariant variables and predicted values of the timevarying variables, setting the value of the shock variables to zero. In the calculation, we assume that a household moves if the cumulative survival probability becomes less than 0.5. 25

It is difficult to compare our results with those from other studies because of the substantial differences in the explanatory variables.

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TABLE 2 Frequencies of Expected Length of Stay Expected length of stay (years)

Percentage of sample

1–5 6–10 11–15 16–20 21–25 26–30 .30 Mean in years Variance

38.5 29.1 20.5 5.1 2.6 2.6 1.7 9.28 6.55

Table 2 reports the frequency distribution of the expected length of stay. The mean value in our sample of owners is 9.3 yr. Chambers and Simonson (1989) report expected lengths of stay derived from a National Association of Realtors’ sample where they find that recently moving households with head age 60 expect to remain in place 10.8 yr, while heads under age 30 expect to stay 5.5 yr. We report in Tables 3 and 4 estimates of log-linear and linear forms of TABLE 3 Housing Demand Estimation (Log-Linear Specification) Variables

Coefficients a

Constant Log transaction-adjusted weighted average cost of owning Log owner cost Log present value of lifetime wealth Head age Children Married Male head African-American Education Year 5 74 Year 5 75 Year 5 76 Year 5 77 Year 5 78 Adjusted R-squared

3.46 (3.3) 20.57 (22.9)

a

0.31 (2.1) 0.10 (1.5) 20.01 (20.2) 0.38 (2.2) 20.02 (20.1) 20.12 (21.0) 0.06 (1.8) 0.07 (0.4) 0.12 (0.7) 20.25 (21.4) 0.11 (0.5) 0.41 (2.6) 0.49

Values in parentheses are t-statistics. N 5 117.

Coefficients a 2.78

Means

(2.7) 2.86

20.44 (22.2) 0.34 (2.3) 0.01 (1.9) 20.00 (20.1) 0.36 (2.0) 0.01 (0.1) 20.17 (21.5) 0.05 (1.4) 0.20 (1.2) 0.27 (1.8) 20.11 (20.6) 0.35 (2.0) 0.54 (3.3) 0.47

2.78 4.97 33.26 1.50 0.79 0.85 0.44 5.25 0.19 0.22 0.14 0.14 0.18

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TABLE 4 Housing Demand Estimation (Linear Specification) Variables

Coefficients a

Constant Transaction-adjusted weighted average cost of owning Owner cost Present value of lifetime wealth Head age Children Married Male head African-American Education Year 5 74 Year 5 75 Year 5 76 Year 5 77 Year 5 78 Adjusted R-squared

28.27 (1.0) 21.57 (22.4)

a

0.35 (6.5) 0.54 (1.5) 0.96 (0.4) 20.08 (1.8) 218.19 (21.4) 211.21 (21.6) 1.70 (0.8) 6.18 (0.6) 6.64 (0.7) 25.68 (20.5) 14.90 (1.2) 27.23 (2.7) 0.62

Coefficients a 11.81

(0.4)

21.29 (21.7) 0.35 (6.2) 0.67 (1.8) 1.24 (0.6) 19.23 (1.7) 214.86 (21.2) 214.53 (22.1) 0.97 (0.5) 13.95 (1.4) 15.57 (1.7) 1.89 (0.2) 26.79 (2.4) 34.77 (3.4) 0.60

Values in parentheses are t-statistics. N 5 117.

housing demand, each with two measures of the owner cost variable. One is the transaction-adjusted weighted average cost of owning; the other is the user cost derived using values of its components at the time of purchase. The focal variable, owner cost, is statistically significant at the 5% level in both cases. Also, in both specifications, the adjusted R 2 is higher when we use the transaction-adjusted weighted average user cost. Permanent income has a higher level of significance in the linear case. Marriage raises housing demand, but race, age, and the number of children are not significant factors. We find that our price and income elasticities (Table 5) lie in the range of previous studies. In the log-linear specification, elasticities can be obtained directly from the estimated coefficients, while in the linear specification, elasticities are calculated at sample means. The absolute value of price elasticity is higher in the log-linear specification (0.57 versus 0.32), while income elasticity is higher in the linear specification (0.65 versus 0.31). Regardless of specifications, the price elasticity of demand for owneroccupied housing is greater when we use the transaction-adjusted weighted average user cost in the estimate and lower when we use the standard measure of the user cost. Specifically, the price elasticity obtained by using the transaction-adjusted weighted average user cost is 34% higher than the value derived using the standard user cost in the linear specification, and 30% higher in the log-linear specification.

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TABLE 5 Comparison of Elasticities Price elasticity

Income elasticity

Owner user cost

Linear a

Log-linear

Linear a

Log-linear

Transaction-adjusted weighted average user cost of owning Transaction-adjusted standard user cost of owning Weighted average user cost of owning, no transaction cost Standard user cost of owning

20.32

20.57

0.65

0.31

20.33

20.54

0.65

0.36

20.24

20.54

0.64

0.28

20.24

20.44

0.65

0.34

a

In the linear demand specification, elasticities are evaluated at sample means.

Table 5 also reports the elasticity estimates for the two intermediate measures of an owner’s user cost that we have previously described. Comparing the first two rows, we find similar elasticities. The implication is that if transaction costs are accounted for in the user cost variable, the price and income elasticities of demand are insensitive to whether the user cost is derived based on contributing factors’ values at the time of purchase or as a weighted average of expected future values.

5. CONCLUSION

This study develops a model of a household’s housing decisions based on an intertemporal optimization framework. In most previous econometric studies of housing demand, the measure of owner cost is static. In a static formulation, the implicit assumption is that a household’s length of stay is proxied by other household characteristics such as age. One approach to overcoming the static assumption is to estimate a structural equation model where the length of stay is included among the endogenous variables. In that framework, length of stay is accounted for by including it in the demand equation for housing as one of the explanatory variables. This study synthesizes these fragmented approaches by consolidating into one measure the standard user cost of owning, transaction costs, and the expected length of stay. From these inputs we develop a transaction-adjusted weighted average cost of owning. Because we observe actual length of stay rather than the expected length of stay, we must account for the occurrence of unexpected events and their impact on the measured length of stay. We project paths of influential variables and compare these paths with subsequent realizations. These

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measures of unexpected shocks are included among the explanatory variables in the length of stay equation. Once measured, we can control for the impact of unexpected, post-planning period events and then compute the planned length of stay at the time of purchase of a home. We estimate the expected length of stay using a parametric hazard rate model that accounts for time-varying covariates such as income, marital status, and number of children. We estimate price and income elasticities of demand for owner-occupied housing using different specifications and different measures of the owner cost variable. In both linear and log-linear forms, we find a superior fit using the transaction-adjusted weighted average user cost variable. Of note is the finding that the price elasticity obtained by using the new measure of user cost is about one-third higher than the value estimated using the traditional specification of user cost. This result suggests that the many prior studies of housing demand may have underestimated the value of the price elasticity of demand. Our results also suggest that this difference is primarily due to the inclusion of transaction costs in user cost, rather than to modification of the user cost to include future values of its components.

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