PREDICTING AGGREGATE RETURNS USING VALUATION RATIOS OUT-OF-SAMPLE*

Transcription

1 PREDICTING AGGREGATE RETURNS USING VALUATION RATIOS OUT-OF-SAMPLE* 93 Ana Sequeira** Aricles Absrac I is well esablished ha valuaion raios (indicaors of he financial marke siuaion) provide, in-sample, relevan signals regarding fuure reurns on asses. Specifically, periods of high prices, relaive o dividends, are proceeded by years of low reurns; and periods of low prices, relaive o dividends, precede years of high reurns. This paern of predicabiliy is pervasive across financial markes. In his paper, we assess he abiliy of valuaion raios o predic ou-of-sample aggregae reurns for he sock and he housing markes in he U.S.. We find ha here is saisical evidence supporing he exension of he in-sample resuls o an ou-of-sample framework. The dividend-price raio and he ren-price raio display a significan abiliy for predicing in real-ime sock and housing reurns, respecively. Neverheless, we noe ha hese findings may be sample dependen. Especially for he sock marke, he sample s ending daa, including he recen financial crisis, may be responsible for he good resuls. 1. Inroducion Predicing reurns is one of he mos discussed opics in he academic financial world. Cochrane (2011) summarizes he evidence ha shows he exisence of a paern of predicabiliy ha is pervasive across markes (socks, bonds, houses, foreign exchange and sovereign deb) and concludes (in-sample) ha valuaion raios predic excess reurns, insead of fuure cashflows. 1 In his conex, for he sock marke, one concludes ha he dividend-price raio predics reurns and does no predic dividend growh. Moreover, low dividend-price raios signal low fuure reurns; and high dividend-price raios signal high fuure reurns. For he housing marke, he argumen is similar: high prices, relaive o rens, imply low reurns; do no signal he permanen increase of rens or prices. In his paper, we inend o verify wheher his pervasive phenomenon holds ou-of-sample, ha is, wheher a forecaser would be able o predic excess reurns sysemaically, if he sood a he forecas momen wihou furher informaion (simulaing he producion of forecass in real-ime). We focus on he housing marke (here are relaively few sudies abou predicing he housing reurns) and use he sock marke analysis as an imporan reference. Among he sudies ha examine he predicabiliy of housing reurns, Case and Shiller (1990) show * This paper is a summary of my maser hesis (which conains all he deails), developed in Economics and Research Deparmen (Banco de Porugal). I would like o hank my advisor João Valle e Azevedo for his permanen availabiliy, ideas and relevan observaions. I am also graeful o Ana Pereira for providing me her Mahemaica programming codes and clarifying some quesions, and Marín Saldías for his imporan suggesions. The opinions expressed in he aricle are hose of he auhor and do no necessarily coincide wih hose of Banco de Porugal or he Eurosysem. Any errors and omissions are his sole responsibiliy. ** Banco de Porugal, Economics and Research Deparmen. 1 See, Fama and French (1988, 1989) for socks; Fama and Bliss (1987), Campbell and Shiller (1991) and Piazzesi and Swanson (2008) for Treasuries; Fama (1986) for Bonds; Hansen and Hodrick (1980) and Fama (1984) for foreign exchange; Gourinchas and Rey (2007) for foreign deb.

2 ha he ren-price raio has a good performance when used o predic (in-sample) he housing excess reurns. Laely, Plazzi e al. (2010) conclude (also, in-sample) ha he ren-price raio predics expeced reurns for aparmens, reail properies and indusrial properies (bu does no predic expeced reurns of office buildings). 94 BANCO DE PORTUGAL ECONOMIC BULLETIN Summer 2013 For he sock marke, he lieraure is voluminous. Goyal and Welch (2003, 2008) explore he exisence of gains when one uses he financial variables wih a reasonable in-sample performance o forecas (ou-of-sample) he equiy premium (sock reurns over he reurn of a shor-erm risk-free ineres rae) and conclude ha almos all models produce poor resuls ou-of-sample. On he oher hand, Rapach and Wohar (2006) find ha several financial variables have a good in-sample and ou-of-sample abiliy o predic sock reurns. Rapach e al. (2010) also find significan ou-of-sample gains using forecas combining mehods. Our resuls show ha here is saisical evidence supporing he exension of he in-sample resuls o an ou-of-sample framework in boh markes. Especially for he housing marke, we conclude ha he ren-price raio has a huge abiliy for predicing reurns. Given he lack of ou-of-sample sudies for he housing marke, hese findings are a conribuion o he lieraure. As Rapach and Wohar (2006), our purpose is esing for he exisence of reurn predicabiliy in populaion. As for his paper, we are no ineresed in exploring wheher a praciioner in real-ime could have consruced a porfolio ha earns exra-normal reurns. The remainder of his paper is organized as follows. In Secion 2 we describe he daa used o obain he empirical resuls. Secion 3 repors he in-sample resuls while Secion 4 exposes he economeric mehodology. In Secion 5 we discuss our main findings and in Secion 6 we presen he conclusions and ideas for fuure research. 2. Daa Sock Marke: 2 As Leau and Ludvigson (2001), we use quarerly daa for he U.S. sock marke. Our sample covers he period 1947:Q1 2010:Q2 (sample wih T 254 observaions) and our dependen variable is he equiy premium from holding socks (represened in an index) from period o h. As usual, we define equiy premium as he reurn on he sock marke minus he reurn on a shor-erm risk-free ineres rae. In our case, we use he Cener for Research in Securiy Prices (CRSP) value-weighed reurn and he 3-monh Treasury bill (as a proxy for he risk-free rae). The dividend-price raio is he financial variable which poenially predics he equiy premium. Housing Marke: In our applicaions for he housing marke, we use quarerly daa from 1960:Q1 o 2010:Q1 (sample wih T 201 observaions). We consider wo house price indexes: he price index compued by Case, Shiller and Weiss and he purchase-only price index released by he Office of Federal Housing Enerprise Oversigh (OFHEO). 3 Since he resuls were very similar, we chose o presen only he conclusions for he daa from Case, Shiller and Weiss. Our dependen variable is he log reurn from holding he house from unil h and he predicor is he respecive ren-price raio. 2 The sock marke daa and he housing marke daa are available a hp://faculy.chicagobooh.edu/john.cochrane/research/index.hm. The housing marke daa is also available a hp:// -values/ren-price-raio.asp. 3 See Calhoun (1996) for deailed informaion.

3 3. In-Sample Fi In his secion, we discuss he resuls obained hrough he in-sample regressions which moivae he ou-of-sample exercise. Le us consider he following regression model: y, 1,, h x u h T (1) where y h is he reurn from holding he financial asse from unil h, h 0 is he forecas horizon, is a disurbance erm. x is he financial variable used o predic y h and To assess he predicive abiliy of x in-sample, we can esimae he equaion (1) using he available T h observaions and hen, examine he -saisic associaed o he OLS esimae of and he goodness-of-fi measure ( R 2 ). 4 When here is evidence o rejec he null hypohesis of 0 and he R 2 is high, we can conclude ha x has predicive power over y h. There are some problems relaed o his perspecive, specifically he small-sample bias (see Sambaugh 1986, 1999) and he dependence beween he observaions for he regressand in (1) (see Richardson and Sock, 1989). The serial correlaion induced in he disurbance erm should be aken ino consideraion when conducing inference. u h 95 Aricles For each marke in quesion, we esimae equaion (1) by OLS and use he Newey and Wes (1987) sandard errors o compue he usual -saisic. Table 1 provides he resuls. Analyzing he resuls, we deec a common paern across he wo markes. The esimae of and he R 2 are higher for longer forecas horizons, and he observed -saisics always rejec he null hypohesis of no predicabiliy. In addiion, he signal of he esimaes is posiive, which confirms he conclusions presened in Cochrane (2011): high prices relaive o dividends (or rens, for he housing marke) can be a sign of low reurns. We mus herefore invesigae wheher he in-sample predicabiliy is kep when subjec o a simulaion exercise of forecass in real ime (ou-of-sample). Table 1 IN-SAMPLE REGRESSIONS (HORIZONS OF 1, 4, 8, 12, 18 AND 24 QUARTERS) Sock Marke Housing Marke Sample period 1947:Q1-2010:Q2 Sample period 1960:Q1-2010:Q1 Case-Shiller-Weiss OFHEO daa Horizon ^ Adj. R² ^ Adj. R² ^ β -sa R² % β -sa R² % β (quarers) % % -sa R² % 1 3,80 (2,89) 2,85 2,46 1,27 (5,24) 22,30 21,91 1,21 (8,47) 32,95 32, ,57 (3,14) 11,23 10,88 5,90 (2,88) 38,80 38,48 5,39 (4,73) 45,47 45, ,08 (3,38) 19,97 19,64 12,86 (3,49) 54,85 54,61 11,41 (5,52) 59,06 58, ,35 (3,97) 25,38 25,07 18,84 (4,62) 64,31 64,12 16,67 (6,81) 67,00 66, ,17 (5,17) 33,05 32,76 25,27 (5,73) 67,05 66,87 22,73 (7,98) 69,76 69, ,28 (6,52) 44,47 44,23 29,68 (5,44) 61,46 61,24 27,18 (7,63) 66,90 66,71 Adj. R² % Source: Auhor s calculaions. Noes: The regression equaion is y h x u, where y +h and x are he equiy premium and he dividend-price raio, respecively, for he sock marke; and he log reurns and he ren-price raio for he housing marke. -sa denoes he Newey and Wes (1987) adjused -saisic. 4 The null hypohesis ( H : 0 ) reflecs he lack of abiliy of x 0 o forecas y h.

4 4. Economeric Procedure In his secion, we discuss he regression models used o produce he ou-of-sample forecass and he saisic ess applied o analyze he resuls. 96 BANCO DE PORTUGAL ECONOMIC BULLETIN Summer Ou-of-sample analysis An ou-of-sample analysis implies he simulaion of forecass for y h, in period. Given he sample in sudy, we should deermine a value for he sample-spli parameer (R), which will mach wih he period of he firs predicion (here is no crierion ha defines how o choose R; one should make a compromise beween he number of observaions used o esimae he coefficiens of he models and he number of available observaions o assess he forecass performance). We use he sample ha includes he firs R h observaions o esimae he model and produce he predicions for R. Then, he nex observaion (which corresponds o he period ( R 1) h), is added o he sample, he model is re-esimaed and a new forecas (for period R 1) is generaed. This process is repeaed unil he end of he sample Predicive regression models We selec several mehods o generae he se of predicions. In wha follows, yˆ h denoes he forecas of y h(he reurn from holding he financial asse from o h ), given he informaion up o period, and x is he valuaion raio ha migh have predicive power for y h. 1 Hisorical mean: y ˆ h y, R,, T. s As Goyal and Welch (2003, 2008) and Campbell s 1 and Thompson (2008), we use he hisorical mean as a benchmark forecasing model, since i represens he hypohesis of no predicabiliy, consisen wih he mos common inerpreaion of he efficien markes hypohesis. 1 Direc auoregressive model: y ˆ ˆ ˆ p h y j j. In our empirical applicaions, we consider j 0 a version wih a fix lag order ( p 2 ) and a version wih he lag order deermined using he AIC Crierion (p ineger [1,4]). 5 The coefficiens are esimaed by OLS. * p1 1 * p2 1 ˆ ˆ j j j j j 0 j 0 * Direc augmened auoregressive model: y ˆ ˆ h y x, p 1*, p 2 inegers [1,4]. The lag orders ( p * 1 and p * 2 ) are deermined using he AIC Crierion and coefficiens are esimaed by OLS. 1 Direc regression model wih or wihou lags: y ˆ ˆ ˆ p h x. i i Here, we also consider a i 0 fix lag order ( p 2 ) and a lag order deermined using he AIC Crierion (p ineger [1,4]). The coefficiens are esimaed by OLS. Univariae and mulivariae low-pass filers: Following he argumen presened in Valle e Azevedo and Pereira (2012), we use his mehod o generae our forecass when i is useful o predic only he low frequencies of y ( w B( L) y, where B L is a band-pass filer eliminaing he flucuaions wih period smaller han 32 quarers, as is usual in business cycle sudies). Thus, we consider he predicions of he low frequencies of aggregae reurns as forecass of aggregae T reurns iself. Since he available sample is finie ( y ) and supposing ha here are c series 1 of covariance-saionary covariaes ( z z 1,, c ) available, we approximae he low frequencies of y (ha is, we approximae w ) hrough a weighed sum of elemens of y and z z 1,, c : p c p ˆ p, f ˆ p, f j j s, j s, j j f s 1 j f (2) wˆ B y R z 5 See Akaike (1974).

5 p and f denoe he number of observaions in he pas and in he fuure, respecively, ha are considered. The coefficiens are esimaed solving a minimizaion problem. 6 To exrac he signal w B( L) y h h for h 0, we should se f h in he soluion of he menioned problem (as a resul, only he available informaion up o period is employed). We obain he univariae filer when we drop he covariaes z,, 1 z from equaion (2). c Forecas evaluaion Aricles As an evaluaion meric o compare he ses of forecass obained hrough he models described before, we chose he raio beween he mean squared forecas error of he compeing model and hose of he benchmark model ( MSFE raio ). When he MSFE raio is lower han 1, he compeing model generaes beer predicions (according o he menioned crieria) han he benchmark model (hisorical mean). We also use a graphical analysis o examine he relaive performance of he forecasing models over he sample. As proposed in Goyal and Welch (2003), we consruc chars wih he difference beween he cumulaive squared forecas errors of he benchmark model and he cumulaive squared forecas errors of he compeing model (hereafer, we will refer o his difference as Ne SSE ). When his difference is posiive, he compeing model ouperforms he benchmark model in he sample beween he firs predicion and he dae in he x axis. 4.4 Ou-of-sample ess We assess he saisical significance of he obained resuls considering equal accuracy ess and forecas encompassing ess. The equal accuracy es allows esing wheher he MSFE raio is saisically equal o 1, agains he alernaive ha he forecass produced by he compeing model are beer (have a lower MSFE ). We apply he es saisics modified MSFE and MSFE F o es his null hypohesis (see Diebold and Mariano, 1995; Harvey e al. 1997, 1998 and McCracken, 2007). Excluding he mulivariae filer, all our models are nesed, herefore he criical values for hese saisics were generaed by simulaion. 7,8 According o Harvey e al. (1997), a se of forecass encompasses a rival se if he laer does no conribue o a saisically significan reducion in MSFE when used in combinaion wih he original se of forecass. So, wih he forecas encompassing es, we assess wheher a given se of forecass generaed by a simpler model embody all he useful predicive informaion conained in anoher se of forecass. Applying his concep o our sudy, if he hisorical mean forecas encompasses he forecas produced by he model wih he valuaion raio, he financial variable does no conain useful addiional informaion for predicing he aggregae reurns. The es saisics employed, modified ENC and ENC F, resul from an adapaion o his problem of he Diebold and Mariano (1995) es saisic. Again, since he es saisics considered do no have a sandard disribuion, he criical values were generaed using a boosrap procedure. 6 More deailed explanaions abou he mulivariae filer can be found in Valle e Azevedo (2011) and Valle e Azevedo and Pereira (2012). 7 Two models are nesed when here is a se of regressors ha is common beween hem (see Clark and McCracken, 2005). In our sudies, we have nesed models due o he consan erm in mos models. 8 We rely on mehods provided by Kilian (1999) and Mark (1995) for developing he boosrap procedure used o generae he criical values.

6 5. Empirical Resuls 98 BANCO DE PORTUGAL ECONOMIC BULLETIN Summer 2013 In his secion, we discuss he main resuls obained using he mehodology described before. Sock Marke: We conclude ha only he direc regression generaes forecass ha can bea he benchmark for all horizons (see Table 2). The MSFE raios are saisically lower han 1 a convenional significance levels, which means ha he forecass from he compeing model have more predicive power han hose from he hisorical mean model. The MSFE raios decrease wih he forecas horizon, which suggess ha he dividend price raio abiliy o predic he aggregae reurns improves when we use longer horizons. These findings are consisen wih he in-sample resuls exposed in Secion 3, where we noe ha he in-sample predicabiliy increases wih he horizon. The univariae filer model failed o ouperform he benchmark model for all horizons, bu he mulivariae filer has MSFE raios lower han 1 for h 20 and h 24 (despie no being saisically lower han 1 when we use he modified MSFE saisic o apply he es). Similar conclusions can be drawn when we analyze he forecas encompassing resuls (Table 3). In paricular, when we use he ENC F o perform he es, we have saisical evidence o rejec he null hypohesis (he hisorical mean forecass encompass hose produced by direc regression model) a a 5% significance level. The following analysis ress on he evaluaion of he Ne SSE chars which display he cumulaive squared forecas errors of he benchmark model (from 1985:Q1 hrough he dae in he x axis) minus he cumulaive squared forecas errors of he compeing model (from 1985:Q1 rough he dae in he x axis), for each horizon. A posiive value means ha he compeing model has ouperformed he benchmark model and a posiive slope indicaes ha he compeing model had a lower forecasing error han he hisorical mean model, in a given quarer. In Char 1 we plo he menioned curves for h 1 and h 24, considering as compeing models he direc regression model (wihou lags) and he mulivariae filer (which uses he dividend-price raio). Considering he shorer forecas horizon (1 quarer), we noe ha he direc regression curve exhibis a volaile paern. This compeing model had a good performance in 1987:Q4 1995:Q4, 2002:Q2 2003:Q3 and 2008:Q2 2010:Q2 and had is poores performance from 1997:Q3 o 2001:Q1 (alhough i begins o recover he curve has a posiive slope from 2000:Q1). For h 1, he mulivariae filer consisenly has a worse performance han he direc regression model. For longer forecas horizons ( h 24 ; see figure 1), he curves are smooher and we can idenify hree disinc periods (which have become more apparen as he horizon increases). Namely: an iniial period Table 2 MSFE RATIOS AND EQUAL ACCURACY TEST RESULTS FOR THE STOCK MARKET Horizon (quarers) Direc regression model (p=0) 0.988* 0.984** 0.976** 0.987* 0.969*** 0.909*** 0.883*** 0.883*** Direc regression model (p=2) Direc regression model (pmax=4) Mulivariae filer wihou indicaors wih dividend-price raio Source: Auhor s calculaions. Noes: For he direc regression (nesed model), we use he es saisic MSFE-F and criical values generaed using a boosrap procedure; for he mulivariae filer (non-nesed model), we consider he es saisic modified MSFE- and criical values from he Suden s disribuion wih (N-1) degrees of freedom (N is he number of forecas errors; see Clark and McCracken, 2001 and McCracken, 2007). Predicions were generaed for he period 1985:Q1 2010:Q2. Significance levels a 10%, 5% and 1% are denoed by one, wo, and hree sars, respecively.

7 Table 3 FORECAST ENCOMPASSING TEST RESULTS FOR THE STOCK MARKET Horizon (quarers) Direc regression model (p=0) ENC-F 0,813 2,224** 1,870** 0,471-0,361 2,615*** 4,375*** 5,892*** Source: Auhor s calculaions. Noes: The es was applied assuming ha he forecass are biased and inefficien (more general case). The criical values are generaed using a boosrap procedure. Predicions were generaed for he period 1985:Q1 2010:Q2. Significance levels a 10%, 5% and 1% are denoed by one, wo, and hree sars, respecively. 99 Aricles when he forecass produced by he compeing models are beer, an inermediae period when he models had a negaive performance and a final period of recovery. We noe ha his final period may be responsible for he good resuls ou-of-sample, meaning ha if we dropped he las observaions of he sample, he direc regression model probably could no bea he benchmark. Addiionally, we decide o plo, in he same char, he Cumulaive SSE Difference curve (he compeing model seleced only conains he dividend-price raio as regressor direc regression model wihou lags) and a price index curve (for he sock marke, we chose he SP500 Index). Wih his exercise, we inend o noe how he curves are relaed and discuss he reasons for his relaionship. The key poin o emphasize is ha he curves exhibi a symmerical behaviour: he peaks in SP500 Index correspond o he roughs in he Ne SSE curve (see Char 2). This means ha he model wih he dividend price-raio produces less accurae predicions for he period in which he sock price is increasing, while is good performance is associaed wih a period in which here is a fall in prices. We noe hese phenomena when, for example, we analyze he exuberan period associaed wih he Do-com, in he lae nineies. During he pre-crash period, he SP500 Index rises and he Ne SSE decreases (he hisorical average is a beer predicor han direc regression model over his period). Neverheless, afer he fall in prices, we idenify he reverse performance: he direc regression forecass are closer o he observed value (he Ne SSE curve has a posiive slope). How can we explain his relaion? Resuming he inroducory discussion low dividend-price raios signal low fuure reurns we can deduce ha when prices increase, relaive o dividends, i can be expeced a reducion of reurns in subsequen periods. Thus, we undersand ha when he price increases (and he dividends remain sable), our model ha is predicing a fall in reurns exhibis a worse performance. This ranslaes ino a negaive Char 1 CUMULATIVE SSE DIFFERENCE CONSIDERING TWO COMPETING MODELS, FOR H=1 AND H= Direc Regression Model (h=1) Mulivariae Filer Model (h=1) :1 1988:1 1991:1 1994:1 1997:1 2000:1 2003:1 2006:1 2009: Direc Regression Model (h=24) Mulivariae Filer Model (h=24) :1 1988:1 1991:1 1994:1 1997:1 2000:1 2003:1 2006:1 2009:1 Source: Auhor s calculaions.

8 Char BANCO DE PORTUGAL ECONOMIC BULLETIN Summer 2013 RELATIONSHIP BETWEEN THE NET-SSE AND THE SP500 INDEX Ne-SSE (Direc Regression Model, h=1) SP500 index (righ-hand scale) :1 1988:1 1991:1 1994:1 1997:1 2000:1 2003:1 2006:1 2009:1 Sources: Federal Reserve Bank of S. Louis (FRED) and auhor s calculaions. inclinaion for he Ne SSE curve and, simulaneously, a posiive inclinaion for he SP500 curve. In he pos-crash period, he reurns descend seeply. A his ime, he compeing model produces good predicions (relaively o he hisorical mean forecass), and herefore he Ne SSE slope is posiive (whereas i is negaive for he SP500 ) Housing Marke: For forecas horizons shorer han 3 years (12 quarers), we find ha all he compeing models produce beer forecass han he benchmark model. However, and imporanly, for longer horizons (over 3 years), only he models ha conain he ren-price raio exhibi MSFE raios lower han 1 (see Table 4). In paricular, he MSFE raio beween he direc regression and he benchmark model decreases as he horizon increases (all he values are saisically lower han 1, a 1% significance level). Comparing wih he resuls obained in Secion 3, we verify ha he predicabiliy paern idenified in-sample holds ou-of-sample for he housing marke. Table 5 displays he forecas encompassing saisics which allow he conclusion ha he hisorical mean forecass never encompass he forecass generaed by he direc regression model (he null hypohesis is always rejeced a a significance level of 1%). Figure 3 conains he chars wih he Ne SSE (for h=1, 12, 18 and 24), considering hree compeing models and he Case, Shiller and Weiss daa. The direc regression model had mild underperformance from 1998:Q1 o 2006:Q4, conversely i had a superior performance in he res of he sample (considering h 1 ). The oher wo models exhibi a really good performance from 2006:Q1 o 2010:Q1 (before ha, he Ne SSE is almos zero for boh models). When we consider he forecas horizon of 12 quarers, he compeing models only bea he hisorical mean model from 2008:Q1 (approximaely) and he models ha include he ren-price raio sar o exhibi a beer performance han he model wih only he auoregressive componen. This paern is obvious when we analyze he horizons of 18 and 24 quarers, where he cumulaive SSE difference beween he auoregressive model and he benchmark model is consanly negaive. From 2008:Q1, he 9 We did he same exercise for he housing marke and he conclusions are similar.

9 Table 4 MSFE RATIOS AND EQUAL ACCURACY TEST RESULTS FOR THE HOUSING MARKET Horizon (quarers) Direc auoregressive model (p=2) , Direc augmened AR (pmax=4) , Direc regression model (p=0) 0.785*** 0,738*** 0.724*** 0.697*** 0.579*** 0.417*** 0,401*** 0.401*** Direc regression model (p=2) , Aricles Mulivariae filer wihou indicaors 0.554** 0, wih ren-price raio 0.541** 0, , Source: Auhor s calculaions. Noes: See Table 2. Predicions were generaed for he period 1998:Q1 2010:Q1. Significance levels a 10%, 5% and 1% are denoed by one, wo, and hree sars, respecively. Table 5 FORECAST ENCOMPASSING TEST RESULTS FOR THE HOUSING MARKET Horizon (quarers) *** Direc regression model (p=0) 3.055*** 2.492*** 3.620*** *** *** *** *** Mulivariae filer wih ren-price raio 3.715*** 2.053** 1.696** 1.607* 1.441* Source: Auhor s calculaions. Noes: See Table 3. Predicions were generaed for he period 1998:Q1 2010:Q1. Significance levels a 10%, 5% and 1% are denoed by one, wo, and hree sars, respecively. direc regression curve grows almos exponenially, evidencing he predicive power of he ren-price raio. Again, i is imporan o underline he imporance of he observaions corresponding o he end of he sample o he good resuls ou-of-sample. 6. Conclusion In his paper, we found evidence ha he known in-sample paern of reurn predicabiliy (using valuaion raios) holds ou-of-sample for he sock marke and, especially, for he housing marke. Considering he sock marke, we show ha a simple regression model ha includes a valuaion raio ouperforms he benchmark (which represens he hypohesis of no predicabiliy of reurns), for all horizons. Addiionally, we noe ha he abiliy of dividend price raio o predic he aggregae reurns improves a longer horizons. For he housing marke, all he models ha conain he ren-price raio consisenly exhibi MSFE raios lower han 1, for all horizons. The sample dependence (significance of he las observaions for he good resuls ou-of-sample) idenified for boh markes deserves furher aenion. I will be ineresing o invesigae his issue in deail, noably by examining he sabiliy of he forecas funcion while linking i o specific evens affecing hese markes or, more generally, he U.S. economy. Addiionally, i would be ineresing o exend his research o oher markes, namely bonds and reasuries since here are relaively few sudies abou predicing reurns on hese markes, ou-of sample. Anoher suggesion would be o reproduce his sudy using daa for European markes, aiming he developmen of overvaluaion indicaors.

10 Char 3 CUMULATIVE SSE DIFFERENCE CONSIDERING THREE COMPETING MODELS (HORIZONS OF 1, 12, 18 AND 24 QUARTERS). 102 BANCO DE PORTUGAL ECONOMIC BULLETIN Summer Auoregressive Model (h=1) Augmened Auoregressive Model (h=1) Direc Regression Model (h=1) Auoregressive Model (h=18) Augmened Auoregressive Model (h=18) 2.00 Direc Regression Model (h=18) Auoregressive Model (h=12) Augmened Auoregressive Model (h=12) Direc Regression Model (h=12) Auoregressive Model (h=24) Augmened Auoregressive Model (h=24) 2.00 Direc Regression Model (h=24) Source: Auhor s calculaions.

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