DOES IT PAY TO GET A REVERSE MORTGAGE

DOES IT PAY TO GET A REVERSE MORTGAGE? VALENTINA MICHELANGELI* (Preliminary and Incomplete) August 6, 2007 Many of today’s households are at risk of having inadequate resources to support their retirement. For most retirees their home is their major asset. The bulk of their savings at retirement is typically locked in home equity and cannot be extracted for old age needs except through selling and moving out. The reverse mortgage allows one to convert equity into an income stream without making periodic loan payments or moving out. But principal plus the accumulated interest has to be paid back if the retiree de…nitively moves. Therefore, assessing the potential of this …nancial instrument requires jointly analyzing consumption, housing and mobility decisions. This paper presents a structural, dynamic model for these decisions. More speci…cally, using the Mathematical Programming with Equilibrium Constraints approach, we estimate elderly preferences for consumption and housing using a subsample of single retirees from the Health and Retirement Study (HRS). We …rst study the optimal choice of consumption and elderly mobility in the absence of a reverse mortgage. Then, we calculate the welfare gain for HRS respondents in taking out a reverse mortgage. This exercise shows that the reverse mortgage eases the liquidity problem and provides longevity insurance. However, it introduces a new risk: the moving risk. The risk of moving and having to repay the cumulated debt on reverse mortgages creates a major welfare loss for those with initial low …nancial wealth. Common belief is that a reverse mortgage bene…ts those with resources tied up in home equity. This paper shows otherwise. KEYWORDS: Housing, Consumption, Elderly Mobility, Dynamic Discrete and Continuous Choices, Constrained optimization approach ———————————————————————————————— *I am very thankful to my advisors Laurence J. Kotliko¤, Kenneth L. Judd and M arc Rysm an for their constant supp ort and guidance. I also thank Francois Gourio, Che-Lin Su, Francisco Gom es, Z. Bodie, A. Verdelhan, Daniele Paserm an, Otto van Hem ert,Gabriel Lop ez Calva, K. Lang, and conference participants at Boston University’s Departm ent of Econom ics and at the 14th International Conference on Com puting in Econom ics and Finance, University of Sorb onne, Paris, for their num erous useful com m ents. I would like to thank Claudio Rebbi, M ike Dugan and the ACES (Advanced Com putation in Engineering and Science) group for providing m e com puting supp ort with the high-end com putational facilities at Boston University and Richard Walthz and Rob ert Fourer for providing supp ort and licensing for the software AM PL and KNITRO. I gratefully acknowledge the …nancial supp ort from IED at Boston University, Hoover Istituion and ICE at the University of Chicago. All errors are my own. Valentina M ichelangeli can b e reached at Boston University, Departm ent of Econom ics, 270 Bay State Road, Boston, M A 02215, Tel. (617) 953 2800, em ail: valem [email protected] 1 1 Introduction Baby boomers started retiring in 2001, signing the beginning of an accelerated rate of growth in the number of retirees in the US. By 2030, one out of …ve people is projected to be 65 years or older. Despite this rapid augment in the retired population, the …nancial situation for future retirees remains uncertain. As a matter the fact, increases in the cost of living and in health care costs, curtailments in medical coverage and other employee bene…ts plans and cutbacks in Social Security bene…ts expose many of today’s households at risk of having to adjust to a decreased standard on living in retirement. An analysis of their …nancial portfolio shows clearly that for most retirees their house is their major asset. More than 80 percent of older households own their homes (Munnel et al, 2007), which are worth approximately $4 trillion. Economists and policymakers look at these …nancial assets as a potential source of savings to …nance consumption in the elderly. According to the "life cycle" hypothesis of saving and consumption, individuals build savings during their working years and divest those savings to support consumption in retirement. However, when it comes to home equity, this pattern is not followed. Typically, older households do not divest home equity. Instead, homeownership rates remain stable until late in life and median home equity increases with age as older homeowners pay o¤ mortgages and home value appreciates. Before the advent of the reverse mortgage, only two alternatives were available to older homeowners to divest their home equity. They could sell and move out or they could borrow against their house property taking a conventional loan, such as mortgage or home equity loan. Traditional loans have to be repaid, either through installments or on maturity and older homeowners are often neither eager nor able to incur in new monthly obligations. Therefore, selling and moving out represented the best way to en-cash the savings locked up in residential property. In the 1990s, reverse mortgages became available and provided a new way to convert home equity into cash. A reverse mortgage is a …nancial instrument that allows borrowers to access the equity in their home that would otherwise not be liquid, by providing income while not requiring payment as long as the borrower lives in the same house. When the retiree moves out or dies, the reverse mortgage lender keeps the minimum between the house value and the outstanding debt. The amount of money that could be borrowed via a reverse mortgage generally depends on the borrower’s age and the value of the home. The minimum age for almost all reverse mortgage programs is 62. A 2005 study by Stucki estimated the potential market at 13.2 million older households. However, at the end of 2007, only 265,234 federally insured reverse mortgages were issued (Department of Housing and Urban Development, 2007b). This represents about 1% of the 30.8 million households with at least one member age 62 and older in 2006 (U.S. Census Bureau,2006). The small percentage of older households with reverse mortgages makes us ask the following question: does it pay to get a reverse mortgage? Will reverse mortgages 2 remain in the future a small niche product in the future or will they become a commonly used tool to …nance consumption in retirement? Is moving out still the best option available to older retirees? Moving for …nancial reasons is only one type of elderly mobility. Other types of elderly mobility include moving for assistance reasons, changes in marital status, climate or weather, health problem or services, desire to change neighborhoods or the location, shopping or other consumption services, and public transportation. This paper presents a structural dynamic model of retiree consumption, housing and moving decisions. More speci…cally, in each period the individual chooses whether to continue living in the same house or move to a new house. If she moves out, she can either buy or rent a new house and she chooses the new house value. The main sources of income during retirement are social security, pensions and investment income, which are assumed not to vary over time for the single retiree. Given our focus on retirement, income from labor is not considered. Data on households with a reverse mortgage are not available, hence we select a sub-sample of single retirees from the Health and Retirement Study (HRS) that could represent a potential target segment for this …nancial instrument. We …rst estimate the preference parameter between housing and consumption and the risk aversion of the households in our sample ,then, simulate how better or worse o¤ they would be if they choose a reverse mortgage contract. We compare this result with the case in which households, instead of closing a reverse mortgage contract, simply liquidate their assets by moving out and renting. Our analysis shows that a reverse mortgage provides liquidity and longevity insurance, however, moving becomes a risky proposition. As a matter the fact, if the homeowners move out, they have to repay the minimum between the house value and the outstanding debt. Both consumption and housing pro…les are affected in the periods following the move. We also show that moving and renting a new house generate the household’s highest welfare gain. This choice provides liquidity, does not increase the household’s level of indebtedness and does not introduce any moving risks. This result might explain why, after almost twenty years from its …rst appearance, the reverse mortgage market is still at 1% of its potential. The solution method is innovative in two main respects. This is the …rst application of the MPEC (Mathematical Programming with Equilibrium Constraints) to an empirical structural model with a life-cycle dynamic programming problem involving a continuous state variable. The current econometric literature seems to dismiss the MPEC approach as computationally infeasible. However, Judd and Su (2008) shows that this approach is feasible if one uses the standard methods in the mathematical programming literature. They apply the MPEC approach to the canonical Zurcher bus repair model (Rust,1987). Second, there is an extensive literature focusing on the solution of discrete choice models. The framework was …rst introduced by Rust (1987,1988). Both the theoretical literature and most of the subsequent applications focus on discrete decision processes. However, given that our study involves both discrete and 3 continuos choices and these data are present in our sample, we extend the existing literature including also continuous choices. The structure of the paper is as follow. First, we present the features of a reverse mortgage contract and evaluate the lender’s expected gain. Second, we present the household’s life-cycle model. Third, we describe the solution method. Fourth, we show the results and the welfare analysis. 2 Reverse Mortgage Reverse mortgages are home loans that do not have to be repaid as long as the borrower lives in the house. The borrower can receive the proceeds in one of the following ways: a lump sum at the beginning, monthly payments until a …xed term or a life-long annuity, by establishing a credit-line with or without accrual of interest on the credit balance, or a combination of the aforementioned. To be eligible for a reverse mortgage, a borrower must be 62 or older, own the home outright (or have a low loan balance) and have no other liens against the home. The retiree does not have to satisfy any credit or income requirements. A reverse mortgage accrues interest charges, beginning when the …rst payment is made to the borrower. When she dies or relocates, the minimum between the house value and the loan plus the cumulated interest has to be repaid. Even if the accumulated loan and interest exceed the realizable value of the house at disposal, the repayment is capped at that value only. The amount of loan is a function of the age of the borrower and any coapplicant, the current value of the property and expected property appreciation rate, the current interest rate and interest rate volatility, closure and servicing costs and other speci…c features chosen. A reverse mortgage is just one of several …nancial instruments that allow a homeowner to secure liquid funds against the equity in a house. In general, Home Equity Conversion Products could be useful to all those who are “houserich but cash-poor” and, together, could enhance social welfare. 1 Most of the empirical estimates fOR the potential of reverse mortgages to meet the …nancial needs of the elderly are from the public policy perspective . They are based on various federally sponsored surveys (American Housing 1 The Home Equity Conversion Products include the following products, in addition to reverse mortgage. Home reversion / sale and lease back allows the homeowner to sell his house outright now, but she keeps the right to live in it for life for a nominal/reduced rent. The sale pro…ts could be paid in a lump sum or as an annuity. The interest-only mortgage allows the borrower to get an immediate lump sum. She is required to make only interest payments during the tenure of the loan and the principal is due only on maturity or death or a permanent move or sale. The mortgage annuity/ home income enables the individual to use the loan amount buy a life annuity. The interest on the mortgage is deducted from the annuity and the balance is paid as periodic income. The principal is repaid on death or sale of the house. The shared appreciation mortgage provides loans at a below market interest rate. The lender obtains a pre-agreed share in any appreciation in the property value over the accumulated value of the loan The loan is due at death or moving or sale. 4 Survey, Survey of Income and Program Participation), rather than on surveys speci…cally to assess the potential for RM. Meyer and Simons (2004) estimated that over six-million homeowners in the United States could increase their effective monthly income by at least 20% by using a reverse mortgage. Of these, more than 1.3 million have no children. In addition, over 1.4 million poor elderly persons could raise their income above the poverty line. They also show that almost …ve million households could receive a lump sum twice as large as their current holdings of liquid assets, giving them access to resources in case of …nancial emergencies without losing their home. Merrill, Finkel and Kutty (1998) use a di¤erent data set and more restrictive criteria to identify the prime target group for RM: the relatively older among the elderly (age>70 yrs); low income (<30,000 $/yr); high home equity (between 100,000 to 200,000 $); and a strong desire to remain in their current home (length of stay >10 yrs). They estimated such households to be 800,000 in late 1980s. Kutty (1994) focused speci…cally on the potential of RM to lift the elderly above the poverty line. Based on a 1991 survey, the author estimated that 621820 such households, constituting 18% of all elderly poor households could be brought above the poverty line. The author advocated strong public policy support for RM. However, when we consider the actual reverse mortgage market, it is very small. At the end of 2007, only 265,234 federally insured reverse mortgages were issued (Department of Housing and Urban Development, 2007b). This represents about 1% of the 30.8 million households with at least one member age 62 and older in 2006 (U.S. Census Bureau, 2006). Stucki(2005) assumed that about half of older persons own homes with su¢cient equity to be considered candidates for a reverse mortgage, which yielded an estimate of only about 2% of the potential market. 3 Is the Reverse Mortgage a fair contract? Lender’s Perspective In our model, we suppose that the reverse mortgage borrower i chooses to receive the proceeds as a lump sum at the closure of the contract in time j. At the closure of the reverse mortgage contract, the lender’s initial cost is the lump sum payment B and the revenue stream includes origination fees and service fees F . If the borrower moves out of the house or dies at time t, that person would be required to repay the minimum between the house value and the outstanding debt: min(Hit ; GRM it ) where GRM is the outstanding debt at time t. It is given by the initial lump it sum payment B and cumulated interests: 5 GRM =B it X (1 + iD )t j (1) j=1::t where iD is the nominal interest rate on reverse mortgage. In present value, the repayment in period t for household i is: min(Hit ; GRM it ) (2) t j R Let ni;t household i’s probability of being alive at time t and mi;t her probability of moving at time t: The expected gain for the lender is: RMit = EGainj;i = F + X t=j+1::T ni;t 1 f(1 ni;t )(1 mi;t ) + ni;t mi;t gRMit (3) A simple calculation, without taking into account the interest rate and the house price risk, shows that a homeowner with a house value equals to $100,000 could borrow about $47,000, $31,000 , or $10,000 respectively if she closes a Monthly Adjusting HECM, a Annually Adjusting HECM, and a Fannie Mae HomeKeeper contract at age 62. This represents the actual cost for the lender. Given women survival probabilities and US mobility rate, the expected gain for the lender is about $74,000, $64,000, $ 30,000. If we assume that F is a cost for the lender and we do not include it in equation (3), the expected gain is $54,000, $63,000, $22,000 respectively. 4 The Model This section describes a model of post-retirement decision making. We consider the optimal consumption and housing decision for an individual from age 64 until age T . The individual dynamically chooses consumption, housing tenure and housing size. We allow the investor to acquire housing services through either renting or owning a house. In each period the individual chooses whether to continue living in the same house or move to a new house. When the individual decides to move, transaction costs are incurred. Consistent with the data, we assume that the household that moves cannot buy a larger house size and can rent any possible house size. 4.0.1 Preferences Individual i’s plan is to maximize her expected lifetime utility at age t, t = 64; :::; T: T is set exogenously and equals 95. In each period she receives utility Uit ; from non-durable consumption Cit and housing services Hit , so that U (Cit ; Hit ). The within-period retiree’s preference over consumption and housing services are represented by the Cobb-Douglas utility function: 6 Uit (Cit ; Hit ) = 1 (Cit ! ! 1 Hit ) 1 + "it (dit ) (4) where Cit denotes consumption, Hit the housing value, ! measures the relative importance of housing services versus numeraire nondurable good consumption, is the coe¢cient of relative risk aversion. We assume that "it (dit ) is independent across individuals and time. It is Extreme Value Type I distributed. It represents housing preference shock. Individuals move out of their home for several reasons, which are explained in detail in our survey. Some households move out for …nancial reasons, looking for a smaller or less expensive house Others because they desire to live near or with their children or other relatives, for health problem, for climate or weather reasons, for reasons related to leisure activities or public transportation and for change in marital status. We model this unobserved utility from moving as housing preference shock. When the individual dies, her terminal wealth, T Wit , is bequeathed according to a bequest function b(T Wit ) : b(T Wit ) = 4.1 B T Wit1 1 (5) Choice set In each discrete period t, the household makes both discrete and continuous choices. Without loss of generality, we assume that the household makes two main decisions, housing and consumption. In particular, the household makes the discrete housing decision and, conditional on the housing decision, she makes the continuous consumption decision. 4.1.1 Discrete Choice: Housing The housing choice is a multi-stage decision. First, the household decides whether to move or stay. The household that moves out makes the choice of owning or renting a new house and the house size of the new house. Consistent with our data, the retiree that moves could not a¤ord a larger house. Let dit de…ne household i’s choice of housing. First of all, the household makes the discrete choice of staying or moving out in period t: d1it = M = 1 if household i moves out of her house in period t Dit M otherwise Dit = 0 Second, if she moves out of the house, she makes the binary choice of owning or renting a new house. 7 d2it jd1it = O Dit =1 O Dit = 0 if household i owns her house in period t if household i rents her house in period t The third stage decision over housing is the house value. d3it jd1it ; d2it = Hit+1 Therefore, the discrete choice set is: dit = fd1it ; d2it ; d3it g 4.1.2 Continuous Choices: Consumption Let Cit be the choice of consumption, conditional on the …rst choice of housing. 4.2 Housing Expenses Per-period housing expenses are a fraction of the market value of the house. We assume that is constant across individuals with the same housing level and deterministic. They depends on DiO , the housing tenure indicator variable M . For which is equal to one for homeowners and zero for renters, and on Dit homeowners, housing expenses represent a maintenance cost, sustained to keep the house at a constant quality. For renters, housing expenses represent the rental cost. The market value of the house is a function of house price per unit and house size. For both homeowners and renters, the housing expenses are assumed to be a constant value over time, denoted by own and rent respectively. it = [DiO own + (1 DiO ) rent M Hit+1 + (1 ][Dit M Dit )Hit ] (6) If the retiree decides to sell her house at time t and move to another house, she pays (receives) the di¤erence in owner-occupied housing wealth. In addition, O she incurs a one-time transaction cost (Dit+1 ). We assume that households could not buy a larger house but could rent a house of any size. The renters that move are allowed only to rent a new house. The cost of moving is: M O O O M O Mit = Dit Dit [Dit+1 Hit+1 Hit +Hit+1 (Dit+1 )]+Dit (1 Dit )Hit+1 rent (7) The transaction cost equals a fraction own ( rent ) of the market value of the new house when the investor moves to an owner-occupied ( a rental) house, i.e. O O (Dit+1 ) = [Dit+1 own + (1 O Dit+1 ) rent ] (8) Typically we have larger moving costs for the case of a retiree that buys a new house, that is own > rent : 8 5 Solution Method The solution method is innovative in two main respects. First, this is the …rst application of the MPEC approach to a dynamic life-cycle model with structural estimation of the preference parameters. Second, we estimate the structural model including both discrete and continuous choices. The traditional approach in solving dynamic structural models consists of repeatedly taking a guess of the structural parameters and solving for the corresponding endogenous variables. This is computationally very demanding. We use the MPEC approach (Mathematical Programming with Equilibrium Constraints) to solve our empirical model. This approach consists in formulating the life-cycle dynamic programming problem and the maximum likelihood estimation of the preferences as a constrained optimization problem. The idea behind the MPEC approach is to choose structural parameters and endogenous economic variables simultaneously and symmetrically, to solve the dynamic programming and the maximum likelihood problems in a one-step procedure. After having formulated the expressions de…ning the objective and the equilibrium equations (the constraints), we submit them to one of the state-of-art optimization solvers. The direct use of the state-of-art algorithms implies that we do not need to make any decision about the algorithmic details (such as, for example, choosing between value function iteration or policy iteration). In several respects, the MPEC approach could not be considered completely new, since it is based on ideas and methods developed in statistics and econometrics literature. Nevertheless, the current econometric literature tends to dismiss this approach as computationally infeasible. Judd and Su (2008) argue that the constrained optimization approach is feasible if one uses standard methods in the mathematical programming literature. They apply the MPEC approach to the canonical Zucher bus repair model. We extend their approach, presenting the MPEC with Dynamic Programming. There is an extensive literature focusing on the solution of discrete choice models. The framework was introduced by Rust (1987,1988) and then extended in Hotz and Miller (1993) and Aguirragabiria and Mira (2002). The focus, both in the theorical literature and in most of the subsequent applications, is on discrete decision processes. However, given that our study involves both discrete and continuous choices and these data are present in our sample, we extend the existing literature including also continuous choices in the model. The continuous state variable is the …nancial assets and the continuous choice variables is consumption. The discrete and discretized variables are the moving decision, the owning decision, and the housing level. The solution method is described in detail in the Technical Appendix. 5.1 State Space The state space in period t consists of variables that are observed by the agent and the econometrician Xit and by variables observed only by the agent "it . 9 O Xit = fWit ; Hit ; Dit ; Aget g where Wit is household i’s non-housing …nancial wealth at time t, Hit the O the beginning of period housing tenure. beginning of period house value, and Dit The term "it references a vector of unobserved utility components determined by the discrete alternative: "it = f"it (dit )g 5.2 Heterogeneity Given the observed realization of household choices and states fdit ; Cit ; Xit g, the objective is to estimate the preferences denoted as = f ; !; g Optimal decisions depend on the state variables, Xit and "it : The constants are denoted as K = f own ; rent ; own ; rent ; CM IN g We allow for heterogeneity in the state variables, Xit and "it , but not in preferences : 5.3 The Household’s Problem The household maximizes the expected lifetime utility over housing tenure M o g; consumption Cit and housing value Hit . ; Dit fDit We assume that the starting period is age 64. The value function of the problem is de…ned as: Vt (Xit ; "it ) = max Et dit ;Cit " T X t 64 # (N (t 1; t)nt U (Cit ; Hit )jXit ; "it ) + b(T Wit ) t=62 (9) s.t Wit+1 = RWit + ss Cit CM IN ; Cit Git Hit it Mit + "it (dit ) HM IN ; (10) (11) where qt denote the probability of being alive at age t conditional on being t Q nk denote the probability of living alive at age (t 1), and let N (t; j) = (1=nj ) k=1 to age t, conditional on being alive at age j. Wit denotes non-housing …nancial wealth after the house loan payment. "it (dit ) is i.i.d. error distributed as a normal with mean 0 and variance 2y which represents out-of-pocket expenses. The choice variables are included in dit :and Cit : 10 Eq. (10) denotes period t retiree i’s budget constraint. Let ss denote the retiree’s income, which includes Social Security bene…ts and pension . Eq. (11) de…nes the retiree i’s constraints on consumption and house size at age t: 5.4 The Bellman Equation The retiree faces two initial alternatives: stay or move: If the individual stays in the previous house, the choice variable is consumption. If the individual moves, the choice variables are consumption, housing tenure and new house value. The value function for period t is given by the following expression: (C 1 Vit (Xit ; "it ) = Wit+1 Cit ! (D M H ! ! 1 M )) )Hit +(1 Dit it it+1 maxdit ;Cit it 1 M (nt+1 E[Vit (Wit+1 ; (Dit Hit+1 + (1 st = RWit + ss Git Cit CM IN ; Hit HM IN ; it + "it (dit )+ s M ); "it+1 )] + b(T Wit+1 )) )Hit Dit Mit The optimal policy function for period t is given by: (C 1 it (Xit ; "it ; )= ! (D M H ! M ! 1 +(1 Dit )Hit )) it it+1 arg maxdit ;Cit it 1 M (nt+1 E[Vit (Wit+1 ; (Dit Hit+1 + (1 + "it (dit )+ M s Dit )Hit ); "it+1 )] + b(T Wit+1 )) The computation of the optimal policy functions is complicated due to the presence of the vector "it (dit ):It enters nonlinearly in the unknown value function EVit+1 Therefore, following Rust (1988), we make the following assumptions: ASSUMPTION 1 (Additivity): The within period utility function has the additively separable representation: U (dit ; Cit ; Xit ; ) = 1 (Cit ! 1 ! 1 ) Hit + "it (dit ) ASSUMPTION 2 (Conditional Independence): The conditional probability function of the state variables is given by: f (Xit+1 ; "it+1 jdit ; Cit ; Xit ; "it ; ) = q("it+1 jXit+1 )g(Xit+1 jdit ; Cit ; Xit ; ) Therefore EVt+1 does not depend on "it and the Bellman equation can be rewritten as: 11 Vt (Xit ; "it ) = = max [U (dit ; Cit ; Xit ; ) + "it (dit ) + qit EV (Xit+1 )] dit ;Cit maxf[U (dit ; Cit ; Xit ; ) + qit V (Xit+1 )]jdit g + "it (dit ) max Cit dit Therefore, the solution of period t’s problem could be divided in two parts without loss of generality. There is an inner maximization with respect to the second stage continuous choices conditional on the …rst stage discrete choices and then an outer maximization with respect to the …rst stage discrete choices. 5.5 Outer Maximization Let r(Xit ; dit ) represent the indirect utility function associated to the …rst stage choice dit : r(Xit ; dit ) = maxf[U (dit ; Cit ; Xit ; ) + nit+1 EVt+1 (Xit+1 )]jdit ] Cit Under the assumption that "(dit ) is distributed as a Type I Extreme Value error, the conditional choice probabilities are given by the following formula: expfr(Xit ; j; )g k2dit (Xit ) expfr(Xit ; k; )g P (jjXit ; ) = P and EVt+1 (Xit+1 ) is given by: 2 EVt+1 (Xit+1 ) = ln 4 X k2dt (Xt ) 5.6 3 expfr(Xit ; k; )g5 Inner Maximization The …rst step in the solution of period t’s involves solving the following choicespeci…c value function: r(Xit ; dit ) = maxf[U (dit ; Cit ; Xit ; ) + nit+1 EVt+1 (Xit+1 )]jdit ] Cit This function has to be computed for each possible dit , subject to the contemporaneous budget constraint and the constraints on consumption and house size. 12 5.7 Estimation of Preferences: Constrained Optimization Approach to Maximum Likelihood O M Given the observed panel data (Cit ; Hit ; Dit ; Dit ) t=1:::T we infer the unknown parameter vector i=1:::N by maximizing the likelihood function: L( ) = T N Y Y i=1 t=1 We estimate the parameter vector problem: P (d j Xit ; ) by solving the constrained optimization max ;fVit (Xit ; )g t=1:::T i=1:::N s:t: (12) log L( ) (13) Equilibrium Constraints where is the sum of Bellman, Euler, Envelope and Policy function errors. See the Technical Appendix for details. 6 The Data The Health ad Retirement Study (HRS) is a US national panel study which covers a wide range of topics. In particular, questions on family structure, employment status, demographic characteristics, housing, stocks, bonds, IRA, other …nancial assets, income, pension, social security, and bene…ts are relevant for our analysis. Questionnaires assessing individual activities and household patterns of consumption were mailed to a sub-sample of the Health and Retirement Study (HRS).The Consumption and Activities Mail Survey (CAMS), the survey including this information on consumption, was …rst conducted in 2001. We select a group of households that is the potential target segment for RM, according to estimates from the public policy perspective. Our sample includes single, retired homeowners, 62 years old or older. They are retiree. Social security is the homeowners’ main source of income. Pensions and earned interests on …nancial assets contribute much less as a source of perperiod income. We eliminate all households with incomplete records or missing information about their consumption and …nancial situation for the years 2000, 2002, 2004. After these cuts were made, a sample of 175 single households remain. The range of their non-housing …nancial wealth is between $0 and $300,000. In our study, we use the net value of non-housing …nancial wealth, which includes stocks, bonds, saving accounts, mutual funds, individual retirement accounts (IRAs), other assets less liabilities, e.g. house loan, credit card debt and other liabilities. It does not include the value of any real estate, vehicles, or 13 business. Consumption includes vehicles, washing machine, drier, dishwasher, television, computer, telephone, cable, internet, vehicle …nance charges, vehicle insurance, health insurance, food and beverages, dining/drinking out, clothing and apparel, gasoline, vehicles, prescription and nonprescription medications, health care services, medical supplies, trip and vacations, tickets to movies, sorting events and performing arts, hobbies, contribution to religious, educational, charitable or political organizations, cash or gift to family, friends outside your household. Housing expenses for homeowners represent the maintenance cost incurred to keep the house at a constant quality, and for renters, represent the rental cost. Table 1 shows the descriptive statistics for house value, …nancial wealth, consumption, social security income and age for the …rst year in the sample. The number of households observed are 175. Table 1. Descriptive Statistics Percentiles 25% 50% H $40,000 $70,000 W $5,000 $17,500 H=W 0.86 2.5 C $6,270 $9,774 ss $6,972 $9,468 Age 69 74 75% $92,000 $63,000 7.5 $15,090 $11,340 79 Min Max Mean $ 2,500 $0 0.11 $110 $0 64 $170,000 $276,548 1500 $84,380 $ 24.701 84 $71,000 $45,950 23.4 $13,873 $9,087 74 Our empirical data show that housing represents an increasing proportion of the total wealth as people age, that is their resources are greatly tied up in their house. In each period, about 10% of the households in our sample moves out of her house. Among those who moved, about 35% decide to rent a new house, while about 65% buy a new house. At the end of the three years of the panel, about 25% of the population moved and about 10% rented a new house. The moving decision does not appear to be strictly related with age. 7 Calibration and Results The subjective discount rate is set at = 0:96: The real interest rate is r = 0:04: Following Yao and Zhang (2005a), the rental rate is rent = 6% and maintenance cost is own = 1:5%: Transaction costs are own = 6% and rent = 1%, respectively, when moving to an owner-occupied house and when moving to a rental house. We estimate the parameter using a grid search approach. Given the parameter , we use the MPEC approach to estimate ! and : Table 2 shows the estimation results 14 Table 2 Parameter ! Estimate 3.87 0.85 0.87 Asymptotic s.e. (1.07e-009) (1.82e-009) (6.82e-004) Bootstrap s.e. (0.04) (0.0002) (0.05) The asymptotic standard errors, calculated using a …nite di¤erence approach, are very small. Several other papers have noted a signi…cant downward bias of asymptotic standard errors of maximum likelihood estimates of non-linear systems. Therefore, in order to reduce the bias, we compute the standard errors using a bootstrap procedure. Resampling was conducted by sampling with replacement across households as is standard practice in panel models. To obtain the global optimum we use di¤erent initial starting points. However, the optimal solution was not in‡uenced by the initial starting points. In total the standard errors are calculated with 30 bootstraps. The results for the standard errors are preliminary and incomplete. Do Reverse Mortgages Pay? A reverse mortgage is a loan against the retiree’s home that does not have to be paid back for as long as the retiree lives there. We assume that the retiree chooses to receive the proceeds as a single lump sum of cash at the closure of the contract. Di¤erently from forward mortgages,the retiree has to pay some start up costs, which we denote as F: They are assumed to be a fraction of the house value plus some additional cash for closing costs f . The up-front costs include an origination fee (2% value of the house), an up-front mortgage insurance premium (2% value of the house), an appraisal fee and certain other standard closing costs (about $2000-4000). F = Hit + f (14) The maximum amount that the household can initially borrow is assumed to be a fraction of the house value and of the borrower’s age: Vit = i Hit (15) We assume that the date t real interest rate on reverse mortgage is equal to the real return on a one-period bond plus a premium D : rD = r + D (16) Let B denote the …xed reverse mortgage payment at time t for household i: This amount depends on several factors including the age of the borrower and the appraised value of the home. In general, the higher the age of the borrower, the larger is the amount that can borrowed. 15 A reverse mortgage accrues interest charges, beginning when the …rst payment is made to the borrower and then the interest is compounded annually. Let GRM be the outstanding debt at time t: it X t j RD (17) GRM =B it j=1::t If the retiree decides to move out of the house, she has to repay the minimum between the value of the house and the accumulated debt plus a one-time O ). The cost of moving is: transaction cost (Dit O M O Mit = Dit Dit [Dit+1 Hit+1 max(0; Hit O GRM it ) + Hit+1 (Dit+1 )] (18) The welfare gain from reverse mortgage is calculated as the percentage increase in the initial …nancial wealth that makes the household without the reverse mortgage as well o¤ in expected utility terms as with the reverse mortgage . For each household in our sample, we calculate the expected lifetime utility from closing the reverse mortgage contract in the …rst year of our panel. Then, we calculate the percentage increase in their initial …nancial wealth that makes them as well o¤ as with the reverse mortgage. Table 3 shows the median …nancial wealth and welfare gain for each house value. Table 3. House Value $40,000 $80,000 $120,000 Financial Wealth $10,300 $26,000 $48,800 Welfare Gain -37.95% 3.16% 25.56% Table 4 shows median …nancial wealth, welfare gain and house value as a function of non-housing …nancial wealth. Let LW denote initial …nancial wealth less than $10,000, MW between $10,000 and $80,000 and HW greater than $80000. Table 4. LW MW HW Financial Wealth $1,540 $26,500 $138,000 Welfare Gain -63.7% 13.44% 34.61% House Value $40,000 $80,000 $80,000 On average, our simulations show that two groups of households have a welfare loss from a reverse mortgage. First, those with house values less than or equal to $40,000. Second, those with low …nancial wealth. The bene…ts from closing a reverse mortgage contract are liquidity and longevity insurance. The consumer can en-cash some of the saving locked in the 16 house and would be able to experience higher levels of consumption than otherwise possible. In addition, she can live in the same house while alive, regardless of the amount of the outstanding debt. However, households with a smaller house experience a signi…cant welfare loss due to the initial high transaction costs, which represent a signi…cant fraction of the consumer’s reverse mortgage payment. In addition, the consumer is facing a new risk, the moving risk. If the household has to move, for any exogenous reason, her future well-being, her future consumption pro…le and housing choices are signi…cantly a¤ected. Home equity is an important component of precautionary savings. If a homeowner has drawn down on his equity through an RM, his ability to meet unforeseen costs or move into alternative housing may be limited. This is speci…cally true for households with initial low …nancial wealth, as a matter the fact that some of the choices over consumption and housing, available before closing the reverse mortgage contract, are not a¤ordable anymore after the closure of the contract. Therefore the common belief is that RM bene…ts those with resources tied up in home equity, those de…ned. "house rich, cash poor". This simulation shows otherwise. Furthermore, the result reveals that some households would be better o¤ in closing a reverse mortgage contract than without. However, even though it is about twenty years that the reverse mortgage is available, its market is still at 1% of its potential. For about 20% of the households that move in our sample, the main motivation was the …nancial one but nobody chooses to have a reverse mortgage. This seems to imply that the traditional way of moving out to encash the savings locked in the house is preferred to a reverse mortgage. To assess whether the households make the right choice of not taking a reverse mortgage we compute the welfare gain from moving out and renting a new house of the same value. Table 5 shows the median …nancial wealth, welfare gain as a function house value. Table 6 reports the median …nancial wealth, welfare gain and house value as a function of the initial wealth. Table 5. Welfare gain from Moving House Value Financial Wealth $40,000 $10,300 $80,000 $26,000 $120,000 $48,800 Table 6. Welfare gain from Financial Wealth LW $1,540 MW $26,500 HW $138,000 and Renting Welfare Gain 498% 366% 205% Moving and Renting Welfare Gain House Value 5200% $40,000 215% $80,000 23% $80,000 Households with very low …nancial wealth have a signi…cant welfare loss from closing a reverse mortgage contract, due to the moving risk, the high transaction 17 costs and the fact that they could borrow only a percentage of the house value. On the other side, if equity in their house is released by moving out and renting, they would be able to en-cash the full amount of the saving locked in their house without increasing their level of indebtedness and without su¤ering the moving risk. 8 Conclusion Increases in the cost of living and in health care costs, curtailments in medical coverage and other employee bene…ts plans, and cutbacks in Social Security bene…ts expose many of today’s households to the risk of having inadequate resources to support their retirement. Empirical evidence shows that for most retirees the majority of resources is tied up in their house. However, tipically retirees do not divest their home equity to support consumption in retirement and, instead, they adjust to a decreased living standard. Before the advent of the reverse mortgage, the best way to divest those savings locked in the residential property was selling and moving out. In the 1990s, the reverse mortgage , a …nancial instrument designed for "house rich but cash poor" households, was introduced as a potential tool able to enhance social welfare. A reverse mortgage is a home loan that allows the retiree to borrow money against the house without having to make periodic loan repayment or moving out. After about twenty years from its …rst appearance, the reverse mortgage market is at 1% of its potentiality. We ask the following question: "Does it pay to get a reverse mortgage?". In order to answer this question, we build a dynamic structural model of housing, consumption and moving decision. We present the …rst application of the MPEC approach to a dynamic life-cycle model with structural estimation of the preference parameters. We allow households to make both a multistage discrete housing choice and a continuous consumption choice, extending the literature on discrete choice processes. We use a sample of single retiree in the HRS that could be a potential target for the reverse mortgage. We …rst calculate their expected life-time utility without reverse mortgage and then we simulate how better o¤ they would be with a reverse mortgage. Our analysis shows that reverse mortgages provide liquidity and longevity insurance, but introduce a new risk, the moving risk. We also show that households are signi…cantly better o¤ by moving out and renting a new house than staying in the same house with a reverse mortgage. This might explains why reverse mortgages are and could remain in the future niche products. Further policy analysis needs to be conducted to design other …nancial instruments, appealing to the older households, that would allow them to easily access the equity in their home. 18 9 References Attanasio, O., Banks, J., Meghir, C. and Weber, G. (1999): "Humps and Bumps in Lifetime Consumption," Journal of Business and Economic Statistics, 17(1),22-35. Cagetti, M. (2003): "Wealth Accumulation Over the Life Cycle and Precautionary Savings," Journal of Business and Economic Statistics, 21(3), 339-353 Campbell, J. Y., and J.F. Cocco (2003): "Household Risk Management and Optimal Mortgage Choice," Quaterly Journal of Economics, 118, 1149-1194 Carroll, C. D. (1997): "Bu¤er-Shock Saving and the Life-Cycle/Permanent Income Hypothesis," Quaterly Journal of Economics, 112, 1-55 Cocco, J.F.(2004): "Portfolio Choice in the Presence of Housing," Review of Financial Studies, 18(2), 535-567 Cocco, J. F., F. Gomes, and P. Maenhout (2005): "Consumption and Portfolio Choice over the Life Cycle," Journal of Financial Studies, 18(2), 491-533 Deaton, A. (1991): "Saving and Liquidity Constraints," Econometrica, 59(5), 1221-1248 Fernandez-Villaverde, J., Rubio-Ramirez, J. F. and Santos, M. S. (2006): "Convergence Properties of the Likelihood of Computed Dynamic Models," Econometrica, 74(1), 93-119 Flavin, M., and T. Yamashita (2002): "Owner-Occupied Housing and the Composition of the Household Portfolio," American Economic Review, 92(1), 345-362 French, E. (2005): "The E¤ect of Health, Wealth, and Wages on Labor Supply and Retirement behaviour,"Review of Economic Studies, 72, 395-427 Gourinchas, P.O. and J.A.Parker (2002): "Consumption over the Life Cycle," Econometrica, 70(1), 47-89 Hotz, J., and R.A. Miller (1993): "Conditional choice probabilities and the estimation of dynamic models," Review of Economic Studies, 60, 497-529 Hubbard, G., J. S. Skinner, and S. Zeldes (1994): "The importance of Precautionary Motives for Explaining Individual and Aggregate Saving," CanegieRochester Conference Series on Public Policy, 40, 59-125 19 Judd, K. L. (1998): Numerical Methods in Economics. The MIT Press, Cambridge, Massachusetts Judd, K. L. and C.L. Su (2008): "Constrained Optimization Approaches to Estimation of Structural Models," working paper Mayer, C. J.,and K. V Simons. (1994) “ Reverse Mortgages and the Liquidity of Housing Wealth”, Journal of the American Real Estate and Urban Economics Association, 22 (2), 235-55 Kutty, N. K. (1998) “The Scope for Poverty Alleviation among Elderly Home-owners in the United States through Reverse Mortgage”, Urban studies, 35 (1), 113-29 Merrill, S. R., M. Finkel and N. K. Kutty (1994) “Potential Bene…ciaries from Reverse Mortgage Products for Elderly Homeowners: An Analysis of American Housing Survey Data”, Journal of the American Real Estate and Urban Economics Association, 22 (2), 257-99 Caplin, A. (2001) “The Reverse Mortgage Market: Problems and Prospects” in Innovations in Housing Finance for the Elderly, edited by Olivia Mitchell, Pension Research Council Rust, J. (1987): "Optimal replacement of GMC bus engines: An empirical model of Harold Zurcher," Econometrica, 55, 999-1033 Yao, R., and H. H. Zhang (2005): "Optimal Consumption and Portfolio Choices with Risky Housing and Borrowing Constraints," Review of Financial Studies, 18(1). 197-239 Zeldes, S. P.(1989): "Consumption and Liquidity Constraints: An Empirical Investigation," Journal of Political Economics, 97(2), 305-346 20 10 Technical Appendix I am currently working on this appendix. MPEC with DP: Discrete and Continuous Choices The panel data used in this study involves 3 years and about 170 individuals. The available data are both continuous and discrete. The continuous data include data on consumption and …nancial wealth. The discrete (or discretized) data are the individual’s housing tenure (own-rent), her moving decision and her house size. We have additional data on the individuals’ demographics, including age. The MPEC with DP approach consists in solving simultaneously the dynamic programming problem and the maximum likelihood estimation of the preference parameters. 11 Dynamic Programming with Approximation of the Value Function Life Cycle Model: One continuous state variable: wealth Two discrete state variables: previous period housing tenure and previous period house size One continuous choice variable: consumption Many discrete choices: Not Move(N ), Move to House size h with housing tenure q (M hq), where q ={Own,Rent} 11.1 Backward Solution from Time T for True Value Functions In each period, the household chooses whether to stay in her house or to move out. If she moves out, she can either buy or rent a new house and she can choose her new house size. Let the subscripts dN , dM hq denote respectively the decision not to move, the decision to move to house size h and housing tenure q. The housing tenure is a binary variable that takes value 1 if the household own the house The last period value function is known and equal to VT (W; H; Q) where W is the household’s …nancial wealth, H her initial period house size and Q her initial period housing tenure.. In periods t = 1::(T 1) we de…ne: 21 VdN ;t VdM hq ;t = u(cdN ; H) + t+1 Vt+1 (RW = u(cdM hq ; h) + + ss; H; Q) + "N t cdN t+1 Vt+1 (RW qh M + ss; h; q) + "M t cdM hq where M is the transaction cost: M = Q(H and qh + own qh + rent (1 q)h) + (1 Q)(1 q) rent h is the per-period housing expense: = [Q own + (1 Q) rent ]H + [q own + (1 q) rent ]h cdN and cdM qh are the consumption levels respectively if the individual does not move and if she moves to house size h choosing the housing tenure q: ss is the household’s per-period income. t+1 is her survival probability. "N t and qh "M are type I extreme value errors. t Following Rust, we assume that the additivity and the conditional indipendence assumptions hold. To simplify the notation, we introduce the following expressions, which are evaluated at the optimal consumption level: VbdN ;t VbdM hq ;t = u(cdN ; H) + = u(cdM hq ; h) + t+1 Vt+1 (RW + ss; H; Q) cdN t+1 Vt+1 (RW cdM hq M + ss; h; q) The extreme value assumption on "t implies that we can reduce the dimensionality of the dynamic programming problem. The Bellman equation is given by the following closed form solution: Vt (W; H; Q) = Pr(N jW; H; Q) VbdN ;t + E("N t jN ) XX hq + fPr(M hqjW; H; Q) VbdM hq ;t + E("M jM )g t h = ( q ln exp(VbdN ;t ) + XX h q ) b exp(VdM hq ;t ) Given Vt+1 , the Bellman equation implies, for each wealth level W , three set of equations that determine the optimal consumption, cdN ,cdM hq , Vt (W; H; Q), and Vt0 (W; H; Q) Euler Equations: 0 cdN u0 (cdN ; H) t+1 RVt+1 (RW 0 0 cdM hq u (cdM hq ; h) t+1 RVt+1 (RW 22 + ss; H; Q) = M + ss; h; q) = 0 0 Envelope Condition: Vt0 (W; H; Q) = Pr(N jW; H; Q) Vbd0N ;t + XX q h Bellman equation: ( Vt (W; H; Q) = ln exp(VbdN ;t ) + Pr(M hqjW; H; Q) Vbd0M hq ;t XX h q ) exp(VbdM hq ;t ) The time t = 1::(T 1) probabilities of not moving and moving to house size h with housing tenure q are: Pr(N jW; H; Q) = exp(VbdN ;t ) exp(VbdN ;t ) = P P exp(Vt (W; H; Q)) exp(VbdN ;t ) + h q exp(VbdM hq ;t ) Pr(M hqjW; H; Q) = exp(VbdM hq ;t ) exp(VbdM hq ;t ) = P P exp(Vt (W; H; Q)) exp(VbdN ;t ) + h q exp(VbdM hq ;t ) 23 11.2 Backward Solution from Time T for Approximate Value Functions Let (W; H; Q; a) and d (W; H; Q; b) be the functions that we use to approximate respectively the value functions, V (W; H; Q): and the policy functions cd (W; H; Q), with d = fdN ; dM hq g: If we assume that they are a seventh-order polynomials centered at W , then (W; H; Q; a; W ) = 7 X W )k ak;H;Q (W k=0 The time t value function is approximated by Vt (W; H; Q) = (W; H; Q; at ; W t ) = 7 X ak+1;H;Q;t (W W t )k k=0 The time t policy functions are approximated by cd;t (W; H; Q) = (W; H; Q; bd;t ; W t ) = 7 X bk+1;H;Q;d;t (W W t )k k=0 where the dependence of the value function on time is represented by the dependence of the a coe¢cients and the center W on time and the.dependence of the policy functions on time is represented by the dependence of the b coe¢cients and the center W : We will choose W t = (Wtmax + Wtmin )=2; the period t average wealth. Note that W t is a parameter and does not change during the dynamic programming solution method. Therefore, we will drop it as an explicit argument of . So, (W; H; Q; at ) will mean (W; H; Q; at ; W t ): We would like to …nd coe¢cients at and bd;t such that each time t Bellman equation, along with the Euler and envelope conditions, holds with the approximation; that is, for each time t < T 2, we want to …nd coe¢cients at such that for all W ( ) XX (W; H; Q; at ) = ln exp(VbdN ;t ) + exp(VbdM hq ;t ) h q where VbdN ;t VbdM hq ;t = u(cdN ; H) + = u(cdM hq ; h) + and for time t = T VbdN ;T VbdM hq ;T t+1 t+1 t+1 (RW + ss; H; Q; at+1 ) cdN t+1 (RW cdM hq M + ss; h; q; at+1 ) 1, we want to …nd coe¢cients at given that 1 = u(cdN ; H) + 1 = u(cdM hq ; h) + T VT (RW T VT (RW 24 cdN cdM hq + ss; H; Q) M + ss; h; q) This is not possible unless the solution is a degree 7 polynomial. We need to approximately solve the Bellman equation. To this end, we de…ne various errors. First, we create a …nite grid of wealth levels we will use for approximating the value functions. Let Wi;t be grid point i in the time t grid. The choice of grids is governed by considerations from approximation theory. Then we create a grid of house sizes. Let Hj;t be grid point j in the time t grid. Next we need to specify the various errors that may arise in our approximation. We will consider three errors and one side condition. First, at each time t and each Wi;t and each initial period house size Hj;t and housing tenure Qt , the absolute value of the Euler equations if consumption is respectively ci;j;dN ;t and ci;j;Q;dM hq ;t , which we denote as ei;j;Q;t 0, satis…es the inequality e i;j;Q;t u0 (ci;j;dN ;t ; Hj;t ) t+1 R e i;j;Q;t t+1 R 0 (RWi;t 0 (RWi;t ci;j;dN ;t e i;j;Q;t +ss; Hj;t ; Qt ; at+1 ) u0 (ci;j;dM hq ;t ; Ht+1 ) ci;j;dM hq ;t M + ss; Ht+1 ; Qt+1 ; at+1 ) e i;j;Q;t where 0 (x; at+1 ) is the derivative of (x; at+1 ) with respect to x. Second, the Bellman equation error at Wi;t with consumption ci;j;dN ;t and ci;j;dM hq ;t is denoted by bj;Q;t and satis…es ( b j;Q;t where Vbi;j;dN ;t Vbi;j;dM hq ;t + (Wi;t ; Hj;t ; Qt ; at ) ln exp(Vbi;j;dN ;t ) + = u(ci;j;dN ;t ; Hj;t ) + = u(ci;j;dM hq ;t ; Ht+1 ) + t+1 (RWi;t Third, the envelope condition errors, env t , env j;Q;t 0 XX h 0 (Wi;t ; Hj;t ; Qt ; at ) ffi;j;dN ;t [fi;j;dM hq ;t 0 (RWi;t ci;j;dM hq ;t q 25 h q ) exp(Vbi;j;dM hq ;t ) ci;j;dN ;t (RW t+1 XX ci;j;dM hq ;t b j;Q;t + ss; Hj;t ; Qt ; at+1 ) M + ss; Ht+1 ; Qt+1 ; at+1 ) satis…es (RWi;t ci;j;dN ;t M hq +ss; Hj;t ; Qt ; at+1 ) M +ss; Ht+1 ; Qt+1 ; at+1 )]g env j;Q;t where 0 (x; at ) is the derivative of fi;j;d;t = Pr(djWi;t ; Hj;t ; Qt ) = 0 (x; at ) with respect to x and exp(Vbi;j;d;t ) P P exp(Vbi;j;dM hq ;t ) exp(Vbi;j;dN ;t ) + h q Fourth, we introduce the policy function errors: cons i;j;Q;d;t 11.3 (Wi;t ; Hj;t ; Qt ; bt ) ci;j;d;t (Wi;t ; Hj;t ; Qt ) cons i;j;Q;d;t Empirical Part In the theorical DP part we obtain the coe¢cients used in the approximation of the value function. In this part, for any individual data of …nancial wealth, initial period house size and age, we calculate the predicted consumption and probabilities of movH N M hq ing. First, the individual makes the housing decision dH g; n;tp , with d = fd ; d then she makes her consumption decision. data Let cpred n;tp and cn;tp denote respectively the predicted and the true value of consumption for household n at time tp: For any given discrete choice on housing dH n;tp , using the real data on consumption, we calculate the measurement error: data data data Pr(cn;t jdH n;tp ; Wn;tp ; Hn;tp ; Qn;tp ) = p 1 e pred cn;tp )2 2 2 (cdata n;tp 2 2 The probability for the discrete choice on housing is given by: eVd;n;tp V m;n;tp me data data data Pr(dH n;tp jWn;tp ; Hn;tp ; Qn;tp ) = P Therefore the joint probability of making the discrete housing choice dH n;t and the continuous consumption choice cn;t is given by: data; data data H data data data H data data data Pr(dH n;tp ; cn;tp jWn;tp Hn;tp ; Qn;tp ) = Pr(dn;tp jWn;tp ; Hn;tp ; Qn;tp ) Pr(cn;t jdn;tp ; Wn;tp ; Hn;tp ; Qn;tp ) The Log-Likelihood is given by: L( ) = TP N X X n=1 tp=1 data data data log Pr(dH n;tp ; cn;tp jWn;tp ; Hn;tp ; Qn;tp ; ) where N denotes the number of individuals in the sample and T P the number of time periods in the panel data. 26 11.4 MPEC With these de…nitions,let XXX XXXX e = i;j;Q;t + t i j t Q j XXX b j;Q;t + t Q j XXXXX env j;Q;t + t Q i j Q d and let P be a penalty parameter. The MPEC approach to the estimation of the preference paramenters is: M axL( ) P ;a;c subject to: Bellman error: b j;Q;t (Wi;t ; at ) Euler error ( ln exp(Vbi;dN ;t ) + e i;j;Q;t uc;i;j;dN ;t + e i;j;Q;t uc;i;j;dM hq ;t + XX h q ) exp(Vbi;dM hq ;t ) b j;Q;t e + i;j;Q;t W ;i;j;dN ;t e + i;j;Q;t W ;i;j;dM hq ;t Envelope error env j;Q;t W ;i;dN ;t ffi;dN ;t XX + + W ;i;j;dN ;t h [fi;j;dM hq ;t + ]g W ;i;j;;dM hq ;t q Policy function error cons i;j;Q;d;t (Wi;t ; Hj;t ; Qt ; bd;t ) ci;j;d;t (Wi;t ; Hj;t ; Qt ) The probability of decision d : fi;j;d;t = exp(Vbi;j;d;t ) P P exp(Vbi;j;dM hq ;t ) exp(Vbi;j;dN ;t ) + h 27 q cons i;j;Q;d;t env j;Q;t cons i;j;Q;d;t 11.5 11.5.1 AMPL Backward Solution from Time T for Approximate Value Functions in AMPL In order to formulate this problem in AMPL, we need to list every quantity that is computed. The time-speci…c wealth grids Wi;t are …xed. We discretize the house size. The parameters are Wi;t ; ; t ; R; own rent ; ; own rent ; bequest ; The basic variables of interest are ci;j;dN ;t ; ci;j;dM hq ;t ak;j;Q;t ; bk;j;Q;d;t e i;j;Q;t ; b j;Q;t ; env j;Q;t ; cons i;j;Q;d;t AMPL does not allow procedure programming; therefore, we need to de…ne other variables to represent quantities de…ned in terms of other variables. We …rst need ui;j;dN ;t u ci;j;dN ;t ; Hj;t uc;i;j;dN ;t u0 ci;j;dN ;t ; Hj;t + Wi;j;d N ;t RWi;t fi;j;dN ;t = ci;j;dN ;t Pr(N jWi;t ; Hj;t ; Qt ) ui;j;dM hq ;t u ci;j;dM hq ;t ; Ht+1 uc;i;j;dM hq ;t u0 ci;j;dM hq ;t ; Ht+1 + Wi;j;d M hq ;t fi;j;dM hq ;t RWi;t = + ss ci;j;dM h ;t M + ss Pr(M hqjWi;t ; Hj;t ; Qt ) We next use those variables to build more variables i;j;Q;t W ;i;j;Q;t + i;j;dM Q ;t + W ;i;dN ;t + i;j;dM hq ;t + W ;i;j;dM hq ;t i;j;d;t (Wi;t ; Hj;t ; Qt ; at ) 0 (Wi;t ; Hj;t ; Qt ; at ) + (Wi;j;d N M ;t ; Hj;t ; Qt ; at+1 ) 0 + (Wi;j;d N M ;t ; Hj;t ; Qt ; at+1 ) + (Wi;j;d M h ;t ; Hj;t+1 ; Qt+1 ; at+1 ) 0 + (Wi;j;d M hq ;t ; Hj;t+1 ; Qt+1 ; at+1 ) (Wi;t ; Hj;t ; Qt ; bd;t ) 28 With these variables de…ned, the Bellman equation error inequality becomes ( ) XX b b b b ln exp(Vi;j;dN ;t ) + exp(Vi;j;dM hq ;t ) i;j;Q;d;t j;Q;t j;Q;t q h where Vbi;j;dN ;t Vbi;j;dM hq ;t = ui;j;dN ;t + + i;j;dN ;t t+1 = ui;j;dM hq ;t + t+1 + i;j;dM hq ;t the Euler equation error inequalities become e i;j;Q;t uc;i;j;dN ;t + e i;j;Q;t uc;i;j;dM hq ;t + e + i;j;Q;t W ;i;j;dN ;t e + i;j;Q;t W ;i;j;dM hq ;t and the envelope error inequality becomes env j;Q;t W ;i;dN ;t ffi;j;dN ;t XX + + W ;i;j;dN ;t h The probability of decision d : fi;j;d;t = [fi;j;dM hq ;t + ]g W ;i;j;;dM hq ;t h exp(Vbi;j;d;t ) P P exp(Vbi;j;dM hq ;t ) exp(Vbi;j;dN M ;t ) + h q The policy function errors are cons i;j;Q;d;t 11.5.2 i;jQ;;d;t ci;j;Q;d;t cons i;j;Q;d;t Empirical Part in AMPL We consider individuals in our sample such that Agedata 2): n;tp = 1::(T data data data data Let Wn;tp , Agen;tp , Hn;tp and Qn;tp denote respectively the data on …nancial wealth, age, house size and housing tenure for household n in year tp in the panel data. Given these data, the variables of interest are: cpred dN ;n;tp = data data data data dN M (Wn;tp ; Agen;tp ; Hn;tp ; Qn;tp ) cpred dM hq ;n;tp = data data data data dM h (Wn;tp ; Agen;tp ; Hn;tp ; Qn;tp ) upred dN ;n;tp data u cpred ; Hn;tp ; Qdata n;tp dN M ;n;tp uc;dN ;n;tp data u0 cdN M ;n;tp ; Hn;tp ; Qdata n;tp Wd+N ;n;tp fdpred N M ;n;tp data RWn;tp = cpred dN ;n;tp data (Hn;tp ; Qdata n;tp ) + ss data data data Pr(N M jWn;t ; Hn;tp ; Agedata n;tp ; Qn;tp ) 29 env j;Q;t upred dM hq ;n;tp data choice choice u cpred ; Hn;tp ; Hn;tp ; Qdata n;tp ; Qn;tp dM h ;n;tp data choice choice u0 cdM hq ;n;tp ; Hn;tp ; Hn;tp ; Qdata n;tp ; Qn;tp uc;dM hq ;n;tp Wd+M hq ;n;tp data RWn;tp fdpred M hq ;n;tp = cpred dM hq ;n;tp choice (Hn;tp ; Qchoice n;tp ) data choice choice M (Hn;tp ; Qdata n;tp ; Hn;tp ; Qn;tp ) + ss data data Pr(M hqjWn;tp ; Hn;tp ; Agedata n;tp ) We next use those variables to build more variables data n;tp data W ;n;t + dN ;n;tp data data (Wn;tp ; Hn;tp ; Qdata ) n;tp ; aAgedata n;tp data 0 data data ) ) (Wn;tp ; Hn;tp ; Qdata n;tp ; aAgedata n;tp ( data data (Wd+N M ;n;tp ; Hn;tp ; Qdata ) n;tp ; aAgedata n;tp +1 + W;dN ;n;tp 0 + dM hq ;n;tp data choice choice (Wd+M ;n;tp ; Hn;tp ; Hn;tp ; Qdata ) n;tp ; Qn;tp ; aAgedata n;tp +1 + W ;dM hq ;n;tp 0 data (Wd+N M ;n;tp ; Hn;tp ; Qdata ) n;tp ; aAgedata n;tp +1 data choice data choice ; aAgedata ) (Wd+M ;n;tp ; Hn;tp ; Hn;tp ; Hn;tp ; Hn;tp n;tp +1 pred data Vbdpred N ;n;tp = u(cdN M ;n;tp ; Hn;tp )+ data (RWn;tp cpred dN M ;n;tp data data data ) ; Qdata (Hn;tp n;tp )+ss; Hn;tp ; Qn;tp ; aAgedata n;tp +1 pred choice Vbdpred ) M hq ;n;tp = u(cdM hq ;n;tp ; Hn;tp + data (RWn;tp cpred dM hq ;n;tp choice (Hn;tp ; Qchoice n;tp ) choice choice data M (Hn;tp ; Qdata n;tp ; Hn;tp ; Qn;tp ) data choice choice +ss; Hn;tp ; Hn;tp ; Qdata ) n;tp ; Qn;tp ; aAgedata n;tp +1 The probabilities of not moving and moving are: data data data data fdpred N M ;n;tp = Pr(HdN ;n;tp jWn;tp ; Hn;tp ; Qn;tp ; Agen;tp ) = fdpred M hq ;n;tp = data data data Pr(HdM hq ;n;tp jWn;tp ; Hn;tp ; Qdata n;tp ; Agen;tp ) 30 exp(Vbdpred N M ;n;tp ) P P exp(Vb pred )+ M hq exp(Vbdpred N ;n;tp h q d ;n;tp ) exp(Vbdpred M h ;n;tp ) = P P b pred exp(Vbdpred N ;n;tp ) + h q exp(VdM hq ;n;tp ) We introduce the following constraints concerning the measurement error in consumption: data data data Pr(cn;t jdH n;tp ; Wn;tp ; Hn;tp ; Qn;tp ) = p KNITRO Problem Characteristics Objective goal: Maximize Number of variables: 72746 bounded below: 23688 bounded above: 0 bounded below and above: 23521 …xed: 0 free: 25537 Number of constraints: 103488 linear equalities: 0 nonlinear equalities: 35280 linear inequalities: 35280 nonlinear inequalities: 32928 range: 0 Number of nonzeros in Jacobian: 960344 Number of nonzeros in Hessian: 287529 31 1 2 2 exp( 2 cpred d;n;tp ) (cdata n;tp 2 2