THE BIAS OF THE RSR ESTIMATOR AND THE ACCURACY OF SOME ALTERNATIVES

Transcription

1 Yale OM Workn Paper No. ICF THE BIA OF THE RR ETIMATOR AND THE ACCURACY OF OME ALTERNATIVE Wllam N. Goetzmann Yale chool of Manaement LIANG PENG Yale Economcs Department JANUARY 00 Ths paper can be downloaded wthout chare from the ocal cence Research Network Electronc Paper Collecton:

2 The Bas of the RR Estmator and the Accuracy of ome Alternatves Wllam N. Goetzmann Yale chool of Manaement Box 0800 New Haven Connectcut Emal: Lan Pen Yale Economcs Department Box 0868 New Haven Connectcut Emal: Frst Draft: May 000 Current Draft: January 00 Ths paper benefted snfcantly from the comments and suestons of John Clapp. We also thank Donald Andrews, tefano Athnasouls, Bradford Case, Davd Geltner, Roer Ibbotson, Peter C B Phllps, Robert hller, Matthew peel, Lu Zhen, and partcpants at the 00 Amercan Real Estate and Urban Economcs meetn for helpful dscussons and comments. We thank Karl Case and Robert hller for provdn us hstorcal data of house prces. All errors are ours alone.

3 Abstract: Ths paper analyzes the mplcatons of cross-sectonal heteroskedastcty n repeat sales reresson (RR). RR estmators are essentally eometrc averaes of ndvdual asset returns because of the loarthmc transformaton of prce relatves. We show that the cross sectonal varance of asset returns affects the mantude of bas n the averae return estmate for that perod, whle reducn the bas for the surroundn perods. It s not easy to use an approxmaton method to correct the bas problem. We suest a maxmum-lkelhood alternatve to the RR that drectly estmates ndex returns that are analoous to the RR estmators but are arthmetc averaes of ndvdual returns. mulatons show that these estmators are robust to tme-varyn crosssectonal varance and may be more accurate than RR and some alternatve methods of RR. Key words: Repeat sales estmators, Real estate ndex, mulaton

4 The Bas of the RR Estmator and the Accuracy of ome Alternatves I. Introducton The repeat sales reresson (RR) and ts varants are wdely used to nfer returns of equalwehted portfolos of assets throuh tme. Most applcatons of RR have been n the area of home prce ndex estmaton. Indeed, local home ndces constructed wth the RR are becomn the benchmarks for home apprasal -- the RR allows a rapd-web-based home prce estmate that can be used for quck home mortae assessment and approval. Althouh t s now becomn a pervasve tool for credt analyss, the RR has some well-known econometrc flaws. One well known problem of the RR estmators s that they are based downwards from actual portfolo returns, whch obvously s not desrable because the most common use of any ndex may be to estmate the current value of ts underlyn portfolo. Whle equal-wehted portfolos of assets have returns that are arthmetc averaes of cross-sectonal ndvdual asset returns, the repeat sales estmators are essentally cross-sectonal eometrc averaes. Because of Jensen's nequalty, the loarthmc transformaton of the prce relatves used as a dependent varable n the repeat-sales reresson results n a bas -- the RR averaes los rather than takes a lo of an averae. Thus after ettn rd of the lo, the RR estmators are eometrc averaes nstead of arthmetc averaes. Three methods have been suested to address the bas problem. hller (99) proposes arthmetc-averae prce estmators for equal-wehted and value-wehted portfolos. Goetzmann (99) proposes a method that approxmates the arthmetc means ven RR estmators, under the assumpton that asset returns n each perod are lonormally dstrbuted and the cross-sectonal varance s constant over tme. In another attempt toward unbased estmators, Goetzmann and

5 Geltner propose a non-lnear method that mnmzes the sum of squared resduals drectly wthout takn los frst. Thouh the bas problem of RR s well known, ts source and mantude may not be well understood by many researchers and practtoners. In ths paper, we nterpret RR estmators as sample statstcs, and show how they are smultaneously determned n the reresson and how they actually mmc cross-sectonal eometrc sample means. pecfcally, we nterpret each RR estmator as a eometrc averae of proxes of ndvdual snle-perod asset returns. As a result, we are able to explctly decompose the bas of RR estmators nto two components and study them separately. Our analyss shows that the two components of the bas are respectvely determned by two dfferent mpacts of the loarthmc transformaton of the prce relatves: the drect mpact and the seral mpact. These two mpacts push RR coeffcents toward opposte drectons. pecfcally, the drect mpact makes RR coeffcents based downwards, whle the seral mpact makes them based upwards. The actual bas of a repeat sales estmator for one specfc tme perod s jontly determned by the sum of these two mpacts n that perod. We show that the mantude of the actual bas of RR estmators may not be unform from perod to perod. In each tme perod, the mantude of both drect mpact and seral mpact of loarthmc transformaton s enerally dfferent. The mantude of the drect mpact s related to the cross-sectonal sample varance of ndvdual asset returns n that perod, whle the mantude of the seral mpact s related to the sample varances n surroundn perods. Therefore, the mantude of the actual bas s enerally dfferent throuh tme, snce the sample varances of ndvdual asset returns are usually dfferent throuh tme. Consequently, mantude of the bas n the RR estmator s predctable to some extent. For example, for tme perods wth larer cross-

6 sectonal varance of ndvdual asset returns, the RR estmator tends to be more downwards based. At the same tme, the performance of the approxmaton method proposed by Goetzmann (99) s also predctable. Ths method would compensate for the bas, nsuffcently for tme perods wth larer varances whle more than enouh for tme perods wth smaller varances. We use smulatons to show such patterns for RR and the approxmaton method. We propose a new approach to mtate the bas problem of the RR estmators. The new arthmetc repeat sales estmators proposed here are unbased, and have a natural nterpretaton as equal-wehted averaes of ndvdual snle-perod asset returns. Wth smulatons, we examne the performance of ths new method toether wth other alternatve RR approaches. The smulaton results suest that the arthmetc repeat sales estmators we propose may be more accurate than RR and other alternatves. The paper s oranzed as follows. ecton nterprets RR estmators as sample statstcs, and shows that they are essentally eometrc averaes of ndvdual returns or ther proxes. ecton decomposes the bas nto two components and nvestates the determnaton of each. It shows that the mantude of the bas of RR estmators s not unform from perod to perod. It also predcts patterns of RR bas and performance of the approxmaton method. ecton proposes unbased repeat sales estmators that are analoous wth RR estmators but are arthmetc averaes. It also provdes comparson between the unbased estmators wth the arthmetc-mean repeat-sale estmators by hller (99), and shows the feasblty of the calculaton of the unbased estmators as well. ecton 4 uses smulatons to test our predctons of the behavor of RR estmators and the performance of the unbased estmators we propose and other alternatves. ecton 5 concludes.

7 II. RR Estmators as Geometrc Means II. RR estmators The repeat sales reresson estmates the return of an equal-wehted portfolo of assets over each perod n tme. Assume n total there are N observatons of repeat sales of ndvdual assets numbered from = to = N. Each observaton, say observaton, conssts of the tme of frst sale b and the prce B, the tme of second sale s and the prce. Denote by H the holdn nterval of observaton, whch conssts of all tme perods later than b and no later than s. Thus the lenth of holdn nterval for observaton, denoted by τ, equals s b. We suppose that there are T + perods numbered from t = 0 to t = T. For each observaton, we defne the compound return and the lo compound return as, and y lo = lo lo B B. Denote by Thus r, t the lo ross return of the asset correspondn to observaton n perod t. P, r, t lo P, t t r t t H y =,. () Denote by P, the value of the portfolo (market) at the end of tme perod t. We defne m t t as the ross return of ndex portfolo for tme perod t, and µ as lo( ). t t P m, t t and t lo t = lo Pm, t lo Pm, t Pm, t µ. The RR assumes that 4

8 r, t µ t + ε, t =, () where the error term s assumed..d. normally dstrbuted. From equaton () and (), y = t + ε, t t H t H µ. () } T t t= Equaton () provdes condtons for dentfyn maxmum lkelhood estmators { µ. The RR estmators are calculated accordn to ( X Ω X ) X Ω Y µ =. (4) where X, Y, and Ω are defned as follows. The X s a N by T dummy matrx whose th row corresponds to the th observaton and t th column corresponds to tme perod t. In the th row, the frst nonzero dummy appears n the column that corresponds to the tme perod mmedately after the buy perod, and the last nonzero dummy appears n ts sale perod. Between are nonzero dummes. Other elements n ths row are zero. For nstance, f a asset was purchased at tme and sold at tme 4, and T = 5, the ts correspondn row s (0,0,,,0). The Ω s a N by N daonal matrx wth th daonal element s τ,.e. the lenth of the holdn nterval. The Y s a N by matrx wth whose th element s y. The bases of RR estmators resultn from the loarthmc transformaton of the prce relatves are dscussed n Goetzmann (99) who uses a one-perod example to show how the loarthmc transformaton makes RR estmators based downwards. I.e. the RR estmator s expressed as: µ = ( ) Y or the smple averae of the elements n the loed prce relatve vector Y. Because the lo functon s concave, Jensen's nequalty mples that the averae of the los s less than the lo of 5

9 the averae, when there s any varance n the data. Thus, f the elements n the Y vector dffer at all, µ s a based estmate of the value lo( + m), where m s the smple return ( Pm, Pm,0) Pm, 0, and m, 0 P and P m, are the ntal and termnal values of the ndex over the snle perod. That s N N + + lo lo. N = B N = B Under the assumpton that the property returns n each perod are lonormally dstrbuted, Goetzmann (99) proposes a method to correct the bas of RR and approxmate the return of market ndex for each perod. lo( ) t µ t + var( εt ) (5) The var( ε t ) term s the cross-sectonal varance of the ross returns of ndvdual assets at tme t. Althouh ths method works well n smulatons, the bas of RR turns out to be more complex when there are more perods. II. Illustraton To nvestate the bas of RR estmators more thorouhly, we nterpret the RR estmators sample statstcs of repeat-sale coeffcent observatons. Consder a data set consstn of three repeat-sale observatons and four tme perods numbered from 0 to. nce perod 0 s the base perod, there are two ndex returns to estmate, correspondn to perod and. The frst two observatons respectvely cover perod and. The thrd observaton covers both perod and. Thus by assumn..d. normally dstrbuted errors, we have 0 X = 0, Y y = y y lo( = lo( lo( B) B ), and B ) Ω =

10 From equaton (4), we have ( X X ) = X Ω Y In the example, we are able to wrte down the equatons explctly. Ω µ, (6) ( + ) µ = y + ( y µ ) ( + ) µ = y + ( y µ ) (7) From equaton (7), we are easly able to nterpret the RR estmators of ndex returns as eometrc averaes of ndvdual snle perod returns or ther proxes. µ = µ = [ y + ( y µ )] ( + ) [ y + ( y µ ) ] ( + ) (8) For example, the RR estmator of lo ndex return for the frst tme perod µ s µ = y + ( y µ ). (9) Obvously t s a wehted averae wth the wehts nversely proportonal to the assets' holdn perods. Ths s the motvaton for the GL verson of the RR, whch wehts observatons by the root nverse of the holdn perod. Therefore the estmator of actual (not-lo) ndex return s = exp = exp y = ( µ ) ( ) exp ( y µ ). (0) The, as defned earler, s = exp( y ) = B, the not-lo compound return for repeat sale observaton. Clearly the RR estmator of actual ndex return n perod s eometrc averae of two numbers. The frst number,, s an ndvdual return n perod. The second number,, 7

11 s a proxy of ndvdual return n perod. The thrd repeat sale observaton covers all two perods and s a compound return. After subtractn the component of the compound return that corresponds to the second perod, one can et a snle perod return n perod. Thouh ths component s unknown, the estmator for ndex return n the second perod obvously s a proxy of t. Thus s a proxy of an ndvdual return n perod. Why s the RR estmator of ndex return n perod an averae of and? Clearly t s because the frst and the thrd repeat sale observaton, and no other observaton, cover perod. Thus, both of these two observatons, and maybe only they, drectly provde useful nformaton about the ndex return n perod. Another queston s why RR ves these two observatons dfferent wehts? Notce that covers only perod, whle covers both perods and then contans both nformaton and nose for all two perods. o ntutvely contans much more nose, and the term s not an actual ndvdual return but just a proxy. Thus, t has smaller weht. From equaton (0), the RR estmator of ndex return n perod,, s a cross-sectonal sample eometrc mean of all avalable ndvdual returns n perod or ther proxes. Actually, all RR estmators can always be wrtten as wehted eometrc averaes of ndvdual snle-perod returns or proxes of them. It s also obvous that RR estmators of ndex returns for dfferent perods always depend on each other so that the loarthmc transformaton at one perod would have drect mpact on that perod's RR estmator and seral mpacts on other perods' estmators. 8

12 III. Bas Decomposton III. Bas components Interpretn the RR estmator as a sample eometrc averae facltates the nvestaton of ts bas. pecfcally, we are able to decompose the bas, defned as the dfference between the } T t t= RR estmator and an unbased estmaton of ndex return, nto two parts. Denote by { the unbased estmators of ndex returns, whch are analoous to the RR estmators but are arthmetc averaes, nstead of eometrc averaes, of ndvdual returns or ther proxes. Therefore they drectly correspond to actual ndex returns. We wll talk about the estmaton of the unbased estmators n secton four. In the example, suppose the unbased estmator for ndex return n tme perod s, we are able to construct a unbased proxy of a snle perod return correspondn to thrd observaton as. Thus the unbased estmator for ndex return n perod,, would be the arthmetc averae of and. ( ) = + () Decompose the dfference between the RR estmator and the unbased estmator nto two parts. = = + + ( ) ( ) ( ) ( ) ( ) + ( ) () Jensen's nequalty mples that 9

13 ( ) ( ) 0 +. () Ths component of the bas s essentally the dfference between a eometrc averae and an arthmetc averae, whch s always postve as lon as all numbers are not the same. It can be corrected usn equaton (5). However, ths s not the end of the story. The other component of the bas s ( ) ( ). (4) It s the dfference between the eometrc mean constructed by subtractn the unbased estmator for perod ndex return and the eometrc mean constructed by subtractn the RR estmator. If the unbased estmator for perod s larer than the RR estmator, ths component of the bas s postve, otherwse t s neatve. Clearly the drecton and the mantude of the bas of RR estmator for perod helps to determne the drecton and mantude of the second component of the bas for RR estmator for perod. At the same tme, for the same reason, the drecton and mantude of the bas for the RR estmator for perod also helps to determne the drecton and mantude of the bas for RR estmators n perod. We call the frst component of the bas for the RR estmator n each perod the drect mpact of the loarthmc transformaton for that perod. We call the second component of the bas n each perod as the seral mpact of loarthmc transformaton for other perods because t s determned by the bas of RR estmators for other perods. These two mpacts tend to offset each other snce RR estmators are smultaneously determned and depend on each other. For nstance, suppose the drect mpact n perod s stron enouh that the RR estmator n that perod s based downward: <, then the second component of the RR bas s neatve. 0

14 ( ) ( ) 0 <. At the same tme, the frst component s always postve. Therefore t s no loner clear f the RR estmator for perod s lower than the unbased arthmetc mean for ths perod or not. In eneral, the loarthmc transformaton's drect mpact n one perod tends to make ths perod's RR estmator lower than the unbased arthmetc mean; at the same tme, ts seral mpact on other perods' RR estmators tend to push them upward. Thus the drecton and the mantude of the bas of RR estmators becomes ambuous n the mult-perod case, whch s obvously dfferent from the one-perod case where RR estmator s smply based down. We expect on averae that the bas s neatve, but t may not be strahtforward to correct. III. Determnaton of the mantude of bas The frst component of the bas of RR s the dfference between a sample arthmetc mean and a sample eometrc mean. Jensen's nequalty tells us that the dfference of E (lo( r)) and lo( E ( r)) depends on the populaton varance of the random varable r. However, when we estmate the market ndex, we always work wth fnte samples and are not able to observe the populaton varance. We want to nvestate what determnes the dfference between the sample arthmetc mean and the sample eometrc mean. pecfcally, we want to make sure whch one actually determnes ths dfference: the populaton varance of the underlyn data eneratn dstrbuton or just the sample varance. uppose we have dfferent samples of cross-sectonal returns enerated from the same dstrbuton. Are the dfferences between the lo equal-wehted ndex returns (lo of arthmetc averaes of ndvdual returns) and the RR estmators (averaes of the lo ndvdual returns or eometrc averaes of not-lo ndvdual returns) unform for all these samples, or they depend on the varance of the sample?

15 Ths queston s mportant because f the dfference between arthmetc mean and eometrc mean s determned by the sample varance, the mantude of the drect mpact component of the RR bas s potentally predctable from the data no matter what the actual underlyn process may be. For perods the varances of sample returns are larer, the mantude of the frst component of the bas for RR estmators tend to be larer. Thus the RR estmators for those perods tend to be based downward. Goetzmann (99) uses a snle number to correct the bas for all perods. Our prevous example shows the correcton would be nsuffcent for the perods wth larer sample varances, and probably would be too much for other perods wth smaller sample varances. We use a smple experment to show that sample varance, not the populaton varance, determnes the dfference between an arthmetc averae and a eometrc averae of sample returns. We randomly sample from a lonormal dstrbuton 00 tmes. Each sample conssts of 0 observatons, and all observatons (say, ndvdual ross returns) are loed terms. The mean of the lonormal dstrbuton we use s (.0 before takn lo). The populaton standard devaton s arbtrarly chosen as 0.8 (. before takn lo). We calculate the RR estmator, whch s just the sample mean of these 0 ndvdual observatons (loed). We can also easly et the arthmetc averae of these 0 ndvdual returns (not loed). Thus we can et the dfference between the loed value of the arthmetc averae and the RR estmator for each sample. Ths dfference s what Goetzmann (99) ntended to adjust wth approxmaton. Fure clearly shows that the dfference s almost perfectly related to sample varance of these 0 observatons, and the slope of the straht lne s just 0.5, whch confrms the formula n Goetzmann (99). Clearly t s the sample varance, not the populaton varance, that determnes the mantude of the frst component of the bas of RR estmators. Thus, for each tme perod, the larer s the cross-sectonal sample varance, the ber s the dfference between eometrc

16 mean and arthmetc mean,.e., the larer the mantude of the bas for RR estmator for that perod tend to be. For the approxmaton method, snce estmators for all perods are adjusted wth one snle number, for perods wth larer sample varances, adjustments may be nsuffcent; for perods wth smaller sample varances, adjustments tend to be too much. Ths phenomenon has been found n our smulaton results. IV. Unbased RR Estmators IV. Unbased RR Estmators Ths secton proposes a method to calculate the unbased estmators that we use to compare wth the RR estmators n earler secton. The unbased estmators have natural nterpretaton as arthmetc means of the cross-sectonal returns of assets or ther proxes. We stll denote by { } T t the unbased arthmetc mean estmators. The unbased estmator of ndex return n tme t = perod t, The t, may be expressed as. (5) t = w s w N t s H, s t N t H s defned as the holdn nterval of observaton, whch conssts of all tme perods later than the purchase tme and no later than the sale tme. Thus s H, s t are all tme perods, except perod t, that belon to the holdn nterval of repeat sale observaton. The N t s the set of all repeat sale observatons that contan perod t n ther holdn ntervals. The n t s the

17 number of observatons that belon to N t. The w term s the weht of the repeat observaton. We could follow the RR and let w equal to. τ Obvously the unbased estmators are analoous to the RR estmators. Frst of all, the unbased estmators are cross-sectonal averaes of all avalable ndvdual returns or ther proxes n correspondn tme perods, just lke RR estmators. econd, they are determned smultaneously and thus depend on each other, also lke RR estmators. However, the unbased estmators are arthmetc averaes of ndvdual asset returns thus strctly correspond to actual ndex returns, whle the RR estmators are eometrc averaes. One specal advantae of the unbased estmators s that when all assets trade n all perods,.e., there s no data mssn, the unbased estmators exactly equal the actual equal-wehted ndex returns. Rearrane expresson (5) and let w equal to, we et τ = N τ t N t τ s H We defne u ( ) = s, and a N by vector ( ) s H U ( ) u u = u N u ( ) ( ) ( ) ( ) N. s. (6) U and a N by vector I., and I =.. Equaton (6) s equvalent to X ( ) t,. Ω I = X t,. Ω U, 4

18 where the X t,. term s the row t of the matrx X. Thus the unbased estmators for ndex returns for all perods are determned by X ( ) I X Ω = Ω U, or Ω ( ( ) I U = 0 X. (7) In equaton (7), there are T equatons and T unknown t. The soluton of these equatons s the unbased estmators. Thouh the equatons are not lnear, t s feasble to search for the soluton va maxmum-lkelhood. At the same tme, the unbased estmators are solutons to the optmzaton problem: ( ( I U ) Ω XX Ω ( I U ( ) { } mn. (8) Thus quadratc-searchn technoloy can help to calculate the unbased estmator. The RR estmators are shown to have other problems. For example, there are the sample selectvty (Clapp and Gaccotto [99]) and unobserved fx-ups between sales (Goetzmann and peel [995] and Clapp and Gaccotto [999]). The RR has many varants to address these problems. Thouh the unbased estmators we propose ntend to address the bas problem only, n prncple they could have varants that address not only the bas problem but also other problems at the same tme. IV. Comparson wth AR by hller We use the same small data set to compare the unbased estmator wth the nstrumental varable arthmetc-mean repeat sale estmator (AR) proposed by hller (99). The AR estmators by hller (99) are for recprocal ndex levels nstead of returns. Here we translate the recprocal ndex level estmators nto return estmators to facltate the comparson. 5

19 6 Denote by t the AR estmator of equal-wehted ndex return n tme perod t. The estmator-determnn equatons of the AR for the smple data set are ( ) ( ) = + + = +. Rearrane the equatons, we have + = + =. We express the AR estmators n an economcally sensble way as follows: ( ) + =, ( ) = + + =. Notce both and are snle perod ndvdual returns, n perod and respectvely, whle the s a two-perod compound return. Obvously the s a proxy of snle-perod return n perod and the s a proxy of snle-perod return n perod. The AR estmators are arthmetc averaes of ndvdual snle-perod returns. pecfcally, the ndex return estmator n perod s averae of and, and the ndex return estmator n perod s averae of and. As mentoned earler, the unbased estmators proposed n ths artcle for the data set are ( ) + =, ( ) + =. Obvously they are also arthmetc averaes.

20 Thouh both the AR estmators and the unbased estmators are arthmetc averaes of ndvdual snle-perod returns or ther proxes, the example shows two nterestn dfferences between them. Frst, n the AR estmators the proxy for perod return s, whle n the unbased estmators t s. If s a more accurate estmator of ndex return n perod than, then we expect to be a better proxy than. econd, the wehts of assets are consstent n the unbased estmators. The frst asset, whch provdes two snle-perod returns and, receves two thrd weht n both perods; whle the second asset, whch provdes one compound return, receves one thrd weht n both perods. The wehts of assets n the AR estmators are dfferent n two perods: n perod two assets receve equal wehts, whle n perod the weht of the frst asset s almost two tmes heaver than that of the second asset. In short, based on the example the unbased estmators seem to be more sensble than the AR estmators are n constructn the proxes of ndvdual returns and wehtn assets. However, the AR estmators have a reat advantae over the unbased estmators: they are easy to calculate. IV. Feasblty of Calculaton Thouh the calculaton of the unbased estmators s not as easy as that of the RR or the AR estmators, t s stll feasble. To show ths, we use the same data used by Case and hller (987, 989) and hller (99) to estmate the equal-wehted quarterly prce ndex for Dallas from 970- to The data have 6,669 repeat sale observatons. Let the ndex value at 970- be, there are 65 ndex returns to estmate. The software we use s -plus, and the computer s a publc-shared Unx server n Yale Internatonal Center for Fnance. The calculaton of the unbased estmators s equvalent to searchn for { } that mnmzes the objectve functon: ( ( I U ) Ω XX Ω ( I U ( ). 7

21 We use an extremely smple but obvously not-effcent search procedure. Frst, we calculate the RR estmators. econd, we use the RR estmators as the startn pont, randomly draw a new pont wthn a small reon around the startn pont. We do so untl the new pont s better than the startn pont,.e. t reduces the value of the objectve functon, then use the new pont as the startn pont for next run of searchn. Ths procedure s repeated untl the value of the objectve functon s small enouh. After several hours' searchn, the value of the objectve functon s reduced from 8.5, whch corresponds to the RR estmators, to.5. Takn account that more effcent search procedure and more powerful computer could be dedcated to the estmaton, the feasblty of the calculaton of the unbased estmators s obvous. Fure shows the RR, the AR, and the unbased estmators for the quarterly equalwehted prce ndex for Dallas. Obvously these ndces are dfferent. However, lttle can be sad about the accuracy of each method, whch s nvestated n next secton. V. mulaton Test V. Methods The oal of the smulaton s twofold. Frst, we want to test our predctons about the RR estmators and the approxmaton method by Goetzmann (99). We predct that the RR estmators tend to be based downward more for perods wth larer cross-sectonal varances of asset returns. We also predct that the adjustment accordn to Goetzmann (99) may be nsuffcent for perods wth larer varances but too much for perods wth smaller varances. econd, we want to test the performance of the unbased estmators proposed here, toether wth some other alternatve estmators for RR. pecfcally, we test performance of fve dfferent 8

22 estmators. They are RR estmators, adjusted RR estmators accordn to Goetzmann (99), AR suested by hller (99), nonlnear drect estmators suested by Goetzmann and Geltner (999), and the unbased arthmetc mean RR proposed n ths artcle. The RR estmators, the adjusted RR estmator by approxmaton, and the AR estmators are well known. The drect estmators suested by Goetzmann and Geltner come from solvn follown problem: mnt { } N w t t = = s s H. (9) Equaton (9) can be rewrtten as mnt { t } t = ( ( I U ) W ( I U ( ). (0) It s nterestn to notce that the optmzaton problem s very smlar to that n equaton (8), whch our arthmetc repeat sales estmators solve. The only dfference s the weht matrx n the mddle. Whle our method uses Ω X X Ω, Goetzmann and Geltner use W. A nce property of the drect method s that, f let the weht ben constant for all observatons, the drect estmators would mnmze the mean-squared-error (ME). In the smulatons, we let the weht for repeatsale observaton be /τ for the drect method. V. mulaton steps In each smulaton, we construct the "actual" market frst. The follown steps are performed:. pecfy the number of assets N, and lenth of tme horzon T.. Randomly draw the underlyn marker returns (lo term) for all perods from a normal dstrbuton. The dstrbuton has mean correspondn to 0% ross return and standard 9

23 devaton correspondn to 7%, whch are reasonable numbers for fnancal market annual returns.. For each and every tme perod, randomly draw N ndvdual asset returns from a normal dstrbuton wth mean equal to that perod's underlyn market return and varance assned by us. After that, we have a N by T panel data set, whch s treated as a perfect T -perod sample data set from a N -asset market. 4. From ths complete data set, construct the actual equal-wehted market ndex, whch s the benchmark used to test estmators' accuracy. After constructn the actual market, we are able to construct repeat sale data from the complete data set. Follown steps are repeated for 00 tmes for each actual market data we create:. Randomly draw two dfferent dates for each asset, and calculate the compound returns between them. Then, nstead of havn the perfect panel data, we now have only one repeat sale observaton for each asset.. Use all fve methods to estmate the ndex returns based on ths repeat sale data set. For two nonlnear estmators, drect estmators and the unbased RR estmators, we use RR estmators as startn pont for search.. For each method, calculate the estmators' percent devatons from the actual ndex return n each perod. For example, f an estmated return s. but the real market return s.0, the percent devaton s 0%. We also calculate all methods' mean squared errors over all perods. We choose N = 0, T =. We use the short tme horzon T = because n each round of smulaton we need to search for two knds of nonlnear estmators 00 tmes. The cross-sectonal varances for three perods are (0.0, 0.0, 0.0), (0.0, 0.08, 0.0), (0.08, 0.0, 0.08), and (0.0, 0

24 0.04, 0.08). We purposely control the varances so that the actual market wll exhbt specfc patterns of tme-varyn cross-sectonal varance. For example, by settn the varances as 0.0, 0.08, and 0.0, we are able to test f RR estmators are more based downward n the second perod. We are also nterested n the performance of fve dfferent methods n each scenaro. For each settn of varances, we run rounds of smulatons. Each round has ts own actual market, and conssts of 00 dfferent repeat sale data sets enerated from the same actual market. Thus there are rounds of smulatons overall. V. mulaton results Table presents smulaton results for four dfferent varances specfcatons. In each settn, the table presents the percentae devaton of each method's estmator n each perod (medan of 00 smulatons), and each method's ME for all three perods (medan of 00 smulatons). In each tme perod, the two smallest percentae devatons are n bold. Fure and fure 4 plot the percentae devatons of RR estmators and adjusted RR estmators; fure 5 and fure 6 plot the percentae devaton of the AR and the unbased RR estmators, all under the settns that cross-sectonal varances are 0.0, 0.08, and 0.0. We have four major fndns from the smulaton results:. As we predct, the RR estmators tend to be more based down n perods wth larer varances. After adjusted accordn to Goetzmann (99), the estmators tend to be based down for perods wth larer varances, but tend to be hher than the actual ndex returns for perods wth smaller varances. These are shown n Table, fure, and fure 4. For example, n fure, the second perod has larer varance. Clearly the RR estmators are based down much more n the second perod than n other two perods. In fure 4, after adjusted accordn to Goetzmann (99), the RR estmators are based down n the second

25 perod, but enerally not based down n other perods. Ths confrms that the adjustment may be nsuffcent for perods wth larer varances, whle may be too much for perods wth smaller varances.. The unbased estmators and the AR estmators seem to be robust to tme-varyn crosssectonal varance. There s no obvous bas-pattern for these two methods, as can be seen n fure 5 and fure 6. At the same tme, the unbased estmators seem to be more accurate than the RR estmators, the adjusted estmators, and the drect estmators, and may be more the AR estmators too. The unbased estmators enerally have smaller percentae devatons from actual ndex returns from perod to perod, whch s shown n Table and the fure 6.. The drect estmators of Goetzmann and Geltner tend to have larer percent devatons from the actual ndex returns. Ths may be partally caused by the dffculty of searchn for the lobal mnmum value. However, ths method tends to have small ME. VI. Concluson We nterpret the RR estmators as sample statstcs and show that they are essentally eometrc averaes of ndvdual returns or ther proxes. At the same tme, t s clear that the RR estmators are jontly determned and depend on each other. We decompose the bas of a RR estmator nto two parts. The two components of the bas are respectvely determned by two dfferent mpacts of the loarthmc transformaton of the prce relatves. One we term the drect mpact and the other we term the seral mpact. These two mpacts push the bas toward opposte drectons. pecfcally, the drect mpact pushes RR estmators downwards, whle the seral mpact pushes them upwards. The actual bas of a repeat sales estmator for one specfc tme perod s jontly determned by the summary of these two mpacts n that perod.

26 We show that the mantude of the drect mpact s postvely related to the cross-sectonal sample varance of ndvdual returns n the prevaln perod, whle the mantude of the seral mpact s neatvely related to the sample varances n surroundn perods. Therefore, the mantude of the actual bas enerally vares throuh tme snce the sample varances of ndvdual asset returns are usually dfferent throuh tme. At the same tme, the bas mantude of the RR estmators and the accuracy of the approxmaton method by Goetzmann (99) are predctable to some extent. The RR estmator tends to be based down more n perods wth larer crosssectonal varances of ndvdual asset returns and less n perods wth smaller cross-sectonal varances. The approxmaton method would nsuffcently compensate for the bas n perods wth larer varances whle more than enouh for tme perods wth smaller varances. We propose unbased repeat sales estmators that are analoous to the RR estmators but are arthmetc averaes of ndvdual returns nstead of eometrc averaes of them. The unbased estmators strctly correspond to the actual ndex returns and there s no bas caused by the dfference between eometrc means and arthmetc means. When there s no data mssn,.e. all assets trade n all perods, the unbased repeat sale estmators would exactly equal the actual ndex returns. We use smulatons to test our predctons of the behavor of RR estmators and the adjusted RR estmators, and the performance of the unbased estmators proposed n ths paper toether wth other alternatve methods. We construct artfcal "actual" market data, and create repeat sale observatons from them. We estmate the "actual" ndex returns wth dfferent methods and calculate the devatons of dfferent estmators from the "actual" ndex returns. The smulaton results confrm our predctons. They show that RR estmators tend to be more based down n perods wth larer sample varances. After adjusted accordn to Goetzmann (99), the

27 estmators tend to be based down for perods wth larer varance, but tend to be hher than the actual ndex returns for perods wth smaller varances. The results also show that the unbased estmators are robust to tme-varyn cross-sectonal varance and may be more accurate than the RR estmators as well as some other alternatves. 4

28 References: Abraham, J.M tatstcal Bases n Transacton-Based Indces. Mmeo, Federal Home Loan Mortae Corporaton. Abraham, J.M. and W.. chauman. 99. New Evdence on Home Prces from Fredde Mac Repeat ales. AREUEA Journal 9(): -5. Baley, M.J., R.F. Muth and H.O. Nourse. 96. A Reresson Method for Real Estmate Prce Index Constructon. Journal of the Amercan tatstcal Assocaton 58: Case, B., H.O. Pollakowsk and.m. Wachter. 99. On Choosn amon House Prce Index Methodoloes. AREUEA Journal 9(): Case, B. and J.M. Quley, J. M. 99. The dynamcs of Real Estate Prces. The Revew of Economcs and tatstcs 7(): Case, K.E. and R.J. hller Prces of nle Famly Homes snce 970: New Indexes for Four Ctes. New Enland Economc Revew: Case, K.E. and R.J. hller The Effcency of the Market for nle Famly Homes. Amercan Economc Revew 79: 5-7. Case, K.E., R.J. hller and A.N. Wess. 99. Index-based Futures and Optons Markets n Real Estate. Journal of Portfolo Manaement 9: 8-9. Clapp, J.M. and C. Gaccotto. 99. Estmatn Prce Trends for Resdental Property: A Comparson of Repeat ales and Assessed Value Methods. Journal of Real Estate Fnance and Economcs 5(4): Clapp, J.M. and C. Gaccotto Revsons n Repeat-ales Indexes: Here Today, Gone Tomorrow? Real Estate Economcs 7(): Dombrow, J., J.R. Knht and C.F. rmans Areaton Bas n Repeat-ales Indces. Journal of Real Estate Fnance and Economcs 4(): Geltner, D Bas and Precson of Estmates of Housn Investment Rsk Based on Repeat- ales Indces: A mulaton Analyss. Journal of Real Estate Fnance and Economcs 4(): Goetzmann, W.N. 99. The Accuracy of Real Estate Indces: Repeat ale Estmators. Journal of Real Estate Fnance and Economcs 5(): 5-5. Goetzmann, W.N. 99. Accountn for Taste: Art and the Fnancal Markets over Three Centures. Amercan Economc Revew 8(5):

29 Goetzmann, W.N. and M. peel Non-temporal Components of Resdental Real Estate Apprecaton. Revew of Economcs and tatstcs 77(): Goetzmann, W.N. and M. peel A patal Model of Housn Returns and Nehborhood ubsttutablty. Journal of Real Estate Fnance and Economcs 4(): -. Kuo, C Housn Prce Dynamcs and the Valuaton of Mortae Default Optons. Journal of Housn Economcs 5(): Mark, J.H. and M.A. Goldber Alternatve Housn Prce Index: An Evaluaton. AREUEA Journal (): Palmqust, R.B. 98. Measurn Envronmental Effects on Property Values wthout Hedonc Prces. Journal of Urban Economcs (): -47. Pollakowsk, H.O. and.m. Wachter Effects of Land Use Constrants on Housn Prces. Land Economcs 66(): 5-4. hller, R.J. 99 Arthmetc Repeat ales Prce Estmators. Journal of Housn Economcs (): 0-6. hller, R.J. 99a. Macro Markets. Clarendon Press: Oxford. hller, R.J. 99b. Measurn Asset Values for Cash ettlement n Dervatve Markets: Hedonc Repeated Measures Indces and Perpetual Futures. Journal of Fnance 48 ():

30 Table Medan of devaton for each perod (n percentae) Methods The varances of ndvdual assets for three perods are 0.0, 0.0, and 0.0 mulaton round mulaton round mulaton round ME Medan of devaton for each ME Medan of devaton for each ME perod (n percentae) perod (n percentae) RR Adjusted RR AR Drect Method Unbased The varances of ndvdual assets for three perods are 0.0, 0.08, and 0.0 mulaton round mulaton round mulaton round Medan of devaton for each perod (n percentae) Methods ME Medan of devaton for each perod (n percentae) ME Medan of devaton for each perod (n percentae) RR Adjusted RR AR Drect Method Unbased The varances of ndvdual assets for three perods are 0.08, 0.0, and 0.08 mulaton round mulaton round mulaton round Medan of devaton for each perod (n percentae) Methods ME Medan of devaton for each perod (n percentae) ME Medan of devaton for each perod (n percentae) RR Adjusted RR AR Drect Method Unbased The varances of ndvdual assets for three perods are 0.0, 0.04, and 0.08 mulaton round mulaton round mulaton round Medan of devaton for each perod (n percentae) Methods ME Medan of devaton for each perod (n percentae) Medan of devaton for each perod (n percentae) RR Adjusted RR AR Drect Method Unbased ME ME ME ME In each round of smulaton, complete data of ndvdual returns are enerated by drawn 00 ndvdual asset returns each perod from a dstrbuton wth mean equal to the ndex return for that perod, whch s randomly enerated too, and wth varances specfed by us. A round conssts of 00 smulatons. In each smulaton, frst we randomly enerate one repeat sale observaton for each "asset" and construct a repeat sale data set. Then we estmate the "actual " ndex returns wth dfferent methods. We calculate the percentae devaton from the "actual" ndex return n each perod for each method. We also calculate the mean squared errors (ME) for each method. The numbers n the table are medans over 00 smulatons. In each round of smulaton, the two smallest percentae devaton numbers are n bold. 7

31 Fure Reresson of Dfference to ample Varance Dfference ample varance We randomly et 00 samples from a lonormal dstrbuton. Each sample conssts of 0 observatons randomly enerated from the dstrbuton. The mean of the lonormal dstrbuton s (.0 before takn lo). The populaton standard devaton s 0.8. The "dfference" for each sample s the dfference between the averae of these 0 lo values and the lo of the averae of 0 not-lo values. The "sample varance" s the sample varance of these 0 observatons. 8

32 Fure Equal-wehted Quarterly Prce Index for Dallas 5 4 Unbased AR RR 5 4 Prce Index Year 970- to Ths fure shows the equal-wehted quarterly prce ndces for Dallas from 970- to 986- calculated wth the RR, the AR, and the unbased method. 9

33 Fure Medan of Pecent Devaton: RR estmators Medan of Percent Devaton cross-sectonal varances are 0.0, 0.08, Ths fure shows the accuracy of the RR estmators measured by the devaton (n percentae) from the "real equalwehted market ndex". Results (three lnes) are for three rounds of smulatons. Three ponts n each lne correspond to the medans of percent devaton of estmators n three perods. 0

34 Fure 4 Medan of Pecent Devaton: Adjusted RR estmators Medan of Percent Devaton cross-sectonal varances are 0.0, 0.08, 0.0 Ths fure shows the accuracy of the adjusted RR estmators, whch correct for the bas caused by Jensen's nequalty wth approxmaton proposed by Goetzmann (99), measured by the devaton (n percentae) from the "real equalwehted market ndex". Results (three lnes) are for three rounds of smulatons. Three ponts n each lne correspond to the medans of percent devaton of estmators n three perods.

35 Fure 5 Medan of Pecent Devaton: AR estmators (hller) Medan of Percent Devaton cross-sectonal varances are 0.0, 0.08, Ths fure shows the accuracy of the AR estmators by hller (99) measured by the devaton (n percentae) from the "real equal-wehted market ndex". Results (three lnes) are for three rounds of smulatons. Three ponts n each lne correspond to the medans of percent devaton of estmators n three perods.

36 Fure 6 Medan of Pecent Devaton: Unbased estmators Medan of Percent Devaton cross-sectonal varances are 0.0, 0.08, Ths fure shows the accuracy of the unbased RR estmators that we propose measured by the devaton (n percentae) from the "real equal-wehted market ndex". Results (three lnes) are for three rounds of smulatons. Three ponts n each lne correspond to the medans of percent devaton of estmators n three perods.

37 For example, Abraham and chauman (990), Case and Quley (99), Case and hller (987, 989), Case, hller, and Wess (99), Goetzmann (99), Goetzmann and peel (997), Mark and Goldber (984), Palmqust (98), Pollakowsk and Wachter (990). There has been a reat deal of research about problems of RR. For example, Abraham (990), Case, Pollakows, and Wachter (99), Clapp and Gaccotto (99, 999), Dombrow, Knht, and rmans (997), Goetzmann and peel (995), Geltner (997), Kuo (996), and hller (99a, 99b). No paper has been wrtten to propose ths method yet. 4