Low-frequency data present significant

Transcription

1 MICKAËL MALLINGER- DOGAN is an assisan vice presiden, Illiquid Asses Analics, a Harvard Managemen Compan in Boson, MA. doganm@hmc.harvard.edu MARK C. SZIGETY is vice presiden and head of Quaniaive Risk Analics a Harvard Managemen Compan in Boson, MA. szigem@hmc.harvard.edu Higher-Frequenc Analsis of Low-Frequenc Daa MICKAËL MALLINGER-DOGAN AND MARK C. SZIGETY Low-frequenc daa presen significan challenges o invesors, because infrequen observaions lead o imperfec inra-period undersandings of marke behavior, which in urn frusrae risk managemen and porfolio monioring. Alhough i is a general feaure of marke dnamics, his phenomenon is a common complain for invesors who oversee porfolios wih significan holdings in illiquid invesmens: endowmens, pension funds, sovereign wealh funds, and oher large insiuional invesors. Consider an insiuion ha invess in socks, bonds, privae equi, and real esae. The sock and bond invesmens are priced monhl, while hose in privae equi and real esae are priced quarerl. This insiuion periodicall calculaes risk measures, such as volaili and VaR, conducs risk facor aribuion, and analzes various sress measures and scenarios, including some ha occur over non-quarerl inervals. To perform hese calculaions, he insiuion requires a comprehensive se of hisorical ime-series reurns. To cover all asses a he porfolio level, he insiuion could conver all relevan reurn series o he highes common frequenc, which in his example would be quarerl. However, here are drawbacks o his approach. Firs, per uni ime, less informaion is used han if he lower frequenc series were convered o higher frequenc. B forcing he sock and bond reurn series o be quarerl, he insiuion discards poeniall useful informaion in undersanding heir facor relaionships. This effec is exacerbaed o he exen ha concerns over relaionship saionari, or he needs of an ad hoc analsis, promp he insiuion o prefer shorer lookback periods. For example, hree ears ma be considered a reasonable period over which o esimae relaionships. On a monhl basis, his represens 36 observaions, bu onl 2 on a quarerl basis arguabl oo few for an inference o be considered meaningful or, a he ver leas, subsaniall increasing he esimaion error. A second drawback is ha, unless he insiuion is comforable working on a lowerfrequenc basis, he informaion gained from he lower-frequenc analsis ma have quesionable relevance. For example, if he bea of he oal porfolio o domesic socks is 0.6 (using quarerl daa), an esimae of porfolio value afer a monhl sock-marke decline of 0% involving simple muliplicaion ma no provide a meaningful answer, because correlaions and beas among series ma be differen over varing observaion frequencies. For hese reasons, man insiuions should prefer o emplo an empirical framework ha mainains he densi of available informaion (b no convering all asses o he highes common frequenc), and ha more closel approximaes he organizaion s decision-making frequenc. For heir lower- IT IS ILLEGAL TO REPRODUCE THIS ARTICLE IN ANY FORMAT FALL 204 THE JOURNAL OF PORTFOLIO MANAGEMENT 2 Coprigh 204

2 frequenc series, hese invesors mus choose beween ) a comprehensive bu ofen difficul o implemen public-markes equivalen, facor-based, or booms-up valuaion approach or 2) an inerpolaion o esimae missing observaions. For cases in which he firs opion is no appealing (or simpl impossible), invesors are lef wih he quesion of which pracical inerpolaion mehodolog o use. Our primar goal in his aricle is o describe wha we ve learned abou wo imporan inerpolaion approaches and o discuss heir effecs on common risk measures. This aricle makes wo primar conribuions. Firs, we presen and evaluae wo inerpolaion mehodologies, one of which we believe is no widel used in he invesmen indusr. Second, we use simulaion o rigorousl compare he wo inerpolaion approaches from he perspecive of an insiuional invesor in illiquid invesmens. Our simulaion environmen is designed o cover a range of pical low-frequenc siuaions and o compare he wo mehodologies along several imporan measures. We discuss he benefis ha a houghful higher-frequenc inerpolaion migh provide and idenif some pifalls. We find ha he decision o use one mehodolog over he oher or wheher o inerpolae a all depends on conex. In cases where fideli o he unobserved series and measures relaed o volaili are imporan, he more modern inerpolaion approach can indeed be helpful. However, boh mehodologies ma have significan rouble preserving relaionships o ouside facors in cerain cases, impling ha care mus be aken when using inerpolaion o enable higher-frequenc risk aribuion and bea-dependen analses. Neverheless, here are man benefis o judiciousl using inerpolaion: Volailiies, disribuional measures, and sresses will hen all be compuable a a higher frequenc, providing a more complee picure of individual asse class and overall porfolio dnamics. In he nex secions, we formall inroduce he wo mehodologies: one somewha familiar (or a leas inuiive) and one relaivel new. We hen evaluae hem wihin a comprehensive simulaion framework, which we believe capures he essence of man risk and porfolio monioring siuaions ha insiuional invesors are likel o encouner. We conclude b providing an example of inerpolaion in a pracical conex, followed b suggesions for fuure work. METHODOLOGIES For man ears, work on he low-frequenc problem has been of grea imporance o economiss and polic makers, who have long regarded higher-frequenc informaion as crucial o suding and managing a modern econom. Since World War II, he single mos imporan economic ime series, gross domesic produc (GDP), has been produced quarerl. Thus, saring wih he pracical work of Chang and Liu [95], economericians have been wresling wih mehodologies o bes derive monhl esimaes of GDP and oher relaed lowfrequenc series. Over ime, wha were iniiall simple mehodologies focused on inerpolaion using a relaed series evenuall evolved o incorporae forecasing and serial correlaion, and now involve relaivel compuaionall inensive algorihms. A leas from an empirical and operaional sandpoin, illiquid asse classes are o insiuional invesors wha GDP is o economiss. Indeed, illiquid invesmens are difficul o manage in a porfolio conex for man reasons. Though progress has been made in modeling heir porfolio dnamics (Takahashi and Alexander [2002]; de Malherbe [2005]; Szige [203]), and while here is work regarding unsmoohing he repored series (Gelner [99]; Fisher e al. [994]; Cho e al. [2003]; Lizieri e al. [202]), we are unaware of an work regarding he higherfrequenc reamen of hese series ha does no presen significan pracical challenges. I s ofen recommended o rel on higher-frequenc public marke or facor analogues, generaed b or enhanced wih cash-flow daa (as in Kaplan and Schoar [2005] and Harris e al. [202]). However, alhough hese and relaed approaches ma heoreicall assis in providing a higher-frequenc series and assessing risk exposures, heir primar drawbacks are heir daa and experise demands, and ha an ime series derived from hese approaches is unrelaed o he realized (likel smoohed) low-frequenc marke value series. 2 Anoher relaed approach is o reproduce and calibrae in-house valuaion models for illiquid funds and properies. Alhough his alleviaes concern over he lack of smoohing and also ma assis in undersanding risk exposures, i suffers from esseniall he same drawbacks as he public marke or facor approach, wih he added complicaion of significanl increased model error. Insead, an inerpolaion approach ma be appropriae, given is relaive simplici and goal: o provide a higher- 22 HIGHER-FREQUENCY ANALYSIS OF LOW-FREQUENCY DATA FALL 204

3 frequenc reurn series ha is as close as possible o wha acuall happened, ha is, smoohed and linked o he low-frequenc observaions. Illiquid invesmens, hough unique in heir funding and reporing characerisics, are neverheless ofen relaed o oher liquid invesmens hrough shared exposures o common facors. A privae equi porfolio shares a relaivel common se of facors wih public equiies, and even an illiquid real esae porfolio is likel o be indirecl relaed o REITs (Ang e al. [203b]). The wo mehodologies presened in his secion rel on he observable low-frequenc relaionship beween he low-frequenc reurn series and he higher-frequenc prox series o infer he low-frequenc series underling high-frequenc behavior, while also ensuring ha he esimaed series maches he acual low-frequenc series over he appropriae ime inervals. Despie his common approach, hese mehodologies address he problem in angenial was. The firs mehodolog, and ha which exemplifies he earlies aemps o solve he low-frequenc challenge, was proposed b Chow and Lin [97], hereafer referred o as CL. Their approach esseniall applies low-frequenc beas o higher frequencies, and has wo main componens. The firs componen applies he observed low-frequenc beas o he proxies higher-frequenc observaions. The second componen is an esimae of he residuals, which are he deviaions from wha migh be expeced, based on he simple low-frequenc bea model. The second approach, and ha which represens he more curren reamen of he problem, is described b Mariano and Murasawa [2003, 200], hereafer referred o as MM. This approach feaures a mixed-frequenc vecor auoregressive (VAR) model ha combines a sandard VAR model for he pariall laen high-frequenc series, wih a sae-space model for he observable mixed-frequenc series. 3 The approach is anamoun o imagining ha he low-frequenc series has a higherfrequenc represenaion ha runs ouside of our abili o observe i, and is akin o aking picures of a sporing even because we are unable o ake video. This saespace represenaion of he high-frequenc VAR model has muliple advanages, he wo primar of which are is simplified maximum-likelihood esimaion and is abili (like all sae-space models) o deal wih missing values. Bu addiionall, and unlike CL, MM can deal wih man low-frequenc variables, as well as wih more complicaed error srucures (e.g., MM can readil handle more han one lag), hough a he cos of he quali of he esimaed coefficiens, as he number of parameers ma increase dramaicall. 4 Exhibi summarizes he main deails of, and differences beween, he wo approaches. E XHIBIT Summar Deails of CL and MM Mehodologies FALL 204 THE JOURNAL OF PORTFOLIO MANAGEMENT 23

4 E XHIBIT 2 Monhl Inerpolaion of a Quarerl Real Esae Index For moivaion, we presen a real-world example of hese wo approaches. In his example, we ransform quarerl reurns from he well-known Cambridge Associaes Real Esae Index (CAREI) 5 o monhl observaions, using informaion from he FTSE NAREIT All-Equi Toal Reurn Index. Exhibi 2 shows resuls for each mehodolog from he fourh quarer of 2002 hrough he hird quarer of One can clearl see a difference beween he wo approaches: MM appears o be a smooher, higher-frequenc represenaion of he series, bu i isn clear ha his reducion in volaili is jusified or even desirable. Given ha he monhl CAREI series is no observable, we have no empirical wa o jusif one approach over he oher, and mus rel on visual preference. An undersanding of he siuaions in which one mehodolog dominaes is somehing we aemp o develop in his aricle. SIMULATION To es he CL and MM approaches wihin a pracical conex, we use a sraighforward simulaion framework. The design of he ess is simple (see Exhibi 3). Firs, we simulae hree relaed higher-frequenc reurn series (op lef). Second, we ransform one ino a lower frequenc b compounding he higher-frequenc reurns (op righ). Third, we use he second higher-frequenc series o inerpolae he low-frequenc series (boom righ; see he Appendix for echnical deails). Finall, we evaluae he abili of each inerpolaor o recover he original higher-frequenc series across several dimensions, including he recovered bea o he hird higherfrequenc series (boom lef). The hree series are composed of one represening a porfolio (P), one represening a prox (X), and one represening a facor (F). Our engine generaes higherfrequenc versions of hese series, based on he following se of equaions: where r P = μ P +γr P + ep P P (a) rx = μ + e (b) X X r = μ + e (c) F F F σ σ σ e j ~ (0, Σ) Σ = σ σ σ σ σ σ PP PX PF PX XX XF PF XF FF In his se of equaions, γ conrols he auocorrelaion of he AR() series P, and σ jk conrols he covariances of (2) 24 HIGHER-FREQUENCY ANALYSIS OF LOW-FREQUENCY DATA FALL 204

5 E XHIBIT 3 Simulaion Design. We Run Each Ccle (from op lef o boom lef) 00 Times for Man Ses of Parameer Inpus he corresponding series j and k. Noe ha he σ jk where j k govern he conemporaneous relaionships beween he hree series. Alhough hese conemporaneous relaionships are no explici in he reduced-form model, he can be made apparen b convering he ssem of Equaions (a c) o a srucural represenaion. Equaions (a c) and (2) are heavil parameerized, and hus highl flexible. To bound our invesigaion, we fix several parameers. Firs, σ PP, σ XX, and σ FF are all se a values ha correspond o an annualized volaili of 5%, assuming monhl simulaed reurns and using he appropriae scaling facor. 7 Second, μ P, μ X, and μ F are fixed a arbirar monhl values, he values of which are irrelevan o he resuls. We var all oher parameers in order o evaluae CL and MM along four dimensions: ) wo differen porfolio auocorrelaion regimes: low and high; 2) he explanaor power of he prox: low, medium, and high; 3) he bea of he porfolio o he facor: from low o high; 8 and, 4) he bea of he facor o he prox: from he minimum o he maximum allowable under our simulaion condiions for a given run. We selec hese comparaive dimensions because he exen of illiquid auocorrelaion, he explan- aor power of he higher-frequenc prox, he general relaionship of he low-frequenc series o an ouside facor, and he facor s relaionship o he inerpolaor s prox are four imporan and somewha observable elemens for invesors in illiquid porfolios. Therefore, CL and MM differeniaion along hese dimensions are useful o know in order o evaluae eiher model s appropriaeness o he inerpolaion ask. For each se of parameers we simulae hree series, discarding he firs several housand observaions o remove he effec of he iniial condiions and leaving approximael,000 higher-frequenc observaions. 9 Because illiquid invesmen reurns are ofen observable quarerl and man insiuions operae a a monhl frequenc, we conver he porfolio higher-frequenc series o a low-frequenc series following a hree-o-one raio. 0 RESULTS For each combinaion of comparaive dimensions, we repor he performance of CL and MM hrough four summar saisics: roo mean squared error (RMSE), vol- FALL 204 THE JOURNAL OF PORTFOLIO MANAGEMENT 25

6 aili, he AR() auocorrelaion saisic, and he bea o he facor. We calculae RMSE as he roo of he average squared error beween he recovered higher-frequenc series and he acual simulaed higher-frequenc series; i measures he accurac of he recovering mehodolog. We look a RMSE because i suggess he overall accurac of each candidae inerpolaion mehodolog. We examine he volaili of he inerpolaed series relaive o ha of he rue higher-frequenc series, because empirical measures of volaili are used in a varie of analical measures of porfolio risk, and a failure o accurael recover volaili could missae risk when aggregaed across he porfolio in he higher-frequenc conex. We look a he AR() saisic because i ma cause a missaemen of annualized volaili hrough he scaling coefficien, mislead he insiuion as o he serial correlaion embedded in he porfolio when used in he higher-frequenc conex, and could affec he usefulness of various downsream unsmoohing approaches. We look a he bea o he facor because i indicaes wheher he inerpolaed series would be more or less useful in preserving relaionships on which invesors heavil rel for facor conribuion o risk, correlaions, and sress esing. Each of he saisics we presen is he average of 00 independen rials, which permis us o assess he saisical significance of he difference beween he wo inerpolaors. The lef and righ panels of Exhibi 4 presen he low auocorrelaion case, where he porfolio s auocorrelaion parameer, γ, equals 0.2. In each panel, moving down he rows corresponds o an increasing higher-frequenc bea of he porfolio (P) o he facor (F). Moving across columns corresponds o he prox s increasing explanaor power, from 30% (low) o 60% (moderae) o 90% (high). The lef and righ panels show he resuls for when he simulaion parameers were chosen such ha he bea of he facor (F) o he prox (X) is he minimum and maximum value possible, respecivel, given he saed explanaor power of he prox and he higher-frequenc bea of he porfolio o he facor. All calculaed saisics are significan from heir counerpar wihin he cell a he 95% confidence inerval, unless highlighed in ialics. The lef panel of Exhibi 4 repors several ineresing findings. To sar, he resuls for RMSE, volaili, and AR() are esseniall he same as one moves down he rows wihin each column. This is inuiivel appealing, because i suggess ha he mehodologies abili (along hese dimensions) does no var based on he simulaed bea of he porfolio o he facor, bu insead on he prox s explanaor power. In oher words, wha ulimael influences he inerpolaed series inernal fideli is he prox s explanaor power. As one increases he prox s explanaor power b moving across he columns, several aspecs of he mehodologies become apparen. Firs, he RMSE of boh CL and MM improve dramaicall: from approximael 2.30% o.6% (a facor of 2) for CL, and from approximael 2.30% o 0.87% (a facor of 2.6) for MM. Again, his makes sense: as he prox approaches he acual reurn series in he limi, he RMSE should end o zero. However, noe ha MM appears o be more efficien han CL on his dimension, especiall as he explanaor power becomes high. Second, he recovered annualized volaili of he MM bu no he CL inerpolaed series improves quickl as he explanaor power of he prox increases. Ineresingl, alhough boh he CL and MM series are respecivel less han and more han he 5% rue volaili b roughl equal amouns, as he explanaor power of he prox increases he MM series quickl converges o 5%, while he CL series acuall becomes even more downwardl biased. Boh CL and MM generae series ha have higher observable AR() coefficiens han he should when he explanaor power is weak. The coefficiens of he CL and MM series are measured a approximael 0.35 and 0.47, respecivel noe CL s considerabl beer performance. However, as he explanaor power increases, he CL series recovered AR() coefficien decreases quickl, and evenuall significanl underesimaes he rue higherfrequenc auoregressive behavior. Meanwhile, i appears MM asmpoes o he rue AR() value. The lef panel indicaes ha boh mehodologies generall lead o significanl underesimaed beas o a facor when he facor has a minimal relaionship o he prox, especiall when he rue porfolio bea o he facor is low. For example, when he bea of he porfolio o he facor is 0.0, boh CL and MM produce higher-frequenc series ha generae beas of abou 0.03 (70% smaller), regardless of he prox s explanaor power. This ma be undersood b considering ha he facor has no relaionship o he prox, so he prox is of minimal value in mainaining he relaionship beween porfolio and facor. As he bea of he porfolio o he facor reaches is maximum in he panel a 0.79, 2 he 26 HIGHER-FREQUENCY ANALYSIS OF LOW-FREQUENCY DATA FALL 204

7 E XHIBIT 4 Low Auocorrelaion (γ = 0.2) Inerpolaion Resuls; Average of 00 Trials per Cell. The Lef/Righ Panel Depics he Case in Which he Bea of he Facor (F) o he Prox (X) Assumes he Minimum/Maximum Value Possible o Calculae Resuls for Tha Cell. In Each Panel, Moving Down Rows Corresponds o an Increasing Bea of he Porfolio (P) o he Facor (F) wihin he Simulaion Framework; Moving Across Columns Corresponds o an Increasing Amoun of Porfolio (P) Variance Explained b he Prox (X) wihin he Simulaion Framework. All Calculaed Saisics Are Significanl Differen from Their Counerpars a he 95% Confidence Level, unless Highlighed in Ialics FALL 204 THE JOURNAL OF PORTFOLIO MANAGEMENT 27

8 recovered series beas are sill underesimaed, excep in he case of a powerful prox. 3 Turning o he righ panel, where we use he maximum possible bea of he facor o he prox, firs noe ha he resuls for RMSE, volaili, and AR() are esseniall he same as in he lef panel. This makes sense, because he relaionship beween he facor and prox (from minimum o maximum possible) should have no relaionship o each mehodolog s abili o recover hese inernal measures. The prox s power is he onl hing ha should maer. The righ panel shows ha as he bea of he facor o he prox shifs from is minimum o maximum, each mehodolog s abili o recover he rue bea of he porfolio o he facor appears o modesl improve. Such improvemen is generall bu no alwas seen when ) he bea of he porfolio o he facor is moderae o high and 2) he prox s power is moderae o high. If one of hese condiions doesn hold, we generall have fairl significan deviaion. For example, when he bea of he porfolio o he facor is 0.0 and he explanaor power is 90%, CL and MM generae series wih overesimaed beas of 0.28 and 0.24, respecivel. When he bea of he porfolio o he facor is 0.79 and he explanaor power is 30%, CL and MM generae series wih underesimaed beas of 0.52 and 0.45, respecivel. The lef and righ panels of Exhibi 5 show he resuls for he high auocorrelaion case, where he porfolio s auocorrelaion parameer, γ, equals 0.7. Saring wih he lef panel, moving down he rows wihin each column does no change he resuls for he RMSE, volaili, and AR() measures. As in he prior case, his suppors he inuiion ha onl he power of he prox maers for hese saisics, no he relaionship of he porfolio o a facor. Turning o RMSE, he resuls sugges ha boh mehodologies are considerabl more effecive a resolving he higher-frequenc series a a given level of prox explanaor power in he presence of high auocorrelaion, wih he sligh excepion of CL wih he mos powerful prox. Remarkabl, CL appears o recover a series ha progressivel becomes more inaccurae as he prox s usefulness increases. MM, however, sars wih a comparaivel low RMSE, a 0.79%, and makes good use of more effecive proxies, lowering he RMSE o 0.30% in he bes case presened here (for a facor of 2.6 reducion in error, as before). MM clearl dominaes CL in his measure for he high auocorrelaion case. The sor is more suble when we look a he volailiies. Compared o he low auocorrelaion case, boh CL and MM are furher under and over 5% when using a weak prox. As he explanaor power of he prox increases, CL acuall generaes a series wih progressivel lower volaili (3.4%, 2.0%, and.6%) furher awa from he rue value of 5%. Meanwhile, MM works wih he more effecive prox o approach 5% from above, alhough apparenl no as quickl as in he low auocorrelaion case. We see a similar sor when we examine he AR() coefficiens of he inerpolaed series. The CL series sars slighl under he rue value, a 0.58, and hen drops dramaicall, producing 0.26 when he prox is a is mos explanaor. MM, on he oher hand, produces a series ha sars slighl more smoohl, wih an AR() of 0.83, hen approaches 0.70 (possibl asmpoicall, as before) as he prox power increases. As in he low auocorrelaion case, he recovered bea of he porfolio o he facor when he relaionship beween he facor and he prox is se o is minimum possible is generall poor, regardless of he mehodolog. The MM series persisenl produces downwardl biased beas, hough less so when he rue bea is moderae o high and he explanaor power is high. The CL series produces higher and lower beas somewha erraicall, onl generaing reasonable beas when ) he rue bea is low o moderae and he explanaor power is high (e.g., 0.5 recovered versus 0.6 rue, as seen in he far righ second row) or 2) when he rue bea is moderae o high and he prox s explanaor power is moderae (e.g., 0.34 recovered versus 0.32 rue, as in he boom cener cell). As before, he righ panel shows esseniall he same resuls as in he lef panel for he RMSE, volaili, and AR() dimensions for all rows and columns. However, seing he bea of he facor o he prox o is maximum possible does have a noiceable effec on he recovered beas. The CL series coninues o produce erroneous beas, hough his ime he are consisenl above he rue value in he wors case b almos hree imes. The MM series are beer behaved, generall saing above and improving as he prox s power increases. We believe ha he akeawas from hese resuls can be srucured b measure. Concerning RMSE, MM is no worse and someimes subsaniall beer up o abou 80% han CL in all circumsances. The sor is similar in recovered volailiies, where MM is usuall no 28 HIGHER-FREQUENCY ANALYSIS OF LOW-FREQUENCY DATA FALL 204

9 E XHIBIT 5 High Auocorrelaion (γ = 0.7) Inerpolaion Resuls; Average of 00 Trials per Cell. The Lef/Righ Panel Depics he Case in Which he Bea of he Facor (F) o he Prox (X) Assumes he Minimum/Maximum value Possible o Calculae Resuls for Tha Cell in Each Panel, Moving Down Rows Corresponds o an Increasing Bea of he Porfolio (P) o he Facor (F) wihin he Simulaion Framework; Moving Across Columns Corresponds o an Increasing Amoun of Porfolio (P) Variance Explained b he Prox (X) wihin he Simulaion Framework. All Calculaed Saisics Are Significanl Differen from Their Counerpar a he 95% Confidence Level FALL 204 THE JOURNAL OF PORTFOLIO MANAGEMENT 29

10 worse and someimes subsaniall beer; MM is onl slighl less accurae on an absolue basis han CL in he circumsance of high auocorrelaion and low prox explanaor power. Noe, however, ha MM overesimaes volaili, while CL underesimaes i. Turning o he auocorrelaion coefficien, we prefer CL if he rue auocorrelaion is low and he prox is poor, in which case MM would produce a series ha is oo smooh b subsaniall overesimaing he AR() parameer. For all oher cases, we prefer MM, especiall when he prox is powerful. Finall, boh mehodologies sruggle o produce a series ha mainains he bea of he porfolio o he facor. However, when he auocorrelaion is high and he power of he prox is high, hen MM appears o perform saisfacoril. SELECTING A PROXY In his secion, we presen a simple heurisic for selecing appropriae proxies. Firs, i s helpful o sar wih a lis of candidae proxies ha have a heoreical relaionship o he low-frequenc asse class, eiher gleaned from he lieraure or hrough exper judgmen. Afer having compiled his lis, invesors mus eliminae candidaes ha are no observable in a imel manner or ha are no available a he higher frequenc. Nex, invesors mus deermine how useful he prox is a explaining he reurns of he low-frequenc series over a varie of differen periods relevan o he inerpolaion, which ma be measured in erms of proporion of remaining porfolio variance ha s explained b he prox. If invesors consider muliple proxies, he ma need a join model of explanaor power. Though ideall his analsis would be done a he higher frequenc, we believe ha he low-frequenc relaionship is a fair indicaor of how relaed he series are. A he end of his sep, invesors can roughl rank candidae proxies (or ses of proxies, when considering join models) in order of usefulness. A his poin, i s helpful o consider how he higher-frequenc series will be used. The simulaion resuls show ha if an invesor cares onl abou overall accurac or volaili, using he op-ranked prox (or se of proxies) makes sense. The sor is more nuanced if he invesor will use he inerpolaed series o derive higherfrequenc beas o risk facors, as in an aribuion. In his case (and as we will see), he invesor should also consider he relaionship of he prox (or ses of proxies) o he facors, in a manner similar o evaluaing he prox o he porfolio alread described. A his poin, each prox (or se of proxies) will have informaion peraining o is explanaor power, boh for he low-frequenc porfolio and for each facor. The prox or ses of proxies ha explain he mos, perhaps weighed b imporance o he evenual resuls, will be he obvious choice. Having more han one prox is probabl advisable when possible, bu for MM enough daa mus be available o accurael esimae he parameers. For his reason, as well as he desire o keep he number of parameers of our comparaive simulaion relaivel low, his aricle presens resuls onl for one prox. Wheher o use muliple proxies is generall dependen on how man observaions are available and which inerpolaion mehodolog is seleced. In he case of MM, muliple proxies quickl presen parameer esimaion issues. In he case of CL, his is far less of an issue and is anamoun o building a simple linear model in muliple regressors. If he concern is covering a large number of poenial facor relaionships, hen invesors migh prefer CL for precisel is abili o incorporae more proxies easil. ORIGIN OF THE DIFFERENCE BETWEEN CL AND MM Our resuls indicae ha MM generall ouperforms CL along several dimensions b an increasing margin as boh he auocorrelaion and he prox s explanaor power increase. Alhough conribuing facors are numerous, we believe ha he main source of his relaive performance is rooed in he difference beween lowerand higher-frequenc beas (o he prox) in he presence of auocorrelaion. In our simulaions, we find ha he discrepanc beween a low- and high-frequenc bea can indeed be quie significan, and he moderaing facor is he exen of auocorrelaion: Higher auocorrelaion in he series leads o more difference beween low- and high-frequenc beas o he prox. Apparenl unknown o is auhors, he CL approach is quie suscepible o his bea bias, because he leas-squares esimae of he lowfrequenc beas are direcl applied on higher-frequenc observaions of he prox. On he oher hand, MM benefis from some buil-in flexibili in esimaing he highfrequenc dnamics. Naurall, bea misesimaion has a greaer effec when he prox explains a large proporion of he variance of he porfolio series. We seleced he as-wrien Chow Lin procedure because of is saus as a common and widel used 30 HIGHER-FREQUENCY ANALYSIS OF LOW-FREQUENCY DATA FALL 204

11 mehod o inerpolae low-frequenc series. However, numerous exensions of he Chow Lin procedure have been proposed ha improve he esimaor in paricular cases. Lierman [983] suggess a generalizaion of he approach described b Fernandez [98] ha is likel o be more accurae when he error erm follows a nonsaionar process. Wei and Sram [990] inroduce a less resricive mehodolog, in which an ARIMA model is esimaed using he low-frequenc residuals. While reducing he subjecivi in he choice of he error erm process, is accurae esimaion requires a large number of observaions, ofen a srong limiaion when dealing wih low-frequenc series. Sanos Silva and Cardoso [200] adap he procedure o inroduce dnamic models ha are paricularl adequae when he series are coinegraed, bu is implemenaion and inerpreaion can be challenging. Depending on usage, one of he exensions ma be beer suied for a specific case a praciioner ma face; however, none of he approaches appears o direcl address he specific issue we ve encounered. DISCUSSION Having reviewed our findings, we firs reurn o he real esae example in Exhibi 2. To he exen ha inerpolaion is considered necessar, he decision regarding which approach o use is largel dependen on he auocorrelaion of he CAREI and he explanaor power of he NAREIT index, as well as wha analsis is being done. We esimae ha he quarerl CAREI series has an auocorrelaion of 0.5, which ranslaes ino a 0.7 auocorrelaion a a monhl frequenc, assuming an AR() process. 4 Quanifing prox explanaor power would require knowing he proporion of remaining CAREI variance explained b he NAREIT index a he monhl frequenc, which is clearl an unobservable quani. Neverheless, he quarerl relaionship beween CAREI and NAREIT suggess ha his proporion is low. Exhibi 5, which summarizes he high auocorrelaion resuls, indicaes ha MM significanl ouperforms CL in erms of RMSE, even when using relaivel weak proxies, bu overesimaes boh he properl annualized volaili and he auocorrelaion coefficien (CL underesimaes boh wih similar magniude). The MM line in Exhibi 2 is probabl closer o he rue monhl series, hough more smoohed and when annualized more volaile, and we prefer i. Furher, if a risk aribuion is aemped along facors ha have lile relaionship o he NAREIT index, boh mehodologies will probabl underesimae he porfolio s monhl relaionship o hose facors. If a risk aribuion is aemped along facors ha have a srong relaionship o he NAREIT, he evidence is more mixed. Nex, we urn o a pracical example of an insiuion ha invess in socks, bonds, privae equi, and real esae and would like o analze oal porfolio risk over he hree ears from 200 hrough 202. In Exhibi 6, we presen a comparison of wha would be repored using he highes common frequenc (HCF) approach, where all daa is quarerl and here are onl 2 observaions, and when inerpolaing privae equi and real esae, so as no o lose daa from he socks and bonds series. Our archepical insiuion s asse allocaion is 45% o socks (Russell 3000), 25% o bonds (Barclas U.S. Aggregae), 20% o privae equi (Cambridge Associaes Privae Equi Index, wih he Russell 3000 as he inerpolaion prox), and 0% o real esae (Cambridge Associaes Real Esae Index, wih he FTSE NAREIT as he inerpolaion prox). 5 For boh privae equi and real esae, we ve inerpolaed using MM. 6 As previousl noed, we believe ha MM probabl overesimaes he volaili and AR() coefficien for real esae, bu given U.S. equiies heoreicall moderae-o-high explanaor power, we expec ha MM will ouperform CL and is generall accurae for privae equi, wih he excepion of a possible sligh overesimaion of he AR() coefficien. 7 For each case, we repor four caegories of risk: disribuional measures (volaili, 8 VaR, 9 and racking error (TE)), facor aribuion (beas 20 and facor percen conribuion o risk (FPCTR)), sresses (facor shocks, he second European Union crisis in lae 20, and max drawdown), and realized correlaions. The differences beween he HCF and inerpolaion approaches are meaningful. Firs, he volailiies and VaRs for all asse classes and he porfolio are higher in he HCF case han in he inerpolaion case. This is no jus because of he mechanics of he inerpolaion: Boh socks and bonds, which consiue 70% of he porfolio, have annualized volailiies ha are higher when we use a quarerl series han when we use a monhl series. In addiion, he impued annualized volailiies of he privae equi and real esae asse classes are lower when we use monhl daa. Largel because of his, he porfolio had an esimaed 7.9% annualized volaili and.86% VaR over he pas hree ears, compared wih 7.63% and 2.59% using he HCF approach. In oher words, FALL 204 THE JOURNAL OF PORTFOLIO MANAGEMENT 3

12 E XHIBIT 6 Comparison of Highes Common Frequenc and Monhl Inerpolaed Approaches Using MM for Boh Privae Equi (prox is Russell 3000) and Real Esae (prox is FTSE NAREIT All Equi), Analics Generaed Using Daa from 200 hrough 202 (2 observaions quarerl, 36 monhl). The Porfolio Is Composed of 45% Socks (Russell 3000), 25% Bonds (Barclas U.S. Aggregae), 20% Privae Equi, and 0% Real Esae, Rebalanced a he Frequenc of he Observaions 32 HIGHER-FREQUENCY ANALYSIS OF LOW-FREQUENCY DATA FALL 204

13 an insiuion focusing on hese merics using he HCF approach ma assume ha heir porfolio is riskier han i acuall has been or ma be. In erms of relaive risk, he racking error o he 60/40 socks/bonds porfolio is esseniall unchanged beween he wo approaches. A slighl more complicaed picure emerges when we urn o facor aribuion. Mos of he asse class beas o socks and bonds remain relaivel unchanged on an absolue basis when we use monhl daa. However, he bea of socks o bonds is more negaive, and consequenl drives a more negaive observed relaionship beween he oal porfolio and bonds. Though similar on an absolue basis, some beas (and correlaions) have changed enough o generae subsaniall differen facor percen conribuion-o-risk resuls. In he HCF approach, he socks conribuion-o-risk o privae equi is onl 47%. Wih he inerpolaion, i is 8%. This increase ma be due in par o he inerpolaion s endenc o overesimae he bea o socks in he case of a lower AR() coefficien, middling rue bea, and a moderael powerful equiies prox, as can be seen in he righ panel of Exhibi 4. I ma also underesimae he absolue value of he bea o bonds, as he facor in he lef panel. A relaed phenomenon ma be even more a pla in he case of real esae, whose conribuion-o-risk profile has similarl changed; here is reason o be skepical of hese real esae resuls, hough wihou a monhl series, we canno be sure. Forunael, he impac of a leas hese privae equi and real esae beas on he porfolio s bea is relaivel small, given heir weigh in he overall allocaion. Thus, a he porfolio level, he increase of he bea o socks and decrease of he bea o bonds is largel driven b a beer undersanding of he relaionship beween socks and bonds ha a monhl reamen affords. There are wo noable comparisons in he sresses panel. Firs, we can now form an educaed opinion abou he porfolio s decline during he difficul wo monhs of he second European Union crisis of lae 20, when socks declined abou 3%. Because his sress occurred over wo monhs, we are unable o produce his hisorical sress resul for he oal porfolio b using he HCF approach. Wih he benefi of inerpolaing he quarerl series, we can now esimae ha he oal porfolio los abou 6% during ha even. Second, our porfolio s hisorical maximum drawdown has increased subsaniall, o 7.23% from 6.35%. This is unsurprising, because monhl peak-o-rough measures will be equal o or greaer han quarerl peak-o-rough measures. The differences in correlaions are suble, bu generall involve less negaive correlaions o bonds a he monhl frequenc han a he quarerl frequenc. This is rue even for socks, which are no subjec o he possible bias inroduced b he choice of inerpolaion prox. Real esae is subsaniall less relaed o bonds on he monhl basis. As previousl noed, his resul ma be due in par o he selecion of REITs as he real esae prox. In summar, Exhibi 6 shows several benefis of using a monhl, inerpolaed approach. I allows more informaion o be used for he socks and bonds asse classes, direcl suggesing a less volaile oal porfolio and one whose relaionship o bonds on a monhl basis ma be more negaive. Inerpolaion also les he insiuion beer undersand possible sress evens, including hisorical evens ha do no fall on quarerl inervals and maximum drawdown experiences. In addiion, inerpolaion can provide insighs on he monhl behavior of he inerpolaed low-frequenc series, especiall when considering measures such as volaili and VaR. A he same ime, we ve seen ha he choice of prox maers for some analses. When inerpolaing, he choice of prox (or proxies) can greal affec wheher recovered beas and risk aribuions are accurae, as in he case of REITs and bonds for real esae. Thus, he decision o inerpolae ma be relaed o he relaive imporance he insiuion places on various risk measures. For an insiuion ha primaril cares abou disribuional measures, such as volaili, VaR, CVaR, and maximum drawdown, inerpolaion probabl makes sense. However, if he insiuion is concerned wih beas, risk aribuion, or risk budgeing, inerpolaion depends cruciall on he proxies power. If a se of proxies can be chosen ha spans he range of risk facors and is highl explanaor, or he inerpolaed series are relaivel small conribuors o risk a he porfolio level, inerpolaion ma be an accepable wa o harves he addiional informaion in he liquid asse classes. Oherwise, he resuls ma be misleading and he bes opion ma be o no perform an inerpolaion a all. CONCLUSION In his aricle, we ve compared wo inerpolaion approaches from economerics ha ma be useful o conver low-frequenc illiquid reurn series o higher-fre- FALL 204 THE JOURNAL OF PORTFOLIO MANAGEMENT 33

14 quenc monhl series in cases where a highes common frequenc or facor-based approach ma no be desirable. Through a simulaion, we ve evaluaed he effeciveness of he wo inerpolaion mehodologies along several imporan dimensions. We ve found ha if he explanaor power of he prox is moderae o high, he mixedfrequenc VAR (MM) approach is significanl more accurae, beer recovers he volaili of he unobservable rue series, and generall beer replicaes he series rue monhl auoregressive behavior (hough i is biased oward overesimaion, which ma be of concern during annualizaion or unsmoohing). This relaive ouperformance in accurac and auoregressive behavior improves when he illiquid series is more seriall correlaed. In cases of low prox explanaor power, boh mehodologies perform similarl when faced wih low serial correlaion, bu he mixed-frequenc VAR approach ofen appears more effecive when faced wih high serial correlaion. On he oher hand, boh mehodologies have significan rouble recovering facor beas when using proxies ha have lile relaionship o ha facor. However, his poenial deficienc ma argue agains using an inerpolaion if he facor beas are he cenral poin of he join analsis of oal porfolio reurns, as in a risk aribuion. Onl in some cases ma i make sense o do, as in our example where real esae represens onl 0% of he oal porfolio and probabl would no bias our overall resuls significanl. Neverheless, here are man benefis o judiciousl using inerpolaion: volailiies, disribuional measures, and sresses will hen all be compuable a a higher frequenc, making use of more informaion among he liquid asse classes and providing a poeniall more complee picure of individual asse class and overall porfolio dnamics. There are several areas for fuure research. Firs, we ve resriced ourselves o one prox for simplici and because of concerns over esimaion error. I is reasonable o hpohesize ha, as he number of proxies increases, he recovered porfolio beas will be more accurae for CL, wih he same number of observaions. The same canno be said for MM: The use of addiional proxies will come a he cos of esimaion accurac and herefore ma affec performance on all imporan merics. Relaedl, we ve limied our analsis of he auocorrelaion o he AR() case. Alhough i appears o be an accurae represenaion for he series we use in his aricle, a more complex auocorrelaion srucure ma be more appropriae for some illiquid asse reurns. Furher, anoher implici assumpion in our work is ha he wo mehodologies comparaive resuls are applicable o inerpolaions wih far fewer observaions, as one experiences in everda usage. Esablishing absolue and relaive performance for differen numbers of low-frequenc observaions would advance our collecive undersanding. Finall, we ve limied ourselves o he mos common case of convering quarerl reurns ino monhl reurns. We expec he accurac and performance of higher-raio conversions o decrease dramaicall, perhaps beond usefulness, bu his is an open quesion. Alhough illiquid porfolios feaure reurn series ha challenge invesors in man was, here are several differen approaches and ools o help. We hope ha our analsis provides useful informaion and resuls for praciioners who wish o supplemen heir porfolio managemen approach oward hese complicaed invesmens. A PPENDIX Bes Linear Unbiased Esimaor (from Chow and Lin [97]) From he low-frequenc regression model P L = CP = CXβ + Ce = X L β + e L (A-) where he subscrip L denoes he low-frequenc series, = is he marix ha ies he C high-frequenc log reurns o he low-frequenc log reurns, assuming a hree-o-one raio and e L N(0, V L ). I can be shown ha he bes linear unbiased esimaor (BLUE) of higher-frequenc P is ˆ ˆ P = X β+ ( ) e P L L L (A-2) where ˆ ( β = V L L L ) X L L PL is he leas-squares esimae of he low-frequenc regression coefficiens and e X ˆ L PL Lβ is he vecor of residuals from he low-frequenc regression, and V P E[ [ L]. L In Equaion (A-2), he ermv P L V L dicaes how he low-frequenc residuals are disribued a he higher frequenc. Assuming ha he higher-frequenc residuals follow an AR() process e = ae + ε ε N(0, σ 2 ) (A-3) 34 HIGHER-FREQUENCY ANALYSIS OF LOW-FREQUENCY DATA FALL 204

15 where he coefficien a can be esimaed from is low-frequenc counerpar, one can easil derive where L is he lag operaor. * * Le μ = E( ) and E ( ). Then for all V V = VC ( ( ) ) = AC ( ( ) V P L L (A-4) μ = H( )( (A-0) * ( L)( μ* ) where V is he auocovariance of e, A = a a a a 2 n a a a a a n a a a n n 2 n 3 and n is he lengh of he high-frequenc series. Finall, hrough subsiuion Equaion (A-2) becomes Pˆ = X βˆ + AC ( ( ) e L (A-5) Mixed-Frequenc VAR Model (from Mariano and Murasawa [2003], [200]) VAR Model For simplici we assume a 3-o- raio o conver highfrequenc series ino low-frequenc ones. Saring wih {x, }, an N -variae low-frequenc reurns series observable ever quarer and {x,2 }, an N 2 -variae highfrequenc reurns series observable ever monh, we have for all * *, = l( ln( + x, ) ln(, + ln( + x, ) ln( * ) l( 2,) * * *, =, +, + 2,,2 + x,2 l( ln( x ) * where {,} (A-6) (A-7) x is a laen monhl series underling {x, }. We observe {, } ever quarer and never observe { }. Le for all =,,2, * = *,,2 (A-8) Assume a Gaussian VAR(p) model for { for all * * φ( L)( ) = w w ~ (0, Σ) Sae-Space Represenaion If p 3, hen we define he sae vecor as s = μ * * μ * * 2 } such ha A sae-space represenaion of he VAR model is (A-) (A-2) s As Bz z ~ (0, I N ) (A-3) =μ+ C s (A-4) φ φ p O ( 3 pn ) where N = N + N 2, A, I 2N O2 N N 2 Σ, and C = [H B = 0 H 2 ]. O2 N N As { } is a mixed-frequenc series, i has missing observaions. If p 3, hen we define he sae vecor as s = μ (A-2a) μ * * * * p+ and le H( ( ) = H + 0 H L H I = 0 N L2 I 0 N 2 + I N 0 I 0 N L L 2 (A-9) A sae-space represenaion of he VAR model is he same φ p p excep ha A = φ φ 2 Σ, B =, I ( p ) N O ( p ) N N O( ) p N N and C = [H 0 H 2 O N (p 3)N ]. FALL 204 THE JOURNAL OF PORTFOLIO MANAGEMENT 35

16 Maximum Likelihood Esimaion Following Mariano and Murasawa [2003], [200], we pu zeroes for missing observaions and rewrie he measuremen equaion as if he are random draws from N(0,) independen of he model parameers, so ha he Kalman filer skips missing observaions. Le for all + = v, i f, oherwise is observable (A-5) where { ν } ~ ( ) whose realizaions are all zeroes. The measuremen equaion is, μ C = + s (A-6),2 μ2 C2 We can wrie for all where + +,,2 = = μ, C, μ 2 + μ μ, = 0 C D Le for all +,,2,, = = 0 C 2 s, + D, 0 if, i sobservable oherwise C if, 0 I N oherwise is observable if, i sobservable oherwise v μ, C,, μ = =, C, D μ 2 C2 = (A-7) Then we have a ime-varing sae-space model for { } such ha for all + D, 0 s + = A s + Bz (A-8) = μ + C s s + D v (A-9) z ~ ( v ( ) (A-20) As { } has no missing observaions, he Kalman filer and smooher appl direcl. We hen obain ˆP, he smoohed esimae of he laen monhl series underling he quarerl series, using he EM algorihm and he smoohing algorihm described in Mariano and Murasawa [200]. ENDNOTES Depending on he approach, daa requiremens include bu are no limied o exensive cash flows hrough ime, full general parner porfolio ransparenc, and availabili of higher-frequenc priced public analogues o compan holdings. And even for simple implemenaions, significan qualiaive experise is necessar. A he oher end of he specrum, some approaches rel on complicaed esimaion echniques (see Ang e al. [203a]). 2 There are man reasons o be skepical of he repored reurn series of illiquid asse classes. However, in man pracical cases he smoohed series a a higher-frequenc is desirable, such as when an insiuion wans o assess is porfolio s esimaed repored hisorical performance during a sress even ha ma no fall on he low-frequenc inervals. Furher, invesors ma emplo downsream daa processing o correc for some undesirable characerisics (smoohing, fee skew, survivorship bias, and so forh). 3 The esimaed high-frequenc VAR model is his approach s ulimae oupu. 4 Mariano and Murasawa somewha miigae his problem b using demeaned series, hus reducing he number of parameers b he number of series. 5 Source: Cambridge Associaes, LLC. Coprigh 204. All righs reserved. The Cambridge Associaes, LLC daa ma no be copied, used, or disribued wihou Cambridge s prior wrien approval. The daa is provided as is, wihou an express or implied warranies. Pas performance is no guaranee of fuure resuls. 6 The models are esimaed using daa from Januar 986 hrough Sepember 202. ρ 7 = ( + [ ρ ρ ρ ] h () 2 ( )( ) ( ) 2 AR Scale Facor h 2 h ( ρ) ) where ρ is he auocorrelaion and h is he number of higherfrequenc reurns per ear. Noe ha his formula collapses o he radiional square-roo-of-ime rule when here is no auocorrelaion. 36 HIGHER-FREQUENCY ANALYSIS OF LOW-FREQUENCY DATA FALL 204

17 8 We repor onl posiive beas because he behavior is smmeric wih respec o sign. In a sense, his dimension can be inerpreed as he absolue value of he bea of he porfolio o he facor. 9 The large number of higher-frequenc observaions was seleced o esablish he pracical limi of comparaive performance of wo inerpolaors. 0 Inerpolaion a a higher raio requires more confidence in he explanaor power of he prox han seems warraned in general insiuional applicaions. Neverheless, such higherraio inerpolaions are possible. The prox canno explain 00% of he remaining variance, because par of he variance, afer adjusmens for he AR() componen, is explained b he error erms in he srucural form for which no adjusmen is made. 2 As he auoregressive componen of he porfolio series increases, he maximum bea of he porfolio o he facor moves furher and furher below one. This is because he porfolio and he facor have he same annualized volailiies, and he porfolio s auoregressive componen ensures ha is monhl volaili is lower han he facor s; hence, he bea is capped a some level less han one. 3 In his case, he facor acuall has a srong relaionship o he prox. I mus, in order o saisf he consrains ha he prox explains he majori of he porfolio reurns and he porfolio has a srong relaionship o he facor. Hence, he minimum bea of he facor o he prox is a + 2a + 3a + 2a + a Solve for a when q = where q = a + 4a + 3 For a derivaion, see Chow and Lin (97). 5 Source: Cambridge Associaes, LLC. Coprigh 204. All righs reserved. The Cambridge Associaes, LLC daa ma no be copied, used, or disribued wihou Cambridge s prior wrien approval. The daa is provided as is, wihou an express or implied warranies. Pas performance is no guaranee of fuure resuls. 6 We use onl one prox here, o be consisen wih he simulaion secion. I is likel ha using muliple proxies can be beneficial (especiall for real esae). As noed in an earlier secion, care mus be aken when adding addiional proxies paricularl when using MM. 7 While an AR() specificaion ma no full accoun for he auocorrelaion ha exiss in illiquid asse reurns, our quarerl daa (2003 hrough 202) indicae ha he parsimonious AR() specificaion is suiable. 8 We annualize volaili, aking ino accoun he observed AR() componen of privae equi and real esae reurns. 9 For 95% VaR, we use a normal disribuional assumpion o find an analical soluion ha depends solel on he sandard deviaion and is cenered b seing he mean o zero. We do his for illusraion onl. 20 We repor conemporaneous beas, hough we are aware ha some mehodologies wihin illiquid asse classes advocae summing lagged beas o public marke facors. We have found (bu do no repor) ha appling hese mehodologies on he inerpolaed series gives resuls similar o hose based on he acual quarerl daa. REFERENCES Ang, A., B. Chen, W. Goezmann, and L. Phalippou. Esimaing Privae Equi Reurns from Limied Parner Cash Flows. Columbia Business School research paper 3-83, 203a. Ang, A., N. Nabar, and S. Wald. Searching for a Common Facor in Public and Privae Real Esae Reurns. The Journal of Porfolio Managemen, Special Real Esae Issue, 6h ed. (203b), pp Chang, C., and T. Liu. Monhl Esimaes of Cerain Naional Produc Componens, The Review of Economics and Saisics, Vol. 33, No. 3 (Augus 95), pp Cho, H., Y. Kawaguchi, and J. Shilling. Unsmoohing Commercial Proper Reurns: A Revision o Fisher- Gelner- Webb s Unsmoohing Mehodolog. The Journal of Real Esae Finance and Economics, Vol. 27, No. 3 (November 2003), pp Chow, G., and A. Lin. Bes Linear Unbiased Inerpolaion, Disribuion, and Exrapolaion of Time Series b Relaed Series. The Review of Economics and Saisics, Vol. 53, No. 4 (November 97), pp Fernandez, R. A Mehodological Noe on he Esimaion of Time Series. The Review of Economics and Saisics, Vol. 63, No. 3 (Augus 98), pp Fisher, J., D. Gelner, and R. Webb. Value Indices of Commercial Real Esae: A Comparison of Index Consrucion Mehods. The Journal of Real Esae Finance and Economics, 9 (994), pp Gelner, D. Smoohing in Appraisal-Based Reurns. The Journal of Real Esae Finance and Economics, Vol. 4, No. 3 (Sepember 99), pp Harris, R., T. Jenkinson, and S. Kaplan. Privae Equi Performance: Wha Do We Know? NBER Working Paper 7874, 202. FALL 204 THE JOURNAL OF PORTFOLIO MANAGEMENT 37