A Note on Robust Estimation of Repeat Sales Indexes with Serial Correlation in Asset Returns

Transcription

1 A Note on Robust Estmaton of Repeat Sales Indexes wth Seral Correlaton n Asset Returns Kathryn Graddy Department of Economcs and Internatonal Busness School Brandes Unversty (kgraddy@brandes.edu) Jonathan Hamlton Department of Economcs Unversty of Florda (hamlton@ufl.edu) September 30, 009 We thank Davd Lng and Andra Ghent and semnar audences at Baruch, McMaster and Waterloo for helpful comments. We also thank Chunrong A and Mark Watson for advce on the statstcal model. Keywords: repeat sales, heteroskedastcty, seral correlaton JEL classfcatons: C3, C9, G

2 Abstract Ths note studes the second stage of the Case-Shller repeat sales method under the assumpton of seral correlaton n the devatons from the mean one-perod returns on the underlyng ndvdual assets. We propose a flexble GLS methodology usng dummy varables for each possble duraton length n the second stage.

3 . Introducton The repeat sales methodology s an mportant technque to determne prce trends and returns for dosyncratc assets, ncludng real estate, art, and antque muscal nstruments. Baley, Muth, and Nourse [963] frst proposed the method, smply usng ordnary least squares. Case and Shller [987] developed a three-stage generalzed least squares (GLS) method. If devatons from the mean sngle-perod returns for the underlyng assets are ndependently and dentcally dstrbuted, the varance of returns grows lnearly when returns are summed over the holdng perod of an asset, whch leads to heteroskedastc errors. To correct for ths, one frst estmates OLS regressons usng dummy varables for tme perods between sales. Then, the squared resduals are regressed aganst the length of the holdng perod. Estmates from the second stage provde weghts for the thrd-stage GLS regressons. Our goal n ths paper s to study the mplcatons of non-..d. errors for the second-stage regresson and to suggest a second stage regresson that s robust to a wde range of errors. Secton detals the Case-Shller methodology and explores prevous research. Secton 3 explores dfferent assumptons regardng the asset return errors. Secton 4 apples our results to a repeat sales dataset of voln prces. Secton 5 dscusses some mplcatons and concludes our analyss.. The Basc Case-Shller Model (..d. errors on ndvdual returns) Each observaton conssts of the purchase (buy) date, b, the purchase prce, B, the sale date, s, and the sale prce, S. Defne the length of the holdng perod as S s b. Let y log be the log of the compound return on property. We can B

4 wrte ths as the sum of the returns to property n each perod between purchase and sale, or y r, t where s tb r t, P t, log P t,, and P t, s the prce of property n perod t (only observed for t = s and b ). The standard assumpton s that rt, t t, where t s ndependent and dentcally normally dstrbuted. Then, y. s s t t tb tb Case and Shller [987] assumed that log( Pt, ) Ct Ht, Nt, where Ct s the value of the ndex n perod t, H t, s the value of a random walk process for property at tme t, and Nt, s the sale-specfc random error. Ths s equvalent to wrtng the prce of property n perod T as T T P T exp t t T where T 0 only f a transacton occurs n perod T. Takng logs and dfferencng prces from two dfferent ln P ln P transactons, we obtan, T T T k t t T T k Tk Tk T. Then let T be the resdual for property n the frst-stage regresson. Hence, t T, Tk Tk E E t where s the expectaton of t,, under the assumpton that the t, are..d. Case and Shller thus suggested frst estmatng an OLS repeat sales regresson. Then, the squared resduals are regressed aganst the length of the holdng perod and a constant n the second stage regresson. Estmates from the second stage provde weghts for the thrd-stage GLS regressons. Below, we explore the theoretcal In the above case, the varance of the error term grows lnearly wth the length of the holdng perod. Under ths assumpton, one can skp the three-stage procedure and smply use s b for GLS. as the weghts

5 mplcaton of droppng the assumpton that the t are..d., and we propose a second stage regresson that s robust to non-..d. errors, usng repeat sales data on fne volns as an example. The Case and Shller method, wth varatons, s wdely used. Both OFHEO (Offce of Federal Housng Enterprse Oversght) and S&P/ Case-Shller house prce ndexes use varatons of the Case-Shller method. The OFHEO approach (see Calhoun [996] for detals) fts a quadratc equaton regressng the squared error on tme between sales and tme squared. Calhoun [996] states that, n practce, the constant term n the second-stage regresson s often negatve, whch s nconsstent wth the Case- Shller explanaton. Calhoun suggests forcng the constant to zero and re-estmatng, whch s OFHEO s approach. Case and Shller drectly estmate an arthmetc ndex but stll use the standard Case-Shller correcton to correct for heteroskedastcty. Other papers that have focused on modfcatons of the Case and Shller method nclude Qugley [995], who fts the squared resduals to a quadratc functon of elapsed tme (wthout a constant), and Qugley and Hwang [004], who model autoregresson n the errors n prce levels rather than returns. 3. Indvdual Asset Errors that Are Serally Correlated Across Perods We now drop the assumpton that return errors are..d. For Goetzmann s [99] study of repeat sales regressons usng stock market data, the..d. assumpton seems approprate. In contrast, for many asset classes studed n repeat sales regressons, prces See also Abraham and Schauman [99]. 3

6 may not adjust quckly. 3 Houses, ndvdual artworks and muscal nstruments have dosyncratc features, makng smple observatons of prces of other assets n the class only sgnals of the true prce of an asset. Tradng costs are also sgnfcant (5-6% commssons plus transactons taxes and other costs for houses n the U.S. and a 0%- 0% buyer s commsson plus a seller s commsson for art sold at aucton), and short sales are essentally mpossble. House prce data are also only avalable wth some lag (the nterval between contract date and closng date at a mnmum). Note that the statstcal ssue s whether the error term on the ndvdual asset returns s correlated between perods. In repeat sales data, only the resduals summed over several tme perods are observed. Ths prevents us from uncoverng much of the fne structure of the tme seres processes of the error returns. In what follows, we shall drop the subscrpt for the ndvdual property snce all calculatons are wth respect to a sngle property. Let the errors follow the general movng average process, k, where k [, ), t s whte nose and 0. 4 t 0 t Then k k t t 0 t t 0 s the sum of return errors over E perods, and. If the process s = k k k 0 E 0 j je j statonary, then k 0 s fnte. Thus, the frst sum equals k 0. Lettng, k, the second sum equals s s j 0 s s, whch s also fnte for 3 Shller [007] dscusses seral dependence n housng prce aggregates. Even when repeat sales ndexes ncorporate a large number of propertes, they combne data on dverse subgroups wthn the asset class (all sngle-famly homes n a large metropoltan area). Snce these submarkets may be qute thn and prces across the submarkets may not be closely lnked, seral dependence n the errors also seems qute lkely. 4 Snce AR and ARMA processes can be represented as nfnte-order MA processes, the case where k = ncludes them. 4

7 statonary processes. E can be wrtten as a term whch s a constant tmes and a term whch s a nonlnear functon of and. For partcular tme seres processes on the errors, we can be more specfc. 3. Specfc examples The MA Process Suppose that n the above general process, 0,, and 0 for >. The errors then follow a frst-order movng average (MA()) process, t t t, where. In ths case by substtuton, E t = ( ) =. Thus, regressng the square of the resdual e t on and a constant yelds ˆ ˆ. Ths provdes a dfferent explanaton for and the constant term than Case and Shller [987]. Here, ˆ 0 s not an anomaly, but arses whenever 0 (unlke frst-order autoregressve processes, there s no presumpton that > 0). Thus, a negatve constant term may be evdence of a non-..d. error process. If we assume that the t follow an MA() process wthout transacton errors, we can dentfy pont estmates of and from ˆ and ˆ. We can extend ths approach to hgher-order MA processes. For the MA() process, (or for the general process, 0, t t t t, and 0 for > ), so by substtuton, E t =. Smlar calculatons reveal that all 5

8 MA(k) processes wth k < have an ntercept term and a constant multplyng, but no terms multplyng hgher powers of. The slope term wll be postve, but the sgn of the ntercept depends on the parameters of the process. For k >, we cannot dentfy the parameters of the MA process snce we observe only a slope and ntercept. The AR Process Suppose nstead that t, t, follows an AR() process, t t t. In the general MA process, ths s equvalent to k = and for = 0, k. Usng the fact that E t tk k, we fnd that E t ( ). See Appendx A for a dervaton. As grows, for 0, ths expresson ncreases at an ncreasng rate and asymptotes to an ncreasng straght lne. For 0, t ncreases at a decreasng rate. Thus, only negatve frst-order seral correlaton s consstent wth a postve coeffcent on tme between sales and a negatve coeffcent on ts square, whch s a common fndng. Snce negatve autocorrelaton s not common n economc data, t seems unlkely that an AR() error process explans the commonly observed pattern. Hgher-order AR processes also result n E t beng a nonlnear functon of the holdng perod. The ARMA Process An ARMA(p, q) process has p th -order autoregresson and q th -order movng average. For p = q =, we can wrte the process as t tt t. In the general 6

9 MA process, ths s equvalent to k =, 0,, and In ths case, for =, k. E t = t ( ). See Appendx B for the dervaton. As wth the MA process, there s an ntercept (whch s easly negatve) and a constant coeffcent on the tme horzon. As wth the AR process, there s also a term whch decays exponentally. For 0 (the normal case), the expectaton of the square of the sum of the resduals ncreases wth the holdng perod length. For, t ncreases as an ncreasng rate, whle for 0, t ncreases as at decreasng rate. Ths last possblty s consstent wth a postve coeffcent on the lnear term and a negatve coeffcent on the quadratc term. It also seems to be the mnmal assumpton on the return error process to generate such concavty. As wth AR processes, hgher-order ARMA processes result n E t t beng a nonlnear functon of the holdng perod. One could ft a nonlnear regresson n to E t. As wth the MA() process, one could only dentfy parameters of the stochastc process condtonal on the assumpton about the order of the ARMA process, but of course the order of the ARMA process cannot be recovered because we do not observe the errors n the ndvdual asset returns, but only the summed resduals. 7

10 3. Flexble GLS In order to correct for heteroskedactcty when the error term on the ndvdual asset returns s correlated between perods, we propose a flexble approach n whch 3 rd stage weghts are constructed by regressng the squared resduals from the frst stage regresson on dummy varables that represent the length of the holdng perod for each asset. Ths s a smple and useful approach that allows for autocorrelaton even when the exact form of the correlaton n the underlyng assets cannot be dentfed. 4. An Applcaton to Repeat Sales of Volns Graddy and Margols [009] study returns to ownng hgh-qualty volns over a long tme perod datng back nto the 9 th Century. The data conssts of 337 repeat sales of fne volns that took place between 849 and 009. The average holdng perod for each voln s 3 years. The shortest holdng perod s 5 years and the longest s 47 years. Columns and of Table report the coeffcents on the OLS (frst stage) of the repeat sales regressons, columns 3 and 4 report the coeffcents usng the Case and Shller method for the 3 rd stage regressons, and columns 5 and 6 report the coeffcents usng the flexble GLS estmator descrbed above. 5 In Table we also present the test results from the Koenker-Basset test for heteroskedastcty. In ths test, the squared resduals from the regresson model ( u ) are regressed on the squared estmated ˆ predcted values of the dependent varable ( Y ˆ ) and a constant: uˆ ( Yˆ ) v. 5 The actual returns n the OLS and standard Case and Shller regressons are calculated and reported n Graddy and Margols (009). 8

11 The null hypothess s that α = 0. If ths s not rejected, then one could conclude that there s no heteroskedastcty. We also report the mean of the standard errors from these repeat sales regressons. The results ndcate that the null hypothess for heteroskedastcty s rejected n both the OLS and the standard Case and Shller regressons. As the standard Case and Shller regresson does not completely correct for heteroskedastc errors, non-..d. errors are suspected. Only n the flexble GLS regresson can we conclude that there s no remanng heteroskedastcty. Furthermore, the mean standard errors are lower wth flexble GLS than n ether of the other specfcatons. To explore for evdence of non-..d. errors, n Table we report the results from varous specfcatons of the second stage regressons. Column reports the regressons from the standard Case and Shller second stage, and column 5 reports the flexble GLS regresson. We also consder other polynomals (wth and wthout constants) and a logarthmc specfcaton. Note that the standard Case and Shller regressons appear to be domnated by the log specfcaton usng the measures of adjusted R-squared, AIC, and BIC. The flexble specfcaton domnates all specfcatons, as ndcated by adjusted R-squared, AIC, and BIC. As ndcated n Table, ths specfcaton both corrects for heteroskedastcty and decreases the errors. In Graddy, Hamlton, and Campbell [009], we test the flexble specfcaton on two larger repeat sales datasets of hstorcal house prces n the Herengracht dstrct of Amsterdam, and on prces of art sold n Amsterdam, wth very smlar results. 6 6 If the errors n asset returns are..d., the varance of the return errors for an ndvdual property should grow lnearly wth tme. The sze of the Herengracht and Amsterdam art datasets allowed us to estmate year, year, 5 year, and 0 year returns. In the Amsterdam art dataset, we could clearly reject lnear growth n returns when estmatng the dfferent return perods. 9

12 5. Implcatons and Conclusons It s well-known that the logarthmc specfcaton of the dependent varable results n the geometrc mean across assets for each tme perod of the ndex. Goetzmann [99] suggested that the coeffcent on the tme between sales should be used as an estmate of the cross secton varance to gve the followng formula for the arthmetc mean, exp, where a g a and g are the arthmetc and geometrc means and errors. s the cross-secton varance. Ths correcton becomes problematc for non-..d. Wthout a specfc assumpton on the errors n ndvdual asset returns, the sngle perod return varance n an asset cannot be dentfed from the second stage of the Case- Shller regresson results. 7 Calhoun [996] proposes usng t At Bt (where A and B are the lnear and quadratc coeffcents from the second stage wth no constant) as the varance n the geometrc to arthmetc correcton formula (n ndex form). There s a problem once the second stage ncludes more than a smple lnear term the estmated varance for any property becomes a functon of the holdng perod. Even usng the varance per perod ( A Bt ) depends on the holdng perod. Any estmate of the arthmetc return depends on the planned holdng perod f the return errors are not..d. In further work, we plan to study two ssues. One s the mpact of seral dependence on the magntude of standard errors. The other s how seral dependence affects the varance n revsons of coeffcent estmates after re-estmaton wth addtonal perods of data. 7 The S&P/Case-Shller prce ndex drectly estmates an arthmetc ndex to crcumvent ths problem. 0

13 References Abraham, J., and W. Schauman, 99, New Evdence on Home Prces from Fredde Mac Repeat Sales, AREUEA Journal 9: Baley, M., R. Muth, and H. Nourse, 963, A Regresson Method for Real Estate Prce Index Constructon, Journal of the Amercan Statstcal Assocaton 58: Calhoun, C., 996, OFHEO House Prce Indexes: HPI Techncal Descrpton, mmeo, Offce of Federal Housng Enterprse Oversght. Case, K., and R. Shller, 987, Prces of Sngle Famly Homes Snce 970: New Indexes for Four Ctes, Cowles Foundaton Dscusson Paper No. 85, Yale Unversty. Goetzmann, W., 99, The Accuracy of Real Estate Indces: Repeat Sale Estmators, Journal of Real Estate Fnance and Economcs 5:5-53. Graddy, K., J. Hamlton, and R. Campbell, 009, Repeat Sales Indexes: Estmaton Wthout Assumng that Errors n Asset Returns Are Independently Dstrbuted, CEPR Dscusson Paper No Graddy, K., and P. Margols, 009, Fddlng wth Value: Volns as an Investment?, forthcomng, Economc Inqury. Hwang, M., and J. Qugley, 004, Selectvty, Qualty Adjustment and Mean Reverson n the Measurement of House Values, Journal of Real Estate Fnance and Economcs 8: Qugley, J., 995, A Smple Hybrd Model for Estmatng Real Estate Prce Indexes, Journal of Housng Economcs 4: -. Shller, R., 007, Hstorc Turnng Ponts n Real Estate, Cowles Foundaton Dscusson Paper No. 60, Yale Unversty (

14 Table Repeat Sales Regressons OLS Case and Shller Flexble GLS perod coef std coef std coef std error error error Koenker Basset α Average std error Obs

15 Table Second Stage Regresson Results coef std coef std coef std coef std coeff std coef std coef std error error error error error error error TBS TBS TBS ln(tbs) Duraton 3 Dummes Cons F Stat * * Prob>F * * Adj R * * AIC BIC Obs

16 Appendx A: Calculatons for the AR Process Usng the fact that E t tk k., we fnd that t t t t tt ( ) t t. Thus, E t ( ) ( )... = ( ) ( )... where k ( ) ( )... ( k) k = k k. The frst term n ths expresson equals, usng Z =... and Z..., whle the second term equals ( ), usng Y ( ) and 3 Y ( ), and ( ) 3... Y Y = N ( N ) N. Hence, E. ( ) t 4

17 Appendx B: Calculatons for the ARMA process For p = q =, an ARMA(, ) process can be wrtten as:, t =,. t t t t The expected values of the varances and covarances of errors equal: E t E tt,, E where k and ˆ t t k ˆ for k. Then, we can wrte the expected value of the square of the sum of the resduals as: E t = E t E tt E tt... E tt t t t t t = E t E tt JE tt J. t t J Takng the expectatons, we obtan: E t = ˆ J J. t J For the last term, let K and. Then let Z K J J. J Now Z 3 K 3... K and 3 K... K, so Z Z Z K 3 K K.... Let 3 K Y... 3 K. Then Y..., so Y Y K, and Y K. Substtutng ths nto the earler formula, 5

18 K K K K K K Z Z K. Hence, K K Z K K Thus, we have E t = ˆ t. Substtutng, E t = t ( ) = ( ). 6