Depreciation of Housing Capital, Maintenance, and House Price Inflation: Estimates from a Repeat Sales Model

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1 Depreciation of Housing Capital, Maintenance, and House Price Inflation: Estimates from a Repeat Sales Model John P. Harding University of Connecticut School of Business Administration 2100 Hillside Road Unit 1041 Storrs, CT johnh@business.uconn.edu Phone: (860) Fax: (860) Stuart S. Rosenthal Department of Economics Syracuse University 426 Eggers Hall Syracuse, New York ssrosent@maxwell.syr.edu Phone: (315) Fax: (315) C. F. Sirmans University of Connecticut School of Business Administration 2100 Hillside Road, Unit 1041 Storrs, CT cf@business.uconn.edu Phone: (860) Fax: (860) June 30, 2006 We thank Dan Black, Jan Brueckner, Karl Case, Richard Green, Barbara Fraumeni, Stephen Oliner and two anonymous referees for helpful comments. Any remaining errors are our own.

2 Abstract The rate at which physical capital depreciates is fundamental to investment in the economy. Nevertheless, although housing capital accounts for one-third of the total capital stock, the rate at which housing capital depreciates has only rarely been directly estimated, in part because prior studies do not control for maintenance. For that same reason, widely publicized measures of house price appreciation overstate the capital gain from homeownership. Using data from the American Housing Survey we examine these issues. Over the 1983 to 2001 period, results indicate that gross of maintenance, housing depreciates at roughly 2.5 percent per year, while net of maintenance, housing depreciates at approximately 2 percent per year. Moreover, although the typical home appreciated at an annual real rate of roughly 0.75 percent, after allowing for depreciation and maintenance, the average homeowner experienced little capital gain.

3 I. Introduction This paper examines two closely related issues. First, the rate at which housing capital depreciates is fundamental to investment in the economy. As shown in Table 1, estimates from the U.S. Bureau of Economic Analysis (BEA) indicate that housing accounts for roughly half of all private fixed assets and one-third of the total capital stock. 1 Nevertheless, the rate at which housing depreciates has only rarely been estimated, in part because prior studies do not control for maintenance. As a result, major federal data agencies such as the Bureau of Economic Analysis have been forced to use measures of housing capital depreciation rates that are based more on approximations than hard evidence in the data. Given the important role that such measures play in calculations of the stock and flow of capital in the economy, obtaining more reliable estimates of the rate at which housing depreciates is of considerable importance. Second, although the belief that homeownership is an excellent investment is deeply entrenched in popular culture, the capital gain enjoyed by homeowners is likely less robust than often portrayed. That is because widely publicized repeat sales measures of house price appreciation do not control for the contribution of maintenance to house price appreciation (Case and Shiller [4], Case and Quigley [3]). 2 Although such indexes are appropriate for valuing portfolios of homes, we show that they overstate the capital gain from homeownership. Understanding the role of depreciation and maintenance is especially important in the current social and political environment. Recent popular accounts, for example, credit house price appreciation with propping up the economy, while new government policy initiatives seek 1 See also, Mills [16]. 2 See, for example, the repeat sales house price indexes at and

4 to boost homeownership, especially among low income and minority households. 3 To illustrate, Figure 1 displays the growth of house prices in both real and nominal terms as reported by the National Freddie Mac Conventional Mortgage Home Price Index (also known as the Freddie Mac repeat sales index). Evident in Figure 1, the growth in house prices has been particularly dramatic since the mid-1990s. Since that time, the Freddie Index has risen more rapidly than inflation at the same time that prices on financial assets have declined sharply. In response to these price patterns, the popular press has routinely touted housing as an outstanding investment. In March 2002, The Economist ran a cover story entitled The Houses That Saved The World in which it argued that rising house prices had sheltered the world economy from deep recession by supporting consumer sentiment and spending. 4 The perception that housing is an excellent investment has also contributed to recent federal policies designed to promote homeownership. 5 While house price appreciation may well have buoyed the economy in recent years and generated capital gains for homeowners, analysis of these issues has failed to consider the role of depreciation and maintenance. Considering the role of depreciation and maintenance, however, is difficult for several reasons. The first challenge is to obtain data on home maintenance. In our case, we address that need by using the American Housing Survey, the only major survey that 3 In various public statements by government officials, it is clear that one of the reasons policy makers hope to boost homeownership is to facilitate the accumulation of wealth among individual homeowners. In addition, however, it should also be emphasized that many other reasons contribute to recent efforts to raise homeownership rates. These include the hope that homeownership helps to stabilize and enhance neighborhoods, and also concerns that a legacy of discrimination has restricted access to mortgage credit among minority families. 4 Note also, that between 1975 and 2002, the Freddie Mac index grew at a compound nominal rate of 5.0% per year and has declined in only six quarters. There has never been a year-over-year decline in the nominal price index. These patterns have contributed to the belief that housing is an outstanding investment for individual homeowners. 5 In 1994, President Clinton instructed HUD to develop programs designed to dramatically increase homeownership in our nation over the next six years. (see In 2002, President George W. Bush set a goal of 5.5 million new minority homeowners by 2010 (for details see while Congress recently passed the American Dream Downpayment Act which provides $5,000 in down payment and closing-cost assistance to low-income and first-time homebuyers. 2

5 provides detailed information on home maintenance. But in addition, maintenance may be endogenous to house value: it is likely that owners of expensive homes will spend more on home maintenance than owners of inexpensive homes. As such, one cannot simply put maintenance into a standard ordinary least squares hedonic regression of house value. Our solution to this problem is to expand the traditional repeat sales model of house price appreciation in two important ways. In the standard specification (e.g. Case and Shiller [4]), the percentage change in house price between sale dates is regressed on a set of annual dummies associated with the dates of sale. Attributes of the home and their shadow prices are assumed to be unchanged between sale dates and, therefore, drop out of the model. We build directly on that feature. Specifically, a home s attributes tend to be correlated with maintenance, but otherwise have no natural role in the standard repeat sales specification. Accordingly, the attributes of a home make excellent instruments for maintenance in a modified repeat sales regression that includes the influence of maintenance efforts between sale dates. To complete our model, we also control for age-related depreciation as this enables us to estimate the independent effects of depreciation and maintenance on house price appreciation. Controlling for depreciation, however, also is not straight forward. That is because the change in age of the home between sale dates is perfectly collinear with the dummy variables that identify the dates of sale. To address this issue, we specify a non-linear rate of age-related depreciation in the model. We estimate our models using a panel of single family home sales from the American Housing Survey that cover the 1983 to 2001 period. Two principal specifications are considered: the standard model that we argue is appropriate for estimating the change in value for an existing portfolio of homes, and our expanded model that controls for both depreciation and maintenance 3

6 using both ordinary least squares (OLS) and a two-stage least squares (2SLS) approach. For the 1983 to 2001 period, the standard model implies roughly a 0.75 percent real annual rate of appreciation. Upon controlling for maintenance and depreciation, the annual real rate of house price inflation is 2.68 percent. 6 Thus, while highly publicized repeat sales measures such as the Freddie Mac and OFHEO (Office of Federal Housing Enterprise Oversight) indexes are appropriate for estimating the change in value for a portfolio of homes, users of these indexes should be aware that they substantially underestimate house price inflation. Additional findings indicate that in the absence of maintenance, housing depreciates at a real annual rate of roughly 3.0 percent for the typical (median) home, referred to hereafter as the gross-of-maintenance rate of depreciation. At this rate, after fifty years the original housing capital would have fallen more than 75 percent in value. Of course, homeowners maintain their homes and this serves to slow the rate of decay. We find that for the typical homeowner, maintenance adds roughly 1 percent real per year to the value of the home, lowering the net-ofmaintenance rate of depreciation to roughly 2 percent. At that rate, after fifty years the typical home would have fallen more than 60 percent in value relative to a newly constructed unit. That rate of depreciation would support longstanding theoretical arguments and recent evidence from Rosenthal (2004) that older homes tend to filter down to families of lower economic status. At the same time, the sizable contribution of maintenance to house price appreciation underscores the importance of ensuring that homeowners and especially low-income homeowners have sufficient means to maintain their homes. 6 That difference is consistent with Hwang and Quigley [13] who emphasize that estimates from standard repeat sales models must be interpreted with care. It should also be noted that whereas our focus on the repeat sales methodology is to control for maintenance and age-related depreciation, the focus on Hwang and Quigley [13] is to control for the selection process governing homes that appear in the repeat sales sample. 4

7 Finally, we also estimate the real annual capital gain enjoyed by homeowners. For the median homeowner, this rate was a negative 0.25 percent over the 1983 to 2001 period. Thus, policy makers and homeowners alike should be aware that the majority of homeowners experienced little capital gain net of maintenance expenditures over the 1983 and We proceed as follows. The following section describes our methodology for building both maintenance and aging of the home into the repeat sales model. Next, we describe our data and summary measures. The remaining sections present our results and conclusions. II. Methodology This section lays out the methodology used to estimate the standard repeat sales model and also our modified model that takes depreciation and maintenance into account. We further show that results from these models can be used to estimate a variety of parameters of interest, including the quality adjusted rate of house price inflation, the gross and net rates at which housing depreciates, the contribution of maintenance to house price appreciation, and more. We begin with the standard specification and argue that this model is appropriate if the goal is to measure the change in value for a portfolio of homes. The Standard Repeat Sales Model As in Bailey, Muth and Nourse [1], Case and Shiller [4] and [5], and Case and Quigley [2], suppose that the price of a home, P, is observed upon purchase and sale in periods t and t+τ, respectively, where and γ t P = e f( X ; β ), (2.1a) t t t 5

8 γ t+ τ P = e f( X ; β ). (2.1b) t+ τ t+ τ t+ τ In equation (2.1a) and (2.1b), f( Xt+ τ ; βt+ τ ) is an unknown and potentially non-linear function of the period-specific characteristics of the home (X) and their shadow prices (β). The elements of X include both structural attributes and characteristics of the neighborhood specific to the house. The terms γ t and γ t+τ represent the influence of period-specific market conditions that are common to all properties in the geographic market from which the sample of homes are drawn. It is these terms that measure the quality adjusted price of housing in periods t and t+τ. Suppose now that both X and β are unchanged between t and t + τ. Substituting (2.1a) into (2.1b) yields, γt+ τ t Pt e γ τ Pt + =. (2.1c) Taking logs and rearranging, Pt + τ log( ) = γ t+ τ γt + ωt+ τ, (2.2) P t where ω is a random error term. Provided X and β are time-invariant and, contingent on the multiplicative structure in (2.1a) and (2.1b), f( X; β ) drops out of the model. This eliminates the need to impose a specific functional form on f ( ) and enhances the reliability of the estimated values for γ t and γ t+τ. The percentage change in the quality adjusted price of housing is then captured by the difference in the constant terms from the underlying hedonic equations (2.1a) and (2.1b). Consider now a sample of properties that are bought and sold at various times. For such a sample, equation (2.3) is given by P t+ τ, i log( ) P ti, τ i γ tdt, i ωt, i for observation i = 1,, n (2.3) t= 1 = + 6

9 where D t equals -1, 0 or 1 depending on whether a given property in period t sells for the first time, is not sold, or sells for the second time, respectively. Note also, that the γ terms are common across all housing units, while the rest of the variables in (2.3) vary across individual homes. Equation (2.3) is the standard repeat sales specification and we refer to it as the Standard model. It is important to note that this specification does not require information on housespecific attributes. Those features difference out of the model. Moreover, the vector of γ parameters can be readily estimated by ordinary least squares (OLS) by regressing the log-price change between sale dates on the vector D. 7 The specification in equation (2.3) depends crucially on two assumptions that tend to be overlooked in the repeat sales literature. First, that the shadow prices β are time invariant, and second, that the attributes of the home, X, are also unchanged between sale dates. In keeping with the literature, throughout this paper we assume that β is unchanged over time. In addition, as a first approximation, it is appealing to assume that most features of a home are also largely unchanged between sale dates, including for example, the number of bedrooms and bathrooms, the size of the home, etc. However, for two very important features of the home the assumption of time invariance cannot hold. The first of these is the level of maintenance, a crucial input into a home s quality. The second feature is the age of the home itself, and more specifically, deterioration that is associated with aging of the structure. To appreciate the importance of allowing for age-related depreciation of the house and related maintenance, one has only to drive through any number of older neighborhoods in cities across the United States. Although there are many exceptions, the norm is that older homes are in worse physical condition than newly built homes of otherwise comparable style and size. 7 Case and Shiller [4] suggest using weighted least squares to estimate equation (2.2) to enhance efficiency because the variance of the error term is likely related to the length of time between sales. 7

10 Indeed, age-related deterioration of the housing stock has long been viewed as the primary mechanism by which markets provide low-income housing: homes built for middle- and higherincome families eventually age, deteriorate, and filter down to families of lower income status. Recent papers by Brueckner and Rosenthal [2], Rosenthal [18] and Gyourko and Saiz [12] confirm this pattern. If one s goal is to appraise the change in market value of a given home or a portfolio of homes, the standard repeat sales index obtained from (2.3) is a useful guide that can be readily estimated. This is especially true if the context does not require intimate knowledge of why a particular home (or set of homes) has appreciated. Applications of this sort can arise in several contexts, but especially in the mortgage industry. It is well known, for example, that default propensities depend on loan-to-value ratios, and that the likelihood of default affects the market price of a mortgage. Accordingly, the standard repeat sales index is a useful tool for valuing portfolios of homes that back pools of mortgage securities. The Inflation Repeat Sales Model Although the standard repeat sales model has many valuable applications, it is not appropriate if one s goal is to evaluate house price inflation, depreciation, or the capital gain associated with homeownership. To measure these factors, it is necessary to control for both maintenance and age-related depreciation. This is not straight forward, but it is feasible. We begin with depreciation. 8

11 Age-Related Depreciation Suppose that housing capital depreciates at a linear rate over time. Then the age of the home, denoted A t, should be included in the hedonic regression (2.1a) as an element of X, and the coefficient on A t would measure the depreciation rate. This specification is common in the hedonic house price literature. This also implies that τ, the change in house age between sale dates, should be included in the repeat sales regression. That is problematic, however, because τ is perfectly collinear with the categorical variables, D t : including D t in the model implicitly includes τ in the model as well. For this reason, Bailey, Muth, and Nourse [1] comment that it is not possible to control for linear age-related depreciation in the repeat sales model. To address this issue, we note that the house price hedonic literature provides evidence that housing depreciates at a non-linear rate (e.g. Shilling et al. [20], Lee et al. [14]). Accordingly, for the repeat sales model, we specify a non-linear depreciation function that avoids the perfect collinearity issue noted by Bailey, Muth and Nourse [1]. To be precise, we include the log of τ in the repeat sales model, and rewrite expression (2.3) as P i t+ τ, i log( ) log P ti, τ γ tdt, i α ( τi) ωt, i. (2.4) = + + t= 1 In this expression, α is the elasticity of house price appreciation with respect to the change in age between sale dates. Maintenance Consider now the role of maintenance, and let the expenditure on maintenance s years in the past be given by m s. In this context, expenditure includes both cash outlays for services rendered and the imputed value of maintenance performed by the homeowner. Our goal is to 9

12 evaluate how this maintenance affects house price appreciation between the purchase and sale dates, t and t +τ. Bearing that in mind, maintenance depreciates over time such that of the maintenance conducted s years in the past, the share that influences house price appreciation between t and t + τ is given by d s, where 0 d s < 1, and d s diminishes with s. Moreover, for maintenance conducted prior to the purchase date (s > τ), d s is equal to zero because the value of that maintenance is already capitalized into P t. Then for house i, the stock of housing capital provided by maintenance that affects appreciation between t and t + τ is, M τ i d m. (2.5) τ, i s t+ τ s, i s= 0 The impact of this capital on the rate of house price appreciation is Mτ b P, i ti,, where b is the rate at which the market values the homeowner s maintenance, and P t,i is the period-t price level. 8 Adding this measure to expression (2.4) we obtain: τ P M ln( ) log( ). (2.6) i t+ τ, i τ, i = γtdt, i + α τi + b + ωt+ τ, i Pti, t= 1 Pti, In this expression, the percentage increase in house price between sale dates is decomposed into τ i three systematic parts: γ tdt, i measures the contribution from house price inflation, log(τ i ) t= 1 controls for age-related depreciation, and change in a home s market value. Mτ b P, i ti, measures the contribution of maintenance to the P P 8 t+ τ t+ τ To clarify why maintenance is normalized by the initial price level, note that ln( ) 1. Substituting into P P t t (2.6) and solving for Pt + τ, we get P [ 1 + ( γ γ ) + αlog( τ) t t t ] P + bm. This says that price in period t + + τ + τ t t, t+ τ τ is approximately equal to the period-t price level, plus the increment from house price inflation, adjusted for depreciation, plus the contribution from maintenance over the holding period. 10

13 Provided one can measure M τ,i it is seemingly straight forward to estimate equation (2.6). In practice, however, because of measurement error and the likely endogenous character of M τ,i, this is not the case. Consider measurement error first. Maintenance performed by the homeowner as opposed to out of pocket expenditures is not reported in the data. This causes us to understate the level of maintenance in the home. We will return to this issue shortly. 9 A second measurement problem would persist even if all maintenance were observed without error. That is because d s between sale dates is unknown and this prevents us from forming M τ,i. In response to this problem it is tempting to re-specify (2.6) by replacing M τ,i with the sequence of annual maintenance efforts, treating each year of maintenance as a separate regressor. The coefficients on m s would then provide estimates of bd s for s = 1, τ max, where τ max is the longest period between sale dates in the sample. Note, however, that τ max in our sample is 18 years (1984 to 2001), so decomposing M τ,i would require us to estimate eighteen separate maintenance parameters. For various reasons, our data do not allow for that level of precision. In part, this is because observations with holding periods longer than ten years are relatively few and this reduces our ability to identify coefficients on maintenance from the distant past. In addition, maintenance from different years is highly correlated and this creates collinearity problems, further reducing our ability to obtain precise estimates. 10 Further, as will be discussed shortly, we instrument for maintenance to allow for the possibility that maintenance may be endogenous. Rank conditions require that there be at least one valid instrument for each endogenous variable, and this further limits the number of maintenance variables that can be included in the model. 9 Examples of owner-provided maintenance efforts include landscaping, gardening, minor carpentry, plumbing, and other fixes around the house. Details of the maintenance data used for the analysis are provided in Appendix A. 10 For example, the correlation between maintenance in the year prior to sale and the undiscounted sum of maintenance in all previous years since the purchase date is roughly 50 percent. 11

14 Our strategy in response to these issues is to strike a balance between maintaining sufficient power to reliably estimate the model while also retaining as much precision as possible. Specifically, we decompose (2.5) into three parts corresponding to maintenance performed in the three years before sale, 3 to 6 years prior to sale, and all other years: (2.7) M d m + d m + d m τ, i s t+ τ s, i s t+ τ s, i s t+ τ s, i s= 1 s= 4 s= 7 Next, we proxy for each of the three terms in (2.7) using the following three measures, 3 1to3, i s t + s, i s= 1 6 M = d m τ, M 4to6, i = d smt + τ s, i, and M 7 18, = d m + τ,. In forming these terms, s= 4 18 to i s t s i s= 7 d s is set to 1 for years between the purchase and sale dates, and to zero for years prior to τ given the assumption above that such maintenance is already capitalized into P t. These terms are then substituted into (2.6) in place of M τ,i to obtain, 11 τ P M M M ln( ) log( ). (2.8) i t+ τ, i 1to3, i 4to6, i 7to18, i = γtdt, i + α τi + b1 + b3 + b3 + ωt+ τ, i Pti, t= 1 Pti, Pti, Pti, Expression (2.8) deviates from (2.6) in that the depreciation factors, d s, are set equal to 1 for the years between the purchase and sale dates (s < τ i ). This causes the maintenance terms in (2.8) to overstate the level of housing capital associated with past maintenance. Moreover, that overstatement increases with s as maintenance decays over time. It is worth emphasizing that the direction of this measurement error is opposite to that of unreported maintenance which causes us to understate the level of maintenance. Both sources of measurement error affect b but in 11 In an earlier version of the paper we estimated the model including the average annual expenditure on maintenance between sale dates in place of the expenditure on maintenance in the year before sale. Results were extremely similar to those reported later in the paper for the specification in (2.8). This suggests that the average annual intensity with which a homeowner maintains the home is a good proxy for the entire history of maintenance efforts. 12

15 different directions, so the net effect is ambiguous, a priori. 12 Nevertheless, including the undiscounted maintenance measures as in (2.8) likely does an excellent job of controlling for the influence of maintenance on house price appreciation and that allows us to obtain consistent estimates of the other model parameters. In addition, we can still estimate the average annual contribution of maintenance to house price appreciation as will be discussed later in this section. We refer to estimates of equation (2.8) as the Inflation model. Consider now the issue of endogeneity. Maintenance could be endogenous to house price appreciation because maintenance yields not only consumption benefits, but also serves as an investment. This latter effect implies that maintenance decisions could be sensitive to anticipated and actual changes in the underlying asset value. 13 The standard repeat sales model outlined in expressions (2.1) to (2.3) provides a convenient solution to this problem. Given the assumptions that underlie that model, physical attributes of the home have no role in the estimating equation. Those attributes, however, are correlated with maintenance and are assumed to be exogenous throughout the vast house price hedonic literature. Accordingly, (2.8) can be estimated by three-stage least squares (3SLS) treating each of the maintenance terms as endogenous and using the structural attributes of the home as instruments. 14 In the empirical work that follows, we estimate both an OLS and a 3SLS version of the Inflation model. 12 That said, it seems likely that maintenance installed in the three years prior to sale would not have depreciated much. In that case, recent maintenance would be understated and b would be upward biased relative to the true 1 value of b. Similarly, for maintenance beyond six years, it seems likely that we overstate this measure, causing b 3 to be downward biased relative to the true value of b. We have no way, however, of checking these assumptions. 13 See Davidoff [7] for a related discussion of the determinants of home maintenance decisions. 14 In the empirical work to follow, we use the age of the home at the initial sale date, number of rooms, bedrooms, bathrooms, central city versus other location, condominium status, and whether the home is single family detached. Each of these variables is also interacted with house age. Results from the first-stage maintenance regressions are provided in Table B-2 of Appendix B. F-statistics indicate that the instruments noted above are jointly and highly significant predictors of maintenance. Reviewing Table B-2, it is also apparent that several of the house-specific attributes are individually significant as well. For maintenance in the 3 years prior to sale, this includes single 13

16 Parameters of Interest Our goal now is to clarify how estimates from (2.3) and (2.8) can be used to calculate a number of parameters of interest, including depreciation rates, capital gains rates, and more. Two definitions will prove useful: and rp = ri rd + rm (2.9a) r = r r. (2.9b) CG I D In (2.9a), the annual rate at which a portfolio of homes appreciate ( r P ) is equal to the rate of house price inflation ( r I ) less the gross-of-maintenance rate of depreciation ( r D ) plus the annual contribution of maintenance to the rate of house price appreciation ( r M ). It is important to note that r P is the appreciation rate implicitly measured by the Standard model in expression (2.3). In (2.9b), the capital gain enjoyed by homeowners ( r CG less the gross-of-maintenance rate of depreciation. ) is equal to the rate of house price inflation Consider now the annual index from the Standard model in (2.3) that ignores both log(τ) and maintenance. In the empirical work to follow, the index value from the first period in the sample horizon is normalized to 0. In addition, let the index value for the last period in the P sample horizon (T) be denoted γ T ; the superscript P is used to emphasize that this model is useful for estimating the appreciation on a portfolio of homes. Drawing on the period-t index family detached, house age, and interactions between house age and some of the other structural attributes. For maintenance 4 to 6 years prior to sale, this includes rooms, bedrooms, single family detached, condominium status, house age, and various interactions with house age. For maintenance more than 6 years prior to sale, this includes bedrooms, central city status, house age, and various interactions with house age. 14

17 value from the Standard model, the annual rate at which a portfolio of homes appreciates (r P ) is given by: 1 P T ( r e γ T = ) 1. (2.10) P An analogous calculation applied to the model in (2.8) which does take log(τ) and maintenance into account provides a measure of the quality adjusted rate at which house prices appreciate on an annual basis (r I ): r I = 1 I T T ( e γ ) 1. (2.11) In this expression, the I superscript is used to convey the idea that this estimate more fully controls for the quality of the housing stock, and in that sense, is closer to a true measure of quality adjusted house price inflation. Pt +τ i Differentiating ln( ) P t i in (2.8) with respect to τ, we obtain r D α =, (2.12) τ where the i subscript is suppressed for simplicity. This measures the annualized gross-ofmaintenance rate at which housing depreciates. The capital gain enjoyed by homeowners can then be obtained by substituting into (2.9b), while the annual contribution of maintenance to house price appreciation (r M ) can be estimated by substituting from ( ) into (2.9a) and rearranging, 15 r = r r + r. (2.13) M P I D 15 Alternatively, one can estimate the annual contribution of maintenance to house price appreciation by annualizing the linear combination of the included maintenance variables and their estimated coefficients. 15

18 Finally, the net-of-maintenance rate of depreciation is readily estimated by forming rnd = rd rm. Substituting for r M from (2.13), this simplifies to rnd = ri rp. (2.14) It is apparent that our estimate of r D is sensitive to τ and this carries over to r CG and r M. This is a necessary outcome of the need to allow for a non-linear rate of depreciation in (2.8). In the empirical work to follow, we evaluate the various parameters above at both the mean and the fiftieth percentile in the sample distribution for τ. In the case of r P, however, we draw upon estimates from the model in (2.3) which are not directly sensitive to τ. 16 Similarly, our estimate of r I, quality adjusted house price appreciation, is by definition unrelated to the aging of the housing stock. III. Data and Summary Measures The data used for the analysis is the America Housing Survey (AHS). The AHS includes roughly 55,000 owner-occupied and rental housing units and gathers data on the occupants of the home, the structure, the neighborhood, the price paid for the home and the costs associated with ownership. A key feature of the AHS is that it follows individual housing units on a biannual basis. Thus, by merging data over the sample years it is possible to observe the original purchase price, the owner s investment over time in the form of maintenance and improvements, and the price the owner obtains when the home is sold. In addition, we also observe an extensive set of 16 Implicitly, ordinary least squares estimates of (2.3) are sensitive, of course, to the underlying distribution of τ. 16

19 attributes of the home that are used as instruments for maintenance in the estimation to follow. Our sample uses nine biannual national surveys from 1985 through In preparing our sample, all dollar valued variables are first converted to year 2001 dollars. We then limit our analysis to owner-occupied units (approximately two-thirds of the total) that were sold at least twice during the period from 1983 through In measuring the sale dates, in most cases we use the move-in year reported by the new occupants. In cases where the move-in year was not reported the purchase year was used if that information was available. 18 Given the sale dates, a sale transaction was coded as having occurred only if each of the following conditions was met: (i) two adjacent surveys indicate that the occupants of a housing unit changed between surveys, and (ii) both the previous occupant and the current occupant are reported as owners. Our initial sample includes 9,392 repeat sales but this is reduced based on several additional filters. First, we restrict the sample to single-family homes, reducing the number of observations to 7,456. This is done primarily to increase the accuracy of the reported maintenance expenditures. We also omit observations for which either the purchase price or the sale price were top-coded. This further reduced the sample to 7,068. Also dropped are observations for which the annual nominal rate of house price appreciation in absolute value was greater than 50 percent between sale dates. This lowers the sample to 6,935. Finally, we drop any observations for which the home was essentially gutted and rebuilt as measured by an unusually high level of maintenance. Specifically, we omit observations for which the average annual level of maintenance expenditures exceeds twenty-five percent of the original purchase 17 Each survey gathers information about the house from the preceding two years. For example, the 1985 survey collects data from 1983 through In each survey year, the AHS drops some housing units and replaces them with new units. Thus, homes enter and leave the sample at various times throughout the sample horizon. 18 If both move-in year and purchase year were unavailable then the observation was omitted. 17

20 price of the home. Additional detail on how the data and especially maintenance was coded is provided in Appendix A. Our final sample includes 6,841 repeat sales. Table 2 provides summary statistics for the final sample of repeat sales transactions. The average purchase price for homes in the sample was $127,588, while the average sale price was $130,498. The average capital gain across all homes was 2.6 percent. Time in the home averaged just shy of 6 years, while the median time between sale dates was 5 years. 19 On average, homeowners report annual maintenance expenditures equal to 1.38 percent of the purchase price, while the median is 0.64 percent. The average home contains roughly 6 rooms, 3 bedrooms, and 1.5 bathrooms. In addition, only 12 percent of homes in the sample are located in the central cities, 96 percent are single family detached (with the remainder single family attached), just 4 percent belong to condominium associations, and the average house age at the time of purchase is 22.2 years. AHS and Freddie Mac Indexes As a check on our AHS data and related estimates it is useful to compare the Freddie Mac repeat sales index available over the web to comparable estimates obtained using the AHS data. The Freddie Mac index is widely cited, is used in a variety of applications, and therefore, serves as a good reference point for our discussion. 20 Figure 2 compares the Freddie Mac repeat sales model to the Standard model (equation 2.3) estimated with the AHS data. In both cases, estimates are provided for the period from 1983 to 2001 with the 1983 value normalized to 19 Previous estimates of time in the home place the average and median durations at 7 and 10 years, respectively (see Rosenthal [17]. The holding periods are somewhat shorter in the AHS sample because the duration of stay among owners who remain in their home beyond 2001 is truncated. 20 For example, Davis and Heathcote [8] use the Freddie index to measure house price changes in the U.S. as part of their analysis of land and housing markets. 18

21 It is important to note that the AHS index largely tracks the Freddie Mac series as is evident from the figure: both indexes show a sharp increase from 1985 to 1989, a period of slower growth from 1989 to 1996, and a period of very rapid appreciation from 1996 to Also apparent, the Freddie index rises at a somewhat faster rate than the AHS index. In real terms, the Freddie index increases 33 percent over the 18 year horizon, equivalent to an annual rate of 1.5 percent. The AHS index, in comparison, rises 14 percent over the horizon, equivalent to an annual rate of 0.74 percent. That difference likely arises because of differences in the sample composition of the Freddie and AHS data. The AHS sample includes only market sales, while the Freddie sample is drawn from Freddie Mac and Fannie Mae loans, including both new mortgage originations and refinances. For the refinance sales, the appraised value at the time of refinancing is used as the second value in the repeat sales model. Moreover, by virtue of its design, the Freddie sample excludes homes purchased without a mortgage. 22 Evidence from the AHS, however, suggests that roughly one-third of homes are purchased without a mortgage (debartolome and Rosenthal [9]). Thus there is a potentially important difference in sample composition between the AHS and Freddie databases. Stephens, et al. [21] investigate the impact of including refinance loans in the Freddie Mac repeat sales data. They find that the presence of refinance loans generally elevates the rate of appreciation in the Freddie index, but the extent to which this occurs varies by region and time period. Refinancing, for example, was especially prevalent in 1991 and 1992, and again in The Freddie Mac index is published in nominal dollars. The real Freddie Mac index was calculated from the nominal index using the CPI-U deflator. 22 Although Freddie and Fannie purchase or insure approximately 55% of all conventional mortgages, that still leaves a large share of the market not accounted for. Moreover, because the GSEs purchase only conforming loans that tend to exclude very low- and very high-valued properties, the set of homes included in the Freddie Mac repeat sales database is not necessarily representative. 19

22 and 2001: both are periods when the deviation between our index and the Freddie index widens somewhat. To explore this issue further, we regressed the difference in the Freddie and AHS Standard Indexes on current and 5-year lagged conventional 30 year fixed mortgage interest rates. It is well documented that refinance activity increases sharply as current interest rates fall relative to past mortgage rates. If the presence of refinance loans in the Freddie data boosts the Freddie index, then higher current mortgage rates should reduce the difference between the Freddie and AHS series while higher lagged mortgage rates should have the opposite effect. Regression results are consistent with those patterns. 23 Moreover, the residual from the regression measures the difference between the Freddie and AHS indexes after controlling for interest rate movements. That residual, along with the unadjusted difference between the Freddie and AHS indexes, is plotted in Figure 3. Observe that after controlling for interest rate movements, the difference in the AHS index relative to the Freddie Mac index is considerably reduced. Moreover, the residual disparity crisscrosses the zero axis on several occasions. On balance, this suggests that after controlling for the select nature of the Freddie sample, especially as relates to the inclusion of refinance loans, the AHS index matches that of the Freddie series to a very close degree. 23 In running the regression, we normalized the interest rate variables by subtracting the 1983 values for current and lagged interest rates from their respective series, comparable to the normalization of both the Freddie and AHS series to 100 in We then ran the regression omitting the constant. The coefficient on current interest rates was with a t-ratio of The coefficient on 5-year lagged interest rates was 0.58 with a t-ratio of The adjusted R-squared was We also ran the regression including a constant and a 1-year lag of the dependent variable. Both had small and insignificant coefficients and their inclusion had little effect on the interest rate coefficients. We also tried adding a linear time trend, and this too was insignificant and had little effect on the interest rate measures. 20

23 IV. Results: The Standard and Inflation Repeat Sales Models This section compares estimates from the Standard and Inflation repeat sales models based on the AHS data. We begin with Table 3. In that table, notice that the first three columns present the coefficients, t-ratios, and index values corresponding to the Standard Model as specified in equation (2.3). That index is plotted in Figure 2 as noted above and is normalized to 100 in The remaining columns in Table 3 present three-stage least squares (3SLS) estimates of the Inflation Model as outlined in equation (2.8). 24 Recall that (2.8) includes controls for log(τ), along with the three maintenance variables, M 1to3, M 4to6, and M 7to18. This model is reported in the last three columns of the table and is labeled Model C. Two restricted versions of the Inflation Model are also reported: Model A, which omits M 4to6 and M 7to18, and Model B, which omits M 7to18. These models are included to address a colinearity problem that is discussed below. In all three of the models, the included maintenance variables are treated as endogenous. Comparing results across Models A, B, and C reveals several patterns that guide our analysis. Notice first that the point estimates for the individual covariates are very similar for the three versions of the Inflation model. For example, the coefficients on log(τ) in Models A, B, and C are , , and , respectively; the coefficients on M 1to3 in Models A, B, and C are 2.59, 2.62, and 2.93, respectively, and the coefficients on the year-2001 index values are 170.7, 160.9, and 154.7, respectively. The signs on these estimates also accord with priors. Consistent with evidence from the hedonic literature, a negative coefficient on log(τ) indicates that housing tends to deteriorate with age, and this reduces appreciation on the home. 24 The 3SLS models were iterated to convergence and yield results similar to 2SLS. Results from the first stage maintenance regressions are presented in Appendix B, Table B-2. 21

24 Analogously, a positive coefficient on M 1to3indicates that maintenance adds value to the home, increasing appreciation. Moreover, in Models B and C, notice that the coefficient on M 1to3 is considerably larger than the coefficients on prior maintenance. In Model C, for example, the coefficients on M 4to6 and M 7to18 are 0.84 and 0.12 compared to 2.93 for M 1to3. This is consistent with the idea that older maintenance has depreciated further and also helps to explain why omitting older maintenance has little effect on the remaining parameters in the model. Observe next that for the Standard model and also for Models A and B, the t-ratios are quite large except for the annual indexes that have values close to zero. This pattern indicates that these specifications have ample power to identify the included parameters. 25 In contrast, the t-ratios in Model C are quite small for nearly all of the coefficients. Together, these patterns and those above are consistent with a colinearity problem: colinearity does not bias the model estimates and hence, the coefficients are similar across the models but it does inflate the standard errors causing the t-ratios to fall in Model C. 26 The colinearity problem appears to arise because of a high degree of correlation between maintenance conducted well before the sale date and the elapsed time between purchase and sale dates. In particular, the correlation between log(τ) and each of the three maintenance variables is 0.2 percent for M 1to3, 29.6 percent for M 4to6, and 38.4 percent for M 7to Our solution to this problem is to drop M 7to18 from the model. Although in principle, this could bias the remaining 25 For example, the t-ratios on the year-2001 index for the Standard Model, Model A, and Model B, are 5.56, 7.37 and 7.37, respectively. 26 Ordinary least squares (OLS) estimates of Models A, B, and C display a similar set of patterns: similar coefficients across models but low t-ratios in Model C. Estimates from those models are reported in Appendix B. 27 The high level of correlation associated with older maintenance occurs, in part, because maintenance prior to purchase is assumed to be capitalized into the purchase price and is set to zero. 22

25 coefficients, in practice, the similarity between coefficients in Models B and C suggest that such bias is relatively small. Accordingly, in the discussion below, we adopt Model B as the preferred specification. Figure 4 now plots the annual indexes associated with the Standard model, an OLS version of Model B, and the 3SLS estimate of Model B. It is immediately apparent that controlling for maintenance and depreciation increases the rate at which the Inflation index rises relative to the Standard model index. Moreover, given the model specifications outlined in expressions (2.3) and (2.8), the pattern in Figure 4 implies that maintenance only partially offsets depreciation. As a result, homes tend to deteriorate with age consistent with filtering models of the housing market and, failing to control for this fact, the standard repeat sales specification understates the quality adjusted rate of house price inflation. Also apparent in Figure 4, the 3SLS Inflation index rises at a faster rate than the OLS Inflation index (for Model B). To decipher this pattern, note that for OLS estimates of Model B, the coefficients on log(τ), M 1to3, and M 4to6 are , 0.95, and 0.99, respectively (see Appendix B, Table B-2). Comparing these values to the Model B estimates in Table 3, it is clear that the OLS estimates on log(τ) and M 1to3 are biased down absolute value, while the coefficient on M 4to6 is similar to the 3SLS estimate. For our sample, therefore, it appears that the tendency of OLS to understate the role of depreciation more than offsets the impact of understating the role of maintenance; on balance, this causes the OLS price index to rise more slowly than in the 2SLS model. 23

26 Inflation, Depreciation, and the Gains from Homeownership Substituting estimates from Table 3 into equations (2.9a) to (2.14), we can now compute several parameters of interest as outlined earlier. These include, on an annualized basis, the real rate at which a portfolio of homes appreciates, the quality adjusted rate of house price inflation, the net- and gross-of-maintenance rates at which housing capital depreciates, and the real capital gain or loss. These values are displayed in Table 4. The annual rate of appreciation for a portfolio of homes (r P ) based on the Standard Model index is 0.71 percent. Taking maintenance and depreciation into account, the quality adjusted rate of house price appreciation (r I ) is 2.68 percent per year. Thus, the standard repeat sales model understates the real annual rate of house price appreciation by approximately 2 percent per year. For applications in which knowledge of the quality adjusted rate of house price appreciation is needed, it is important to take this discrepancy into account when interpreting results from widely reported standard repeat sales specifications such as the Freddie Mac index discussed earlier. The difference between r I and r P appears to arise largely because the Standard Model does not control for age related depreciation. In particular, our model estimate of the gross-ofmaintenance rate of depreciation is 2.49 percent when evaluated using the mean value for τ (5.99 years) and 2.93 percent when evaluated using the median (5 years). It should be emphasized that despite a vast literature on house price hedonics, estimates of the gross of maintenance rate at which housing depreciates are scarce, presumably because most datasets do not provide information sufficient to control for both aging of the housing stock and maintenance. Nevertheless, the rate at which housing depreciates is an important driver of investment in the 24

27 economy and feeds directly into estimates of the stock and flow of capital federal data agencies such as the Bureau of Economic Analysis (BEA) and the Bureau of Labor Analysis (BLS). 28 Of course, age related depreciation can be offset by homeowner maintenance, and this is partly the case. We estimate that maintenance contributes 0.55 percent to house price appreciation using the mean value of τ and 0.99 percent using the median. This implies an annual net-of-maintenance rate of depreciation (r ND ) equal to 1.94 percent per year (where r ND is equal to r D - r M and is independent of τ). Finally, our model also allows us to estimate the net capital gains enjoyed by homeowners after accounting for maintenance expenditures. That capital gain is 0.19 percent year when evaluated at the mean of τ and percent when evaluated at the median: after controlling for maintenance, the typical homeowner does not appear to earn a capital gain. It is safe to say that this assessment is less upbeat than many advocates of homeownership might hope for. V. Conclusions This paper addresses two closely related issues. First, we estimate the rate at which housing capital depreciates gross of maintenance. Although that rate is fundamental to investment in the economy, we are not aware of other micro-data based estimates in the literature, in part because prior studies fail to control for both maintenance and age-related 28 It is interesting to consider our estimate of the rate at which housing capital depreciates in the context of Federal tax guidelines imposed with the Tax Reform Act of 1986 (TRA86). Specifically, TRA86 stipulates that owners of residential income properties (rental housing) can deduct depreciation from their ordinary pretax income. For these purposes, TRA86 set the depreciable life of residential property at 27.5 years and specified a straight-line depreciation method (Gitman and Joehnk [10], pp 661). At an annual real rate of 2.5 percent, half of the original housing capital would have depreciated after 27.5 years. 25