Age, time, vintage, and price indexes: measuring the depreciation pattern of houses.

Syed, Iqbal A. ; De Haan, Jan

I. INTRODUCTION

It is well known that time of sale, product age, and cohort--three temporal variables--cannot all be included linearly or as dichotomous variables in hedonic regressions because of the identity: Age + Cohort = Sale time. This is problematic in the housing context because these variables are regarded as important determinants of house prices, and the exclusion of one of the variables potentially biases the estimates of the other variables (Bailey, Muth, and Nourse 1963; Chau. Wong, and Yiu 2005; Englund. Quigley, and Redfearn 1998; Goodman and Thibodeau 1995; Harding, Rosenthal, and Sirmans 2007; Hill, Knight, and Sirmans 1997; Knight and Sirmans 1996).

Perhaps two of the most important measures in the housing context are housing inflation and ageprice profile of houses. In hedonic regressions, it is customary to include the time of sale as dummy variables to capture the movement of house prices. The age variable, on the other hand, is included in a nonlinear form, such as log(age), squared-age. or squared- and cubic-age, in order to obtain estimates of depreciation rates. This approach averts the perfect collinearity problem and provides an algebraic solution to the ordinary least squares problem. However, forcing the measure of the age effect to follow a predefined functional form may produce biased estimates of the depreciation pattern of houses (Coulson and McMillen 2008; Francke and van de Minne 2016; Malpezzi, Ozanne, and Thibodeau 1987). Malpezzi, Ozanne, and Thibodeau (1987), surveying the empirical literature of housing depreciation, found a large variability in the estimates of depreciation rates, ranging from 0.5% to 2.5% per year. They make the following observation: One shortcoming of the first two methods [the observed age and perpetual inventory methods] and of most hedonic studies, which may be a source of variability of results, is that they restrict functional form in a manner which arbitrarily imposes a particular depreciation pattern, (p. 373)

In this article, we introduce a method that provides estimates of age, cohort, and time effects emerging directly from the variation in the data without requiring us to specify a predefined functional form for any of these variables. The method follows the logic of the hedonic imputation approach and uses state-of-the-art index number formulae in the housing context. The advantages of the hedonic imputation method in the context of measuring inflation have been discussed in the literature (e.g., de Haan 2010; Diewert, Heravi, and Silver 2009; Hill 2013; Hill and Melser 2008; Pakes 2003; Rambaldi and Fletcher 2014; Silver and Heravi 2007; Syed 2010; Triplett 1996); these include the method's flexibility in terms of its treatment toward the regression parameters of the models and, when the double imputation is applied, its ability to reduce the omitted variable bias in estimated price indexes. However, the use of the hedonic imputation method in disentangling the age, cohort, and time effects, and measuring the depreciation pattern of houses remains unexplored. We show how the method can address one of the challenging econometric tasks of separating the effects of these three highly correlated temporal variables in product prices. The method, in contrast to conventional practice, enables us to obtain a measure of each of the temporal effects while holding the other temporal variables constant (e.g., a separate depreciation profile of houses for each cohort and time period).

Applying our method to data for a city in the Netherlands, we find that the functional forms typically used in the literature to represent the depreciation pattern do not provide reasonable approximations of the actual pattern of depreciation over the life of a house. We also find that omitting the cohort effect in hedonic regressions significantly overestimates the aging effect on house prices. Although we apply our model to the housing market, its application extends to many other durable products, such as furniture, used cars, electronic items, machineries, and nonresidential structures.

In the next section, we briefly discuss the hedonic method typically used in the literature for measuring depreciation rates, including the approaches pursued so far to disentangle the age, time and vintage effects in house prices. Section III provides a detailed description of our method. In Section IV, we apply our method to Dutch data and compare our measures with those obtained from standard approaches. Section V concludes the paper.

II. HEDONIC APPROACHES TO MEASURING THE DEPRECIATION OF HOUSES

The economic depreciation of assets is typically defined as the decline in asset prices due to the aging of assets (Fieldstein and Rothschild 1974; Hall 1968; Hotelling 1925; Hulten and Wykoff 1981b), and the measurement of economic depreciation centers around establishing an empirical relationship between price and age of assets (Clapp and Giaccotto 1998; Jorgenson 1996). (1) The most widely used method to obtain a measure of economic depreciation is the hedonic method, where data are pooled across time, and the natural log of prices for houses is hypothesized as a function of time dummies; characteristics of houses, including their age; and a random error term (e,), as follows: (2)

(1) ln[p.sub.i] = [T.summation over (t=1)] [[delta].sub.t][d.sub.t,i] + [C.summation over (c=1)] [[beta].sub.c][z.sub.c,i] + [gamma]f([a.sub.i]) + [[epsilon].sub.i], i = 1, ..., I;

where In [p.sub.i] refers to the natural log of prices of house i, for i = 1, ..., I. [d.sub.t,i] refers to time dummies taking the value of 1 if the period of sale of house i is t, and 0 otherwise, for t = 1, ..., T, with the exponentials of the coefficients, [[delta].sub.t], providing a quality-adjusted price index. The contribution of each characteristic of a house to its ln(price) is given by the coefficient [[beta].sub.c], where [z.sub.c,i] is a measure of characteristic c for house i, c = 1, ..., C. A nonlinear function of age, f([a.sub.i]), is included as one of the characteristics in the hedonic regression, and [gamma] is the coefficient corresponding to the age function. The depreciation rate of house i at age a, measured as the percentage change in the price of the house due to one unit change in its age, is obtained from 100 x [??]f'(a), where [??] is the estimated coefficient of y and f'(a) is the first derivative of f(a) with respect to a evaluated at age a. Equation (1) shows that the conventional method provides us with only one measure of depreciation profile that applies to all cohorts of houses and all time periods in the sample. (3)

Different nonlinear specifications of f([a.sub.i]) have been used in the literature, which might have influenced the estimates of depreciation rates obtained in different studies. For example, Malpezzi, Ozanne, and Thibodeau (1987) specify a cubic age function and a dummy variable if the dwelling belongs to the oldest cohort, while Lee, Chung, and Kim (2005) specify cubic and log age functions. Smith (2004) specifies a squared age function and interaction terms of age with location and period of sale, and Fletcher, Gallimore, and Mangan (2000) and Fisher et al. (2005) use a quadratic age function. Knight and Sirmans (1996) and Wilhelmsson (2008) specify a quadratic age function and interaction variables of age with maintenance, and Clapp and Giaccotto (1998) apply a nonlinear weight to the age variable and allow the age coefficient to change with shifts in supply and demand over time. Ong, Ho, and Lim (2003) and Harding et al. (2007) specify a log age function, the latter within a repeat-sales regression framework. If the age of houses is included in the hedonic regression of house prices, typically the cohort of houses is excluded, as revealed from a survey of 78 hedonic studies conducted by Sirmans et al. (2006). (4)

Sirmans et al. (2006) conducting a survey of 78 hedonic studies finds that age and cohort are not typically included simultaneously in the hedonic regressions of house prices. However, the problem that this may cause has been noted at least since Bailey et al. (1963). Case and Quigley (1991), Hill et al. (1997), and Englund et al. (1998) approached the perfect collinearity issue by jointly estimating hedonic and repeat-sales models. A common feature of these models, the hybrid models, is that the single and repeat prices are combined in a single regression through some explicit assumptions about the error structures of the single and repeat prices, where the details of the error structures vary across the models. Intuitively, these models exploit the cross-sectional variation of price levels to obtain estimates of the aging effect which are then augmented in the repeat-sales prices to obtain estimates of the inflationary effects (see also Yiu 2009). The findings of these papers indicate that both age and time effects should be accounted for in the regression of housing values.

In a more recent work, and following a different approach, Coulson and McMillen (2008) find that the age and cohort effects on the house prices in Chicago are different and argue for "treating cohort and age effects separately and more flexibly than is possible in a standard hedonic [model]" (p. 148). Following McKenzie (2006), they assume that each price consists of additive components of age, time, and cohort effects. They take the second-difference of house prices in a particular order to nullify two temporal effects and, with some normalization assumptions, identify the third temporal effect on the prices. Recently, Karato, Movshuk, and Shimizu (2015) have constructed bivariate splines of age and cohort effects on house prices in the framework of the generalized additive model and find that the depreciation patterns vary across different cohorts of houses in the city of Tokyo.

III. DISENTANGLING THE AGE, COHORT, AND TIME EFFECTS IN HOUSE PRICES

Our objective is to obtain separate measures of age, cohort, and time effects, which are not convoluted by other effects including the quality change effect on houses prices. The time effect measures the change in the general market conditions affecting the value of the land and structure of houses (Diewert, de Haan, and Hendriks 2015; Francke and van de Minne 2016). The age effect, applied on the total value of houses, is typically interpreted to measure the economic depreciation of houses caused by wear and tear of the structure, and functional and external obsolescence (Knight and Sirmans 1996; Smith 2004; Wilhelmsson 2008). The cohort or vintage effect pertaining to certain construction period measures the preference over particular housing characteristics, such as distinct architecture (Coulson and McMillen 2008; Englund et al. 1998). Since each of these effects originates from separate sources, omitting one of the variables would bias the estimates of the other effects. For example, Francke and van de Minne (2016) argue that since older homes are appreciated more for their distinctive architecture, ignoring the vintage effect may lead to the estimation of a positive age effect on house prices.

In order to achieve our objective of obtaining separate temporal measures, we run hedonic regressions separately for each age-cohort pair (or combination) of houses. A cohort is typically represented by a decade or longer period of time during which the house was constructed. The length of the period corresponding to a cohort depends on the purpose of the study and the availability of the data set (e.g., Coulson and McMillen assigned a cohort for each decade). The age of a house, on the other hand, is typically documented in number of years since the house was constructed. Hence, depending on the time of sale, the houses belonging to a particular cohort would be sold at their different ages. Suppose we consider all houses constructed during 1961-1970 that belong to a particular cohort. Among these houses, consider two houses, h1 and hi, which were constructed in 1961 and 1968, respectively. Suppose hi was sold in 2000 and h2 in 2009, implying that both the houses were sold at the age of 40. Hence, house h 1 and hi, sold in different periods, belong to the same cohort-age pair of houses. The houses belonging to each age-cohort pair would be of different quality as indicated by their locations and physical characteristics. Each of the age-cohort hedonic regressions includes time dummies to control for the difference in the time of sale, and the characteristics of houses to control for the difference in the quality of houses. There are no restrictions on how these variables and their interactions are included in the hedonic regressions, and whether the hedonic models of log(price) or price level are specified. (5)

Once the hedonic models have been estimated for each age-cohort pair of houses, the price of a house of a given cohort sold at one age in a particular period is imputed if the same house in the same period was sold at another age. Suppose a house was sold in 2010, 10 years after its construction. We would impute the price of this house sold in the same period but when it was 9 and 11 years old. The imputation process provides us with estimates of price ratios, referred to as price relatives in the index number literature, which compare the prices of houses sold at two different ages, holding the time of sale, cohort, and characteristics constant. These price relatives are aggregated using index number formulae in order to obtain price indexes measuring the price changes due to the aging of the houses. In a similar way, we can measure the cohort effect by estimating the price relatives, comparing the prices of houses belonging to different cohorts, holding the age, time, and characteristics constant. The inflationary effects, on the other hand, are obtained directly from the estimated time dummy coefficients in the regressions.

A. Hedonic Imputations

For the purpose of illustration, let us consider two ages, j and k, and two cohorts, l and m, of houses. This gives us houses belonging to four age-cohort pairs: (j, l), (k, l), (j, m), and (k, m). We estimate hedonic regressions of the following form for each age-cohort pair of houses:

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where In[p.sup.a,v.sub.i] denotes the price of house i belonging to cohort v and sold at age a, for i = 1, ..., [I.sup.a,v], where [I.sup.a,v] is the number of houses of age a and cohort v in the entire sample, with (j, k) G a and (Z, m) [member of] v. [d.sup.a,v.sub.t,i] takes the value of 1 if house i is sold in period t and 0 otherwise. [z.sup.a,v.sub.e,i] refers to the value of characteristic c = 1, ..., C of house i, and the [u.sup.a,v.sub.i] are i. i. d. error terms. The periods t = 1, ... ,T are those in which transactions took place. The exponential of 5"'v provides a measure of price change of the houses of cohort v sold at age a in period t holding other characteristics constant, and [[beta].sup.a,v.sub.c] refers to the implicit value of the characteristic c of the (a, v) pair houses.

Consider a house, h, which belongs to cohort l and was sold when it was at age j. Then the price of house h reaching age k (i.e., when house h becomes older) can be imputed from Equation (2) as follows, for [x.sup.j,l.sub.h] = ([d.sup.j,l.sub.l,h], ..., [d.sup.j,l.sub.T,h], [z.sup.j,l.sub.l,h], ..., [z.sup.j,l.sub.C,h]) and where a hat denotes an estimated parameter: (6)

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, in order to obtain the imputed price of house h of cohort I as it reaches age k (originally sold at age j): (1) we insert the characteristics of house h into the corresponding estimated implicit values obtained from the hedonic regression for the houses of the (k, l) pair and (2) we stick the period of the sale of house h with the estimated coefficient for the same period obtained from the hedonic regression for (k, l) houses. Therefore, [[??].sup.k,l.sub.h] ([x.sup.j,l.sub.h]) provides the imputed price of house h if this house was sold in period t, but instead of being sold at age j, the house was sold at age k.

In a similar way, using the estimated coefficients obtained from the (j.m) hedonic regression, we can impute the price of a house sold at age j, originally belonging to cohort l, had this house belonged to cohort in:

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can use the estimated hedonic regression of the (j, l) pair houses to impute a price for the same house sold at age j and cohort l:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, the imputation process provides us with three imputed prices for each house in the sample. This is shown in Figure 1. For example, if a house is in the (j, l) age-cohort pair, as in Figure 1A, then the three imputed prices are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; if a house belongs to the (k, l) age-cohort pair, as in Figure IB, then the three imputed prices are [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; and so on. (7)

A price relative measuring the change in the price of house h as it ages from j to k--holding the cohort, time of sale and the characteristics constant--is obtained from dividing the imputed price of the house at age k, shown in Equation (3), by its actual price:

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Equation (6), referred to as single imputation (SI) price relative, uses the imputed price only when the price is unobserved. An alternative is to use the double imputation (DI) method where we replace the observed price in Equation (6) with the imputed price obtained from Equation (5). This provides the following measure of price change between age j and k:

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can likewise obtain two price relatives measuring the change in price of house h if the cohort changes from l to m, while the age, time of sale, and other characteristics remain the same. The SI price relative divides the imputed price in Equation (4) by its actual price to obtain the following measure of cohort effect:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The corresponding DI price relative, which replaces the actual price with the imputed price shown in Equation (5), is the following:

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, we can obtain the aging and cohort effects for each house in the sample belonging to (j, l), (j, m), (k, l), and (k, m) pairs of houses. Table 1 shows the imputed price relatives measuring these effects for the houses belonging to our four age-cohort pairs. For example, suppose now that house h belongs to the (k, m) pair. The effect of the change in the cohort from m to l on the price of house h can be obtained from the price relative, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the imputed price of the house at its original cohort m and [TEXT NOT REPRODUCIBLE IN ASCII] is the imputed price of house h if the house had belonged to cohort l. (8)

[FIGURE 1 OMITTED]

B. Constructing Price Indexes

In order to obtain aggregated measures of the aging and cohort effects of houses, we use the resulting price relatives as inputs in index number formulas. We calculate the Fisher Ideal index because it falls in the class of "superlative" index numbers used for measuring price changes (Diewert 1976) and also satisfies the largest number of desirable axiomatic properties of index numbers (see Balk 1995). It is argued that, if data permit, superlative indexes should be used in order to measure price changes (e.g., Triplett 1996; Hill 2006). (9)

The Fisher index can be obtained by taking the geometric mean of the Laspeyres and Paasche indexes. The latter two indexes are probably the two best-known price index formulas, whose inception dates back to the nineteenth century. A Laspeyres index based on the houses in the (j, v) pair with (l, m) [member of] v and measuring the price changes of houses as they age from j to k is as follows:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where [w.sup.j,v.sub.i] = [p.sup.j,v.sub.i][q.sup.j,v.sub.i]/[[I.sup.j,v].summation over (i=1)] [p.sup.j,v.sub.i][q.sup.j,v.sub.i], (l, m) [member of] v.

Here the price relative is obtained from Equation (7); [q.sup.j,v.sub.i] refers to the quantity of house i belonging to cohort v sold at age j, and [w.sup.j,v.sub.i] reflects the corresponding expenditure share. In constructing consumer price indexes, the weighting of items according to their expenditure shares in the basket of goods and services consumed provides the best representation of the average price movements faced by households. However, the case of housing is different to the regular type of goods and services consumed by households. This is because each house is different and, therefore, irrespective of the price, only one house of a particular type is bought. This means that, in Equation (10), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This implies that [w.sup.j,v.sub.i] gives more weight to more expensive houses in the construction of price indexes, which is not the same as giving more weight to items which account for larger expenditure shares in household consumption. Since each house is somewhat different from other houses, it is more reasonable to give equal weight to each house in the sample, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Hill 2013). Hence, our Laspeyres-type index that measures the price change due to houses aging from j to k, with holding cohort, time of sale, and other characteristics constant, is as follows:

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which is the arithmetic mean of the imputed price relatives corresponding to the houses sold in the (j, v) pair. A Paasche index based on the houses in the (k, v) pair measuring the price changes due to houses aging from j to k is as follows:

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where [w.sup.k,v.sub.i] = ([p.sup.k,v.sub.i][q.sup.k,v.sub.i]/[[I.sub.k,v].summation over (i=1)][p.sup.k,v.sub.i][q.sup.k,v.sub.i]), (l, m) [member of] v.

Here [q.sup.k,v.sub.i] refers to the quantity of house i of cohort v sold at age k. Following the argument as in the Laspeyres case above, we set [w.sup.k,v.sub.i] = 1/[I.sup.k,v]. Hence, our Paasche-type index that measures the price change due to houses aging from j to k is as follows:

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which is the harmonic mean of the imputed price relatives corresponding to the houses in the (k, v). We take the geometric mean of the Laspeyres- and Paasche-type indexes to obtain the Fisher-type index as follows:

(14) [P.sup.(j,k),v.sub.F] = [square root of ([P.sup.(j,k),v.sub.Las] x [P.sup.(j,k),v.sub.Pas])], (l, m) [member of] v.

In a similar way, we construct the price indexes measuring the cohort effect of the houses in (a, 0 and (a, m) pairs for a [member of] (j, k). A Laspeyres index based on the houses in the (a, l) pair and measuring the price change if the cohort had changed from l to m is the following:

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The corresponding Paasche index based on the houses in the (a, in) pair is the following:

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The corresponding Fisher index is the following:

(17) [P.sup.(l,m),v.sub.Fis] = [square root of ([P.sup.a,(l,m).sub.Las] x [P.sup.(l,m).sub.Pas])], (j, k) [member of] a.

An alternative to the Fisher index is the Tornqvist index which is also a widely used superlative index. The Tornqvist-type index for our purpose can be calculated by taking the geometric mean of the geometric analogues of the Laspeyres- and Paasche-type indexes. Diewert (1976) shows that superlative indexes approximate each other to the second order, and thus empirically it should not matter which one is used. In fact, conforming to Diewert's theory we find that our measures of Fisherand Tornqvist-type indexes are very similar (see Table 4). (10)

The above illustration is shown for two ages and two cohorts, whereas in most cases we would construct indexes covering many ages, a = 1, ..., A, and, at least, a number of cohorts, v = 1, ..., V. This extension can be attained by constructing direct and chained indexes based on the constructed superlative indexes. In constructing the direct index measuring the aging effect, one particular age is taken as the base age, and the prices corresponding to other ages are compared with the prices of the base age. Suppose the base age is I, then in Equations (11), (13), and (14), we set j = 1, and k = 2 for obtaining the price indexes between age 1 and 2, k = 3 for obtaining indexes between age 1 and 3, and so on. The direct index measuring the aging effect between age 1 and any arbitrary age a, and for (Z, m) [member of] v, is the following:

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, a direct index measuring the cohort effect between cohort 1 and any arbitrary cohort k, and for (j, k) [member of] a, can be obtained by setting l = 1 and m = k in Equations (15-17) as follows:

Although the chained indexes circumvent the comparability problem inherent in the direct indexes, and may reduce the Paasche-Laspeyres spread (see Hill 2006), the chained indexes have a shortcoming in that they are not transitive,

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

two adjacent cohorts. The chained index measuring the aging effect between age 1 and a for (l, m) [member of] v is as follows:

(20) [P.sup.(1,[alpha]),v.sub.FC] = [P.sup.(1,2),v.sub.F] x [P.sup.(2,3),v.sub.F] x ... x [P.sup.([alpha]-2,[alpha]-1),v.sub.F] x [P.sup.([alpha]-1,[alpha]),v.sub.F],

where the indexes on the right-hand side are the Fisher indexes shown in Equation (14). In a similar way, the chained index measuring the cohort effect between cohort 1 and k, using the Fisher index in Equation (17) and for (j, k) [member of] a, is the following:

(21) [P.sup.a,(l,k).sub.FC] = [P.sup.a,(1,2).sub.F] x [P.sup.a,(2,3).sub.F] x ... x [P.sup.a,(k-2,k-1).sub.F] x [P.sup.a,(k-1,k).sub.F].

A problem with the direct index is that it makes the price comparison dependent on the choice of base age or cohort. Furthermore, as the distance between the comparison ages and, similarly, the comparison cohorts get larger, the price comparisons may become less reliable. For example, houses constructed in two adjacent periods would probably entail more "like with like" comparisons than the houses constructed in longer periods apart. An alternative to constructing the direct indexes is to chain the bilateral indexes for two adjacent ages and, similarly, for that is, [P.sup.1,3.sub.F] [not equal to] [P.sup.1,2.sub.F] x [P.sup.2.3.sub.F], even when they are based on superlative index number formulae. The chaining may introduce a drift in the price comparison causing the chained index to deviate from the direct index counterpart (Fox and Syed 2016; Ivancic, Diewert, and Fox 2011). This would make the measurement of price changes dependent on the selection of base age or cohort. A solution to this problem is to apply the GEKS formula on the Fisher indexes in Equations (14) and (17). The GEKS index is the geometric mean of the ratios of the Fisher indexes between a number of entities, where each entity is taken as the base. Let [P.sup.(j,[alpha]),v.sub.F]. and [P.sup.(k,[alpha]),v.sub.F] be the indexes shown in Equation (14) measuring the aging effect between j and [alpha], and k and [alpha], respectively, where a = 1, ..., A. The GEKS index measuring the aging effect between j and k is the following:

(22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the second expression holds because the bilateral Fisher index satisfies the entity reversal property of indexes, so that [p.sup.(k,[alpha]),v.sub.F] = 1/[P.sup.([alpha],k),v.sub.F]. Unlike the Fisher chained indexes, it can be shown that the GEKS indexes are transitive, i.e., [P.sup.1.3.sub.geks] = [P.sup.1,2.sub.GEKS] x [P.sup.2,3.sub.GEKS], making the price comparison independent of the choice of the base.11 In a similar way, we can obtain the GEKS index measuring the cohort effect between l and m using the Fisher index specified in Equation (17) in the following way, where v = 1, ... ,V:

(23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

The price indexes [P.sup.(j,k),v.sub.Z] for Z [member of] (FD, FC, FGEKS) shown in Equations (18), (20), and (22), respectively, measure the aging effect of the houses belonging to a particular cohort, v, as these houses age from j to k. In order to obtain an overall measure of the aging effect for all houses in the sample, we aggregate [P.sup.(j,k),v.sub.Z] across all cohorts, v = 1, ..., V. as follows:

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [S.sup.j,v] and [S.sup.k,v] are the proportion of houses in the (j, v) and (k, v) pairs in the sample. The overall measure of the cohort effect is obtained by aggregating [P.sup.(l,m),a.sub.Z] indexes shown in Equations (19), (21), and (23) across all ages as follows:

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [S.sup.a,l] and [S.sup.a,m] are the share of houses in the (a, l) and (a, m) pairs in the sample. Similarly, we construct the Tornqvist-based GEKS index measuring the aging and cohort effects of houses. (12)

The time effects are obtained from the estimated coefficients of the time dummies in Equation (2). Let exp ([[??].sup.a,v.sub.t-1,t]) provide the estimate of the price change from period t - 1 to t for a particular age-cohort pair of houses. Aggregating these estimates across all age-cohort pairs of houses provides the measurement of housing inflation as follows:

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [S.sup.a,v.sub.t-1] and [S.sup.a,v.sub.t] are the shares of houses in the (a, v) pairs in period t - 1 and t, respectively. In the second stage of aggregation, [S.sup.a.sub.t-1] and [S.sup.a.sub.t-1] refer to the shares of houses sold at age a in period t - 1 and t, respectively.

It should be noted that in the above framework, the aging and cohort effects are estimated using the hedonic imputation method and the inflationary effects are estimated using the time dummy method. Both methods allow these effects to emerge directly from the variation in the data rather than being forced to take a pre-determined functional form. While de Haan (2010) derives the conditions required for the hedonic imputation and time dummy price indexes to be equivalent, the hedonic imputation method outperforms the time dummy method in a number of different aspects. Evidence shows that the implicit values of the characteristics vary across different parts of a data set belonging to a market (Hill 2013; Hill and Melser 2008; Hill et al. 1997; Pakes 2003). That is, in our context, the estimated hedonic coefficients of, for example, the physical attributes are expected to vary across different ages and cohorts of houses.

The hedonic imputation method would allow this variation to take place. (13)

Silver and Heravi (2007) show, through formal algebraic exposition, that two factors lead to the difference between the hedonic imputation and time dummy indexes, which are parameter instability and changes in the value of characteristics. Diewert, Heravi, and Silver (2009), Eurostat (2013), and Rambaldi and Fletcher (2014) argue that the parameter flexibility is a significant advantage of the hedonic imputation method and favor using hedonic imputation price indexes unless degrees of freedom are very limited. This implies that if the interest of a study is, for example, to measure the age-price profile of houses, one should set the framework so that the aging effect is one of the two temporal effects estimated through hedonic imputations while the control for the third temporal effect is attained through the dummy variables in the regressions. (14)

Another advantage of the hedonic imputation method lies in the flexibility in its application. This arises from the fact that the regression and compilation stages are separate in the hedonic imputation method. The method provides separate estimates of price relatives for each observation, which essentially adds new columns to the data. Once the price relatives are obtained, one is free to compile these using index number formulae in order to obtain different aggregated measures of choice, such as for different sections of the market and periods in the sample. The double imputation method has an added advantage because it has the potential for correcting for omitted variable bias incurred in the estimated hedonic regressions (Hill 2013; Hill and Melser 2008; Syed 2010). Hill argues that since houses are so heterogeneous, there is likely to be serious omitted variable bias problems in hedonic regressions of house prices. It can be shown that under certain plausible assumptions the omitted variable bias in the numerator and denominator of the double imputation price relative tend to cancel each other out. (15)

In addition, our method provides an in-built mechanism to correct for sample selection bias occurring when depreciation is estimated only on the surviving houses. The issue of potential sample selection bias has been raised, among others, by Hulten and Wykoff (1981a), Dixon, Crosby, and Law (1999), and Coulson and McMillen (2008). The problem is that the houses which have retired early might have depreciated faster than the average. Hence, if this attrition is not accounted for, it would lead to an underestimation of the measure of depreciation rates. In our method, the prices of houses are compared between two adjacent ages; between age 1 and 2, 2 and 3, and so on. Therefore, in each comparison. the houses would have the same or a similar survival rate. This implies that the sample selection bias incurred in each regression would tend to cancel each other out when constructing the indexes between two consecutive ages. We obtain estimates of the depreciation rates between ages of larger gaps by chaining the price indexes obtained for consecutive ages, implying that the depreciation pattern covering the life of houses would also tend to be free of sample selection bias. (16)

IV. EMPIRICAL RESULTS

We apply our framework to housing data consisting of 6,348 observations of the quarterly sales of detached houses for a city, "Assen," in the Netherlands, covering the period between 1998:1 and 2008:2. (17) Assen is a small city with a population of around 60,000. Each observation in the data contains information on the address, sale price, quarterly period of sale, lot size, floor space, number of rooms, number of toilets, construction period, house type, maintenance indicator, and whether the house has a garage, balcony, dormer, and roof terrace. The construction period of the houses are available in tens of years as follows: 1960-1970; 1971-1980; 1981-1990; 1991-2000; and 2001-2008. By taking the midpoint as the construction year and subtracting it from the year of sale of the house, we obtain the age at which the house was sold. We group the calculated age into five groups: age 1-10 years: Age1; age 11-20 years: Age2; age 21-30 years: Age3; age 31-40 years: Age4; and age 41-50 years: Age5. This grouping will enable us to show the results of all hedonic regressions (corresponding to Agel-Age5) and help in our illustration of the method. Figure 2 provides the location plots of the sample houses in the city and Table 2 provides some summary information about the data.

Note that the coverage of the data is 1998-2008 and the last identified construction period is 2001-2008. The fact that these two periods nearly overlap makes the age and cohort of houses indistinguishable from each other. This would not, however, be the case in data sets where the actual construction time is known (e.g., in years) and sample covers a longer period of time (e.g., for 20 years). (18) However, the data set corresponds to a small and homogenous town which works as a natural control to location-specific heterogeneity of prices. This lets us use the locational information to construct a cohort-specific variable, which somewhat counteracts the above shortcoming. The cohort or construction vintage effect measures the separate impact attributed to the period of construction of a house. When a new area is opened up for residential development, the houses in the area are built around the same period. Therefore, these houses tend to share similar features in terms of style, and use of technology and materials which are specific to the construction period, making region and cohort effects correlated with each other.

We divide the city into three regions, shown in Figure 2B, based on whether the postcodes are old or new as indicated by the construction period of the sample houses. We do the division by ordering the postcodes in ascending order in terms of the average construction period and then combining the adjacent postcodes in the order to create old, mid-age and new regions. These three regions comprise of 1,619 (three postcodes), 2,000 (three postcodes), and 2,729 observations (two postcodes), respectively, and their average construction periods are 1972, 1983, and 1994, respectively (the average construction period for all houses is 1985). The dummy variables corresponding to these construction period-specific regions are used as proxies to control for the cohort effect in our hedonic regressions. (19)

Following Equation (2), we estimate hedonic models of log prices separately for each age(a) of houses as follows:

(27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In Equation (27), [d.sup.a.sub.t,i] refers to a dummy variable taking the value of 1 if the period of sale of house i is f, and 0 otherwise, for all a = 1, ..., 5. r, running from 1 to 11. indexes the year of sale, 1998, ... , 2008. [[delta].sup.a.sub.t] is the time dummy coefficient corresponding to period t and age a, and exp ([[delta].sup.a.sub.t+1]) provides a measure of price change between period t + 1 and t. [d.sup.a.sub.r,i] refers to a dummy variable taking the value of 1 if house i is located in region r. and 0 otherwise, where r denotes old, mid-age and new regions in the city. The dummy variables--hereafter, called the cohort-region dummies--are constructed for the old and new regions, with the mid-age region being treated as the base. In(lotsize) and ln(flrspace) refer to the natural log of lot size and floor space measured in square meters, respectively. [d.sup.a.sub.c,i] refers to a dummy variable taking the value of 1 if the physical attribute c is present in house i of age a, and 0 otherwise, [[epsilon].sup.a.sub.i] is assumed to be i. i. d. error terms with a zero mean and constant variances.

[FIGURE 2 OMITTED]

Table 3 provides the results of the hedonic regressions specified in Equation (27). The adjusted-[R.sup.2] ranges from .90 to .94, with the average being .92. The estimated time dummy and, in most cases, the region dummy coefficients are found to be significant at the 5% level. The estimated coefficients corresponding to the physical attributes have the expected sign in most cases, particularly for the ones which are significant. While the estimates are quite stable across the regressions for different ages of the houses, their differences are significant at the 5% level in many cases. An interesting result corresponds to the estimated coefficients of ln(lotsize) and ln(flrspace). The average of the sum of these elasticity measures is around 0.5, indicating that a 1 percentage increase in the land and floor area leads to a 0.5 percentage increase in the value of the sample houses. (20)

Before we move to our measures of depreciation, we construct price indexes measuring inflation using the estimated time dummy coefficients provided in Table 3. Taking 2001 as the base year, we obtain a measure of price change between t and 2001 from exp ([[??].sup.a.sub.t] - [[??].sup.a.sub.2001]), for a = 1, ..., 5. Figure 3 shows the plots of these price indexes measuring the inflation of houses, which are aggregated using Equation (26) to obtain an overall measure of inflation of the sample houses.

Table 4 shows the median and double imputation GEKS indexes measuring the age-price profiles for our houses. The median index shows a much larger depreciation rate than the quality-adjusted indexes. This is because the old houses are in general of lower quality than the new houses and, therefore, the quality adjustments in the construction of the GEKS indexes reduce the aging effect on house prices. The Fisher-GEKS and Tornqvist-GEKS indexes are, as expected, very similar, prompting us to focus on the former in the subsequent discussion. From the resulting age-price profile, we find that the houses in our sample depreciate by 20.7% over 40 years of life. They depreciate by 16.1% in the first 20 years and by 4.6% in the next 20 years of life. The estimated depreciation pattern implies that the simple annual average depreciation rate is 0.52% per year, given that in the calculation we take the average age of Agel as 5 years and Age5 as 45 years, giving us a total of 40 years of life of the houses.

The finding that the depreciation rate is greater in the early years of life than in the later years is in concordance with the expectation of the general shape of the depreciation pattern of houses (Francke and van de Minne 2016; Hulten and Wykoff 1981a, 1996; Jorgenson 1996; Karato et al. 2015). For example, Hulten and Wykoff (1981a), using the Box-Cox power transformation, find that the value of used buildings approximately follows a geometric pattern of depreciation rather than a linear or one-horse shay pattern. However, the depreciation pattern in our data does not follow a geometric rate of decline over the life of the houses. If our finding of 20.7% depreciation over 40 years was incurred through an annual geometric rate, then it would imply a decline in value of the houses by 0.58% per year. This annual geometric rate would imply a depreciation of 11.0% in the first 20 years of life and 9.7% in the next 20 years of life. This is much lower than our measure of depreciation in the first 20 years (which is 16.1%) and higher than our measure of depreciation in the next 20 years of life of the houses (which is 4.6%). Hence, our measure of depreciation pattern--though indicating a decline in the rate of decay of the houses as they age--deviates for the depreciation pattern implied by the geometric rate of decline of houses.

[FIGURE 3 OMITTED]

Our measures of annual depreciation rate, 0.52%-0.58% applied on the total value of houses falls within the range reported by many authors. Malpezzi, Ozanne, and Thibodeau (1987) find that the annual depreciation rate in 59 metropolitan areas in the United States ranges from 0.43% to 0.93%. Cannaday and Sunderman (1986) estimate the depreciation rate of single-family homes in Champaign, Illinois, ranging from 0.38% to 0.75%. Wilhelmsson (2008) estimates a depreciation rate of 0.77% per year for well-maintained properties in a municipality of Stockholm, Sweden (see also Chinloy 1979; Fletcher et al. 2000; Smith 2004; and Chau et al. 2005).

We compare our measures of the depreciation pattern with the estimates obtained from Equation (1). In Equation (1), data are pooled across all ages of houses and a nonlinear function of age is included in the regression. The regression includes all variables, including the cohortspecific regional dummies, that were included in the regressions corresponding to Equation (27). We consider four different nonlinear specifications of the age function in Equation (1) by setting [gamma]f(a) equal to (1) [[gamma].sub.1]ln(a), (2) [[gamma].sub.1][a.sup.2], (3) [[gamma].sub.1][a.sup.2] + [[gamma].sub.2][a.sup.3], and (4) [[gamma].sub.1][e.sup.-a] ("a" denotes age). The regression results are provided in Table 5, and they show that different age functions do not impact the overall performance of the models; the adjusted-[R.sup.2]-s are around .92 for the four regressions. The estimated coefficients, including the age coefficients, have the expected signs and are significant at the 5% level (with the exception of one coefficient). The estimated coefficients of the physical attributes are around the same across regressions. We find that the price indexes constructed from the estimated time dummy coefficients from the regressions are virtually identical.

The only estimated coefficients that are showing some difference across the regressions correspond to the cohort-region dummies, though the difference is small. The positive estimated coefficients for the new region and negative estimated coefficients for the old region (with the mid-age region as the base) indicate that newer houses are in general perceived to be of higher quality. This may be because houses built later in the period embody technical progress which make them more valuable (Englund et al. 1998; Hulten and Wykoff 198 lb, 1996). (21) However, the regression results corresponding to different age specifications do not provide any indication that would let us select one from the four regressions. The question is: do they imply a similar depreciation pattern of houses?

The age-price profiles obtained from the estimates of the age functions are shown in Figure 4. (22) The solid blue line shows the FisherGEKS index obtained from our method. The ln(a) and [e.sup.-a] specifications show that the houses depreciate by 23.5% and 24.2% over 40 years, respectively, which exceed our estimates. The depreciation patterns of the ln(a) and [e.sup.-a] specifications deviate from our estimates from the very beginning of the life of the houses. The [a.sup.2] specification, on the other hand, underestimates the depreciation rate, and the estimated depreciation pattern exhibits an increase in the depreciation rate as the houses age, which is exactly the opposite of what our estimates exhibit. The [[gamma].sub.1][a.sup.2] + [[gamma].sub.2][a.sup.3] specification also provides a poor approximation to our measure of depreciation pattern of the houses, estimating a depreciation of 18.8% over 40 years, with only 10.9% in the first 20 years (lower than our estimate of 16.1%) and 7.9% in the next 20 years (higher than our estimate of 4.6%). These results provide strong evidence that (1) the choice of functional form for the age variable has a significant impact on the estimation of depreciation rates and patterns of houses and (2) the functional forms which are typically used do not provide a reasonable approximation of the actual pattern of depreciation over the life of houses. This finding is in concordance with Coulson and McMillen (2008) who report that the linear and quadratic functional forms do not accord well with their estimates of the depreciation profile of houses.

[FIGURE 4 OMITTED]

We also obtain depreciation patterns with two modifications to the specification of the regression models. First, we exclude the cohort-region dummies without any replacements. The idea is that since cohort and age variables are highly correlated, if the cohort effect is not controlled for in hedonic regressions, it would affect our measure of the aging effect of house prices. Second, we replace the cohort-region dummies with the regional dummies constructed from combining adjacent postcodes (postcode-region dummies) as shown in Figure 2C. This is probably typical for how regional heterogeneity is accounted for in house prices. However, as discussed earlier, since houses in one area tend to be built around the same period, the postcode-region dummies are expected to, at least, partially control for construction vintage effects. Hence, our base case and two additional scenarios differ in terms of the extent to which the cohort effect is controlled for in hedonic regressions of houses prices. Our interest is to see how this variation affects the measured depreciation pattern. (23)

The estimated depreciation patterns are shown in Figure 5. The omission of cohortregion dummies without any replacement swings the depreciation index downward in all our models and, hence, increases the aging effect on the price of houses (Figure 5A). This larger aging effect is expected because the estimated cohort-region dummy coefficients shown in Table 5 imply that older houses on average are valued less, ceteris paribus, in the market. However, the extent to which the changes take place differs across models. For example, the impact is the largest for the [e.sup.-a] specification, with the average decline of house prices increasing from 0.61% to 0.98% per year (by 0.37% points). In the log(age) specification, the annual average decline increases from 0.59% to 0.85% (by 0.26% points). The impact of this omission is the lowest in the Fisher-GEKS index where the average price fall increases from 0.52% to 0.57% (by 0.05% points). The lower impact in the Fisher-GEKS index might be because of the inherent correction to the omitted variable bias that takes place in the hedonic imputation methods. The depreciation indexes similarly exhibit a larger fall in prices when cohort-region dummies are replaced with postcode-region dummies in the models (Figure 5B). However, in line with our expectation, the extent to which the changes take place in each model is lower than the corresponding model when no region dummies are included. (24)

[FIGURE 5 OMITTED]

Figure 5 shows that despite the changes in the depreciation patterns obtained from different models due to differing levels at which the construction vintage is controlled for, none of the depreciation patterns obtained from the nonlinear age functions provide a reasonably good approximation to the depreciation pattern exhibited by the Fisher-GEKS index. Referring to Figures 4 and 5, the closest approximation is probably provided by the squared- and cubicage function ([[gamma].sub.1][a.sup.2] + [[gamma].sub.2][a.sup.3]), yet this approximation is inconsistent; while in Figure 4, [[gamma].sub.1][[alpha].sup.2] + [[gamma].sub.2][a.sup.3] underestimates the overall depreciation rates, in Figure 5 it overestimates the depreciation rates. All in all, our empirical results provide a strong support in favor of estimating the depreciation pattern of houses in a flexible manner, such as the way we have described in Section III.

V. CONCLUSION

Age, cohort, and time cannot be included in a flexible manner in hedonic regressions because of the identity, age + cohort = sale time. The typical procedure in hedonic regressions is to include time-dummies in order to measure inflation flexibly and a nonlinear age function in order to obtain a measure of depreciation rate, while not accounting for the potential cohort effect on the price of houses. In support of the concerns raised previously about this approach in the literature, we find that this procedure distorts the measurement of depreciation pattern of houses. We find robust evidence that the commonly used functional forms provide poor approximations of the actual depreciation patterns of houses.

We introduce a method where each of these temporal effects on house prices can be accounted for within a hedonic regression framework in a flexible manner. Our method uses the hedonic imputation approach and state-of-the-art index number formulae. We show how the method provides an estimate of the aging effect, controlled for time and cohort effects, in the form of a separate price relative for each dwelling in the sample, which is then combined using superlative-GEKS index number formulae in order to obtain an aggregated measure of the depreciation pattern of houses. Our method includes an in-built mechanism that reduces the omitted variable bias and corrects for the sample selection bias.

Applying our method to a housing data set for a town in the Netherlands, we find that the houses depreciate by around 20.7% over the life of 40 years, with 16.1% depreciation in the first 20 years and by 4.6% over the next 20 years (0.52% per year). If the cohort effect is not controlled for, the depreciation index swings downward exhibiting a larger annual depreciation rate for our houses. In either case, the estimated depreciation pattern does not follow the pattern implied by the commonly assumed geometric rate of decline in housing value. We also find that the ln(age) and exp(-age) specifications overestimate the depreciation rates, and the squared-age specification underestimates the depreciation rates of our houses. The results obtained from the cubic-age function are also found to be unreliable, providing an incorrect pattern of the age-price profile and sometimes overestimating and at other times underestimating depreciation rates.

While we apply our method to obtain the age-price profile of houses, it would be interesting to undertake a similar exercise for a broad category of asset classes (such as those listed in Hulten and Wykoff 1981b; Jorgenson 1996) and see the implications the results would have on national accounts and productivity measurements. ABBREVIATIONS DI: Double Imputation SI: Single Imputation

doi: 10.1111/ecin.12383

Online Early publication June 13,2016

REFERENCES

Bailey, M. J., R. F. Muth, and H. O. Nourse. "A Regression Model for Real Estate Price Index Construction." Journal of the American Statistical Association, 58(304), 1963, 933-42.

Balk, B. M. "Axiomatic Price Index Theory: A Survey." International Statistical Review, 63, 1995, 69-93.

Cannaday, R. E., and M. A. Sunderman. "Estimation of Depreciation for Single-Family Appraisals." Real Estate Economics, 14(2), 1986, 255-73.

Case, B., and J. M. Quigley. "The Dynamics of Real Estate Prices." The Review of Economics and Statistics, 73(1), 1991,50-58.

Chau, K. W., S. K. Wong, and C. Y. Yiu. "Adjusting for Non-Linear Age Effects in the Repeat Sales Index." Journal of Real Estate Finance and Economics, 31(2), 2005, 137-53.

Chinloy, P. T. "The Estimation of Net Depreciation Rates on Housing." Journal of Urban Economics, 6, 1979, 432-43.

Clapp. J. M., and C. Giaccotto. "Residential Hedonic Models: A Rational Expectations Approach to Age Effect." Journal of Urban Economics, 44, 1998, 415-37.

Coulson, N. E., and D. P. McMillen. "Estimating Time, Age and Vintage Effects in Housing Prices." Journal of Housing Economics, 17,2008, 138-51.

Diewert, W. E. "Exact and Superlative Index Numbers." Journal of Econometrics, A, 1976, 114-45.

Diewert, W. E., S. Heravi, and M. Silver. "Hedonic Imputation Versus Time Dummy Indexes," in Price Index Concepts and Measurements, Chapter 4, edited by W. E. Diewert, J. S. Greenlees, and C. R. Hulten. Chicago: LTniversity of Chicago Press, 2009. 161-96.

Diewert, W. E., J. de Haan, and R. Hendriks. "Hedonic Regressions and the Decomposition of a House Price Index into Land and Structure Components." Econometric Reviews, 34(1-2). 2015. 106-26.

Dixon. T. J., N. Crosby, and V. K. Law. "A Critical Review of Methodologies Measuring Rental Depreciation Applied to UK Commercial Real Estate." Journal of Property Research, 16(2), 1999, 153-80.

Englund, R, J. M. Quigley, and C. L. Redfearn. "Improved Price Indexes for Real Estate: Measuring the Course of Swedish Housing Prices." Journal of Urban Economics, 44. 1998, 171-196.

Eurostat. "Handbook on Residential Property Prices Indices (RPPIs)", Eurostat Methodologies and Working Papers, Cat. No. KS-RA-12-022-EN-N, Publications Office of the European Union, Luxembourg, 2013.

Ferreira, F. and J. Gyourko. "A New Look at the U.S. Foreclosure Crisis: Panel Data Evidence of Prime and Subprime Borrowers from 1997 to 2012." Paper presented at the University of New South Wales, Sydney, Australia, June 9, 2015.

Fieldstein, M. S., and M. Rothschild. "Towards an Economic Theory of Replacement Investment." Econometrica, 42(3), 1974, 393-423.

Fisher, J. D., B. C. Smith, J. J. Stern, and R. B. Webb. "Analysis of Economic Depreciation for Multi-Family Property." Journal of Real Estate Research, 27(4), 2005, 355-69.

Fletcher, M.. P. Gallimore. and J. Mangan. "The Modeling of Housing Submarkets." Journal of Property Investment and Finance, 18(4), 2000, 473-87.

Fox, K. J., and I. A. Syed. "Price Discounts and the Measurement of Inflation." Journal of Econometrics, 191(2). 2016. 398-406.

Francke, M. K., and A. M. van de Minne. "Land, Structure and Depreciation," Real Estate Economics 2016 Jan 21. doi: 10.1111/1540-6229.12146

Goodman, A. C., and T. G. Thibodeau. "Age-related Heteroskedasticity in Hedonic House Price Equations." Journal of Housing Research, 6(1). 1995, 25-42.

de Haan, J. "Hedonic Price Indexes: A Comparison of Imputation, Time Dummy and Other Approaches." Journal of Economics and Statistics, 230(6), 2010, 772-91.

de Haan, J., and F. Krsinich. "Scanner Data and the Treatment of Quality Change in Nonrevisable Price Indexes." Journal of Business and Economic Statistics, 32(3), 2014. 341-58.

Hall, R. E. "Technical Change and Capital from the Point of View of the Dual." Review of Economics and Statistics, 35(Jan), 1968, 35-46.

Harding, J. P, S. S. Rosenthal, and C. F. Sirmans. "Depreciation of Housing Capital, Maintenance, and House Price Inflation: Estimates from a Repeat Sales Model." Journal of Urban Economics, 61(2), 2007, 193-217.

Hill, R. J. "When Does Chaining Reduce the Paasche-Laspeyres Spread? An Application to Scanner Data." Review of Income and Wealth, 52(2), 2006, 309-25.

--. "Hedonic Price Indexes for Residential Housing: A Survey, Evaluation and Taxonomy." Journal of Economic Surveys, 27(5), 2013, 879-914.

Hill. R. J., and D. Melser. "Hedonic Imputation and the Price Index Problem: An Application to Housing." Economic Inquiry, 46(4), 2008, 593-609.

Hill, R. C., J. R. Knight, and C. F. Sirmans. "Estimating Capital Asset Price Indexes." Review of Economics and Statistics, 79(2), 1997, 226-33.

Hotelling, H. S. "A General Mathematical Theory of Depreciation." Journal of American Statistical Society, 20(Sep.), 1925, 340-53.

Hulten, C. R., and F. C. Wykoff. "The Estimation of Economic Depreciation Using Vintage Asset Prices: An Application of the Box-Cox Power Transformation." Journal of Econometrics, 15, 1981a, 367-96.

--. "The Measurement of Economic Depreciation," in Depreciation, Inflation, and the Taxation Income from Capital, edited by C. R. Hulten. Washington, DC: Urban Institute Press. 1981b.

--. "Issues in the Measurement of Economic Depreciation: Introductory Remarks." Economic Inquiry, 34(1), 1996, 10-23.

Ivancic, L., W. E. Diewert, and K. J. Fox. "Scanner Data, Time Aggregation and the Construction of Price Indexes." Journal of Econometrics, 161, 2011,24-35.

Jorgenson, D. W. "Empirical Studies of Depreciation." Economic Inquiry, 34(1), 1996, 24-42.

Karato, K.. O. Movshuk and C. Shimizu. "A Semiparametric Model of Hedonic Housing Prices in Japan," 1RES Working Paper Series IRES2015-005, National University of Singapore, Singapore, 2015.

Kennedy, P. E. "Estimation with Correctly Interpreted Dummy Variables in Semilogarithmic Equations." American Economic Review, 71(4), 1981,801.

Knight, J. R., and C. F. Sirmans. "Depreciation, Maintenance, and Housing Prices." Journal of Housing Economics, 5, 1996,369-89.

Lee, B. S., E. Chung, and Y. H. Kim. "Dwelling Age, Redevelopment, and Housing Prices: The Case of Apartment Complexes in Seoul." Journal of Reed Estate Finance and Economics, 30(1), 2005, 55-80.

Malpezzi, S., L. Ozanne, and T. G. Thibodeau. "Microeconomic Estimates of Housing Depreciation." Land Economics, 63(4), 1987, 372-85.

McKenzie, D. "Disentangling Age, Cohort and Time Effects in the Additive Model." Oxford Bulletin of Economics and Statistics, 68, 2006, 473-95.

Ong, S. E., K. H. D. Ho, and C. H. Lim. "A Constant-Quality Price Index for Resale Public Housing Flats in Singapore." Urban Studies, 40(13), 2003, 2705-2729.

Pakes, A. "A Reconsideration of Hedonic Price Indexes with an Application to PCs." American Economic Review, 93(5), 2003, 1578-1596.

Rambaldi, A. N., and C. S. Fletcher. "Hedonic Imputed Property Price Indexes: The Effects of Econometric Modeling Choices." Review of Income and Wealth, 60(S2), 2014, S423-S448.

Redfeam, C. L. "How Informative are Average Effects? Hedonic Regression and Amenity Capitalization in Complex Urban Housing Markets." Regional Science and Urban Economics, 39, 2009, 297-306.

Rubin, G. M. "Is Housing Age a Commodity? Hedonic Price Estimates of Unit Age." Journal of Housing Research, 4(1), 1993, 165-84.

Shilling, J. D., C. F. Sirmans, and J. Dombrow. "Measuring Depreciation in Single-Family Rental and Owner-Occupied Housing." Journal of Housing Economics, 1, 1991, 368-83.

Silver, M. and S. Heravi. "The Difference between Hedonic Imputation Indexes and Time Dummy Hedonic Indexes." Journal of Business and Economic Statistics, 25(2), 2007, 239-46.

Sirmans, G. S., L. MacDonald, D. A. Macpherson, and E. N. Zietz. "The Value of Housing Characteristics: A Meta Analysis." Journal of Real Estate Finance and Economics, 33(3), 2006, 215-40.

Smith, B. C. "Economic Depreciation of Residential Real Estate: Microlevel Space and Time Analysis." Real Estate Economics, 32(1), 2004, 161-80.

Syed, I. A. "Consistency of Hedonic Price Indexes with Unobserved Characteristics," Australian School of Business Research Paper No. 2010 ECON 03, University of New South Wales, Sydney, 2010.

Triplett, J. E. "The Importance of Using Superlative Index Numbers," Paper prepared for CSO Meeting on Chain Indexes for GDP, London, UK, April 26, 1996.

Wilhelmsson, M. "House Price Depreciation Rates and Level of Maintenance." Journal of Housing Economics, 17. 2008, 88-101.

Yiu. C. Y. "Disentanglement of Age, Time, and Vintage Effects on Housing Price by Forward Contracts." Journal of Real Estate Literature, 17(2), 2009, 273-91.

SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article:

Figure S1. Depreciation Patterns, with and without Controlling for Cohort Effect. (A) Ln(age) Specification, (B) Squared-age Specification, (C) Squared-plus Cubic-age Specification, (D) Inverse of Exponential-age Specification, and (E) Hedonic Imputation Method

Table S1. The Impact of Omitting Temporal and Physical Attributes on the Estimated Age Coefficient of Hedonic Regressions of House Prices

Table S2. Hedonic Regression Results for Each Age of Dwellings with Postcode Specific Regional Dummies included in the Regressions

Table S3. Hedonic Regression Results for Each Age of Dwellings without any Regional Dummy Variables

Table S4. Impact of Different Regional Divisions on the Time-dummy Hedonic Regression with Nonlinear Specification of Age

* We would like to thank seminar participants at the Fall 2012 Midwest Macroeconomics Meetings (UC Boulder), the 2013 Society for Nonlinear Dynamics and Econometrics (Milan), the 2013 Society for Computational Economics (Vancouver), and the University of Miami. We would also like to thank Bill Dupor, Jesus Fernandez-Villaverde, Dirk Krueger, Frank Schorfeide, and Felipe Saffie for insightful comments and discussions.

Airaudo: Associate Professor, Department of Economics, LeBow College of Business. Drexel University, Philadelphia, PA 19146. Phone 215-895-6982, Fax 215-5714670, E-mail ma639@drexel.edu

Bossi: Senior Lecturer, Department of Economics, University of Pennsylvania, Philadelphia, PA 19104-6297. Phone 215-898-7779, Fax 215-573-2057, E-mail boluca@upenn.edu TABLE 1 Price Relatives Measuring the Aging and Cohort Effect of Houses (a) Aging Effect of a House due to Change in Age from j to k Age-Cohort Imputation Price Relative for a (l, m) Method Single House ([dagger]) [member of] v (j, v) Single [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (j, v) Double [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (k, v) Single [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (k, v) Double [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (b) Cohort Effect of a house due to change in cohort from l to m Age-Cohort Imputation Price Relative for (j, k) Method a Single House [member of] a (a, l) Single [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (a, l) Double [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (a, m) Single [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (a, m) Double [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([dagger]) [x.sup.a,v.sub.h] = ([d.sup.a,v.sub.1,h], ..., [d.sup.a,v.sub.T,h], [z.sup.a,v.sub.1,h], ..., [z.sup.a,v.sub.C,h]), a [member of] (j, k), v [member of] (l, m) and t = 1, ..., T. TABLE 2 Data Description Agel Age2 Age3 Age4 Number of observations 432 2169 1529 1418 1998 -- 154 164 165 1999 -- 176 181 123 2000 -- 218 153 125 2001 4 227 140 145 2002 12 230 144 135 2003 36 198 146 143 2004 51 235 117 128 2005 75 217 148 121 2006 83 217 136 140 2007 108 193 134 139 2008 63 104 66 54 Median price ('000 Euros) 230.13 159.28 127.06 130.00 Mean price ('000 Euros) 249.96 178.14 136.23 140.44 Median lot size ([m.sup.2]) 274.00 244.00 189.00 190.00 Mean lot size ([m.sup.2]) 314.58 263.26 226.63 243.79 Median floor space ([m.sup.2]) 140.00 120.00 115.00 125.00 Mean floor space ([m.sup.2]) 148.57 129.55 119.34 128.02 Median number of rooms 5.00 5.00 5.00 5.00 Mean number of rooms 4.75 4.61 4.67 4.79 Median number of toilet-baths 2.00 2.00 2.00 2.00 Mean number of toilet-baths 1.92 2.00 1.97 1.97 Ages All Number of observations 800 6348 1998 66 549 1999 72 552 2000 68 564 2001 64 580 2002 77 598 2003 79 602 2004 77 608 2005 61 622 2006 77 653 2007 108 682 2008 51 338 Median price ('000 Euros) 122.00 142.94 Mean price ('000 Euros) 137.88 159.44 Median lot size ([m.sup.2]) 215.00 209.50 Mean lot size ([m.sup.2]) 291.63 257.16 Median floor space ([m.sup.2]) 105.00 120.00 Mean floor space ([m.sup.2]) 113.28 125.99 Median number of rooms 4.00 5.00 Mean number of rooms 4.59 4.67 Median number of toilet-baths 2.00 2.00 Mean number of toilet-baths 1.82 1.96 Notes: The age variable is defined as follows: age 1-10 years: Agel; age 11 -20 years: Age2; age 21-30 years: Age3; age 31-40 years: Age4; and age 41-50 years: Age5. TABLE 3 Hedonic Regression Results for Each Age of Dwellings * Agel Variables (a,b) Coefficients Standard Deviation 1998 dummy -- -- 1999 dummy -- -- 2000 dummy -- -- 2001 dummy 2.240 0.155 2002 dummy 2.243 0.156 2003 dummy 2.258 0.155 2004 dummy 2.328 0.154 2005 dummy 2.342 0.153 2006 dummy 2.402 0.154 2007 dummy 2.427 0.153 2008 dummy 2.448 0.154 Region-old 0.067 0.039 Region-new -0.052 0.036 Semi-detached 0.260 0.037 Comer-house 0.029 0.015 Oneside-duplex 0.195 0.017 Detached 0.383 0.027 Log(lotsize) 0.222 0.019 Log(flrspace) 0.354 0.031 Maintain-med -- -- Maintain-high 0.011 0.002 Room4 dummy -0.075 0.035 Room5 dummy -0.059 0.035 Roomo dummy -0.025 0.037 Toiletbath2 -0.022 0.011 Toiletbath3 0.010 0.016 Balcony-yes 0.007 0.013 Garage-yes 0.021 0.011 Dormer-yes 0.029 0.017 Roofter-yes 0.016 0.015 Adjusted [R.sup.2] .940 Degree of freedom 406 Age2 Variables (a,b) Coefficients Standard Deviation 1998 dummy 1.648 0.081 1999 dummy 1.818 0.081 2000 dummy 1.934 0.081 2001 dummy 2.048 0.081 2002 dummy 2.072 0.081 2003 dummy 2.114 0.081 2004 dummy 2.158 0.081 2005 dummy 2.198 0.081 2006 dummy 2.239 0.081 2007 dummy 2.279 0.081 2008 dummy 2.291 0.081 Region-old -0.085 0.017 Region-new -0.014 0.006 Semi-detached 0.156 0.020 Comer-house 0.010 0.009 Oneside-duplex 0.152 0.010 Detached 0.326 0.017 Log(lotsize) 0.258 0.012 Log(flrspace) 0.268 0.016 Maintain-med 0.032 0.008 Maintain-high 0.041 0.008 Room4 dummy 0.044 0.009 Room5 dummy 0.057 0.009 Roomo dummy 0.096 0.011 Toiletbath2 0.019 0.007 Toiletbath3 0.018 0.010 Balcony-yes 0.041 0.009 Garage-yes 0.053 0.006 Dormer-yes 0.001 0.011 Roofter-yes 0.057 0.008 Adjusted [R.sup.2] .931 Degree of freedom 2139 Age3 Variables (a,b) Coefficients Standard Deviation 1998 dummy 2.108 0.114 1999 dummy 2.265 0.114 2000 dummy 2.403 0.114 2001 dummy 2.523 0.115 2002 dummy 2.585 0.114 2003 dummy 2.608 0.114 2004 dummy 2.661 0.114 2005 dummy 2.713 0.114 2006 dummy 2.726 0.114 2007 dummy 2.763 0.114 2008 dummy 2.779 0.115 Region-old 0.083 0.015 Region-new 0.097 0.006 Semi-detached 0.034 0.015 Comer-house 0.038 0.008 Oneside-duplex 0.214 0.013 Detached 0.472 0.022 Log(lotsize) 0.156 0.014 Log(flrspace) 0.266 0.023 Maintain-med 0.013 0.005 Maintain-high 0.026 0.005 Room4 dummy 0.017 0.016 Room5 dummy 0.036 0.016 Roomo dummy 0.079 0.019 Toiletbath2 -0.011 0.010 Toiletbath3 -0.001 0.014 Balcony-yes 0.071 0.020 Garage-yes 0.055 0.010 Dormer-yes 0.021 0.013 Roofter-yes 0.191 0.039 Adjusted [R.sup.2] .906 Degree of freedom 1499 Age4 Variables (a,b) Coefficients Standard Deviation 1998 dummy 2.027 0.107 1999 dummy 2.164 0.107 2000 dummy 2.318 0.107 2001 dummy 2.411 0.107 2002 dummy 2.493 0.107 2003 dummy 2.535 0.106 2004 dummy 2.547 0.107 2005 dummy 2.621 0.106 2006 dummy 2.621 0.107 2007 dummy 2.661 0.106 2008 dummy 2.681 0.107 Region-old -0.107 0.007 Region-new -0.032 0.021 Semi-detached 0.103 0.012 Comer-house 0.031 0.009 Oneside-duplex 0.181 0.013 Detached 0.449 0.023 Log(lotsize) 0.218 0.012 Log(flrspace) 0.215 0.024 Maintain-med 0.012 0.004 Maintain-high 0.021 0.004 Room4 dummy 0.088 0.015 Room5 dummy 0.100 0.015 Roomo dummy 0.145 0.018 Toiletbath2 0.019 0.010 Toiletbath3 0.018 0.014 Balcony-yes 0.049 0.015 Garage-yes 0.077 0.009 Dormer-yes 0.010 0.014 Roofter-yes 0.066 0.022 Adjusted [R.sup.2] .897 Degree of freedom 1388 Age5 Variables (a,b) Coefficients Standard Deviation 1998 dummy 1.442 0.131 1999 dummy 1.592 0.130 2000 dummy 1.662 0.132 2001 dummy 1.792 0.129 2002 dummy 1.886 0.129 2003 dummy 1.933 0.130 2004 dummy 1.953 0.130 2005 dummy 2.035 0.130 2006 dummy 2.020 0.129 2007 dummy 2.053 0.130 2008 dummy 2.061 0.131 Region-old -0.093 0.023 Region-new -0.023 0.034 Semi-detached 0.229 0.056 Comer-house 0.004 0.015 Oneside-duplex 0.114 0.018 Detached 0.317 0.027 Log(lotsize) 0.253 0.017 Log(flrspace) 0.306 0.027 Maintain-med 0.018 0.003 Maintain-high 0.037 0.005 Room4 dummy 0.021 0.020 Room5 dummy 0.054 0.021 Roomo dummy 0.077 0.026 Toiletbath2 0.036 0.011 Toiletbath3 0.118 0.022 Balcony-yes -0.014 0.012 Garage-yes 0.051 0.013 Dormer-yes 0.033 0.017 Roofter-yes 0.030 0.026 Adjusted [R.sup.2] .912 Degree of freedom 770 Notes: Dependent variable is the natural log of prices where the prices are expressed in 1000 euros. (a) The base case for the dummy variables corresponding to the physical attributes are the following: terraced house for the type of houses, low maintenance for the maintenance indicator, number of toilets and bathrooms is 1 for the toilet-bath dummies and number of rooms is 3 and less for the room dummies. The roomo dummy indicates houses with the number of rooms 6 and above. The dummy variables corresponding to whether a house has a balcony, garage, dormer, and roof terrace take the value 1 for "yeso and 0 for "no.o lotsize and flrspace are in square meters. (b) Most of the estimated coefficients are significant at the 5% level (128 out of 147 coefficients). Therefore, we have not separately indicated whether an estimated coefficient is significant or not. * Correction added on 10 August 2016, after first online publication: The headers for rows 12 to 17 have been corrected. TABLE 4 Age-Price Profiles: Median and GEKS Indexes Age of Median Fisher-GEKS Tornqvist-GEKS Houses Index Index Index (a) Agel 100.00 100.00 100.00 Age2 69.21 92.46 92.45 Age3 55.21 83.94 83.93 Age4 56.49 81.25 81.32 Age5 53.02 79.32 79.31 Depreciation rates (%): Annual geometric (b) 1.57 0.58 0.58 Annual average0 1.17 0.52 0.52 (a) Age 1-10 years: Agel; age 11-20 years: Age2; age 21-30 years: Age3; age 31-40 years: Age4; and age 41-50 years: Age5. (b) Example: Let dr be the geometric depreciation rate, then 79.32=100[(1 - [d.sub.r]).sup.40] [??] [d.sub.r] = 0.58%. (c) Calculated by dividing the cumulative depreciation by 40. TABLE 5 Time-dummy Hedonic Regressions with Nonlinear Specifications of Age* Variables f(age) = I (age} (a,b) Coefs Std f(age) -0.123 0.005 1998 dummy 1.823 0.050 1999 dummy 1.978 0.050 2000 dummy 2.110 0.050 2001 dummy 2.223 0.050 2002 dummy 2.279 0.049 2003 dummy 2.316 0.050 2004 dummy 2.354 0.050 2005 dummy 2.403 0.050 2006 dummy 2.426 0.049 2007 dummy 2.461 0.049 2008 dummy 2.471 0.050 Region-old -0.043 0.004 Region-new 0.047 0.004 Semi-detached 0.121 0.008 Corner house 0.024 0.005 Oneside-duplex 0.176 0.006 Detached 0.386 0.010 ln(lotsize) 0.226 0.006 ln(flrspace) 0.283 0.010 Maintain-med 0.017 0.002 Maintain-high 0.028 0.002 Room4 dummy 0.029 0.007 Room5 dummy 0.040 0.007 Room6 dummy 0.080 0.008 Toiletbath2 0.010 0.005 Toiletbath3 0.036 0.007 Balcony-yes 0.028 0.006 Garage-yes 0.059 0.004 Dormer-yes 0.022 0.007 Roofter-yes 0.065 0.007 Adjusted [R.sup.2] 921 Variables f(age) = [age.sup.2] (a,b) Coefs Std f(age) -0.007 0.001 1998 dummy 1.700 0.050 1999 dummy 1.856 0.049 2000 dummy 1.989 0.050 2001 dummy 2.103 0.049 2002 dummy 2.161 0.049 2003 dummy 2.200 0.049 2004 dummy 2.239 0.049 2005 dummy 2.291 0.049 2006 dummy 2.314 0.049 2007 dummy 2.352 0.049 2008 dummy 2.363 0.050 Region-old -0.025 0.005 Region-new 0.061 0.004 Semi-detached 0.120 0.008 Corner house 0.022 0.005 Oneside-duplex 0.173 0.006 Detached 0.383 0.010 ln(lotsize) 0.231 0.010 ln(flrspace) 0.289 0.010 Maintain-med 0.016 0.002 Maintain-high 0.029 0.002 Room4 dummy 0.033 0.007 Room5 dummy 0.038 0.007 Room6 dummy 0.077 0.008 Toiletbath2 0.007 0.005 Toiletbath3 0.033 0.007 Balcony-yes 0.037 0.006 Garage-yes 0.060 0.004 Dormer-yes 0.234 0.007 Roofter-yes 0.072 0.007 Adjusted [R.sup.2] .919 Variables f(age) = [age.sup.2], [age.sup.3c] (a,b) Coefs Std f(age) -0.029 0.002 0.004 0.001 1998 dummy 1.802 0.050 1999 dummy 1.956 0.049 2000 dummy 2.088 0.049 2001 dummy 2.201 0.049 2002 dummy 2.257 0.049 2003 dummy 2.296 0.049 2004 dummy 2.332 0.049 2005 dummy 2.383 0.049 2006 dummy 2.406 0.049 2007 dummy 2.442 0.049 2008 dummy 2.452 0.049 Region-old -0.042 0.005 Region-new 0.037 0.004 Semi-detached 0.129 0.008 Corner house 0.027 0.005 Oneside-duplex 0.179 0.006 Detached 0.389 0.010 ln(lotsize) 0.221 0.006 ln(flrspace) 0.296 0.010 Maintain-med 0.017 0.002 Maintain-high 0.028 0.002 Room4 dummy 0.027 0.007 Room5 dummy 0.039 0.007 Room6 dummy 0.079 0.008 Toiletbath2 0.010 0.005 Toiletbath3 0.037 0.007 Balcony-yes 0.025 0.006 Garage-yes 0.058 0.004 Dormer-yes 0.024 0.007 Roofter-yes 0.062 0.007 Adjusted [R.sup.2] .921 Variables f(age) = [e.sup.-age] (a,b) Coefs Std f(age) 0.454 0.022 1998 dummy 1.665 0.049 1999 dummy 1.820 0.049 2000 dummy 1.953 0.049 2001 dummy 2.066 0.049 2002 dummy 2.122 0.049 2003 dummy 2.158 0.049 2004 dummy 2.200 0.049 2005 dummy 2.245 0.049 2006 dummy 2.268 0.049 2007 dummy 2.302 0.049 2008 dummy 2.312 0.049 Region-old -0.066 0.004 Region-new 0.057 0.004 Semi-detached 0.115 0.008 Corner house 0.024 0.005 Oneside-duplex 0.179 0.006 Detached 0.388 0.010 ln(lotsize) 0.222 0.006 ln(flrspace) 0.286 0.010 Maintain-med 0.018 0.002 Maintain-high 0.030 0.002 Room4 dummy 0.027 0.007 Room5 dummy 0.038 0.007 Room6 dummy 0.077 0.008 Toiletbath2 0.012 0.005 Toiletbath3 0.037 0.007 Balcony-yes 0.022 0.006 Garage-yes 0.059 0.004 Dormer-yes 0.020 0.007 Roofter-yes 0.066 0.007 Adjusted [R.sup.2] .919 (a) Dependent variable is the natural log of prices where the prices are expressed in thousand euros. No. of observations: 6348. (b) With the exception of the estimated coefficient corresponding to toiletbath2 in the /(age) = [age.sup.2] regression, all estimated coefficients are significant at the 5% level. (c) The numbers corresponding to row/(age) are for [age.sup.2], and the row below/(age) are for [age.sup.3]. * Correction added on 10 August 2016, after first online publication: The headers for rows 1 to 11 have been corrected.