The estimation of economic depreciation using vintage asset prices

Journal

of Econometrics

15 (1981) 367 396. North-Holland

THE ESTIMATION

OF ECONOMIC VINTAGE ASSET

An Application

Charles

Frank Pomona

College

Received

and the Claremont

December

Company

DEPRECIATION PRICES

of the Box-Cox

The Urban Jnstirute,

Publishing

USING

Power Transformation*

R. HULTEN Washington,

DC 20037, USA

C. WYKOFF Graduate

School,

1979. tinal version

Claremont,

received

CA 91711,

December

USA

1980

This paper applies the BoxCox power transformation to the problem of estimating the rate and form of economic depreciation from data on used asset prices. The use of the BoxCox model to statistically discriminate between geometric, linear and ‘one-hoss-shay’ depreciation patterns (which are special cases of the BoxCox form) is analogous to the use of the CES production function to discriminate between CobbDouglas and fixed proportion technologies. We apply the BoxCox model to a sample of used building prices, and find that the appropriate depreciation pattern is approximately geometric.

1. Introduction In their studies of investment behavior and of productivity change, Dale W. Jorgenson and his collaborators, make extensive use of the assumption that economic depreciation can be characterized by a single geometric rate. [See, for example, Hall and Jorgenson (1967), Christensen and Jorgenson (1969, 1970, 1973), Jorgenson and Griliches (1967), the literature summarized in Jorgenson (1971), Hall (1977), and Christensen, Cummings and Jorgenson (1980).] The geometric depreciation hypothesis has, however, been the focus of considerable debate. Feldstein and Foot (1971), Eisner (1972), and Feldstein and Rothschild (1974) challenged the geometric assumption as theoretically untenable and as inconsistent with the McGraw-Hill surveys on *The research reported in this paper was financially supported by the Office of Tax Analysis, U.S. Department of the Treasury. We gratefully acknowledge this support, and wish to thank Emil Sunley, Harvey Galper, Larry Dildine, and Seymour Fiekowski of that Offtce for their comments and help on earlier drafts. We also thank Dale W. Jorgenson, Robert Eisner, and Paul Taubman for their comments. Opinions and remaining errors are solely our responsibility.

0165-7410/81/0000-0000/$02.50

0 North-Holland

Publishing

Company

368

C.R. Huh

and F.C. Wyko$ Estimation of economic depreciation

anticipated replacement investment. 1 Furthermore, Taubman and Rasche (1969) estimate an age-price profile of office buildings which is strongly inconsistent with geometric depreciation. In this paper, we test the geometric assumption using a sample of used asset transaction prices. If the critics of the single parameter (geometric) approach are correct in assuming depreciation is not approximately geometric, then a large body of empirical results is called into question and further empirical work is rendered considerably more difficult. First, with non-geometric depreciation, the aggregate rate of depreciation for a stock of assets and the quantity of replacement investment depend upon the age structure of the capital stock. This fact alone greatly complicates empirical analysis. Second, a nonconstant rate of depreciation suggests the need to consider the endogenous determination of the rate of depreciation. While this treatment of the subject is more realistic, it further complicates econometric work on investment and production behavior.2 If, on the other hand, geometric depreciation can be shown to provide a reasonable approximation to actual depreciation patterns, then researchers can continue to characterize depreciation as a single rate and thus achieve a considerable degree of simplification.3 ‘In discussing the various uses of depreciation estimates and in discussing the debate over the geometric hypothesis, a number of complicated and subtle issues must be borne in mind. One of the most important distinctions to be made in this area is between physical depreciation and economic (or price) depreciation. Physical depreciation, which we prefer to call mortality, refers to the loss in productive capacity of a physical asset due either to loss of in-use efficiency or to retirement. Corresponding to this quantity concept of mortality is the price concept of economic depreciation. Economic depreciation refers to the asset’s loss in monetary value with age at a point in time. These two concepts of depreciation are closely related to one another but are by no means identical. Mortality is relevant to the analysis of physical investment, replacement requirements, and capital stock estimation, while economic depreciation is relevant to analysis of taxes, asset prices, and the measurement of income. However, in one special case, mortality and economic depreciation are congruent; i.e., in the case of geometric depreciation. When economic depreciation is geometric then the duality relation implies that mortality is geometric as well (and vice versa). This unique feature of geometric depreciation lends it additional importance. Replacement investment, in the sense used by Jorgenson in his investment analysis, is that quantity of investment required to replace the reduction in capacity due to retirement of assets from service and due to the in-place loss in productive efficiency; in other words, mortality. The McGraw-Hill surveys appear to report investment plans which are intended to replace and modernize worn out or obsolete assets. These two concepts of replacement are not necessarily identical. For more on the replacement debate see Jorgenson (1973) and Feldstein and Rothschild (1974) in which these issues are discussed in detail. Yet another important distinction lies in the difference between the purchase price of a capital asset and the rental cost of using the asset for one time period. In equilibrium, the former is equal to the expected present value of the latter. Economic depreciation refers to the decline in purchase price because of age, while mortality is related to the rental price. In equilibrium, the productive efficiency of an s year-old asset relative to a new asset is equal to the relative rental price ratio. Mortality is the capacity change in efficiency, implying a link between mortality and economic depreciation through the present value relationship. 2Jorgenson (1973) provides a detailed description of problems and procedures associated with a non-constant rate of depreciation. Feldstein and Rothschild (1974) explore some of the implications of endogenous depreciation. 3The importance of estimating the rate and pattern of economic depreciation is not confined to investment and production theory. Taxpayers generally select one of three patterns in computing

CR. Hulten and F.C. Wykoff, Estimation of economic depreciation

369

In this paper, we describe a framework for simultaneously estimating the form and the rate of economic depreciation. We use this framework to test the geometric depreciation hypothesis, as well as other forms discussed in the literature. Like recent studies of economic depreciation, such as those by Ackerman (1973) Cagan (1973), Griliches (1970), Hall (1973), Hulten and Wykoff (1975, 1977, 1978) Lee (1978), Ramm (1970), and Wykoff (1970), we employ actual market transaction prices of used capital assets to obtain our estimates. However, our approach differs from other studies of vintage asset prices in two important respects: (1) we recognize that price data for used assets only sample surviving assets, and therefore, do not represent all assets in an original investment cohort; and (2) we use the BoxCox power transformation regression model rather than the analysis-of-variance model. The BoxCox power transformation is a highly flexible functional form which is ideally suited for analyzing depreciation. It contains the functional forms most often discussed in the depreciation literature (geometric, straightline, and one-horse shay) as special cases. We apply our model to a large sample of prices for used commercial and industrial structures, and we find that the pattern of depreciation does not strictly follow any of the common forms. We do find, however, that the ‘BoxCox pattern of depreciation is accelerated with respect to a straightline form and that on average the geometric pattern provides a reasonably close approximation. These results for buildings appear to provide some support for the single parameter approach. As to specific parameter values, we find that the average rates of depreciation implied by our estimates are considerably slower than those used in the official capital stock studies of the Bureau of Economic Analysis (BEA). The BEA estimates for nonresidential structures in manufacturing average 6+ percent whereas the estimates of this study suggest rates closer to the 13 to 34 percent range.4 The remaining sections of the paper have the following organization. Section 2 contains five subsections in which we develop, justify, and illustrate the BoxCox model as an appropriate device for extracting depreciation estimates from vintage asset prices. We also analyze, in section 2, several their depreciation deductions for tax purposes: straight-line, declining balance, or sum-of-theyears digits. They also select a tax life over which their asset may be depreciated. The corresponding rates and patterns of economic depreciation are important for tax policy because they provide a basis for evaluating the ap‘propriateness of these tax depreciation practices [see Break (1974) and Samuelson (1964) and for an opposing view, Smith (1963)]. ‘Our model also provides estimates of asset inflation for each type of building. As with our estimates of depreciation, our inflation estimates include a quality and compositional change component. We obtain various rates of inflation falling in the 2 percent to 4 percent range. These are roughly consistent with the corresponding Boeckh construction cost indexes. To the extent that the Boeckh price indexes measure a pure inflation component, our finding may be interpreted as suggesting a relatively low rate of qualitycomposition change in non-residential structures. The relatively large standard errors on the BoxCox inflation coefficients, do, however, make this result somewhat problematic.

370

C.R. Hulten and F.C. Wykofi Estimation of economic depreciation

potential sources of bias in vintage asset price data: the ‘lemons’ problem and the ‘secondary markets’ problem. In the first three subsections of section 3, we present and analyze our empirical results which include point estimates of depreciation rates by age and hypothesis tests of alternative depreciation patterns. In subsection 3.4, we study the intertemporal stability of our depreciation estimates from pooled price data for different years. We are concerned that non-systematic jumps in variables, such as tax policy or interest Me:, over time may significantly alter the estimated depreciation rates. We find, however, that when the data are decomposed into annual components and analyzed separately, the resulting estimates show a high degree of stability over time. The summary and conclusions are in section 4. 2. The use of vintage asset prices to measure economic depreciation 2.1. Economic depreciation is defined in this paper as the rate of change of asset price with age at a point in time.5 In the absence of inflation, this definition corresponds to the widely accepted view that economic depreciation is the value of the capital stock which must be replaced in order to maintain an initial investment. During periods of inflation, this traditional view of depreciation holds only when the age effect is separated from the effect of year-to-year changes in the price level. Separation of effects may be shown by calculating the total differential of the price with respect to time. Letting q(s, t) denote the price of an s-year-old asset in period t and letting q denote the total differential of q(s, t) with respect to time, we have tj 8q ds -=-._+_.__. 4 8s 4

i?q

dt

at

4

(1)

We define the first term on the right-hand side of eq. (1) to be the rate of economic depreciation and the second term to be the rate of asset inflation. [The latter term is sometimes called the rate of revaluation, the capital gain (loss) term, or the rate of change of replacement cost.] An intuitive interpretation of eq. (1) can be obtained for discrete values of s and I by viewing q(s, t) as an element of a matrix of asset prices whose rows range over values of s (ages) and whose columns range over values of t (years), t= 1981 1982 ... 1980

5This definition

is fairly conventional

and goes back at least as far as Hotelling

(1925).

C.R. H&en

and F.C. Wykoff, Estimation

of economic

depreciation

371

Each column traces the change in asset value with age in a given year and thus defines a depreciation pattern or an ‘age-price profile’ for that year. The term aq/ds corresponds to the rate of change along the profile of a given year, while inflation can be thought of as a shift in the entire age-price profile from year to year. The term dq/dt can thus be interpreted as the rate of change along the profile of a given age (row), or as the inflation component at a given point on the age-price profile. (Of course, a ‘timeprice profile’ could also be constructed, and depreciation would be identified as the shift in this profile; however, our focus is on alternative patterns of depreciation.) If depreciation is in fact geometric, then the age-price profile would be a geometric function conditional on year t. The matrix interpretation provides a framework for simultaneously estimating the rate of economic depreciation and asset inflation. Data on used asset prices can be arranged into a matrix [q(s, t)], and the age-price profile can be estimated econometrically either for each year separately or for all years jointly. Explicit models for carrying out the estimation will be discussed shortly. We note here, however, that a major inferential leap is required to link the discrete changes, Aq/As and Aq/At of [q(s,t)], to the corresponding derivatives of eq. (1). Discrete changes refer to the price differentials between different assets at different ages and dates, while the derivatives refer to the components of the changes in price of the same asset over time. Specifically, the age effect Aq/As refers to the difference in price between assets of age s and s + 1 at time t, and the price effect Aq/At refers to the difference in price between years t and t + 1 for an asset of age s.~ When we use the price matrix by age and date to estimate economic depreciation and asset inflation for any given type of asset, therefore, we must assume that the discrete formulation, Aq/As, Aq/At, is equivalent to the continuous formulation, dq/&, dq/&. What does this assumption imply? First of all, it implies that a given category of assets (e.g., turret lathes or office buildings) are sufficiently similar that across-asset price comparisons will permit meaningful inferences about the price performance of any one asset. Second, a distinction must be made between successive generations (or ‘vintages’) of assets. The term 4(s, t) of eq. (1) refers to the price changes within a given vintage; in the matrix [q(s, t)], such a total change appears as a movement along the relevant left-hand diagonal of the matrix, e.g., from q(O,l980) to q(l,l981). Each successive / diagonal of the matrix is indexed by the variable u= t-s and represents the total price history of that particular vintage cohort. A distinction must be 6To illustrate, the discrete age effect would refer to the difference in price between a tive-yearold asset and a six-year-old asset in 1975, while the discrete time effect would refer to the difference in price between a five-year-old asset in 1975 and a live-year-old asset in 1976. On the other hand, the total differential of the q(s,t) term in (1) would refer to the difference in price between a five-year-old asset in 1975 and the price of the same asset in 1976 when it is six years old.

312

C.R. Hulten and F.C. Wyko& Estimation of economic depreciation

made between successive vintages because improvements in the design and construction (or reductions in the quality of construction) will generally cause asset prices to differ systematically across vintages for reasons not related to the general price level effects. This vintage-price effect reflects the fact that the superiority of one model in a given year will persist over time. These diagonal effects are termed ‘vintage’ effects and are associated with obsolescence. These vintage effects must be taken into account when considering the determinants of used asset prices. However, in his 1968 paper, Hall showed that the three independent trend effects of age (s), date (t) and vintage (u) on asset prices cannot be separately identified econometrically. Thus, if we are to use the price matrix [q(s, t)] to estimate economic depreciation and asset inflation, we must assume either that there is some arbitrary level of net quality change (e.g., zero) or that our estimates of depreciation and inflation include a vintage effect component.7

2.2. The ageedate price matrix described in the preceding subsection can be used to estimate the composite age and date effects by using separate dummy variables for each age s, date t and vintage v. This method is the analysis-ofvariance approach (ANOVA) used by Hall (1973) and Lee (1978). The coefficients of the vintage dummy variables are normalized so that no net quality change is permitted, but allowances are made for annual variations in quality. The ANOVA approach is useful in some contexts. However, when the range of ages and dates over which the sample varies becomes large, as is the case with our sample of commercial and industrial structures, the dummy variable approach becomes cumbersome to implement. With buildings up to 110 years old and a time span covering 80 years, even a large sample (1500 observations per class of structure, in our case) leaves the majority of agedate cells empty. Furthermore, even if data were to exist for each cell, the number of dummy variables would be intractably large. Another approach to this problem still within the ANOVA framework, is to group observations into broad age intervals. This grouping of observations is conceptually equivalent to fitting a broad step function to the data, with the width of the steps determined by the length of the age and date intervals. A major difficulty with this approach is that in order to ‘The problem can be illustrated algebraically by assuming that vintage asset prices are determined by pure trend terms in age s, date C, and vintage t-s. Letting a, p, and y, denote trend rates of depreciation, asset inflation, and obsolescence respectively, Inq(s,r)=~+as+Bt+v(t-s)=cl,+(a-jl)s+(p+~)t It is evident that a, /I, and y, cannot be separately identified by analyzing the variation of q(s,t) with respect to s and t. Any such analysis will yield an age effect which combines obsolescence and deterioration, and a time effect which combines asset inflation and quality change.

C.R. H&en

and F.C. Wykoff, Estimation of economic depreciation

373

reduce the problem to a manageable number of dummy variables, the width of the steps must be quite broad and a good deal of sample information is lost, A different solution is to employ highly flexible functional forms which ‘bridge’ the missing observations and which allow alternative depreciation patterns to be discriminated statistically. A number of econometric techniques are suitable for this purpose the polynomial model, spline regression, Bayesian techniques. We have selected one, the Box-Cox power transformation, as the basis for our empirical analysis.* We selected the BoxCox model as our basic econometric framework because it is particularly well suited to the problem of sorting out alternative depreciation patterns. The geometric, straight-line, and one-horse shay patterns of depreciation are all special parametric cases of the Box-Cox model, and can therefore be discriminated using standard hypothesis testing techniques. This approach conforms to the conventional approach of comparing alternative models by finding a more general model which includes the alternatives as special parameter cases. Just as the CobbDouglas and fixed proportion technologies are special cases of the CES production function, so are geometric, straight-line, and one-horse shay depreciation patterns special cases of the Box-Cox depreciation function.

2.3. The BoxCox power transformation involves jointly estimating the parameters which determine specific functional forms within the Box-Cox class, and the parameters which determine the slope(s) and intercept of the equation.’ Unlike the standard regression model, the BoxCox model assigns two parameters to each regressor. Letting qi represent the market transaction price of an asset of age si in year ti, we apply the BoxCox model to the used asset pricing problem in the following way: q; = CI+ fls: + yt* + ui,

i= 1,.

.,N,

(2)

where 4: = (4’ - 1 )P, 2

s*= (SF -1)/O,,

ti”=@

-1)/O,,

(3)

and where the subscript i indexes observations from 1 through N, and the ui are N independent random disturbance terms which are assumed to be normally distributed with zero mean and constant variance 02.ro The *In addition to estimating the parameters of the BoxCox model, we also carried out experiments with the analysis-of-variance and polynomial regression models. We investigated, in addition, the possibility of using the error components model. ‘This transformation was developed by Box and Cox (1964). Zarembka (1974) provides further discussion and a survey of the literature. “As will be seen in a subsequent section, the independence of the error terms will turn out to be a key assumption in relating (3) and (4) to a more general model which permits the parameters to change from year-to-year.

374

C.R. Hulten and F.C. Wykoff; Estimation of economic depreciation

unknown parameters 0 = (e,, /32,6,) determine the functional form within the BoxxCox power family, whereas the unknown parameters (M,j, y) determine the intercept and slope(s) of the transformed model. As noted above, the BoxXox transformation is particularly useful for the analysis of depreciation because as the &vector takes on different values, the form of eq. (2) changes: The linear, decelerated and geometric forms are among those which eq. (2) can attain.” For example, the restriction 13= (1, 1,1) implies that (2) becomes linear i = 1,. . ., N.

(4)

The limits of q*, s* and t* as 0,, e2 and e3 each approach and In t, where In denotes a natural logarithm. Thus, =(O, 1,1) implies that (2) is semi-log,

zero are lnq, Ins the restriction 0

4i=(“-B-r+1)+Bs,+yti+ui,

lnq=(cr-_-_Y)+BSi+yt,+ui,

i=

1,. . ., N.

(5)

If 6= (O,O,O), then (2) is the log-log form. The ability to distinguish between linear and semi-log forms is of particular importance in depreciation theory. The semi-log form corresponds to Jorgenson’s geometric decay assumption; the linear form is a well-known alternative hypothesis widely believed to be accurate because of its frequent application in accounting and tax practice. Depreciation patterns in which the age-price profile declines slower than a straight-line (concave) are called decelerated and are also special cases of eq. (2). The decelerated pattern most commonly discussed in the depreciation literature is the one-horse shay form in which an asset maintains its full productive efficiency until the end of its life. The price of such an asset declines gradually in the early years of asset life and accelerates rapidly as the date of retirement approaches. This type of pattern can be characterized in the Box-Cox model as the case 8= (1,3,1). This restriction on 8 results in a cubic regression of price on age, and is essentially the form used by Taubman and Rasche in their study of ofhce buildings.” There is, of course, no a priori reason that the Box-Cox 8 parameter should take on any of the commonly discussed depreciation patterns. It is therefore useful to analyze the general relationship between the vector 0 and the form of depreciation.

“Recall from our discussion above that the analysis of the age and time components of used asset prices yields an estimate of the rate of depreciation which includes the effects of quality change (if it is present). We will, nevertheless, refer throughout the paper to ‘the rate of depreciation’. “Taubman and Rasche, however, start with rental prices rather than asset values and their framework, as well as their results, differs significantly from the framework of this paper. It is also useful to note that a cubic regression is only an approximation to the one-horse shay depreciation pattern and that other decelerated forms will also approximate this pattern.

C.R. Hulten and F.C. Wykoff, Estimation of economic depreciation

This may be done by examining derivatives of (2) with respect to s,

the

first-

and

second-order

315

partial

aq

z’pq(l-elp-l),

and

a24 (e2-i)aq

-zz

a?

-as+-

When (6) is divided depreciation,

s

(i-e,) 4

aq 2

0-as

by the asset price q, the result

(7)

is the rate of economic

(6’) Since q and s are positive, the rate of depreciation will be negative (i.e., price will decline with age, holding time constant) if p is negative. Furthermore, when o1 < 1 and 8,~ 1, then the Box-Cox age-price profile is accelerated with respect to stright-line depreciation (i.e., convex). Fig. 1 illustrates a convex age-price profile normalized on 1 for a new asset. The convex pattern is generated using the parameter values: fl= -0.05, (!I1=O.l, 8, =0.8. Conversely, whenever 8, > 1 and Q2> 1, the age-price profile is decelerated (concave). The one-horse shay form of depreciation follows a decelerated pattern. Fig. 1 illustrates the concave depreciation pattern using the parameter values: fi = -0.01, 8, = 1.3, 8, = 1.01. An interesting intermediate case occurs when 8, < 1 and 0,> 1 in which case the age-price profile is shaped like a backward S. I3 This case is also illustrated in fig. 1, as are the linear case (/?= -0.014286, 0, = 1, o2 = 1) and the semi-log case (/I= -0.03, 8, +O, e2 = 1). An important implication of this Box-Cox model is that even if the hypotheses of geometric, straight-line or one-horse shay depreciation are statistically rejected, the Box-Cox point estimates can indicate whether the pattern of depreciation is accelerated relative to straight-line (convex) or whether it is decelerated (concave). Accelerated is closer to geometric and decelerated is closer to one-horse shay. r3The possibility of a reverse S-shaped pattern of depreciation may be particularly relevant for buildings and other long lived assets. New relatively young buildings may be virtually indistinguishable from the buyer’s point of view. After this initial period, however, buyers may require substantial discounts in order to acquire the asset. These discounts may be caused by uncertainty about repair requirements, or by obsolescence causing older buildings to be simply less efficient in generating income (relative to newer, and perhaps more technically effkient, structures). JOE-C

C.R. Hulten and F.C. Wykofi Estimation of economic depreciation

316 1 Price .8

60 Age Fig.

1. Illustrative age-price profiles from the BoxXox model [normalized on 4(0, t)=l]. values for each form: Normalized Box-Cox equation: 4 = 1+ (0, pSe2/0,) *‘@I.Parameter

Form

B

01

02

Linear Geometric Convex Concave Backward

-0.014 -0.030 -0.050 -0.010 0.020

1.00 0.00 0.10 1.30 0.10

1.00 1.00 0.80 1.01 1.30

S

2.4. The use of the flexible BoxXox power transformation distinguishes this paper fro& earlier analyses of vintage asset prices. An even more important difference, however, arises from our recognition of the need to allow for asset retirement in estimating depreciation. Implicit in all previous studies using vintage price data is the assumption that all assets of a given type and vintage are retired at the same time. In fact, individual assets in a given cohort are typically retired from service at different points in time; there will be, for example, fewer twenty-year-old office buildings in 1990 than there were ten-year-old offices in 1980. And, therefore, any sample of used asset prices has been ‘censored’ by the retirement process, since assets taken out of service are no longer available for sampling. Or to put the matter differently, the prices of surviving assets of vintage u do not represent the average experience of the initial cohort of assets put in service in year u.

C.R. Hulten and F.C. Wykoff, Estimution of economic depreciution

371

It is the depreciation experience of the typical (average) asset in the original cohort rather than that of the longest survivors which is relevant for investment theory, tax policy, and capital measurement. The relationship between retirement and censored sampling can be illustrated by the following example. Suppose that one wishes to forecast the 1990 gasoline consumption of all cars built in 1980. Suppose, also, that ten year old cars consume X gallons of gas a year. If there are Y 1980 automobiles placed in service, can we conclude that the contribution of 1980 autos to gas consumption in 1990 is the product of X and Y? No, because many 1980 autos will have been removed from service by 1990. If, for example, 50 percent have been retired, our estimate of 1990 gas consumption would be fXY In other words, the average 1990 gas consumption of a 1980 car is 4X, even though a statistical analysis of survivors indicates that gas consumption is X gallons a year. Similar examples can be constructed from demography, since human mortality also introduces a wedge between the characteristics of those who survive to a given age and the original population as a whole. In the current context, the retirement from service of business structures causes survivors to have fundamentally different characteristics than nonsurvivors. While a surviving structure would presumably sell for the present value of its expected net income, a building being removed from service is worth only the value of scrappage less demolition costs. This net amount is a component of the remaining value of the initial investment cohort and should be included in the value of the cohort even after the building is demolished. The net scrap value can, however, be a positive or negative amount, depending on whether demolition costs are less or greater than the scrap value of the building. For this reason, and because we do not have information on net scrap values, we have assumed that this value is zero. As a result, the total used value of structures put in place in any given year includes, in our analysis, both the positive value of surviving buildings and the zero value of retired buildings.14 Our treatment of retirement and scrap value implies that at any point on the age-price profile the average value of the initial investment cohort is different from ‘the average value of surviving structures. Any analysis based only on survivors will therefore tend to overstate both the value and productivity of estimated capital stocks. On the other hand, the use of survivors to evaluate tax policy will tend to understate the anticipated true depreciation path of an average asset. A method of correcting for the presence of non-survivors is thus necessary and, while several methods may 14To the extent that buildings drop from service because they enter a secondary use, they do not enter the sample. Our treatment corresponds to valuing such structures the same as those still in the primary use. They are not treated as retired, however, by our average retirement lives from Bulletin F. We discuss the secondary use problem below.

378

be used following any date price of surviving to age s

C.R. Hulten and F.C. Wykofi

Estimation

of economic depreciation

to correct for this potential sampling bias, we have adopted the approach. We assume that the average price of the initial cohort at is the sum of the weighted average of the price of survivors and the non-survivors, where the respective weights are the probabilities of and not surviving. In symbols, if f, is the probability of surviving and if q(s) is the average price at age s of the original cohort, then

4(s)=Ms)+

(1-f,) ‘0,

(8)

where t(s) is the average age s price from the sample of survivors. In order to estimate q(s) from observations based on G(s), eq. (8) indicates that we must deflate each observation by the probability of survival to that observation’s age. l5 This approach is a non-stochastic correction which we have shown in (1977) may be fully integrated into the theory of replacement and depreciation presented in Jorgenson (1973).16 The average prices q(s) calculated using eq. (8) can be thought of in the following way: if all M assets in a given cohort were pooled to form a company in which M stock certificates were issued, then the certificate price would equal the average asset price. Average economic depreciation is the rate of change of this average price with respect to age at a point in time. Our survivor probabilities are based upon the set of retirement distributions developed by Winfrey (1935). We chose the &-distribution for our analysis, and fig. 2 illustrates the L, survivor curve and frequency curve. While the choice is somewhat arbitrary, the &-distribution has the virtue of allowing for gradual asset retirement with the probability that a few assets survive to very old ages. (This pattern is thus especially appropriate for the analysis of structures.) If the mean asset life of a cohort is 50 years, then the &-distribution implies that 92 percent of the assets survive 10 years, 69 percent survive 30 years, 45 percent survive 50 years, and 8 percent survive IsThe assumption that the error term is normally distributed should be considered in light of the censored sample problem: if price per square foot is distributed normally for the new assets in each cohort, retirements from the cohort will censor the normal distribution, so that strict normality will not hold for used assets in the cohort. In the model of this paper, however, we assume that the error term ui is dominated by the heterogeneity of the buildings within any class, and that each observation is on the deterministic average price path for its homogeneous sub-class. Since the average price path for each sub-class already accounts for the effect of retirements, the distribution of ai is not censored by the retirement process. An alternative to the approach of this paper is to specify a censored normal likelihood function [a discussion of the censored sample approach is contained in Heckman (1974)]. This approach substitutes assumptions about the price at which the asset is retired (and censored from the sample) for assumptions about the form of the probability of survival. We selected the survival probability approach because the precedent for assuming survival probabilities is more firmly established (due to the procedure of the Bureau of Economic Analysis discussed later). We do, however, regard the censored likelihood approach as a useful direction for further research. t6The assumptions and analysis underlying this approach are discussed in more detail in Hulten and Wykoff (1977).

C.R. H&en

and F.C. Wykoff, Estimation of economic depreciation

379

a0 frequency

60

.a

4

_I 300

200

100 % of average life Fig. 2. Winfrey

L, distribution.

Source:

Marston,

Winfrey

and Hempstead

(195_3, p. 419).

100 years. Other retirement patterns could clearly be used; BEA, for example, uses a symmetrical Winfrey distribution. While the La-distribution is plausible, it is not based on direct evidence of structure retirement, but on other assets studied by Winfrey. The results presented here must be interpreted with this caveat in mind. In order to construct survival probabilities for various types of structures from a retirement distribution, we need to determine the central location of the distribution. We take the mean asset lives from Bulletin F published by the U.S. Treasury in 1942. Bulletin F is a compendium of asset lives which was assembled as a guideline for tax depreciation calculations. The Bureau of Economic Analysis uses a similar procedure in their capital stock studies.17 “BEA uses the Winfrey S, distribution and 85 percent of Bulletin F lives. We adopted the ‘slower’ Winfrey La-distribution and 100 percent of Bulletin F lives for our analysis because the BEA assumptions resulted in a situation in which some observations in our sample had ages older than the age at which all assets are assumed to be retired.

380

C.R. Hulten and F.C. Wykoff, Estimation of economic depreciation

2.5. Finally, since our price data consist of market transaction prices on used assets, we consider the validity of using vintage price data to infer the depreciation of assets which are never sold. If there are systematic differences between capital goods that enter used markets and those that do not, these differences may result in biased estimates of the depreciation of assets as a whole. There are two arguments suggesting the possibility of such a bias. First, Ackerlof (1970) presents a ‘lemons’ model in which the existence of ‘lemons’ (poorly built assets) in the used market drives out higher quality assets. If the market consists mostly of lemons, the average used asset market price would be a downward biased estimate of in-use asset values. Second, assets, when resold, may be placed in different and less productive uses. The used market prices may only reflect productivity of the secondary uses, not the productivity of assets still in their primary use. In this case, as we shall show below, prices may falsely produce a geometric decay pattern. The fundamental assumption of the lemons model is that owners (potential sellers) have more information about asset quality than prospective buyers. This asymmetrical information produces a situation in which the optimal strategy on the part of each owner is to offer only a lower quality asset in the organized market and either sell the better assets in private transactions or hold these assets longer. Ackerlof illustrates his argument with the used consumer automobile market, and it is tempting to extend his analysis to business assets. However, while it may characterize the consumer auto market, it is less reasonable to suppose that the Ackerlof model characterizes markets for other types of assets. Purchasers of business assets (used commercial and industrial buildings, machines, trucks, etc.) are quite likely to be sophisticated in their assessment of used assets, since the purchase of such assets is often a routine part of their business. Furthermore, ignorance of potential buyers seems to be particularly implausible in the commercial and industrial structures market where the size of investments justifies extensive market and structural analysis before purchase, and where buyers are likely to be specialists. The second potential source of sampling error, that of secondary uses, is illustrated in fig. 3. Line AB depicts the age-price profile of an asset in its initially most productive use, and line CD reflects the age-price profile for the asset in a secondary, less productive use. (Date is suppressed for simplicity of exposition.) Some assets are first placed in service in the most productive use, and at age s* transferred to the secondary use. Market price observations capture the price performance only of these marketed assets, and thus observations lie on the segments AED (or possibly at point A and along segment ED).A regression on these observations will yield a distorted picture of the average depreciation experience of the cohort and will lead to the conclusion that the age-price profile is close to geometric (curve GG), whereas the true relationship is linear in the early years of asset life. This

C.R. H&en

and F.C. Wykoff, Estimation of economic

381

depreciation

Price

B

S’ Fig. 3. Age-price

profile for an asset with primary

D

and secondary

Age

uses.

argument implies that the regression estimates are biased in favor of the geometric depreciation hypothesis. Secondary market bias may indeed be a problem for some types of assets (e.g., machine tools), but it is not necessarily a problem for all types of assets. In particular, business structures are sold for a variety of reasons unrelated to primary and secondary uses: certain types of buildings are sold after the benefits of accelerated depreciation are exhausted; other buildings are sold by speculators moving into alternative investments; and still others are sold because business conditions have changed for reasons not systematically related to the quality or location of the structure. In each of these cases, the building may return to its primary use after it is sold, or it may be put into a different, but equally productive, use. Thus, we cannot see a priori grounds for believing that secondary market bias is a major problem for analysis of used business structures.

3. Empirical results from the Box-Cox 3.1.

The Box-Cox

model,

model

eqs. (2) and

(3), was estimated

using

maximum

382

C.R. H&en

likelihood techniques. implies that the natural

and F.C. Wykoff, Estimation of economic

depreciation

The assumption that the error term is N(O,o’Z) logarithm of the associated likelihood function is

(9)

= (0, - 1) 5 lnq,-Gln2rr i=l

-~lnf?-&~~l

[4;“-cr-psi*--ytT]‘. L

Letting L(o) denote the maximand of the likelihood function over the unrestricted parameter set 52 and letting L(h) denote the maximand over a restricted set o, the asymptotic likelihood ratio test of the restrictions can be performed using the statistic A= - 2 In [L (&3)/L (fi)], since 1, is approximately distributed as chi-squared when N is larger and WCQ. The restrictions tested, using the asymptotic likelihood ratio test, are listed in table 1.

Table Constraints

1

tested using likelihood

ratio test.

Case

Constraint”

No. of restrictions imposed on (9)

I II III IV

t$=& t11=02=e3 Q1+O, ez = & = 1 (semi-log) Q1= e2 = 8a = 1 (linear)

1 2 3 3

“While the first two cases are not particularly interesting as depreciation patterns, cases III and IV are. If e1 -+O and O2 = BJ = 1, then eq. (2), the Box-Cox model, becomes lnqi=a+fisi+yti+ui. If f?,=8,=6’3=1, then eq. (2) becomes qi=(a+l-fi-y)+&+yfi+ui. See Box and Cox (1964) and Zarembka (1976) for details.

The data for our analysis comes from a sample, collected by the U.S. Treasury’s Office of Industrial Economics in 1972, of 8066 observations on 22 types of buildings. The owner of a building was asked when it was constructed, when he acquired the building, and the price he paid for it exclusive of the value of the land. From the responses one can compile a matrix of market transaction prices by age and date of purchase. This data is summarized in Business Building Statistics, published in 1975 by the Office of Industrial Economics. We report in this paper on results for the four classes

C.R. Hulten and F.C. Wykofi Estimation of economic depreciation

383

with the largest amount of data: factories, office buildings, retail trade stores, and warehouses.i8 The unknown parameters of the BoxZox model were estimated using both price data adjusted for retirement by Winfrey’s &-distribution and the Bulletin F lives, and data that was not adjusted for retirement. (The latter might correspond to a model in which retired assets are assumed to have the same average value as survivors, the opposite case from our zero value assumption for non-survivors.) We present in table 2 a summary of the hypothesis tests on restrictions on the unknown parameters of eqs. (2) and (3). These tests were all carried out at the 95 percent level of significance. Briefly, the linear form (case IV: 6i = e2 = /33= 1) was rejected for every class of assets studied, using retired and unretired prices. The geometric (semi-log) form was rejected in each of the four classes, the restriction O2= 8, could not be rejected in two classes (warehouses and factories), and 8, = 0, = 0, could not be rejected for unretired warehouse prices.” Table 2 Constraints

accepted

at the 95 percent significance Cox statistical analysis.

Class

Unretired

price data

Retail trade Office building Warehouses Factories

None None Cases I and II Case I

level in the Box

Retired

price data

None None Case I Case I

Tables 3 and 4 contain the maximum likelihood point estimates for the unknown Box~Cox parameters, 8,, 02, Q3, c(, fi, and y as well as the loglikelihood estimates for retired and unretired price data respectively. As we noted earlier, the estimates of fi turn out to be small and negative (- 0.03 to -0.13). The estimates of 0i are uniformly positive and significantly less than “These classes fall in the category of commercial and industrial structures. In 1975, commercial and industrial structures accounted for 39 percent of the purchases of new private non-residential structures and over 10 percent of all private non-residential fixed investments (as reported in Tables 1.1, 5.5, and 5.6 of National Income and Product Accounts). “Tests on the individual f!Iiwere also carried out using the asymptotic normal test at the 95 percent level of significance. The results support the conclusions of the likelihood ratio test: (a) 0, is significantly different from zero in every case and (b) 6, is significantly different from one in every class except factories. A test of the hypothesis 0i = 1 is rejected in every class. Also a simultaneous test of the hypothesis 0, -+O and e2 = 1 at the 95 percent level of significance was carried out by allocating the probability of a Type I error equally between the separate hypotheses. This tests specifically for geometric depreciation and is a weaker hypothesis than the semi-log constraint. The results of the simultaneous test indicate geometric depreciation is rejected in all classes.

C.R. Hulten and F.C. Wykoff, Estimation of economic depreciation

384

Table 3 BoxCox

parameter

estimates

Class

(retirement

adjusted

price data).

a

B

Y

Loglikelihood

Retail stores

0.104 (0.015)

0.802 (0.071)

2.149 (0.561)

1.111 (0.174)

-0.051 (0.011)

0.00003 (0.00007)

- 4387.9

Offices

0.212 (0.011)

0.714 (0.060)

3.154 (0.634)

2.340 (0.165)

- 0.094 (0.017)

0.000006 (0.000016)

- 5220.4

Warehouses

0.168 (0.027)

0.658 (0.102)

1.265 (0.669)

0.349 (0.617)

-0.091 (0.027)

0.010 (0.027)

- 1378.1

Factories

0.104 (0.030)

0.983 (0.114)

1.651 (0.673)

0.517 (0.368)

- 0.038 (0.014)

0.002 (0.006)

- 1242.7

“Standard

errors in parentheses.

Table 4 BoxCox

parameter

estimates

(unretired

price. data). Log. likelihood

Class

6,

6,

9,

a

B

Y

Retail stores

0.128 (0.014)

0.370 (0.125)

2.870 (0.581)

1.160 (0.177)

- 0.098 (0.031)

o.OOc019 (O.OQOO4)

-4701.9

Offices

0.147 (0.011)

0.384 (0.098)

3.289 (0.659)

2.452 (0.178)

-0.158 (0.039)

O.OOOOO4 (O.wOOl)

-5481.49

Warehouses

0.138 (0.029)

0.688 (0.210)

1.754 (0.717)

0.582 (0.368)

-0.049 (0.030)

0.002 (0.004)

- 1360.3

Factories

0.187 (0.027)

0.400 (0.152)

1.314 (0.686)

0.384 (0.613)

-0.130 (0.049)

0.009 (0.023)

- 1428.3

“Standard

errors

in parentheses.

one (0.014 to 0.247); only in the case of retirement adjusted factory prices is the 13~estimate not significantly less than one (point estimates from 0.37 to 0.98). Recalling our discussion of the first- and second-order partials of eq. (2) above, small negative /I, 8, < 1 and O2< 1 imply convexity of the age-price therefore, indicate a strictly profile. The Box-Cox parameter estimates, convex age-price profile and provide no support for the conventional wisdom that depreciation of buildings is one-horse shay, nor for the more moderate view that the age-price profile is shaped like a backwards S (indicating one-horse shay behavior in early life followed by a period of sudden deterioration). Because of the comparative novelty of the Box-Cox procedure, we have plotted age-price profiles in fig. 4 both for Box-Cox fitted and actual prices. The actual prices are averages for each five-year age interval adjusted only

C.R. H&en

und F.C. Wykoff, Estimation

ofeconomicdepreciation

0 0

20

40

5a

WI

80

385

i 0

Age

Fig. 4. BoxZox fitted vs. (average) actual age-price profiles. interval, adjusted for retirement is compared to the Box-Cox for average date of purchase of assets in that age interval. profiles, and the lines connecting the dots refer to

50

100

Ace

Mean price per square foot by age fitted price for that mean age and Smooth lines refer to fitted price the actual price profiles.

for retirement. It is evident from visual inspection of the actual prices (the lines connecting the dots) that the actual age-price profiles are uniformly convex. The convexity result is thus inherent in the data and is not imposed by any peculiarities of the Box+Cox model. Note that some of the age-price curves appear to rise in the early years of asset life. This rise in price should not be, interpreted as a negative rate of depreciation (i.e., as appreciation in asset value with age), but rather as a consequence of the fact that we did not deflate these prices to correct for inflation. (We did not deflate because we wanted to demonstrate the ability of the Box-Cox results to track price variations both due to depreciation and to inflation.) Because we did not deflate these prices and because the average date of purchase varies across age intervals, some intervals represent more recent dates of purchase and thus more expensive assets. For example, the average date of purchase for new office buildings is 1957, while the average date of purchase for offices in the one to five year age interval is 1962. For this reason, the curves rise in

386

C.R. Hulten and F.C. Wykoff, Estimation

of economic depreciation

early years of life. It is evident from the graphs in fig. 4 that the Box-Cox fitted prices track the data reasonably we11.20 The convexity result is further supported by analysis of the data using a polynomial regression model. The polynomial tests indicate that the most common functional form is the quadratic, which is similar in shape to the geometric form. The polynomial analysis also included variables related to the characteristics of buildings in the sample, such as construction material. These additional variables contributed little to the analysis’l

3.2. Given the Box-Cox parameter estimates from tables 3 and 4, what can one say about depreciation? Implicit rates of depreciation can be calculated from the BoxxCox estimates by applying the definition of depreciation as the rate of change of asset price with respect to age in a given year. This definition implies that we calculate fitted Box-Cox prices by age and date, q(s, t), and compute [q(s, t)-q(s+ 1, t)]/q(s, t), which is the age s depreciation rate in period t. Table 5 contains selected rates of depreciation based on the BoxxCox fitted prices for the year 1970 for the four classes of structures: retail stores, offices, warehouses and factories. Rates for both retired and unretired price data are reported. The figures represent average annual percentage rates of decline at selected ages. For example, 5-year-old retail stores depreciate at 2.77 percent per year (including allowance for retirement), whereas 15-year-old retail stores depreciate at 2.32 percent per year. The depreciation rates are not uncreasonable, usually varying from l$ percent to 34 percent. With the exception of the factory rates based on retirement adjusted prices, the rates tend to decline with age. Since a constant percentage rate of decline per year is implied by geometric depreciation, the table 5 patterns imply a depreciation process that is even faster than geometric. This accelerated pattern should not, however, be interpreted as implying that structures depreciate rapidly, since the estimated rates 13 percent to 31 percent are low compared to conventional wisdom. (BEA average rates are around 6$ percent and tax depreciation allows deductions within a 5 percent to 7 percent range.) loThe BoxCox fitted curves were calculated by taking the average age and date in each interval and transforming them according to the last two terms in eq. (3). The implied value of the price was then calculated using (2) and the first term of (3). *IThe data was rather limited in distinguishing the characteristics of the various structures, but the sample did report primary structural materials, construction quality (presumably closely correlated to the availability and quality of ancillary equipment) and postal zip code areas. Structural material and construction quality were entered directly as dummy variables in the polynomial analysis. From postal zip codes, one can obtain a measure of population using data reported by the Internal Revenue Service in Supplemental Report, Statistics of Income-1969, ZIP Code Area Data from Individual Income Tax Returns. These variables may serve as rough proxies for the value of the land on which the structures are located. The inclusion of these characteristic variables had little influence on the age coefficients from which depreciation is determined, but tended to increase the overall explanatory power of the equation.

C.R. Hulten and F.C. Wykoff, Estimation

of economicdepreciation

387

Table 5 Selected rates of economic Box-Cox

(retired prices)

Age

Retail

Office

Warehouse

lb 5 10 15 20 30 40 50 60 70

3.54 2.17 2.47 2.32 2.22 2.10 2.03 1.99 1.96 1.94

4.32 2.85 2.64 2.43 2.30 2.15 2.08 2.04 2.02 2.02

BGA’

2.02

(R’)

(0.993)

_

depreciation. Box-Cox

(unretired

prices)

Factory

Retail

Office

Warehouse

Factory

5.51 3.68 3.05 2.74 2.55 2.32 2.19 2.11 2.05 2.01

3.02 2.99 3.01 3.04 3.07 3.15 3.24 3.34 3.45 3.57

5.39 2.41 1.63 1.29 1.09 0.86 0.73 0.64 0.57 0.53

5.72 2.66 1.84 1.48 1.27 1.02 0.88 0.79 0.72 0.66

6.81 3.23 2.26 1.83 1.57 1.27 1.10 0.98 0.90 0.83

3.00 2.02 1.68 1.50 1.39 1.25 1.17 1.11 1.06 1.03

2.47

2.73

3.61

0.82

(0.985)

(0.995)

(0.997)

(0.997)

“Percentage decline. “New asset. ‘Best geometric approximation

and the R* of the approximation

1.05

1.22

1.28

(0.971)

(0.979)

(0.995)

in parentheses.

The bottom two rows of table 5 contain the results of calculating the ‘average’ rate of depreciation based on the Box-Cox fitted prices. The average is obtained by regressing the natural logarithm of the fitted price on age and time. This single rate of depreciation is the best geometric approximation (BGA) to the implicit Box~Cox rates. The BGA rates are on average quite close to the Box-Cox fitted prices resulting in R2 values (last row of table 5) which are uniformly greater than 0.97. The important conclusion which emerges from this analysis is that a c~rzstant rate of depreciation can serve as a reasonable statistical approximation to the underlying Box-Cox rates even though the latter are not geometric. This result, in turn, supports those who use the single parameter depreciation approach in calculating capital stocks using the perpetual inventory method.22 “The perpetual inventory method of calculating the capital stock deducts depreciation from gross investment and then adds the result to the previous period’s capital stock. When depreciation is geometric, the amount of depreciation deducted is independent of the age composition of the capital stock carried forward. This simplifying feature accounts for the enormous popularity of geometric depreciation in empirical work on productivity, growth, and investment. See, for example, the capital stock studies of Christensen and Jorgenson (1969, 1970, 1973) the productivity study of Jorgenson and Griliches (1967) and the growth study of Christensen, Cummings and Jorgenson (1980). We do not mean to suggest, by our support of this empirical simplification, that more complex formulations which would allow for asset heterogeneity, variations in the age structure of the capital stock, and, in general, the endogeneity of depreciation, might not advance economic research. For example, in calculating present values of tax depreciation deductions one may prefer the full detail of BoxCox

388

C.R. Hulten and F.C. Wykoff; Estimation of economic depreciation

Comparison of the rates of unadjusted prices to those for prices adjusted for retirement indicates that the allowance for retirement produces depreciation rates which are nearly double the unretired estimates. To the extent that the Bulletin F lives and &-distribution accurately correct for retirement, the difference between the retired and unretired estimates provides a feeling for the magnitude of the censored sample bias associated with ignoring retirement altogether. This magnitude is, in turn, suggestive of the size of the censoring bias present in previous studies of used asset prices.

3.3. The unknown parameter estimates for the time variable coefficients, y than those associated with age. The and es, are much less satisfactory estimates of 7 are extremely small and statistically insignificant, while the estimated values for e3 are quite large and clearly non-zero. These results may imply that the depreciation estimates are affected by the time process. We shall examine this possibility in subsection 4 below. We note here, however, that the combined effects of y and 8, lead to very reasonable estimates of the rate of inflation for buildings. Rates of inflation, calculated from the BoxCox fitted prices for new assets, are remarkably close to the Boeckh construction cost indexes published in The Construction Review and to the non-residential structures deflator in The National Accountsz3 These comparisons are shown in table 6. Since the standard errors of the estimates of y and e3 are so large, interpretation of the relative sizes of the Box-Cox and Boeckh growth rates is hazardous. Notice, however, that the Boeckh rates are 1 percent to 2 percent larger for each building class except retail trade, where the difference is negligible. This observation is consistent with the fact that the Boeckh indexes are fixed-weight indexes of the prices of labor and material inputs, and therefore tend to overstate the change in the market value of new structures. Thus, to the extent that the Boeckh indexes measure the cost-inflation component of changing market values, the closeness of the BoxCox and Boeckh rates may be interpreted as implying only a limited role for changes in structure quality. 3.4. All of the results presented price data from various years assumed that the effects on asset smooth, systematic process. This

in the previous section were obtained using pooled into one set of observations. We prices of time could be characterized by a assumption is not necessarily appropriate,

estimates to the BGA rates because the BGA method tends to underestimate the present value of depreciation as a result of its relatively low early year values compared to the BoxCox rates. 231t must be recalled that the rate of inflation includes a quality change component, as does the rate of depreciation. It is also worth noting that the rates of inflation are generally larger than the rates of depreciation, implying a rising price for used buildings. The increasing price of buildings has quite likely been responsible for the erroneous, but widely held, belief that buildings depreciate as one-horse shays.

389

C.R. Hulten and F.C. Wykoff; Estimation of economic depreciation Table 6 Selected rates of new building

inflation.

Average

annual

percentage

increase

1950-70

1955-70

196G70

3.70 2.87 2.32 2.60

3.92 3.06 2.32 2.65

4.13 3.26 2.33 2.70

3.70 2.9 1 2.45 3.58

3.93 3.12 2.45 3.65

4.16 3.32 2.45 3.71

Boeckh index Apartments, hotels, and of&es

3.89

3.99

4.75

Boeckh index Commercial bldgs. and factories

4.02

3.96

4.35

National Accounts Non-residential structures implicit deflator”

2.99

2.94

3.39

Box-Cox

Estimates

Transformed Retail Oflice Warehouse Factory Untransformed Retail Ofhce Warehouse Factory Construction Reuiew

“Table 7.1 of the National Income and Product Accounts.

because year-to-year shifts in the age-price profile may well arise from sudden non-systematic changes in variables such as interest rates, tax policy, and expectations about the future. The large standard errors of the estimates of the time variable coefftcients may reflect such non-smooth changes. In order to explore the possibility that the parameter estimates upon which by the non-systematic depreciation depends, p, 8, and 8,, were distorted effects of time and the interdependence of time and age, we broke the pooled sample into annual components and analyzed each year’s age-price profile separately. We then tested for the stability of the estimates over time. The test for stability of successive cross-sections can be based, in principle, on a generalised form of the pooled likelihood function equation (9). Just as the linear and geometric forms are special cases of (9), (9) is itself a special case of a likelihood function in which the BoxxCox parameters are allowed to change from year to year. Let N, denote the number of observations of buildings sold in year t, then N, + N, + . . . + N,= N, and In L* =t$l i$l 2

f (&I.

I

(10)

390

C.R. Hulten and F.C. Wykoff, Estimation of economic depreciation

Asset price is related

to age in each year by

(11) where f%f -

cq=Clt+ytUnder the assumption likelihood function is

lnL,=

1

(12)

e321 . that

ui,* is distributed

as N (O,o:),

the

associated

2 %f(ui) i=idqi =(0,,,-1)

z

lnqi-~[ln2n+lno~+l].

(13)

i=l

Thus, lnL*=lnL,+lnL,+... +ln L,. Under the assumption that the errors ui are independently distributed, maximization of each L, with respect to a:, is analyzed Pt, %, and (32,t maximizes L*. Thus, if each annual cross-section separately, the sum of the maximized annual log-likelihoods provides a maximized likelihood for the entire sample under conditions in which the sample is not constrained by the assumption that the parameters are the same in every year. The likelihood function for the pooled sample (9) can be derived from (10) under the following 6T restrictions: ~=a,; p=p,; y=yt; 8, =Q1,t; &=Q,,,; and e3 = &. The statistic A= -2[lnL-lnL*]

(14)

is asymptotically chi-square with 6T degrees of freedom, and may be employed to test the stability of the parameters over time. (If the restrictions are valid, then LE L* and J.2 0.) Because of computer costs, the stability test was carried out for only one class of assets, office buildings.24 Computational costs also prevented our estimating (11) for each year; instead, we adopted two less general procedures, but procedures which are still derived from (11). First we fitted semi-log regressions for contiguous years for which data existed (1955-71).25 24This class was selected because it was studied by Taubman *sThis time interval was chosen because of data limitations; too few observations to support a cross-sectional analysis.

and Rasche (1969). years before 1955 generally

had

C.R. Hulten and F.C. Wykoff, Estimation of economic

391

depreciation

The semi-log form imposes the restriction 19= (0,1,1) for all years, and involves 3T (= 51) additional restrictions in deriving (9) from (10). The resulting test statistic (14) was 64.7 for the price data adjusted for retirement and 64.1 for the unadjusted price data. Since the critical value of a chi-square variable with 51 degrees of freedom is approximately 68 at the 95 percent level of significance, we accept the null hypothesis of no structural change over the period 1955-71. I The second procedure involved fitting the Box~Cox cross-sectional model for every third year in the interval 1955-71; 1956, 1959, 1962, 1965, 1968 and 1971. The unrestricted Box-Cox parameter estimates for these years are shown in tables 7 and 8. The functional form parameter estimates are shown

Table 1 Cross-sectional Unrestricted

cross-sectional

estimates

(unretired

estimates

%, and 8, restricted”

Loglikelihood Year 1956 1959 1962 1965 1968 1971

0.55 0.25 0.20 0.40 0.40 0.40

aThe restricted

0.20 0.40 0.60 0.00 0.60 0.60

5.673 3.511 3.397 4.577 5.582 5.617

parameter

-0.528 -0.131 -0.062 -0.406 -0.167 -0.158

price data).

(5)

b (6)

- 149.53 - 178.96 -221.36 - 365.46 - 349.89 -238.29

-0.188 -0.136 -0.118 -0.123 - 0.208 -0.217

values are 0, =0.247,

h, =0.234,

%I>%,> B restricted”

Loglikelihood

Loglikelihood

(7)

(8)

Obs.

- 152.05 - 179.35 -221.45 - 367.78 -351.28 - 239.42

-152.33 -179.63 -221.81 - 368.37 - 352.83 -240.31

49 57 64 108 104 72

fi= -0.158.

Table 8 Cross-sectional Unrestricted

estimates

cross-sectional

Year 1956 1959 1962 1965 1968 1971

0.45 0.20 0.20 0.35 0.30 0.30

aThe restricted

0.40 0.80 1.20 0.80 0.80 0.80

4.813 3.259 3.356 4.143 4.666 4.690

parameter

-0.357 -0.067 -0.018 -0.081 -0.105 -0.101

(retirement

estimates

adjusted

price data).

0, and O2 restricted”

%I>0, B restricted”

Loglikelihood (5)

p (6)

Loglikelihood (7)

Loglikelihood (8)

Obs.

~ 144.65 - 171.28 -211.53 - 349.18 - 333.56 -221.15

-0.120 - 0.088 - 0.079 -0.081 -0.114 -0.113

- 147.05 - 171.31 -211.63 - 350.83 - 334.06 -221.73

-147.19 -171.42 -212.02 -351.41 - 335.92 -222.70

49 57 64 108 104 72

values are 0, = 0.212, %2= 0.714, /3= - 0.094

C.R. H&en

392

and F.C. Wyko& Estimation of economic depreciation

in columns (l)-(4) and range between zero and one for every year except 1962 (for retired price data). These values imply the age-price profiles are convex. However, it must be noted that the likelihood function was fairly flat in the vicinity of the point estimates, so that non-convexity cannot be ruled out. The likelihood (13) is shown in column (5) of tables 7 and 8, and it forms the basis for a modified form of the stability test. If the pooled restrictions of (11) subject to are true, then Or = 8,,, and OZ= &, and the maximization these constraints results in values of the log-likelihood which equal In&. Letting L,(Oi, 0,) denote the restricted log-likelihood, the statistic $ = -2[lnL,(O,,&)-lnL,] is distributed Furthermore,

(15)

as asymptotically chi-square with two degrees when the maximization also restricts p = &, then

A:= -2[lnL,(e,,e,,P)-InL,]

of freedom.

(16)

is asymptotically chi-square with three degrees of freedom [L,(e,,e,,p) denotes the maximized value of (13) subject to the constraint that 01, 02, and /I are equal to the pooled sample values]. The log-likelihood Z,,(e,, (3,) is shown in column (7) of tables 7 and 8, and L,(e,,e,,~) in column (8). The corresponding 2 statistics are shown in table 9. These statistics are uniformly less than the critical values, implying that structural stability cannot be rejected. Again, however, the flatness of the likelihood function is an obvious factor in interpreting the results. Thus, while the stability tests of this section are somewhat limited, they provide no evidence for the hypothesis that the pooled Box-Cox models (2) and (3) represent a major misspecification of the time variable. Furthermore,

Table 9 Test statistics

for stability

in selected years.

Retirement

adjustment

prices

Unretired

prices

Year

i:

I.:

i.f

E.f

1956 1959 1962 1965 1968 1971

4.80 0.06 0.20 3.30 1.00 0.84

5.08 0.28 0.98 4.46 4.12 2.78

5.04 0.78 0.18 4.64 2.78 2.26

5.60 1.34 1.02 5.82 5.88 4.04

Critical value

5.99

7.81

5.99

7.81

C.R. Hulten and F.C. Wykoff, Estimation

of economic depreciation

393

the stability of the coefficients over time lends support to the hypothesis that be characterized by a single number, which is depreciation can approximately invariant over time as well as over different asset ages.

4. Summary and conclusion In this paper we have developed an econometric framework for inferring economic depreciation from vintage asset prices. This framework deals with the censoring bias introduced by ignoring asset retirement and provides an alternative to the analysis-of-variance approach (an alternative which we believe is superior when the ranges of the independent variables are large, relative to the number of observations in the sample). This framework is based on the Box-Cox power transformation, and permits classical testing of the hypotheses that depreciation is geometric, linear, or one-horse shay. We then applied the BoxCox framework to a large sample of transaction prices of used buildings. While the data and retirement adjustments leave something to be desired, some tentative conclusions seem warranted: (1) It appears that depreciation patterns are accelerated vis-a-vis straightline, and perhaps also vis-a-vis the geometric form. Certainly, the pattern of depreciation does not appear to be linear or decelerated, and obviously, our results, if correct, rule out one-horse shay depreciation as well. These results for structures confirm similar findings for specific assets ~ automobiles, trucks, and tractors and for the general category of assets in manufacturing. 26 The main conflicting result to date is the decelerated pattern suggested for office buildings by Taubman and Rasche in (1969). That study assumes perfect foresight asset pricing, groups the data into only four age intervals, and makes no allowance for retirement. The Taubman and Rasche results conform to the conventional wisdom that buildings retain their full efficiency until retired, and thus depreciate like a one-horse shay. Our results, if confirmed by further research, will therefore constitute a significant departure from the prevailing point of view. (2) The depreciation rates produced by this study center around 1; percent to 33 percent per year. While the actual estimates are sensitive to the form of the retirement distribution, most of the estimates do in fact fall within thisrange which is below the 5 percent to 7 percent range employed in the tax treatment of assets, and the @ percent rate implicit in the BEA Capital Stock Studies. Since the classes of assets under consideration accounted for nearly 26Coen (1976) reports non-structures.

accelerated

depreciation

for manufacturing

structures

as well as for

394

C.R. Hulten and F.C. Wykoff, Estimation of economic depreciation

40 percent of new private non-residential potential practical importance.

structures

in 1975, this finding

is of

(3) The cross-sectional analysis suggests that there is a reasonable stability of depreciation rates over time. This result, in turn, suggests that the hypothesis of a constant rate of depreciation may be a reasonable approximation for certain types of empirical work (the estimation of net investment and capital stocks, for example). This conclusion may be surprising in light of the apparently large and abrupt changes in the tax code and in other variables affecting building values (e.g., the rate of interest). Such changes do undoubtedly take place, but our results indicate that they do not have a significant effect on depreciation. One possible reason for this is the slowness with which investors react to changes in economic variables. For example, in his study of the 1959 change in the tax code permitting accelerated depreciation, Wales found that businesses were remarkably slow in adopting the new and more advantageous depreciation practices. One final remark is in order. We intentionally have not tried to explain the reasons for the observed patterns of economic depreciation. We do not, in other words, attempt to explain why the markets for used buildings yield an approximately geometric pattern of depreciation rather than a straightline or one-horse shay pattern. We have let the market speak for itself, and merely measured market results. This is appropriate because the issue at hand, i.e., whether or not depreciation is approximately geometric, is essentially an issue of market outcome.

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