The Housing Decisions of Older Households: A Dynamic Analysis

JOURNAL OF HOUSING ECONOMICS ARTICLE NO.

7, 21–48 (1998)

HE980221

The Housing Decisions of Older Households: A Dynamic Analysis* Peter G. VanderHart Bowling Green State University, Bowling Green, Ohio 43403 Received October 6, 1997

The housing decisions of older households are best modeled as dynamic rather than static decisions. A dynamic discrete choice econometric technique is applied to data on households whose heads are aged 50 years or older to estimate older households’ preferences for living in homes with large and small amounts of equity, rental units, and dependent arrangements. The dynamic technique provides a better fit than a static analysis, and the results suggest that economic factors such as income and financial assets are more important to the decision than previous conventional analyses have suggested.  1998 Academic Press

1. INTRODUCTION

The decision of whether to remain in a given housing situation or move to an alternative arrangement is one of the most dramatic decisions that a household makes. This is particularly the case for older households, which may have a significant amount of their wealth and their members’ sense of self invested in their homes. Factors such as health, marital status, life expectancy, and the possible desire to tap home equity to finance retirement make an older household’s housing decision very complex. Previous studies of older households’ housing decisions have generally found that housing changes among the aged are not very frequent, and that when they do occur they are most likely for noneconomic reasons. From these observations it is typically concluded that older households’ housing decisions do not fit the typical economic models of consumer choice and wealth accumulation. However, these analyses do not explicitly take account of the dynamic forward-looking nature of the housing decision: Decisions made today dictate the housing arrangement for the near future, and thus must consider the expected future path of economic and noneco* The supercomputer time that made this research possible was provided through an inkind grant from the Ohio Supercomputer Center. I thank John Rust for his help with the initial stages of this research. Any errors are my own. 21 1051-1377/98 $25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.

22

PETER G. VANDERHART

nomic factors. When modeled in this way it may be found that infrequent housing changes are actually optimal, and that the effect of noneconomic factors on future economic variables may create a spurious relationship between noneconomic factors and housing choice. This paper is intended to provide a dynamic empirical investigation of the housing decisions of older households. The following section describes in more detail the past research on this topic. The third section explains the static choice models on which most of this research is based, and makes a case for a dynamic empirical analysis. A fourth section describes the dynamic discrete choice technique used here and explains how it compares to conventional techniques. A fifth section describes data drawn from the Panel Study of Income Dynamics, the categorization of the data, and the specification of the model. A sixth section presents the results of both the dynamic and static models, and a final section critiques the results and the dynamic estimation technique, and offers some conclusions.

2. LITERATURE REVIEW

2.1. Mobility Much previous work in this area has examined the mobility of older households, and the most common observation is that it is quite low. Reschovsky (1990) finds that households with heads over 65 move only 6.3% of the time in a given year, compared to 22.1% for their younger counterparts. Slightly higher rates are found when households with heads in their 50’s are included (VanderHart (1995) and Venti and Wise (1989)), but a large difference between the groups remains. One striking observation in these studies is the difference between the mobility rates of older homeowners versus renters and those in other arrangements. Venti and Wise (1989) find that homeowners are only about one third as likely to move as those in other tenures. Similar results are found in Feinstein and McFadden (1987) and VanderHart (1995). This result has two possible interpretations: Either older homeowners are content with their arrangements and wish to ‘‘age in place,’’ or high transition costs prevent them from making otherwise beneficial moves. 2.2. Home Equity Decisions This literature has also recognized that there is much more to the elderly’s housing decisions than the decision to move. After all, moves do not necessarily change the tenure or the amount of housing consumed, and changes in tenure, mortgage status, and even dependency can be accomplished without moves. Much of the literature concentrates on the question of

HOUSING DECISIONS OF OLDER HOUSEHOLDS

23

whether or not older homeowners desire to reduce their home equity (as implied by some versions of the permanent income and life-cycle hypotheses). This change could be accomplished by selling the home and moving to a ‘‘smaller’’ home or a rental unit or dependent arrangement, or without moving by tapping the home equity through second mortgages, reverse mortgages, sale leasebacks, and other arrangements. There is conflicting evidence regarding the desire of older homeowners to reduce their home equity. Analyses of tenure transition matrices by both Poterba (in his comment on Venti and Wise (1989)) and Feinstein and McFadden (1989) indicate a net movement from home ownership to rental units by the elderly. Other work by Stahl (1989) indicates that households that move tend to occupy less housing (measured by rooms per person) than those that do not move. Although this result is suggestive, it is unclear whether movers actually reduce their housing or merely tend to hold less of it (both before and after the move) than nonmovers. Other authors dispute the notion that older homeowners wish to reduce their home equity. Venti and Wise (1990) determine that older households that move from one owner-occupied unit to another are just as likely to increase home equity as to decrease it. Their analysis implies that if everyone in their sample were to move, the average level of home equity would increase slightly. Moreover, the tenure transition matrices that indicate a net movement from homeownership also indicate that a significant number of households become homeowners at advanced ages. 2.3. Causes There is conflicting evidence regarding the causes of housing changes by older households. In previous literature these factors are usually discussed in the context of the housing decisions of older homeowners, to the exclusion of those moving from other tenures. For the most part this precedent is followed in what appears below. Some authors have tended to concentrate on economic causes (amount of home equity, financial assets, income, and housing cost) while others have emphasized noneconomic factors (marital status, physical limitations, children, and retirement status). These factors are discussed in turn below. Among the economic factors, several authors have examined the role of wealth in housing decisions. Venti and Wise (1990) find a negative relationship between wealth and desired home equity, while Feinstein and McFadden (1987) find a weak positive relationship. However both of these studies combine financial assets with home equity to define wealth, and the two components may have different effects. Several works have attempted to isolate the effect of home equity on the propensity of homeowners to undertake various housing changes. Re-

24

PETER G. VANDERHART

schovsky (1990) finds mixed results: Households with high home value are determined to have a high potential benefit from moving to other owneroccupied housing, but there is a weak negative relationship between a home’s value and the potential benefit from entering a rental unit. Merrill (1984) uses a multivariate logistic analysis to find that home equity has negative effects on the probability of a homeowner moving to a different home and on the probability of a homeowner becoming a renter. Using similar techniques, VanderHart (1994) finds that high levels of home equity are negatively associated with moves to situations with increased equity, but are positively associated with same-tenure moves that reduce equity. Further mixed evidence is found for the effect of financial assets. Venti and Wise (1989) find very little difference in mobility rates when they separate homeowners into financial asset quartiles. Merrill (1984) finds that liquid assets increase the probability that a homeowner becomes a renter. VanderHart (1994) finds the opposite, and also finds that assets increase the likelihood of a move to higher equity arrangements. There does seem to be some consensus with regard to the effect of income. Generally it appears that higher levels of income deter moves away from home ownership (Feinstein and McFadden (1987), Merrill (1984), and Reschovsky (1990)) and that income increases the desired level of home equity (Venti and Wise (1989, 1990)). Several authors have examined the relationship between income and the propensity to enter dependent living arrangements, and generally they find a negative relationship (Beland (1984), Boersch-Supan (1989), Boersch-Supan et al. (1988), and Elwood and Kane (1989)). The effect of housing costs is less clear. Although gerontological work links high housing costs with homeowner dissatisfaction (O’Bryant and Wolf (1983)), other work appears to dispute this notion (Venti and Wise (1989) and Ai et al. (1990)). While there is not much agreement in this literature regarding the effects of economic variables, there is much more agreement as to the effect of noneconomic variables. A significant number of articles examine the effects of martial status on housing changes. Reschovsky (1990) finds that married homeowners have less to gain by moving from their homes or switching tenure. Feinstein and McFadden find that the exit of a wife from a maleheaded, home-owning household has a large positive effect on the household’s mobility, but a negative effect on a moving household’s propensity to remain an owner. Similar results are found by Venti and Wise (1990) and VanderHart (1994). Additionally, many studies have found that being married greatly reduces the elderly’s chances of institutionalization. (See Garber and MaCurdy (1989), Ellwood and Kane (1989), and BoerschSupan (1989).) Having a spouse also tends to reduce the propensity to live

HOUSING DECISIONS OF OLDER HOUSEHOLDS

25

dependently with others or with children, (See Kotlikoff and Morris (1988) and Boersch-Supan (1989).) Physical limitation and bad health appear to have a dramatic effect on moves to dependent arrangements, but little effect on independencepreserving moves. Merrill (1984), VanderHart (1994), and Venti and Wise (1990) all find insignificant effects of physical limitation or bad health on own-to-own moves or moves to rental units. Other authors concentrate on the effect of health on living independently, and not surprisingly, most agree that bad and deteriorating health tend to precipitate moves into institutions. (See, for example, Garber and MaCurdy (1989), and Ellwood and Kane (1989).) There is mixed evidence on health’s effect on an older household’s propensity to live with others: Boersch-Supan (1989) finds that health limitations are negatively associated with the elderly’s odds of living dependently with others. However, Kotlikoff and Morris (1988) find that the elderly who are comparatively more healthy are actually less likely to live with their children. Children also seem to have some effect on the housing decisions of older home owners. Venti and Wise (1990) find that the presence of children in the household deters owners from moving. Feinstein and McFadden (1987) report that changes in the number of children in a household are negatively related to both the elderly’s mobility and the mobile homeowner’s propensity to remain an owner. Boersch-supan et al. (1988) find that after leaving the home, children appear to have a positive effect on the likelihood that parents live dependently with their children and a negative effect on the likelihood that the parents live dependently with others. Several authors have examined the relationship between retirement and housing change. Venti and Wise (1989) report that being retired tends to increase elderly homeowners’ mobility. Feinstein and McFadden (1987) concur, and also find that the same relationship holds for wives’ retirement status. They also find that the head of household being retired also makes it more likely that owners become renters. VanderHart (1994) finds the same result, and also finds that becoming retired is associated with moves into dependent arrangements. 2.4. Conclusions of the Literature Taken as a whole, these articles suggest that noneconomic factors are much more important to the housing decisions of older households than are economic factors. The effects of noneconomic factors are almost always substantial and unambiguous, while the effects of the economic factors often are not consistent across studies and are often small in magnitude. Jones (1996) summarizes this literature by stating ‘‘. . . when housing wealth liquidation does occur, it is triggered by noneconomic events rather than by the economic logic of life cycle theory.’’

26

PETER G. VANDERHART

This is a convincing case, but one concern remains: What if the ‘‘noneconomic events’’ have a strong effect on housing decisions not because of their direct effect, but because of the indirect effect that these events might have on future economic variables? For example, retirement not only is a significant change by itself, but also signals that a household’s income will be lower in the future. The onset of a physical limitation will undoubtedly have a direct effect on housing decisions, but may also have an indirect effect via the expectation of higher health care costs, lower future income, faster depletion of financial assets in the future, and higher owner-occupied maintenance costs from the discontinuation of do-it-yourself repair. If these indirect effects on future economic variables are more important than the direct effects of the noneconomic variables, then the literature is erroneously concluding that the noneconomic factors are more important. One way around this difficulty is to employ a dynamic discrete choice empirical model. Unlike the conventional discrete choice models employed by the articles described above, the dynamic discrete choice model can discern among the direct and indirect effects of a change in any relevant variable. The next sections explain.

3. CONVENTIONAL DISCRETE CHOICE MODELS

3.1. Background Many economic decisions require a discrete action rather than choice along a continuum. In these contexts least-squares estimation of the determinants of a decision is not appropriate, and a limited-dependent-variable technique must be used. The first techniques of this kind were the binomial logit and the binomial probit due to Goldberger (1964). In short order these were extended to cover the multinomial case, as well as sequential, nested, and other variations. The multinomial probit, while perfectly acceptable in theory, proves to be more computationally burdensome in practice, and thus most analyses involving choice sets greater than 2 have utilized the logit form. In general, the explanatory variables of a multinomial logit analysis can include both characteristics of the economic agent and characteristics of the choices he or she confronts. In the context of the housing choice problem considered here, characteristics of the alternatives are unlikely to be observed (unless they are chosen), and therefore explanatory variables are intentionally restricted to include household characteristics only. Given this restriction, the probability of household i choosing alternative j can be written (Maddala, (1983))

HOUSING DECISIONS OF OLDER HOUSEHOLDS

Pi j 5

exp(b9j Xi )

i 5 1, . . . , N, j 5 1, . . . , M

O exp(b9 X ) M

k

27 (1)

i

k

where X is a vector of explanatory variables, b is a vector of parameters of interest, N is the number of observations, and M is the number of alternatives. Using these probabilities, the log-likelihood function can be written

ln(L) 5

O O d · ln M

M

ij

i

j

exp(b9j Xi )

3O M

4

,

(2)

exp(b9k Xi )

k

where di j is an indicator variable equal to 1 if person i chooses alternative j. All of the parameters in Eq. (2) are not identified. To estimate the model the b vector must be normalized by setting one alternative’s parameters equal to 0. The other parameters are then estimated relative to the parameters for the null alternative. The maximization of Eq. (2) has no closedform solution for b, so an iterative approach must be used. 3.2. Random Utility Models McFadden (1973) showed that the multinomial logit can be derived from a model of random utility. The unobserved latent variable Y i*j is used to denote the indirect utility that person i receives from choice j. Thus, Y i*j 5 Uj (Xi ) 1 «i ,

(3)

where Xi is a vector of attributes of the decision maker, and «i includes unobserved individual-specific variations in tastes and possibly errors in optimization by the decision maker. Although Y i*j is not observed, it is possible to observe the choice made. If the decision maker is rational in that he or she chooses the alternative with the highest utility, then Yi j 5 1

* , Y *i2 , . . . , Y *iM ] if Y *i j 5 Max[Y i1

Yi j 5 0

otherwise.

(4)

If it is further assumed that the error terms in Eq. (3) are independently and identically distributed, and that they come from an extreme-value distribution, then it is possible to show that

28

PETER G. VANDERHART

Prob(Yi j 5 1 u Xi ) 5

exp(Uj (Xi ))

3O M k

exp(Uk(Xi ))

4

(5)

(See Maddala (1983, pp. 59–61) for a complete derivation.) If it is further specified that Uj (Xi ) 5 b9j Xi for j 5 1, 2, . . . , M, then the log-likelihood function produced by Eq. (5) is precisely what appears in Eq. (2). 3.3. Interpretation Because the random utility model specifies utility to be a function of the dependent variables, the parameter vector b can be interpreted as made up of parameters of a utility function. For situations in which the decision can be assumed to affect current utility only, the interpretation is straightforward. However, when the decision can be expected to have long-lasting effects on the decision-maker’s well-being, the interpretation is more problematic. Most commonly the parameters are interpreted as if they were derived from a value function representing the summation of all discounted future realizations of a single period utility function. However, if the variables used in the regression influence one another in future years, this interpretation may lead to erronrous conclusions: A variable that plays a minor role in a person’s utility function but that influences other variables that play major roles will be found to have a large parameter and thus would be interpreted as affecting utility directly.

4. DYNAMIC DISCRETE CHOICE MODELS

This problem of interpretation has led several authors to attempt to disentangle direct and indirect effects by explicitly modeling forward-looking intertemporal decision making. Applications have included job search (Wolpin, 1987), patent renewal (Pakes, 1986), the retirement decision (Phelan and Rust, 1991), bus engine replacement (Rust, 1987), and aircraft engine maintenance (Kennet, 1994). (See Eckstein and Wolpin (1989) for a survey, and Kapteyn et al. (1995) for further recent examples.) These applications use a wide variety of specifications and solution methods. The technique described here most closely resembles the formulation used by Rust (1990) and Phelan and Rust (1991). This section begins with a description of dynamic programming and explains how it is used in the estimation procedure.

HOUSING DECISIONS OF OLDER HOUSEHOLDS

29

4.1. The Theory of Dynamic Programing In its simplest form, dynamic programing can be thought of as the optimal choice of a control variable ct , given stochastic state variables st , for all t. In utility analysis, the economic agent selects ct from time 0 to terminal time T to maximize expected lifetime utility. Mathematically this is written (Bertsekas, (1976))

FO T

Max E ct

G

d tU(u, ct , st ) s.t. st11 5 f (st , ct , wt ),

0

(6)

where d is an intertemporal discount factor, u represents utility function parameters, and wt is a disturbance. A problem such as this almost never has an obvious solution. This is true both because of the randomness of the state variables and because the choice of ct that maximizes the single period utility U(u, ct , st ) will not in general maximize the entire sum of utilities from 0 to T: The choice of ct must be made considering its effect on future utility via the constraint. A common solution to this sort of problem is known as backward induction, which uses Bellman’s (1957) ‘‘principle of optimality’’: In the final period the economic agent chooses cT to maximize utility in time T given sT . The solution is generally not a specific value but rather a rule that specifies an optimal value of cT for every sT . Once this rule is determined, a value function can be defined, VT (u, s) 5 Max U(u, cT , sT ),

(7)

cT

where u represents the underlying preference parameters. Once the value function is defined for time T, the optimization problem for time T 2 1 can be addressed: Max [U(u, cT21 , sT21) 1 d · E[VT (u, s)usT21 , cT21]] given sT21 .

(8)

cT21

Here the control variable will be chosen not only for its effect on current utility, but also for its effect on the value function. This process of updating the value function backwards through time is continued until the formula for time 0 is derived, Max [U(u, c0 , s0) 1 d · E[V1(u, s) u s0 , c0)]] given s0 , c0

(9)

30

PETER G. VANDERHART

where the value function is the sum of the discounted utilities from time 1 to T. 4.2. Estimation of Preferences Using Dynamic Programing The technique described above shows how one can model the intertemporal decisions of an economic agent. In this setting it is assumed that agents behave as if they solve the dynamic programming problem, and that by observing their behavior one can estimate the preferences that cause them to take their actions. Thus the estimation procedure can be thought of as a complex intertemporal revealed preference problem. The analysis begins with a description of the control and state variables, ct and st . A framework described in Rust (1988) is used, where st is decomposed into observed state variables xt and unobserved state variables «t . It is assumed that the evolution of the unobserved state variables is independent of contemporaneously observed state variables and can be described by the probability equation

P (xt11 , «t11 u xt , «t , ct ) 5 p(xt11 u xt , ct ) · q(«t11 u xt11),

(10)

where q(?) is a nonidentical multivariate extreme value distribution. It is also assumed that the realizations of «t are independent across persons, time, and choice of value for control variable, and that «t has dimension equal to the dimension of the choice set for the control variable. These assumptions allow the specification of an empirically tractable likelihood function over the estimated utility parameter vector u (Rust, (1988)), I

T

L(u ) 5 p p P(c ti u x ti , u ) · p(x ti u x ti21 , c ti21),

(11)

i 51 t 51

where (because of the assumption of the extreme value distribution, with t and i notation suppressed) P(c u x, u ) 5

exp[U(u, c, x) 1 d · E(V(u, c, x))] exp[U(u, d, x) 1 d · E(V(u, d, x))]

O

(12)

d

and where E(V(?)) is the solution to the functional equation (Phelan and Rust, 1991) E(V(u, c, x)) 5

E ln FO exphU(u, d, y) 1 dE(V(u, d, y))jG p(dy u c, x), y

d

(13)

HOUSING DECISIONS OF OLDER HOUSEHOLDS

31

where y and d index the future values of the state and decision variables respectively. Solving Eq. (13) is equivalent to solving the implicit dynamic programing model by backward induction. Expression (12) is the dynamic version of Eq. (5), in that it includes the entire discounted stream of utilities embodied in the value function. Note that if d is set equal to zero, Eq. (13) becomes irrelevant and Eq. (12) collapses to the conventional discrete choice expression. The estimation procedure proceeds as follows: (1) The p(?) transitions are estimated separately, and are taken as given in the likelihood function. (2) Given an initial value of the vector u, the derivatives of the log-likelihood function with respect to the u vector are calculated. (3) Using these derivatives and an approximation of the Hessian matrix, a new value of u is calculated. The process is repeated until a convergence criteria is met. In this respect the estimation technique is completely analogous to Newton’s well-known iterative estimation method (Judge et al. (1982)).

5. MODEL SPECIFICATION

5.1. Data The data used in this analysis come from several waves of the Panel Study of Income Dynamics (PSID, 1989). Although the PSID is not a panel of the elderly per se, it does contain a fairly large number of the aged, including some who are very old. The sample is restricted to contain households whose head is at least 50 years of age in the first year of the survey (1968). Observations are formed by comparing two adjacent waves to determine housing change. Some waves lack the necessary housing data and are therefore excluded from preference estimation, although they do contribute to the expectation estimation. In most years the PSID contains good information on housing status. In all years households are asked if they own, rent, or have other housing arrangements. If the response is ‘‘other,’’ they are asked if it is paid for by others, is some sort of payment for services, or takes a number of other forms. If the housing arrangement is owner-occupied, the owner is asked the value of the home, whether there is a mortgage, and if so the remaining mortgage principal. In some waves it is also possible to identify households that live in nursing homes or live dependently with relatives using a procedure developed by Ellwood and Kane (1989). Other variables measure various aspects of housing costs and the type of structure, but are not used in this paper. As its name implies, the PSID has excellent variables describing respondents’ income. Broad categories of income are available, including taxable

32

PETER G. VANDERHART

income, asset income, and transfer income. In this paper, nonasset taxable income is combined with transfer income to form the income measure. The asset data are more problematic. Except for a few years, data on assets are unavailable. Asset amounts are therefore imputed using the following procedure: All households are assumed to have $200 in non-interest-bearing financial assets. The first $100 in asset income is assumed to come from assets with a 5% return. Asset income above this amount is assumed to come from assets achieving a 10% rate of return. The PSID also has a wealth of demographic information. The variables used here are the head of household’s age, retirement status, marital status, and a subjective measure of the head’s physical limitation. Other variables including sex, number of children, and education are also available. 5.2. State Variables The computational complexity of the dynamic programing estimation technique requires economy regarding the factors included as state variables, and the number of categories for each variable is restricted to be rather small. The observed state vector is specified as xt 5 (hst , ast , yst , rst , lst , mst , at , dst ) where hst is initial housing status (1987$), with hst 5 1 if own a home with equity ($44,000, 1y) 2 if own a home with equity (2y, $44,000] 3 if rent 4 if live in a dependent living arrangement ast is imputed financial asset level (1987$), with 1 if financial assets (0, $200] ast 5 2 if financial assets ($200, $20,000] 3 if financial assets ($20,000, $40,000] 4 if financial assets ($40,000, $60,000] 5 if financial assets ($60,000, 1y) yst is total non-asset income (1987$), with yst 5 1 if income (2y, $5,000] 2 if income ($5,000, $10,000] 3 if income ($10,000, $15,000] 4 if income ($15,000, $20,000] 5 if income ($20,000, 1y) rst is retirement status, with

(14)

HOUSING DECISIONS OF OLDER HOUSEHOLDS

rst 5

33

1 if the head is retired or permanently disabled 0 if not

lst is limitation status, with lst 5 1 if the head is physically limited 0 if not mst is marital status, with mst 5 1 if the head is married 0 if not at is age, with at 5 the age of the head minus 50: [0, 50] dst is death status, with 1 if the head is deceased dst 5 0 if not. Even this coarse specification allows xt to take on 40,000 non-degenerate values. 5.3. Control Variable The control variable is one-dimensional, and describes the housing choice made by the household: hdt is the housing decision, with hdt 5 1 if stay in or move into a home with equity ($44,000, 1 y) 2 if stay in or move into a home with equity (2y, $44,000] 3 if stay in or move into a rental unit 4 if stay in or move into a dependent living arrangement. There are many cases in which a homeowner would be classified as moving from one home equity bracket to another even though no move or mortgage action was taken. This might occur because of appreciation or depreciation of the home’s value, the gradual payment of mortgage principal, or simple reporting errors. It would be inappropriate to classify these households as making the same changes as those that move or acquire mortgages. It is therefore prudent to specify an own-to-own housing change to take place only if the homeowner moved to another home or acquired a mortgage during the previous year and if the action placed them in a different home equity bracket. All other continuing homeowners are assumed to stay in the housing state that they occupied in the previous period. This specification of the control variable may not capture all housing

34

PETER G. VANDERHART

changes of interest. For example, a homeowner who sells her $200,000 home to move into a different $60,000 home is making a significant housing change, but the specification above classifies her as choosing to remain in the same housing state. Likewise, households may move between different renting arrangements, or different dependent arrangements, and be classified the same as those making no change. Additionally, there is no explicit housing state describing mortgage behavior. These problems could be dealt with by enriching the descriptions of the housing states. However, the addition of more housing states would greatly increase the computational complexity of the problem, and it is not attempted here. 5.4. Utility Function Specification To complete the specification of the dynamic model, the following functional form of the single-period utility function is specified: U(hs, hd, as, ys, rs, ls, ms, a) 5

O [Ihhd 5 ij · a 1 Ihhd 5 ij 4

i

i 51

· [bi1(as) 1 bi2(ys) 1 bi3(rs) 1 bi4(ls) 1 bi5(ms) 1 bi6(a)/10]] 1

(15)

O O Ihhs 5 j, hd 5 ij · c . 4

4

ij

j 51 i 51

In the first line of (15), the parameters of interest are the four ai ’s, which represent the baseline utilities associated with the four housing states defined above. Each of these measures is ‘‘shifted’’ by the parameters of the six other state variables in the second line of (15). These parameters represent how economic and noneconomic variables affect the utility received from each of the housing alternatives. For instance, if b16 , 0, then there is evidence that advanced age decreases the utility received from living in a home with equity greater than $44,000. The last line of equation (15) contains 16 c parameters, which are meant to capture the change in utility that occurs when moving from one housing state to another.

6. RESULTS

6.1. Estimation of Expectations The first task in undertaking the dynamic discrete choice analysis is to estimate the transition probabilities of the state variables. This amounts to an estimation of the households’ beliefs about the evolution of their state variables. Implicit in this procedure is the assumption that the households’

HOUSING DECISIONS OF OLDER HOUSEHOLDS

35

expectations are equal to the empirically observable transition probabilities for the state variables. Because xt may take on a great number of values, direct estimation of the probability transition matrix with even the largest of data sets is infeasible. Instead the analysis takes advantage of various properties of the state variables and a well-known law of probability to simplify the estimation of the transition matrix. First, it is noted that death is an absorbing state and that the value of other state variables is irrelevant after death. Furthermore, the age variable is not random, because at11 5 at 1 1 with probability equal to 1. It is also assumed that the housing state variable is perfectly controlled by the household, so that hst11 5 hdt with probability equal to 1. It is also assumed that the housing state plays no role in the determination of the future values of the other variables. This simplifies the remaining problem by reducing the dimensionality of the matrix to be estimated. The remaining state variables are assumed to depend on one another and make up the probability matrix to be estimated: p(ast11 , yst11 , rst11 , lst11 , mst11 , dst11 u ast , yst , rst , lst , mst , at , hdst 5 0j). (16) This expression can be rewritten by decomposing it into the product of each argument’s marginal probability. Except for the probability of death (where severe spurious correlation with some variables is present), each marginal probability is posited to depend on all other state variables: p(ast11 u ast , yst , rst , lst , mst , at , hdst11 5 0j) ? p(yst11 u ast , yst , rst , lst , mst , at , hdst11 5 0j) ? p(rst11 u ast , yst , rst , lst , mst , at , hdst11 5 0j) ? p(lst11 u ast , yst , rst , lst , mst , at , hdst11 5 0j) ? p(mst11 u ast , yst , rst , lst , mst , at , hdst11 5 0j) ? p(hdst11 5 0j u lst , at ).

(17)

Each of these marginal probabilities are estimated with separate binomial or multinomial logit regressions, using all cross-sections and all observations in the PSID that contain the necessary data. The results are presented with brief explanation in Appendix A. 6.2. Preference Estimation As with the case of standard multinomial logit regression, all of the utility parameters will not be identified. Thus it is necessary to constrain the coefficients for one housing category to be 0, and to interpret the other

36

PETER G. VANDERHART

TABLE I Results from Dynamic Programming Estimation Coefficient a b*1 (assets) b*2 (income) b*3 (retired) b*4 (limited) b*5 (married) b*6 (age) c*1 (to high equity) c*2 (to low equity) c*3 (to rental) c*4 (to dependent)

Low equity 0.0741 (0.720) 20.0414 (22.22) 20.0487 (21.74) 0.0532 (0.629) 20.0837 (20.879) 20.0405 (0.813) 20.0829 (23.51) 29.41 (235.4) — 24.36 (216.9) 24.45 (214.4)

Renting

Dependent

0.0424 (0.404) 20.0634 (23.41) 20.0545 (21.97) 0.132 (1.53) 0.00670 (0.071) 20.0812 (21.64) 20.0647 (22.86) 28.79 (234.3) 23.38 (213.3) —

20.201 (21.42) 20.103 (22.87) 20.116 (22.35) 0.106 (1.02) 0.178 (1.51) 20.116 (21.51) 0.0142 (0.495) 27.89 (220.6) 22.50 (27.61) 22.34 (28.15) —

23.37 (212.1)

Discount factor 5 0.95 Number of observations 5 12,323 2log likelihood 5 2636.161 Gradient direction 5 2.45 3 1025 Note. t-statistics are in parentheses.

coefficients as relative to what they are for the constrained category. In the results reported here, the highest home equity category is omitted for this reason. Additionally, the c terms for situations in which the housing state is unchanged are also set equal to zero, so that the other c terms can be interpreted as the utility cost of making a housing change relative to making no change at all. Table I presents the results. In each column of the table the utility parameters associated with a particular housing state are presented. The first line of each column represents the baseline utility received from living in that housing alternative. The next six values represent the coefficients for the variables that shift the utility for each housing alternative. The final three values represent the utility cost of switching housing arrangements. The reader should not interpret the coefficients as describing directly

HOUSING DECISIONS OF OLDER HOUSEHOLDS

37

observable levels of utility. In this context it is not possible to say anything about the absolute level of utility for any housing arrangement, only that the utility received is greater or smaller than the utility in another arrangement. Moreover, one should not interpret the baseline utilities for the three estimated housing alternatives as an indication that these alternatives provide more or less satisfaction than the housing alternative that is omitted. The utility levels are appropriately evaluated only in the context of realized values for the economic and noneconomic variables. Likewise, it is not appropriate to interpret a negative coefficient as an indication that a higher value for a variable reduces the utility from a given housing alternative. Rather, it reduces the utility relative to the utility for the omitted alternative. The most dominant coefficients in Table I are the ones that measure the utility costs of making a transition. This may include the monetary costs such as down payments, security deposits, closing costs, and moving expenses, but also include the psychic costs of leaving a house or neighborhood to which the household has become attached. The most negative of these values are for moves to the high equity category. This likely reflects the higher down payment and other costs necessary to move to high equity arrangements. Note also that the transition parameters for moves from a low equity arrangement are somewhat larger than those for moves from a non-ownership arrangement, possibly reflecting the costs of selling a home. Perhaps the most striking results involve the coefficients on income and assets. The coefficients on both these factors are generally of high statistical significance. Their consistently negative signs indicate that relative to the omitted category, high levels of both income and assets reduce the utility derived from other arrangements. This suggests that high levels of income and assets make (high levels of ) owner-occupied housing more affordable, and thus cause that arrangement to provide more satisfaction compared to other arrangements. It appears as though advancing age causes a polarization in the elderly’s choice of housing arrangements: Age tends to decrease the utility received from owning at low levels of equity and renting relative to owning in the highest equity bracket. Age appears to have very little effect on the utility from dependency (relative to the omitted category). This suggests that as age increases, households tend to enjoy high-equity home ownership and dependent living conditions more than the other alternatives. The coefficients on other noneconomic characteristics for the most part have the signs one would expect. For instance, being married appears to decrease the utility garnered from either living in a dependent arrangement or renting. This may occur because a couple can care for each other and for their (owner-occupied) home more easily than can a single person. Physical limitation increases the utility received from renting or living dependently, which indicates that disability adversely effects the happiness

38

PETER G. VANDERHART

that a household receives from owner-occupied housing. Retirement status shows a similar pattern, especially for the rental alternative, which may indicate that retirement causes households to desire to liquidate their home equity, or to desire less permanent living arrangements. While these coefficients have intuitively understandable signs, they do not enter with statistical significance for any of the housing alternatives. Thus the dynamic analysis does not suggest a significant effect of noneconomic variables on older households’ housing decisions. 6.3. Conventional Estimation To make a comparison between the dynamic estimation described above and more conventional techniques, results from a conventional estimation are needed. This is accomplished simply by setting the discount factor equal to zero. This has the effect of making the transition probabilities irrelevant and reducing the probability described in Eq. (12) to that which appears in Eq. (5). The same data are used and the results are presented in Table II. The most obvious distinction between the two tables is that the coefficients for the conventional analysis tend to be much larger in magnitude and more often statistically significant. While the more frequent statistical significance of the coefficients might tempt one to believe that this is a better specification, a comparison of the log likelihood values for the two regressions indicates that the dynamic analysis provides a better fit. Compared to the dynamic estimation results, the coefficients for the economic variables increase in magnitude and maintain their statistical significance. The coefficients for retirement increase in magnitude, and for the rental alternative become statistically significant. The limitation coefficients switch sign for the low-equity alternative, but become larger for the other two and achieve significance for the dependency alternative. The magnitude of the coefficients on marital status increases, with the renting and dependent alternatives becoming statistically significant. The coefficients for the age variable are much more erratic and actually lose statistical significance for two alternatives. Taken as a whole, these results suggest a much more important role for noneconomic variables than is suggested by the dynamic analysis. These changes are due to the fact that the coefficients in Table II capture both the direct effect of the variable and its indirect effects on future values of the variable itself and other related variables via the transition probabilities. For instance, the increase in magnitude of the marital status coefficient for the renting alternative (from 20.08 to 20.52) reflects the fact that marital status is very stable over time (see Table A-VI) and that the conventional coefficient includes the effect of current marital status on the stream of future marital states, which in turn affects utility. Indirect

HOUSING DECISIONS OF OLDER HOUSEHOLDS

39

TABLE II Results from Conventional Estimation Coefficient a b*1 (assets) b*2 (income) b*3 (retired) b*4 (limited) b*5 (married) b*6 (age) c*1 (to high equity) c*2 (to low equity) c*3 (to rental) c*4 (to dependent)

Low equity

Renting

Dependent

23.22 (28.67) 20.204 (23.29) 20.177 (22.88) 0.132 (0.707) 0.0158 (0.0809) 0.114 (0.544) 20.167 (21.49) 29.40 (235.6) —

22.70 (26.97) 20.284 (24.74) 20.253 (23.38) 0.363 (1.96) 0.249 (1.29) 20.519 (22.56) 20.0480 (20.451) 28.80 (234.4) 23.37 (213.3) —

23.24 (27.77) 20.364 (24.45) 20.382 (23.85) 0.300 (1.46) 0.575 (2.67) (20.562 (22.37) 0.182 (1.60) 27.86 (220.5) 22.48 (27.59) 22.32 (28.02) —

24.34 (216.8) 24.44 (214.5)

23.40 (212.0)

Discount rate 5 0 Number of observations 5 12,323 2log likelihood 5 2642.361 Gradient direction 5 3.71 3 1025 Note. t-statistics are in parentheses.

effects may also be at work: One possible reason that the age coefficients lose their significance in the rental alternative is that the conventional coefficient includes both the direct (negative) effect of aging and an indirect effect via age’s positive effect on the onset of retirement, which has a direct positive effect on utility from renting relative to the omitted category. These results cast some light on the relative importance of the economic and noneconomic variables. While the conventional analysis presented here suggests that both types of variables play an important role, the dynamic analysis suggests that only economic variables are statistically significant determinants of the housing decisions of the elderly. The dynamic estimation results are somewhat at odds with previous conventional analyses which have tended to emphasize the importance of noneconomic factors. It is possible that these previous studies find noneconomic factors to be

40

PETER G. VANDERHART

important due to their indirect effects on the financial factors that truly matter. This is not to say that the demographic factors are not important to the elderly’s housing decisions, only that they are important for their indirect, rather than their direct effects on homeowners’ preferences.

7. CRITIQUE

Before a chorus is recruited to sing the praises of the dynamic estimation technique, it should be pointed out that this technique suffers from a number of shortcomings relative to conventional technique. This is true both generally and for the specific example described above. Most of these shortcomings are related to the computational complexity of the technique and are described below. 7.1. State Variables The state variables included in this analysis are by no means exhaustive. Sex of the head of household, number of children in the family unit, and length of time in the home, all of which may play a significant role in a household’s taste for housing, are omitted. Also ignored are potentially significant measures of housing cost, such as property tax, mortgage payment, and the cost of utilities. Some of the state variables included in the analysis may suffer from an oversimplified specification of their transition probabilities. Specifically, while the coefficients in income and asset regressions seem quite reasonable, they may not adequately describe the empirically observable log-normal distribution of income or the left-truncated distribution of financial assets. Moreover, categorization of each of these variables into five groups is very coarse and may miss significant effects within each group. Inclusion of these factors would greatly increase the computational burden of the dynamic programming estimation problem. One advantage of the conventional technique is that it is quite easy to include a large number of variables and that the ability to include continuous variables makes the use of categorical variables unnecessary. 7.2. Housing Status The housing status variable is also somewhat lacking. Using only four states to describe all possible housing arrangements seriously understates the diversity of housing alternatives to which the elderly have access. A more complete specification would have more than two levels of home equity in its categories and would include states that specify the mortgage status of the arrangement. The dependent classification may also be disag-

HOUSING DECISIONS OF OLDER HOUSEHOLDS

41

gregated to provide for separate treatment of nursing homes and living with relatives. Augmentation of the housing state variable as described above would increase the computational complexity of both the dynamic and conventional techniques. For both methods the number of utility parameters to be estimated increases. The effect is much more dramatic for the dynamic technique, however, because the dimensionality of the dynamic programing problem increases greatly as the number of choice alternatives increases. 7.3. Exclusion Restrictions The specification of transition probabilities according to Eq. (17) is very complete in that all state variables are allowed to influence the future realization of each of the state variables. This is done primarily to avoid any appearance of excluding certain variables to manipulate the outcomes in Table I. However, in certain situations it may be prudent to exclude certain variables in some estimations of transition probabilities. For instance, in Table A-III the coefficient indicating retirement’s effect on limitation status is positive and significant. This suggests that retirement ‘‘causes’’ one to become or remain physically limited. This is very unlikely and is probably due to the fact that the household first becomes slightly limited (although not enough to report itself as such), then retires while still relatively healthy, and finally reports the limitation once retirement has occurred. Excluding variables in situations such as these may actually improve the specification presented here. 7.4. Inflated t-statistics The reader should be aware that the t-statistics in Tables I and II may be overstated. Both the dynamic and conventional analyses make use of pooled cross-sectional data, where a single household often contributes more than one observation. If there are unobserved individual-specific effects, the variance of the sample will be unnaturally low and t-statistics will be too high. 7.5. Computational Considerations The final advantage of the conventional technique over the dynamic technique is computational. The conventional estimation described above can be accomplished in only a few minutes of time on an ordinary personal computer. Even more complicated versions of the estimation (more variables, more choice alternatives) can be accomplished within an hour. The dynamic estimation described above would prove difficult on even a highperformance personal computer, but was accomplished with only a few

42

PETER G. VANDERHART

minutes of CPU time on a Cray Y-MP8 supercomputer. In this context the use of the Cray was necessary not primarily for the speed of its calculations, but for its relative abundance of high-speed memory. 7.6. Conclusion This paper has described how dynamic discrete choice estimation techniques can be applied to elderly housing decisions. It is argued that dynamic estimation is conceptually superior to conventional techniques when the decision maker faces an intertemporal optimization problem. The results from the dynamic estimation are significantly different from those derived from conventional techniques and suggest that economic variables play a relatively large role in the choice of housing arrangements by the elderly. While in theory the dynamic technique may be superior, in practice it may be much less flexible than conventional techniques. The computational complexity of dynamic estimation requires that explanatory variables be limited to categorical versions and be few in number. Furthermore, for situations of the scale described here, the dynamic estimation requires a great deal more computational power. Even with the shortcomings of the analysis, dynamic discrete choice estimation is a promising technique. No other technique is as consistent with the intertemporal optimization that is assumed by so many economic models. As powerful computers become more available, this technique will become more practical and its limitations will diminish.

APPENDIX A Expectation Estimation Results Remaining Alive

The first transition estimated is whether the head of household continues to live from one period to the next hdst11 5 0j. The results of a binomial logit regression are presented in Table A-I. The negative coefficients on the physical limitation dummy and age indicate that limitation and advancing age reduce the probability that the head remains alive. Marital Status For the prediction of marital status, a binomial logit regression is again specified with the dependent variable indicating whether or not the head of household is married in the following year. The dependent variables now include all other state variables. Squared terms for the income and

43

HOUSING DECISIONS OF OLDER HOUSEHOLDS

TABLE A-I Survival Probability Estimation Results (Dependent Variable 5 1 If Alive in Next Period) Variable Constant Limited Age

Coefficient

t-statistic

5.35 20.988 20.0355

23.411 27.821 25.574

Number of observations 5 19,439 2log likelihood 5 1608.648

asset variables are also included to account for nonlinear effects. (Using dummy variables for each income and asset category would have been preferable, but creates convergence problems in some regressions.) The sample size decreases because some newly included dependent variables have missing values. The results of this regression are presented in Table A-II. The results indicate that change between marital states is infrequent: Households that start off married tend to continue to be married, and those that are not married are even more likely to remain not married. The income, retirement, and age variables also enter significantly. Limitation Status The dependent variable in this regression is equal to 1 if the head of household is physically limited in the following year, 0 if not. The results are presented in Table A-III. Nearly all variables enter with statistical TABLE A-II Martial Status Estimation Results (Dependent Variable 5 1 If Married in Next Period) Variable Constant Assets Assets squared Income Income squared Disabled Retired Married Age Number of observations 5 18,855 2log likelihood 5 1858.493

Coefficient

t-statistic

24.60 20.129 0.0246 20.472 0.137 0.0161 1.23 7.91 20.0535

213.200 20.670 0.748 22.273 4.211 0.147 10.027 54.668 27.473

44

PETER G. VANDERHART

TABLE A-III Limitation Status Estimation Results (Dependent Variable 5 1 If Physically Limited in Next Period) Variable Constant Assets Assets squared Income Income squared Disabled Retired Married Age

Coefficient

t-statistic

20.772 20.285 0.0265 20.332 0.0144 2.92 0.568 0.133 0.0111

25.971 23.613 1.974 24.294 1.161 71.802 12.773 2.735 3.990

Number of observations 5 18,855 2log likelihood 5 8105.333

significance. The results indicate that physical limitation is somewhat persistent, that high levels of assets and income may ward off disability, and that age and retirement seem to induce it. Retirement Status The final demographic factor explored in this manner is the retirement status of the head of household. Again the problem is specified as a binomial logit regression, with the dependent variable equal to 1 if the head is retired or permanently disabled in the following year. The results of this regression are presented in Table A-IV. Being retired in the current year makes it very likely that a head will be retired the following year. Income, physical limitation, being married, and age also seems to encourage retirement. Income Because income status is specified to assume five different values, a multinomial logit specification is estimated. The results appear in Table A-V. Recall that the coefficients should be interpreted as relative to the null group, income category 5. Generally the asset and income variables perform as expected, with coefficients that imply high probabilities of staying in the same income category or moving to adjacent categories. Disability and retirement raises the probability that a household will be in the lower income categories, as does advancing age. Being married reduces the probability of being in lower income categories.

45

HOUSING DECISIONS OF OLDER HOUSEHOLDS

TABLE A-IV Retirement Status Estimation Results (Dependent Variable 5 1 If Retired in Next Period) Variable Constant Assets Assets squared Income Income squared Disabled Retired Married Age

Coefficient

t-statistic

23.18 20.0146 0.00875 0.571 20.0988 0.556 3.279 0.607 0.0452

222.075 20.170 0.596 6.985 27.619 11.846 72.798 11.010 15.417

Number of observations 5 18,855 2log likelihood 5 7216.804

TABLE A-V Income Estimation Results (Dependent Variable 5 Income Category in Next Period)

Constant Assets Assets squared Income Income squared Disabled Retired Married Age

Category 1

Category 2

Category 3

Category 4

9.42 (19.3) 21.45 (28.18) 0.162 (5.36) 23.79 (213.7) 0.284 (7.08) 0.844 (8.44) 0.769 (7.12) 21.78 (216.7) 0.0830 (11.9)

3.62 (7.60) 20.728 (24.80) 0.0581 (2.28) 1.08 (3.95) 20.499 (212.5) 0.514 (5.72) 0.989 (10.0) 21.41 (215.1) 0.0738 (11.3)

24.28 (28.65) 20.366 (22.59) 0.0199 (0.839) 5.33 (19.4) 21.00 (226.1) 0.242 (2.80) 0.904 (9.58) 20.808 (28.93) 0.0421 (6.62)

29.55 (215.6) 20.151 (21.16) 20.00575 (20.265) 6.86 (21.2) 21.06 (225.0) 0.130 (1.56) 0.906 (10.2) 20.589 (26.65) 0.0278 (4.55)

Number of observations 5 18,855 2log likelihood 5 18030.613 Note. t-statistics are in parentheses.

46

PETER G. VANDERHART

TABLE A-VI Asset Estimation Results (Dependent Variable 5 Asset Category in Next Period)

Constant Asset Asset squared Income Income squared Disabled Retired Married Age

Category 1

Category 2

Category 3

Category 4

9.08 (29.6) 22.72 (217.3) 0.164 (6.47) 20.361 (22.30) 20.0269 (21.10) 0.411 (4.91) 20.321 (23.62) 0.148 (1.60) 20.0575 (210.5)

1.08 (3.46) 2.13 (13.2) 20.611 (222.2) 0.275 (1.75) 20.0718 (22.96) 0.129 (1.58) 20.314 (23.63) 20.0598 (20.658) 20.0102 (21.91)

22.37 (26.77) 3.57 (20.2) 20.711 (225.7) 0.082 (0.49) 20.0386 (21.51) 0.112 (1.31) 20.166 (21.84) 0.0263 (0.275) 20.0116 (22.05)

23.58 (28.61) 3.48 (16.4) 20.600 (219.4) 20.0571 (20.33) 20.0165 (20.622) 0.138 (1.56) 20.244 (22.62) 0.0988 (1.00) 20.0149 (22.53)

Number of observations 5 18,855 2log likelihood 5 16107.372 Note. t-statistics are in parentheses.

Asset Level Asset transitions are estimated analogously to those for income, in a multinomial logit framework. The results are presented in Table A-VI. The asset variables imply that households are most likely to remain in the same asset category or move into one directly adjacent. Being retired seems to reduce the probability of being in low asset categories, while being physically limited increases it. No clear pattern emerges from the coefficients on other variables.

REFERENCES Ai, C., Feinstein, J., McFadden, D., and Pollakowski, H. (1990). ‘‘The Dynamics of Housing Demand by the Elderly: User Cost Effects,’’ in Issues in the Economics of Aging (D. Wise, Ed.). Chicago: Univ. of Chicago Press. Beland, F. (1984). ‘‘The Decision of Elderly Persons to Leave Their Homes,’’ The Gerontologist 24, 179–185. Bellman, R. (1957). Dynamic Programming. Princeton NJ: Princeton Univ. Press.

HOUSING DECISIONS OF OLDER HOUSEHOLDS

47

Bertsekas, D. (1976). Dynamic Programing and Stochastic Control. New York: Academic Press. Boersch-Supan, A. (1989). A Dynamic Analysis of Household Dissolution and Living Arrangement Transitions by Elderly Americans. NBER Working Paper No. 2808. Boersch-Supan, A., Kotlikoff, L. J., and Morris, J. N. (1988). The Dynamics of Living Arrangements of the Elderly. NBER Working Paper No. 2787. Eckstein, Z., and Wolpin, K. (1989). ‘‘The Specification and Estimation of Dynamic Stochastic Discrete Choice Models.’’ J. Human Res. 24, 562–598. Ellwood, D., and Kane T. (1989). The American Way of Aging: An Event History Analysis. NBER Working Paper No. 2892. Feinstein, J., and McFadden, D. (1987). The Dynamics of Housing Demand by the Elderly: Wealth, Cash Flow, and Demographic Effects. NBER Working Paper No. 2471. Garber, A. M., and MaCurdy, T. (1989). Nursing Home Utilization among the High-Risk Elderly, Hoover Institution Working Paper E-89-1. Goldberger, A. (1964). Econometric Theory, New York: Wiley. Jones, L. D. (1996). ‘‘Housing Tenure Transition and Dissaving by the Elderly,’’ Can. J. Econ. 29, 505–509. Judge, G. C., Hill, R. C., Griffiths, W. E., Lu¨tkepohl, H., and Lee, T. (1982). Introduction to the Theory and Practice of Econometrics. New York: Wiley. Kapteyn, A. (1995). ‘‘The Microeconometrics of Dynamic Decision Making,’’ J. Appl. Econometrics 10, 1–7. Kennet, D. M. (1994). ‘‘A Structural Model of Aircraft Engine Maintenance,’’ A. Appl. Econometrics 9, 351–368. Kotlikoff, L. J., and Morris, J. (1988). Why Don’t the Elderly Live with Their Children? A New Look. NBER Working Paper No. 2734. Maddala, G. S. (1983). Limited Dependent and Qualitative Variables in Econometrics. Cambridge, UK: Cambridge Univ. Press. Merrill, S. (1984). ‘‘Home Equity and the Elderly,’’ in Retirement and Economic Behavior (H. Aaron and G. Burtless, Eds.). Washington, D.C.: Brookings Institution. O’Bryant, S. L., and Wolf, S. M. (1983). ‘‘Explanations of Housing Satisfaction of Older Homeowners and Renters,’’ Res. Aging 5, 217–233. Pakes, A. (1984). ‘‘Patents as Options: Some Estimates of the Value of Holding European Patent Stocks,’’ Econometrica 54, 755–784. Phelan, C., and Rust J. (1991), U.S. Social Security Policy: A Dynamic Analysis of Incentives and Self-Selection. Unpublished Manuscript, University of Wisconsin—Madison. Reschovsky, J. D. (1990). ‘‘Residential Immobility of the Elderly: An Empirical Investigation,’’ J. Amer. Real Estate Urban Econ. Assoc. 18, 160–183. Rust, J. (1987). ‘‘Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher,’’ Econometrica 55, 999–1033. Rust, J. (1988). ‘‘Maximum Likelhood Estimation of Discrete Control Processes,’’ SIAM J. Control Optim. 26, 1006–1024. Rust, J. (1990). ‘‘Behavior of Male Workers at the End of the Life Cycle: An Empirical Analysis of States and Controls,’’ in Issues in the Economics of Aging (D. Wise, Ed.). Chicago: Univ. of Chicago Press. Stahl, K., (1989). ‘‘Housing Patterns and Mobility of the Aged: The United States and West Germany,’’ in The Economics of Aging (D. Wise, Ed.). Chicago: Univ. of Chicago Press. VanderHart, P. (1994). ‘‘An Empirical Analysis of the Housing Decisions of Older Homeowners,’’ J. Amer. Real Estate Urban Econ. Assoc. 22, 205–233.

48

PETER G. VANDERHART

VanderHart, P. (1995). ‘‘The Socioeconomic Determinants of the Housing Decisions of Elderly Homeowners,’’ J. Housing Elderly 11, 5–35. Venti, S., and Wise, D. (1989). ‘‘Aging, Moving, and Housing Wealth,’’ in The Economics of Aging (D. Wise, Ed.). Chicago: Univ. of Chicago Press. Venti, S., and Wise, D. (1990). ‘‘But They Don’t Want to Reduce Housing Equity,’’ in Issues in the Economics of Aging (D. Wise, Ed.). Chicago: Univ. of Chicago Press. Wolpin, K. I. (1987). ‘‘Estimating a Structural Search Model: The Transition from School to Work,’’ Econometrica 55, 801–818.