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1、Stock Return Predictability:Is it There?Andrew Ang?Columbia UniversityGeert Bekaert?Columbia University and NBERFirst Version: 4 March 2001JEL Classification Codes: C12, C51, C52, E49, F30, G12Keywords: present value model, predictability, international predictability,short rates, dividend yield, ea

2、rnings yield?We thank Xiaoyan Zhang for useful discussions and for help with data. We thank Charles Jones,Tano Santos, Ken Singleton and seminar participants at Columbia University and Stanford Universityfor helpful comments. Andrew Ang thanks the Chazen Institute at Columbia University for financia

3、lsupport. Geert Bekaert thanks the NSF for financial support.?Columbia Business School, 805 Uris Hall, 3022 Broadway, New York, NY 10027; ph: (212) 854-9154; fax: (212) 662-8474; email: aa610columbia.edu; WWW: http:/www.columbia.edu/?aa610.?Columbia Business School, 802 Uris Hall, 3022 Broadway, New

4、 York, NY 10027; ph: (212) 854-9156; fax: (212) 662-8474; email: gb241columbia.edu.AbstractWe ask whether stock returns in France, Germany, Japan, the UK and the US are predictableby three instruments: the dividendyield, the earnings yield and the short rate. The predictabilityregression is suggeste

5、d by a present value model with earnings growth, payout ratios and theshort rate as state variables. We use this model imposing a constant risk premium to examinethe finite sample evidence on predictability. Not only do we find the short rate to be a relevantstate variable theoretically, it is also

6、the only robust short-run predictor of equity returns. Theevidence in Lamont (1998) on earnings and dividend yield predictability is not robust to ourincreased sample period, does not survive finite sample corrections and does not extend to othercountries. We find no evidence of long-horizon predict

7、ability once we account for finite sampleinfluence. Finally, cross-country predictability appears stronger than predictability using localinstruments.1IntroductionAlarge bodyofempiricalwork hasaccumulateddocumentingexcessstockreturnpredictability.Among the most popular predictors are the nominal int

8、erest rate and the dividend yield.1Thedividend yield appears to be the most popular stock return predictor used in applied work, butmore recently Lamont (1998) and Campbell and Shiller (1988) argue that the earnings yield,has independent forecasting power for excess stock returns in addition to the

9、dividend yield.The debate on what drives the predictability continues. It may reflect irrational investor be-havior and hence be exploitable in trading strategies (see Cutler, Poterba and Summers (1989);it may reflect time-varying risk premiums (Kandel and Stambaugh (1990), Campbell andCochrane (199

10、9), Bekaert and Grenadier (2000); or it may simply not be present in the data.This last possibility gains credulity considering the long list of authors criticizing the statisticalmethodologies in the predictability literature. The coefficients on the predictor variables arebiased, since these varia

11、bles are typically persistent, endogenous regressors correlated with re-turns innovations (Stambaugh (1999). The standard focus on long-horizon regressions is prob-lematic from a number of different perspectives. The distributions of the?(Kirby (1997)and the t-statistics on the coefficients, (Richar

12、dson and Stock (1989), Richardson and Smith(1991), Hodrick (1992) and Valkanov (2000) in long-horizon regressions are severely shiftedto the right, leading to over-rejection of the no-predictability null. Researchers often forgetto properly interpret various tests over different horizons by providin

13、g joint tests (Richardson(1993). Finally, the possibility of decades of data mining clouds any inference regarding pre-dictability for US stock returns (Lo and MacKinlay (1990), Foster, Smith and Whaley (1997)and Bossaerts and Hillion (1999).In this paper, we re-examine the case for the predictabili

14、ty of short and long-horizon stockreturns. We start by proposing a simple price earnings model, in which the variation in theprice-earnings ratio and expected returns on equities is drivenby three stochastic state variables,the payout ratio, earnings growth and the short rate. In this model, the ear

15、nings yield, thedividend yield and the short rate jointly capture any potential predictability, motivatinga simplemultivariate regression of excess stock returns over various horizons on these three variables asthe main predictability regression. When certain parameter restrictions are met, the mode

16、l hasa (near) constant expected excess return variant, in which the expected gross return on equityequals a constant multiple of the short rate.1For predictability of excess stock returns by the nominal interest rate see, among others, Fama and Schwert(1977), Campbell (1987), Breen, Glosten and Jaga

17、nnathan (1989), Shiller and Beltratti (1992), and Lee (1992).Among those examining the predictive power of the dividend yield on excess stock returns are Fama and French(1988), Campbell and Shiller (1988, 1989), Goetzmann andJorion (1993, 1995), Hodrick (1992), Goyaland Welch(1999) and Valkanov (200

18、0).1Given the considerable statistical challenges in establishing predictability, we preceed ouranalysis of the data with an extensive Monte Carlo analysis under the null of no predictabil-ity. Our analysis incorporating earnings yields largely corroborates the results of Boudoukhand Richardson (199

19、3), Hodrick (1992), and Richardson and Smith (1991) who suggested thatthe finite sample properties of the long-horizon regression t-statistics improve dramatically byremoving the moving average structure in the error terms, induced by summing returns overlong horizons, in constructing the standard e

20、rrors. For brevity, we will refer to these alternativestandard errors as Hodrick (1992) standard errors. As Goetzmann and Jorion (1993, 1995) pointout, standard Monte Carlo analysis ignores the fact that yield variables involve the inverse ofprice, an endogenous variable, which is also present in th

21、e denominator of the return on theleft hand side. They conduct bootstrap exercises that impose this constraint, but their boot-strap keeps dividends non-stochastic at their data levels and therefore ignores the cointegrationrelation between dividends and price levels that characterizes rational pric

22、ing. By simulatingthe constant expected return variant of our earnings model, we accommodate this endogeneityconstraint in an entirely coherent way.2It remains true that the Hodrick standard errors are farsuperior in conducting inference and have negligible size distortions, whereas Ordinary LeastSq

23、uares (OLS) or Hansen-Hodrick (1980) standard errors lead to severe over-rejections of theno predictability null at long horizons.Armed with well-behaved t-statistics, we establish that the predictability evidence for USreturns is surprisingly weak. In fact, the only variable retaining significance

24、is the short rate,and it is only significant at short horizons. To mitigate data snooping concerns, we investigateanalogous predictability regressions for four other countries, France, Germany, Japan and theUK. Interestingly, we find that the predictability coefficients are not robust across countri

25、esin sign or magnitude, except for the short rate effect. When we pool the regression acrosscountries, the short rate remains the only significant predictor of excess stock returns.Finally, we also investigate a number of cross-country predictability regressions, examiningwhether any predictors have

26、 predictive power across countries. Unlike Bekaert and Hodrick(1992) and Ferson and Harvey (1993), we only find evidence of strong predictability when wepool across countries. With cross-sectional information from international data we find that USinstruments are strong predictors of foreign equity

27、returns, unlike local instruments. The localshort rate effect is subsumed by the predictive power of the US short rate. We also confirmand extend Bekaert and Hodrick (1992)s finding that yield variables have predictive powerfor excess returns in the foreign exchange market. We conclude that the curr

28、ent predictability2Bollerslev and Hodrick (1996) provide a detailed Monte Carlo analysis in the context of a present value modelwith constant and time-varying expected return variants which also imposes this constraint, but their solution tothe present value model is only approximately true.2debate

29、focuses on the wrong horizon (long-run instead of short-run), the wrong instruments(yield variables instead of interest rates) and the wrong setting (US segmented market insteadof a globally integrated market).The remainder of the paper is organized as follows. Section 2 sets out the empirical frame

30、-work, including the present value earnings model and the predictability regressions. Section 3describes the econometric estimation, the Monte Carlo analysis and describes the data. Section4 considers the predictability in US returns, whereas Section 5 investigates and compares pre-dictability in al

31、l 5 countries. Section 6 investigates predictability across countries. Section 7concludes and offers an interpretation of our results.2Theoretical and Econometric Framework2.1A Simple Present Value ModelModern predictability regressions consider the predictability of excess stock returns, the return

32、on equity overand abovethe return on anominallyrisk-free security of the same holding period,which is known one period in advance. Since there is substantial time-variation in interest rates,and it is likely that expected stock returns vary with the interest rate, the hypothesis of interestis the co

33、nstancy of the conditional equity premium, not the constancy of expected stock returns.Building present value models that imply constant excess stock returns, but allow time-varyinginterest rates is a non-trivial matter. Most of the recent work on present value models with time-varyingdiscount rates

34、 builds on Campbell and Shiller (1989) who, by linearizing returns aroundsteady state log price dividend ratios, obtain a tractable linear present value model in which it isstraightforward to impose the constancy of expected excess returns while allowing for variationin interest rates. More recently

35、, the term structure models in the affine class (see Duffie and Kan(1996) have been applied to stock pricing to yield tractable pricing equations in many settingswithout linearization (see Ang and Liu (2001) and Bekaert and Grenadier (2000). We deviatefrom this literature by presenting a model for p

36、rice earnings ratios.Stock returns from time?to?can always be decomposed as:! whereis the stock price at time?,is the dividend paid at time?#,! $&%(isthe payout ratio of dividendsto earnings,is the price-earnings ratio. Defining)aslog growth in earnings)+*-,/.102(3%/2(54687and9;:as the log payout ra

37、tio9:+*?ACBD0E)F7DGH?&CBD0E9;:87(1)In this pricing framework, we have decomposed dividend growth into earnings growth and apayout ratio. From a theoretical perspective, dividend growth should suffice to price stocks, butour decomposition may yield more accurate pricing formulas in finite samples. Fi

38、rst, in finitesamples using dividends may be problematic, since they are often manipulated, smoothed, orset to zero, making them poor indicators of the true value-relevant cashflows in the future. Itis no surprise that in the real world analysts almost entirely focus on earnings growth. Second,the d

39、ecomposition simply increases the information set for prediction, and will include a modelthat features only dividend growth as a special case.The model has three state variables, the short rateI, log earnings growth)and the logpayout ratio9;:. DenoteJ0EI)9;:=7=Kwhich we assume to follow a first-ord

40、er VectorAutoregression:JMLNJ546HO(2)whereOQPSRRUTV05W/XZY7. To price equity, we use the Dividend Discount Model:_X(3)whereis the stochastic discount factor applying to payoffs at time?aHb. To ensure theabsence of arbitrage (Harrison and Kreps (1979) we model the one-period log pricing kernelc, such

41、 that:cdeIdfhgKYggKOX(4)wheregis a 3i1 vector containing the prices of risk and the discount factor can be written as:?ACBj8cjkWe also impose conditions on the parameterslmnLKXvec0o7=KXvech0Y7=KXgKqprKso that thetransversality condition*dH?ACBD0EI37FpU05Qd7G?&CBD0EI37(10)where$?ACBD0d0|?|7KYg7. The

42、unconditional expected simple return is given by:ondH?ACBD0EI37FpU05Qd7G?&CBD0Lf?7(11)whereLand?are the unconditional mean and variance ofIrespectively.The simple expected excess return is a multiple of the nominal rate. Hence, a regression ofd$?ACBD0EI7on the nominal rate would actually yield a pos

43、itive coefficient equal tod.However, the scaled expected return,n%?ACB0xI37Fpis constant and equal to. The constantis a function of the correlation between dividend growth innovations (the sum of the earningsgrowth and payout ratio innovations) with the pricing kernel. The predictability regressions

44、typically run in the literature do not correspond to any of these two concepts, since they use logreturns,a*-,/.1087dI. It is straightforward to show that up to second order terms, theexpected log risk premium will be constant in this homoskedastic model. We also verified thisresult by simulation.It

45、 turns out we can also solve for the conditional and unconditional volatility of equityreturns and the risk premium, under the restrictions of equation (9).Corollary 2.2 If the companion matrixtakes the form in equation (9) then the conditionalvolatility of the simple risk premium is a multiple of t

46、he gross short rate and given by:var0dH?&CBD0EI377S?Gn?ACBD0F05|?H|7KY!0|?|7F7dp?ACBD0fI7(12)where?&BD05d05|?|7=KYg7. The unconditional variance of the simple risk premium is givenby:var0dH?ACBD0xI37F7var07S0df7?ACBD0fLQ$?705?&BD0&?7d7(13)whereLand?are the unconditional mean and variance ofIrespecti

47、vely, and the uncondi-tional variance of the simple gross return var0=7is given by:?G?&BD0fL#?7n?ACBD0&?0|?|7KY!0|?|7F7dp6Equation (12) shows two characteristics of the conditional volatility of equity returns. First,the conditional volatility of equity returns is related to the level of interest ra

48、te. Many studieshave empirically documented a strong link between the conditional volatility of equity returnsand interest rates (see for example, Glosten, Jagannathan and Runkle (1993). However, notethat this effect would disappear if we measure the return as%?&CB0EI37or in logs. Second,the conditi

49、onal variance is positively related to the conditional covariance between the pricingkernel and dividendgrowth, and to the conditional variance of dividend growth0|?N|7KY!0|?|7.The model with the restrictions in equation (9) will serve as the data generating process(DGP) under the null and will also

50、 help us interpret our empirical results. If the restrictionsare not imposed, expected returns vary through time, but since the model is homoskedastic, thetime-variation is likely to be modest in magnitude. Note there are no restrictions onL.What do the restrictions in equation (9) actually mean? Im

51、posing the restrictions fromequation (9) allows the conditional mean for log earnings growth and log payout ratio to bewritten as:n)FpHL?I?3?)?=9;:n9;:FpHL&0Fd?87Id?3?)S0Fd?79;:(14)Note that log dividend growth)U)?9;:. Hence under this economy,expected conditional dividend growth is equal to a const

52、antLNL&plus the current short rate:n)p#n)8pHn9;:8pd9;:L(L&Ik(15)Normally,wheninterestratesmoveawayfromtheirunconditionalmeantheresultingchangein discount rates and prices would induce predictable components in returns. The restriction onthe companion formengineers an opposite cashflow effect that ne

53、utralizes the price-decreaseinduced by the interest rate change. This effect would also happen in a standard equilibriumLucas (1978)-type economy. In an equilibrium setting the constantLaL&would be relatedto the degree of risk aversion of the representative agent.2.2Predictability RegressionsThe mai

54、n regression we consider is:/$KHO/(16)where/U0f%/h70F0dI37GGG0/dI/46F7F77is the annualized-month excess return for the aggregate stock market, anddIis theexcess 1 month return from time?to?a. All returns are continuously compounded. OurPresent Value Model implies that0F/7is constant and henceis zero

55、 for all. The errortermO/follows a0d7process under the null of no predictability because of over-lapping observations. The instrumentsconsist of the log dividend yield?, the log earningsyield|?, and continuously compounded monthly short rateI. The superscript?indicatesthat the dividend (earnings) yi

56、elds use dividends (earnings) summed over the past 12 months intheir construction.The log payout ratio9:?is linearly related to the dividend yield and earnings yield9:?d|?. The three predictive instruments are endogenous instruments in our Present ValueModel. However, they should capture the predict

57、ability present under the null of the presentvalue model, because there is a one-to-one (albeit non-linear) mapping between earnings yield,dividend yield and the short rate and our three state variables. One reason variables such asdividend yields may predict future returns more generally is the pre

58、sence of price in the de-nominator. On the one hand, the presence of price on both sides of the regression may worsensmall sample biases in the regressions (see Goetzmann and Jorion (1993). On the other hand,since price reflects all information about future expected returns and cashflow growth rates

59、, itspresence may capture genuine predictability. If the Present Value Model we present is truth,price would not be necessary to capture time-variation in expected returns, and all informationshould be captured by the three state variables, or transformations of them.In a globally integrated world,

60、predictability is likely also to extend across borders. InSection 6, following Bekaert and Hodrick (1992), we consider cross-country regressions of theform:/0|/7$K/(17)where/are-period annualized excess equity returns in local currency for countryb, and|/arek-periodannualizedexchangeratereturnsUSDpe

61、rforeigncurrencyforforeigncountryb. The instruments inwe consider are log dividend yields for the US, log earnings yields forthe US, and one-month risk-free rates for the US, and the foreign country counterparts of thesevariables. Note that as we use continuously compounded returns|.The regressions

62、in equations (16) and (17) can be estimated by OLS. We consider threeestimators of the standard errors. First, OLS standard errors are appropriate if there is no serialcorrelation of the error term and the error terms are homoskedastic. These are the standarderrors used by Lamont (1998) and we use t

63、hem as a benchmark even when, in whichcase they will likely underestimate the true sampling error. Second, to account for the over-lap in the residuals forand to capture potential heteroskedasticity in returns, we use a8heteroskedastic extension of Hansen and Hodrick (1980) standard errors. Using GM

64、M the pa-rametersl00E7K7Kin equation (16) have an asymptotic distribution (see Hodrick (1992)0zlvd#l7PTV05W/XZ7where$m4646,mo0K7,0K7Kandis estimatedby:05W746j8n0x70E7Kp(18)where0E7/j0x/K/4j7and/O/. This estimator ofis not guaranteed to be positive semi-definite. If itis not, we use a Newey-West (198

65、7) estimate ofwithlags. We will refer to these standarderrors as Robust Hansen-Hodrick (1980) standard errors.Finally, we report what we will call Hodrick (1992) standard errors. This estimator exploitscovariance stationarity to remove the overlapping nature of the error terms in the standard errorc

66、omputation. Instead of summingO/into the future to obtain an estimate of, Hodrick(1992) sumsK54jinto the past:K(19)where$O&46_54kIn our Monte Carlo analysis we run a horse race between these three estimators. We findHodrick standard errors to be far superior, and most of our results will exclusively

67、 focus ont-statistics computed with the Hodrick standard errors. Readers not interested in the details ofthe Monte Carlo analysis can skip Section 3, although we feel that it contains some importantresults.Apart from running univariate regressions, we are mindful of Richardsons (1993) critiqueof pre

68、dictability tests testing for only one particular horizonand we provide a number of jointtests across horizons. To test if the predictability coefficients are statistically significant acrosshorizonskkkwe set up the simultaneous equations:/K/./K/(20)9Denote the vector of coefficients0KkZkkK7K. In pr

69、actice, an estimateofisobtained by performing OLS on each equation. Appendix E details the construction of jointtests across horizons accomodating Hodrick standard errors.When we consider predictability in multiple countries, we also provide joint tests of nopredictability across countries and we es

70、timate pooled coefficients across countries. Such apooled estimation mitigates the data mining problem plaguing US data and increases efficiencyand power under the null of no predictability. Here we estimate the system:/SK/(21)forbmkZkkTcountries, subject to the restrictionb, but imposing no restric

71、tions onacross countries. We takeb= US, UK, France, Germany, Japan. The econometrics underlyingthe pooled estimation is detailed in Appendix F.2.3Data DescriptionOur data set consists of equity total return (price plus dividend) indices from Morgan StanleyCapital International (MSCI) for the US, Jap

72、an, UK, Germany and France. The short-terminterest rates we use are 1 month EURO rates from Datastream. The sample period is fromFebruary 1975 to December 1999 for the US, UK, France and Germany and from January 1978to December 1999 for Japan. MSCI provide dividend and earnings yields which use divi

73、dendand earnings summed over the past 12 months. Although this is restrictive, monthly earningslevels are impossible to use because they are dominated by seasonal components, given thatmost firms have a December-end calendar year.Table (1) reports summary statistics of returns and instruments. The h

74、istorical log equitypremium is around 7.5% in the Anglo-Saxon countries, 6.9% in France and Germany and only3.6% in Japan. Excess return volatility is over 18% in all countries, except for the US where itis only 15%. All three instruments - the log dividend yields?, log earnings yield|?andshort rate

75、sIare highly persistent.As Bekaert and Hodrick (1992) discuss, MSCI report price (capital appreciation) and totalreturns (including income). The total return is an estimate constructed from annualized divi-dends (summed over the previous twelve months). For some countries there is a discrepancybetwe

76、en the MSCI dividend yield?and the implied annualized dividend yield constructedfrom the price and total return indices due to a differential tax treatment across the two series.33For a discussion on how taxes are treated see MSCI Methodology and IndexPolicy, 1999. The MSCI dividendyieldA-series and

77、 the implied annualized dividend yield from the price and total return indices are identical forthe US and Japan, but differ for the UK, France and Germany.10We checked our results by re-constructing the total return using the MSCI dividend yield?with the price return and found them to be unchanged.

78、3Finite Sample Properties of Various EstimatorsOur analysis of the small sample properties of the estimators proceeds in two steps. In thefirst sub-section, we describe the results of a Monte Carlo analysis on returns, earnings yields,payout ratios and short rates that suffices to mimic the regressi

79、ons in Lamont (1998). SinceLamont (1998)used OLSstandard errors toestablish strong predictability results, itis importantto ascertain that they survive small sample biases. In the second sub-section, we calibrateour Present Value Model of the price-earnings ratio and use it as the DGP for a Monte Ca

80、rloanalysis. We solely use US data for the Monte Carlo analyses, but the results are so clear-cut that there is little reason to suspect they would not extend to DGPs calibrated using othercountrys data. All Monte Carlo experiments use 5,000 replications.3.1An Empirical Trivariate ModelLamont (1998)

81、 regresses excess stock returns on dividend yields, earnings yields and payoutratios, both univariately and bivariately since the three variables are totally linearly dependent.In some regressions, he adds the Treasury bill rate as a regressor, but does not find it to besignificant. To examine the s

82、mall sample performance of these regression tests, we investigatethe following DGP:4HL(OIHLM?3?I546O?|?HLHe3|?546Me9:?546O9;:?HLMQ|?546HQ39;:?546O(22)whereis the 1 month excess return,Idenotes the annualized short rate,|?is the logearnings yield and9:?is the log payout ratio. The errorsO0OXO8?XOXDO7

83、=KPIIDTV0WXZ7,with correlation matrix.We estimate this model on US MSCI data and use it to generate small samples of the samelength (299) as the data sets used in the empirical work in the other sections. We then runregressions of the form:/SKH|4We also examined a simpler DGP eliminating the short r

84、ate equation. The small sample properties of thevarious estimates and the OLS biases were similar to what is reported here.11where/are the cumulated and annualized-month ahead excess returns, andare predic-tor instruments. We set, 12 and 60. We take?(dividend yield regression),|?(earnings yield regr

85、ession),05?|?7K(Lamont (1998)s regression) and0?|?I7K(trivariate regression). To conserve space, we relegate the tables whichcontain the estimation results for the DGP, the simulation results on the coefficients (to examinefinite sample bias) and the simulation results on the t-statistics (to examin

86、e size distortions) inthe above regressions to an Appendix, which is available upon request. Here we report the mainresults.Since our instruments and returns are likely to be negatively correlated, regressions of ex-cess returns on any of our instruments would suffer from the well-known persistent r

87、egressorbias (see Stambaugh (2000), which will bias the regression coefficients upward. This is mostobvious for the yield instruments because of the presence of price in their denominator but shortrate innovations and return innovations are also negatively correlated (correlation = -0.1107).However,

88、 in multivariate regressions, it is no longer possible to sign the bias and it may well bethe case that the bias is less severe.Our results reveal that the coefficients on all regressors in virtually all regressions are up-wardly biased. The univariate upward bias is larger for the earnings yield th

89、an for the dividendyield, as the earnings yield is slightly more persistent than the dividend yield, and it variesbetween 0.07 and 0.11 depending on horizon. In bivariate regressions, the earnings yield biasworsens but the dividend yield bias becomes tiny. When the short rate is added, the roles are

90、reversed with large upward biases for the dividend yield coefficient (between 0.07 and 0.16),virtually no bias for the earnings yield, and a substantial upward bias for the short rate coeffi-cient (between 0.09 and 0.17 depending on horizon).Second, all three estimators show very small size distorti

91、ons for. Hence, the use ofthe heteroskedasticity-correction, unnecessary given the homoskedastic DGP, does not lead tosize distortions. Forfand/W, the OLS estimator, which fails to correct for serialcorrelation, not surprisingly performs very poorly. For example, forf, more than 50%of the simulated

92、samples yield OLS t-statistics higher than the 5% asymptotic critical value,with that number going up to 78% or more for/W. The heteroskedasticity-consistentHansen-Hodrick standard error estimator behaves better forUfwith empirical sizes for 5%tests varying between 12.6%, and 15.9% but for/W, the em

93、pirical sizes exceed 49%. Incontrast, the Hodrick standard errors show virtually no size distortion forandf,but are slightly conservative for$/W. The worst size distortion occurs for the earnings yieldcoefficient in the trivariate regression, where only 2.2% of the experiments yield t-stats higherth

94、an the 5% critical value.Taken together, our Monte Carlo analysis of the different estimators overwhelmingly sug-12gests using Hodrick standard errors. It also suggests that the use of OLS or Hansen-Hodrickstandard errors may frequently lead to the wrong inference, in particular their use may lead t

95、othe false rejection of the null of no predictability.3.2The Present Value Model Empirical ResultsThe DGP in the previous section ignores the fact that price appears both on the right hand sideand the left hand side of the regression. Allowing for correlation in the innovations is helpful,but does n

96、ot fully capture the relation between the left hand and right hand side variables. ThePresent Value Model presented in Section 2 under the parameter restrictions that ensure theabsence of predictability (see Corollary 2.1), accomplishes the full endogeneity of price.To perform the Monte Carlo analys

97、is we have to estimate the parameters0xLXXYeXg7. Weproceed in two steps. First, we estimate the VAR parameters0xLXXZY7. Second, we calibrategto fit the observed equity premium in the data. We can do so becausegg&(See Observation2.1) and we set the price of interest rate risk to zero, so thatgis a sc

98、alar. The latter assumptionis motivated by the failure of most term structure studies to reject that prices of risk for interestrate factors are zero (see the discussion in Ang and Piazzesi (2000).We turn first to the VAR parameter estimation, which is complicated by the fact that ourMSCI data uses

99、earnings and dividends summed up over the past year (we observe earningsgrowth rates)?and log payout ratios9;:?) but our Present Value Model requires monthlyearnings and dividends (we do not unobserve monthly growth rates)and log payout ratios9;:).5If2(denotes monthly earnings, then)?is related to)b

100、y:)?$*-,/.2(546GGG2(54632(54654?GGG2(546?$*-,/.2(54630FQH|8-GGG|8-&;72(546?0FH|8_GGG|8_;87M)5463H*-,/.10FQ|F-GGG|8-&;q7dH*-,/.10H|_GGGH|-_;-7(23)In addition, ifdenotes monthly dividends then9;:?is related to9;:by:9;:?*-,/.H546GGG5463546GGG2(5463*-,/.5463nr|&-_|&-|8-GGGH|&|8-&8;p(5463n-Q|8-GGGH|F-&;p

101、*-,/.10|&-_|&-&F-GGGH|&8-&8;7d*-,/.10FQH|8-GGG|8-&;Eat a monthly frequencyfrom earnings yields-using-NZ-2-1o3.13Equations (23) and (24) show that the relation between monthly growth rates and payout ratiosand their counterparts using earnings and dividends summed over the past year is highly non-lin

102、ear. Inparticular, theuseofsummingpastearningsanddividendsoverthepasttwelvemonthswill potentially induce very high autocorrelation up to 11 lags.To estimate the VAR onJwe use Simulated Method of Moments (SMM) (Duffie andSingleton (1993). We use a two-step SMM procedure and impose a restricted compan

103、ion formwhereo?oW. The latter assumption is motivated by an analysis of a VAR on0xI)?9:?7, in which we fail to reject that no variables Granger-cause interest rates.6In thefirst step, we estimate the equation forIon US EURO 1 month rates since we have monthlydata on interest rates. In the second ste

104、p, holding the parameters forIfixed, we estimate theremaining parameters in,LandYusing the first and second moments of)?and9;:?. Wealso use the moments inonJ?J?546?prelating to)?and9;:?. The lag length is set at 12 sincethe first 11 lags are affected by the autocorrelation induced by the non-linear

105、filters in equations(23) and (24). We compute the weighting matrix using the data, so we need not iterate on theweighting matrix.Table (2) reports our results. We report two estimations, an Alternative Model which isexactly identified, and the Null Model which is estimated subject to the restriction

106、s ensuringconstant expected returns in Corollary 2.1. Focusing first on the Alternative Model, payoutratios are close to a random walk with no other significant feedback coefficients. Earningsgrowth on the other hand shows little persistence with the coefficient on past earnings growthbarely signifi

107、cantly different from zero. However, high current payout ratios predict high futureearnings growth, perhaps because they reflect permanent rather than transitory earnings. Highshort rates significantly reduce future expected earnings growth. A 1% increase in interest ratesleads to a 25 basis point d

108、ecrease in expected earnings growth.When we estimate the covariance parameters of the Null Model on the moments of theAlternative Model, we obtain a singular covariance matrix, as the correlation of)and9:approaches -1. Therefore we hold the covariance matrixYfrom the Alternative Model fixed,and esti

109、mateLand a restricted companion matrixusing the first the and cross-moments ofthe Alternative Model estimation. Using a(?test, we fail to reject the Null Model versusthe Alternative Model with a p-value of 0.9252. The Null Model retains the important feedbackfrom interest rates to earnings growth bu

110、t makes the interest rate feedback to payout ratios muchlarger than before. The feedback from payout ratios to earnings growth rates remains intact aswell, but the earnings growth rate is no longer persistent.The estimation of the covariance matrixYhas two important features. First, the conditionalv

111、olatility of innovations to)is much more volatile than innovations to)?(0.0647 versus6This preliminary data analysis is reported in an unpublished Appendix table available upon request.140.0233 from an unreported VAR on0xI)?9;:?7). The filter in equations (23) and (24) smoothsout some of the volatil

112、ity in earnings growth by summing earnings over the past twelve months.This implies that equity returns will be more volatile using monthly earnings than summedannual earnings. Second, shocks to)and9;:are negatively correlated (-0.8496), which is truein the annual data as well. This might be due to

113、the unusual smoothing that occurs in mostcorporate dividend policies. High temporary earnings are not paid out, so they decrease thepayout ratio.The constrained VAR ties down most of the parameters, but we still have to determine theprice of risk for dividend growth. Figure (1) calibratesggg&to the

114、observed equitypremium in the sample. The log risk premium is produced by simulation using 100,000 obser-vations, while the simple risk premium is calculated using equation Corollary 2.1 (and checkedby simulation). Settinggdk/, we match both the log and the simple risk premium. Theannualized volatil

115、ity of the log (simple) excess return corresponding togUd!k/is 0.1367(0.1387), which is slightly below the annualized volatility in the sample 0.1477 (0.1472). Themain source of equity return volatility in the model is the volatility of payout ratios and earningsgrowth rates, since the risk premium

116、is constant. In fact, the implied estimate of the conditionalvolatility of dividend growth is0|?|7KY!0|?|7, which is 0.1368 annualized.With the fully calibrated model we simulate 5000 samples of 299 observations to examinethe empirical distribution of the various test statistics. We construct divide

117、nd yields?andearnings yields|?using summed dividends and earnings over the past year as in the actualdata:?546GGG5463?546GGG2(5463(25)Focusing first on the earnings yield:?2(2(546546546GGG2(546354635463mX(26)we note that%/can be evaluated using Proposition 2.1, and5454(5454(2(54n?ACBD0E)54GGG)7Fp46(

118、27)allowing us to evaluate each term in equation (26). The dividend yield can be written as:?S(5462(5462(546546546GGG54632(54635463546354632 (2 546546546GGG2 546354635463(28)15where! ?&CBD0E9;:7and54%can be evaluated using equation (27). The dividend andearnings yields used in the regressions are?*-

119、,/.10?%7and|?*-,/.102?%7respectively.In Table (3) we report the mean and standard deviation of the empirical distribution for thecoefficientsinthe variousregressions weconsider. First, inthe dividendregression, wecontinueto find substantial upward bias. Interestingly, the bias is not worse than what

120、 we find for theDGP that does not explicitly consider the endogeneity of price in Section 3.1. Goetzmannand Jorion (1993) claim that considering price endogeneity explicitly considerably worsens thebias. However, their comparison is strained since in their DGP with explicit endogeneity, thedividend

121、yield implicitly follows a random walk and Hodrick (1992) also finds stronger biasesfor a DGP in which dividend yields follow a random walk. Our results indicate that modelingthe price endogeneity explicitly does not lead to larger biases.Second, in the earnings regression we now find a downward bia

122、s, instead of the upwardbias we reported above. Why might this be the case? In our price-earnings model with constantexpected returns, the variationin returns is dominated by variation in earnings growthrates. Theprice-earnings ratio is not constant but is completely driven by the payout ratio (see

123、Proposition2.1and Corollary 2.1). Thisimplies thatthe earnings yield, even when summed over12 periods,is likely to be primarily negatively correlated with the payout ratio, but in the DGP (see Table(2), earnings growth rates and payout ratios are highly negatively correlated, so that returnsand earn

124、ings yield innovations end up being positively correlated, reversing the sign of the bias.This is not true for dividend yields, since the log dividend yield can be written as the sum ofthe log payout ratio (negatively correlated with the earnings growth rate and hence with returninnovations) and the

125、 log earnings yield, and the first effect dominates. Note that the smallsample coefficient in the dividend yield regression has the same positive sign that the dividendyield predictability literature finds (Fama and French (1988) and Hodrick (1992), while thenegative small sample coefficient on the

126、earnings yield is what Lamont (1998) finds.Third, in the bivariate regressions, the univariate biases are accentuated, with the bias onthe dividend yield coefficient reaching 0.19 for. The biases decrease slightly with thehorizon. Lamont (1998) finds positive signs on the dividend yield and negative

127、 signs on theearnings yield in a bivariate regression. Our biases are exactly the same sign as his result.Finally, in the trivariate regression, the biases for the earnings and dividend yield coefficientsbecome larger still in absolute magnitude (0.24 for the dividend yield coefficient; -0.11 for th

128、eearnings yield coefficient). However, the bias on the short rate coefficient is negligible.In Table (4) we examine the size properties of significance tests. We report empirical sizesfor tests of size 10% and of size 5%, but we focus our discussion on the 5% tests. The priceendogeneity also alters

129、the absolute performance of the various test statistics, but not their16relative performance. In particular, at, the univariate regressions display negligiblesize distortions, but for the bivariate and trivariate regressions, all tests slightly over-reject atasymptotic critical values and the empiri

130、cal sizes exceedthe nominal sizes. The worst distortionoccurs for the dividend yield coefficient in the trivariate regression, with the nominal size being8.6% for the OLS estimator, and 8.9% for the robust Hansen-Hodrick and Hodrick estimators(which coincide for). For longer horizons, the performanc

131、e of the OLS estimator rapidlydeteriorates with the empirical size exceeding 54% for a 5% test in the dividend regression andexceeding 72% in all other regressions at/W. Accounting for the overlap in the error termsusing Hansen-Hodrick standard errors improves the small sample performance but not en

132、oughto yield a reliable test. The empirical size for a 5% test atmWis at least 39.1% in the caseof the dividend regression.Table (4) shows that Hodrick standard errors are far superior to OLS and Hansen-Hodrickstandard errors. Forf, there is negligible size distortion for the univariate regressions,

133、whereas for the bivariate and trivariate regressions, the size distortion is at most 4.9% for thedividend yield coefficient in the trivariate regression (a 9.9% empirical size). Note that theHodrick test now also over-rejects. For/W, the size distortions actually become smaller,with the worst size d

134、istortion occurring again for the dividend yield coefficient in the trivariateregression (a7.9% empiricalsize for the5% test). For the earnings yieldregression, the Hodricktest is conservative with an empirical size of 3.6%. In sum, the Hodrick standard errors displayoverall far superior small sampl

135、e properties, and using the asymptotic p-values is unlikely todramatically affect statistical inference. For that reason, we will report the Hodrick standarderrors for all of our empirical tests.4Predictability in US Excess Stock ReturnsTable (5) contains our main results regarding the predictabilit

136、y of US excess stock returns.To allow comparison with Lamont (1998)s article, we report univariate and bivariate yieldregressions in addition to our main trivariate specification. We report t-statistics in parenthesesbased on Hodrick standard errors. For the full sample and looking over three horizo

137、ns, the yieldregressions produce only one significant coefficient, the dividend yield. This appears to be asignificant predictor of excess stock returns in the bivariate regression, but its sign is negative,not positive! Lamont on the other hand finds positive coefficients for both yield variables i

138、nunivariate regressions, but a positive coefficient on dividend yields and a negative coefficienton the earnings yield in the bivariate regression. His main point is that the predictive power ofthe dividend yield stems from the role of dividends in capturing the permanent component ofprices, whereas

139、 the negative coefficient on the earnings yield is due to earnings being a good17measure of business conditions which captures counter-cyclical risk aversion. According toLamont there is information in earnings and dividends over and above price. Unfortunately, ourresults appear rather inconsistent

140、with this story. Only for#/Wdo the regression coefficientshave the sign predicted by Lamonts analysis, but for long horizons Lamont finds that onlyprice mattered. In the trivariateregression, the same sign pattern appears for the yield variables:negative on dividends and positive on earnings which r

141、everses for/W.Note that accounting for biases would not help recover the Lamont pattern. In the univariateregressions, the bias-corrected dividend yield coefficient would be even more negativesince thebias is positive, and the negative bias in the earnings yield regression is too small to reverse th

142、esign of the coefficients. For the bivariate and trivariate regressions, accounting for biases wouldmake the pattern we observe (and which is opposite to what Lamont finds) even more striking.Figure (2) shows the pattern over different horizons more clearly for both the univariate andbivariate regre

143、ssions (the inclusion of the interest rate does not change the pattern very much).In both univariate regressions, the coefficients on the dividend or earnings yield turn increas-ingly more negativeuntil around the 30 month horizon after which they start to increase withoutbecoming positive. In the b

144、ivariate regression, we clearly see how the “Lamont-pattern” of pos-itive dividend yield coefficients and negativeearnings coefficients requires settingequal to 50months or higher. The right hand side of the three panels shows the corresponding t-statisticsfor OLS, Robust Hansen-Hodrick and Hodrick

145、standard errors. Given the general lack of sta-tistical significance, it is clearly pointless to try and interpret the sign changes. Interestingly, ifwe had relied on the OLS or Robust Hansen-Hodrick standard errors, we would have concludedthere was significant predictive power in the dividend yield

146、 variable, which again drives homethe importance of the simulation experiments in Section 3.Goyal and Welch (1999) point out that the dividend yield predictability is not robust to theaddition of the last decade of high returns coinciding with low dividend yields and is actuallyhighly unstable in ge

147、neral. Therefore, two additional panels in Table (5) report results forshorter sub-samples eliminating either the last 5 or the last 8 years in the sample. Whereasthe sign of the coefficients now better corresponds to what Lamont (1998) finds, statisticalsignificance is still lacking. Lettau and Lud

148、vigson (2001) also find that the Lamont-patterndoes not hold with late 1990s data with a quarterly sampling frequency.The results described so far are pretty bleak if finding predictability were the objective ofthis study. However, there is yet another regressor in our fundamental regression, the sh

149、ortrate. Whatever the sample period we use, the short rate enters significantly at the 99% levelin theregression and its impact on the equity premium is remarkably robust in termsof magnitude across the various sample periods. A 1% increase in the annualized short ratedecreases the equity premium by

150、 about 3.7%. The predictive power of the short rate dissipates18quickly for longer horizons. The coefficient slowly becomes less negative and eventually evenslightly positive.We also conduct joint tests across regressors. Joint tests across the?and|?regressorsin the Lamont regression yield p-values

151、of 0.2294, 0.3760 and 0.1788 for horizons, 12and 60 respectively. Joint tests across?,|?andIin the trivariate regression yield p-valuesof 0.0139, 0.3723 and 0.0985 for horizonsXfXZW. In this regression for, thedividend yield and earnings yield are also individually significant at the 5% level.Finall

152、y, following the lead of Richardson (1993), Table (6) reports tests of predictabilityover three horizons, 12 and 60, simultaneously. The table also lists results for four othercountries, which will be discussed in the following section. If we now focus on the first column,the US results, we see that

153、 there is only strong evidence for yield predictability, if we considerthe joint predictability of earnings yields and dividend yields in the trivariate specification.Despite the short-lived nature of the predictable patterns, the short rate still significantly (at the5% level) predicts excess retur

154、ns at the one month, 12 month and 5 year horizons.5Predictability of Excess Stock Returns in Five CountriesOur previous results suggest that the predictability patterns formerly found in US data appearnot to be robust to the addition of the last few years and that statistical significance only occur

155、swhen the short rate is used as a predictor. It is conceivable that the lack of predictive poweris simply a small sample phenomenon, due to the very special nature of the last decade for theUS stock market. It is equally probable that the previous results of strong predictability andinteresting pred

156、ictability patterns are a statistical fluke. International evidence should help ussort out these two interpretations of the data. If we cannot confirm the previous predictabilitypatterns for any countries, and if yield variables do not appear to predict stock returns in othercounties, it seems likel

157、y that data mining and other statistical problems have led researchers inthe US astray regarding the predictive power of the yield variables. We can also increase or de-crease our confidence in the short rate as a robust predictor of equity returns using internationalstock return data. Finally, pool

158、ed estimation across countries can lead to powerful evidence onthe robustness and magnitude of predictability patterns across countries.We begin by summarizing the patterns in univariate and bivariate yield regressions. Figure(3) displays the dividend yield coefficients and their t-statistics using

159、Hodrick standard errorsThe UK coefficient pattern is strikingly similar to that of the US but, as in the US, not a singlet-statistic reaches higher than 2.00 and it is not surprising that the joint tests for, 12 and 60in Table (6) fail to reject the null of no predictability. The coefficient pattern

160、s in the three othercountries are more akin to these prevalent in the US in earlier samples, in that the coefficients19increase with horizon, but in both France and Germany they start out negative. In Japan, theindividual coefficient for?is borderline significant in the univariate regression and the

161、 jointtest across horizons in Table (6) rejects at the 10% level.InFigure(4), werepeatthesamegraphsfortheearningsyieldregression. Theearningsyieldpredictability coefficients again are similar for the US and the UK, followinga reverse J-pattern.For France and Germany they have a U-shaped pattern, whe

162、reas they increase monotonicallywith the horizon for Japan. Individual significance again only occurs for certain horizons inJapan, but nowhere else. Joint tests in Table (6) fail to reject the null of no predictability inFrance and Germany but there is a borderline rejection in Japan (at the 5.7% l

163、evel) and a rejec-tion at the 5% level in the UK. Inspection of the UK coefficients in Figure (4) reveals that the, 12, 60 choice very fortunately avoids the lowest spot in the t-statistics curve and mightsomewhat overstate the true predictability. We conclude that the univariate yield predictabilit

164、ypatterns are not terribly robust across countries and not very significant statistically.Do we observe the Lamont pattern of positive dividend yield and negative earnings yieldcoefficients in international data? Figure (5) simply shows the coefficient patterns. Again (notreported), the individualt-

165、statisticsusingHodrickstandarderrorsbarely everreachsignificance.Two countries, Japan and France, show a pattern somewhat reminiscent of the Lamont findings,with negative coefficients for the earnings variable and positive ones for the dividend yield atshort horizons. In France, the coefficients fur

166、ther diverge as the horizon lengthens, whereasin Japan they converge. The UK pattern of the coefficients is similar to that in France but theinitial earnings yield coefficient is actually slightly positive. In Germany, the sign pattern and itsevolution over horizons matches the one we found for the

167、US, with positive (negative) earningsyield (dividend yield) coefficients eventually switching sign, but the switch occurs much earlierthan in the US. Joint tests across horizons reveal no significant rejections of the null of nopredictability, except in the case of Japan, where earnings and dividend

168、 yields jointly predictstock returns at the 5% level (see Table (6). However, this may be due to a multicollinearityproblem in the regression for Japan (see below).The coefficient patterns for the yield variables that we observe for the bivariate regressionsqualitatively persist for the trivariate r

169、egressions (except for Japan) and hence we do not showthem in a figure. However, the t-statistics are generally somewhat larger, resulting in jointrejections of the null of no predictivepower forthe yield variablesin threecountries, the US,theUKand Japan atthe 1%level. Sothe yieldvariablesappeartoha

170、vesomepredictivepowerwhenconsideredtogetherand overthreehorizonssimultaneously. However,the patternincoefficientswe see is very different across countries and differs from what Lamont found, except in Japan.77For Japan, we observe a Lamont pattern at short horizons (less than 15 months) but at horiz

171、ons longer than 15months the earnings yield coefficients become positive. The dividend yield coefficients are less than the earnings20Figure (6) displays the coefficient patterns for the annualized short rate and its associatedt-statistics in the trivariate regression. Strikingly, this coefficient p

172、attern is much more robustacross countries and a similar shape for the coefficient patterns appear for univariateregressions(not reported). For all countries, the one-month coefficient is negative between -3.73 for the USand -0.97 for Japan. For 4 of the 5 countries, the coefficient increases monoto

173、nically withhorizon, leveling off at around 0.55 for the US, France, and Germany and at slightly less thanzero for the UK (-.55). In Japan, the coefficient never reaches zero, but the horizon dependenceis not monotonically increasing. The t-statistics are generally larger in absolute magnitude forsh

174、ort horizons with the exception again being Japan. At the one-month horizon the short ratecoefficients are statistically significant only for the US and UK. When we pool across horizonsin Table (6), only the US short rate retains significant predictive power at the 5% level, whichis not surprising g

175、iven the pattern of the coefficients and t-statistics in Figure (6). What issurprising is the very high p-value for Japan. Inspection of the graph reveals than the joint testhappens to select twos out of only a small set ofs that yield coefficients close to zero.We conclude that the international ev

176、idence on predictability does not support Lamontsfindings regarding the predictability of earnings and dividend yields. There is only weak ev-idence in favor of it in three countries, and the international data do not reveal a consistent,interpretable data pattern. However, the short rate robustly p

177、redicts excess stock returns, butits effect is limited to the short forecasting horizon (mostly one month). The short rate coef-ficient is not significantly different from zero for all countries. Table (6) reports joint tests ofpredictability for the three coefficients. The null of no predictability

178、 is rejected at the 10% levelin Germany, and at the 5% level in the US and UK. It is also rejected at the 1% level in Japanwhich is surprising given the lack of significance of the individual coefficients. However, this ismainly due to multicolinearity in the regressions, and a near-singular covaria

179、nce matrix of theregressor coefficients.Onewaytocometomoreclear-cutconclusionsregardingthemagnitudesandsignificanceofthe coefficients is to pool the estimation across countries. Under the null of no predictability,thepooled estimation should enhance efficiency considerably given that the correlation

180、 of returnsacross countries is not very high. Unfortunately, we have to exclude Japan, because of theconsiderable difference in data coverage both in terms of sample size and in terms of variables(we use levels of earnings yields rather than log earnings yields, because of the presence ofnegative ea

181、rnings in the Japanese series). Table (7) reports pooled predictability coefficients,t-statistics and joint tests. We also report a test of the over-identifying restrictions, describedyieldcoefficientsbetweenhorizons 20and 40months. Atlonghorizons (40-60 months)bothdividendand earningsyields are pos

182、itive, with the dividend yield coefficients greater than the earnings yield coefficients. This figure isavailable upon request.21in Appendix F. For all of our specifications, this test fails to reject the restrictions imposedby equal coefficients across countries. In the Lamont regressions with only

183、 the yield variables,we fail to find statistical significance for the yield variables, both when tested individually andjointly. The coefficients get closer to significant levels at longer horizons. In terms of signat, the dominant pattern appears to be negative dividend yield coefficients and posit

184、iveearnings yield coefficients, a pattern opposite to that found by Lamont (1998).In the trivariate system, both the dividend yield and earnings yield coefficients are positivefor, but the earnings yield coefficient turns negativeat longer horizons. Consequently, theLamont pattern again fails to sho

185、w. Moreover, none of the coefficients is individually signifi-cant, although the t-statistics increase with horizon. A joint test on the yield variables also failsto reject the null of zero coefficients. On the other hand, the short rate coefficient is -1.7334at the one month horizon and significant

186、 at the 5% level. The coefficient increases to -0.34 at/Wbut loses statistical significance. Joint tests across the three variables fail to reject thenull of no predictability for all three horizons at the 5% level, but the test would reject at the10% level form. We conclude that the only robust and

187、 significant predictor of excess stockreturns in 5 countries appears to be the short rate.6Cross-Country PredictabilityThe previous discussion implicitly considered our 5 countries to be segmented markets and didnot allow the possibility of cross-country influences. However, in an integrated market,

188、 globaldiscount rates should price equity returns on all markets, and we may find common componentsin the predictable components of returns. To investigate this, we extend our trivariate regressionto a 6 variable regression, looking at pairs of countries. This set-up is an extension of theregression

189、s run by Bekaert and Hodrick (1992), who regressed equity and foreign exchangereturns on the dividend yields in two countries and the forward premium. Since the forwardpremium is the interest differential through Covered Interest Parity, our regression frees up animplicit constraint on the interest

190、rate coefficients in the Bekaert and Hodrick regressions. LikeBekaert and Hodrick, we also investigate the predictability of foreign exchange returns. Ourmain contribution here is to examine the added role earnings yields may play.Table (8) reports the main results for the predictability of equity e

191、xcess returns by local andUS instruments. A number of striking results emerge. First, there is not a single significantcoefficient for the 12 and 60-month horizons, confirming once again that predictability is not along-horizon phenomenon. Second, the only significant coefficients we report are US d

192、ividendyields significantly predicting returns in France at the 5% level and in Germany at the 1% level.Hence, the strongest predictability pattern is a cross-country effect. The sign of the coefficient22is negative as it is in the other countries. Moreover, the US earnings yield consistently carrie

193、sa positive sign but fails to reach statistical significance. The local yield variables mostly havepositive but insignificant coefficients. Third, the local short rate is no longer significant andincreases in value, relative to its value in the domestic trivariate regression we studied before.For Fr

194、ance and Germany, it even becomes positive. However, the US short rate enters with anegative sign in every regression and the magnitude of the coefficient is large, varying between-2.00 in Japan to -4.17 in Germany. Although the coefficients are never significant at the 5%level, the t-statistics are

195、 all 1.00 or larger.These coefficient patterns suggest that cross-country predictability may be stronger thandomestic predictability. This may be the case in an integrated world where perhaps shocks af-fecting global discount rates are best reflected in the instruments of the dominant stock market,t

196、he US. To examine this further, Table (9) reports p-values from a series of joint tests of pre-dictability. The first set, labeled with US as ”Base,” concerns the regressions of Table (8). Thebase predictability column contains a test of the null of zero coefficients for the base instrument,in this

197、case, the US instruments. The local predictability column reports p-values for tests ofpredictability using only the local instruments. This column is likely to confirm the results ofthe trivariate regressions we reported earlier. The US instruments jointly only significantly pre-dict German excess

198、returns. In the other sub-panels, we make other countries the base country.For example, in the second set we look at excess returns in the US, France, Germany and Japanand our predictor instruments comprise local and UK instruments.We find a number of interesting significant predictability patterns

199、in Table (9). First, noforeign country instruments predict US returns. However, in the presence of foreign instru-ments which have no predictive power, there is still significant predictivepower of US domesticinstruments, in particular the short rate, for US excess returns. Second, we find only two

200、signif-icant base country predictors: US instruments predict German returns, and German instrumentspredict Japanese returns. Finally, the local predictability results confirm the rejections we foundfor the US in Table (6), no matter which foreign instruments are included in the regression. Forthe UK

201、, we no longer reject, which may reflect a loss of power due to the introduction of threenew regressors. For Japan, we also no longer reject, which simply indicates that the inclusion ofthe foreign instruments resolved the singularity problem we faced with the regular estimation.The results in Table

202、 (9) may be weak because of the inclusion of too many highly corre-lated regressors. Therefore, Table (10) reports predictability results using only US instruments,including a pooled estimation. The results are indeed stronger than what we reported in Table(8). First, long-horizon predictability rem

203、ains rather weak to non-existent. Second, the yieldvariables retain their sign patterns (a reverse Lamont pattern) and are now significant in bothFrance and Germany. The US short rate consistently has a negative sign and is now significant23in the UK and Germany. Overall, the US instruments predict

204、excess equity returns in thesecountries better than the local instruments! When we pool the estimation across countries, wefind very strong results, with all three coefficients being significant at the 1% level.8A 1%increase in the US dividend yield reduces the equity premium in other countries by a

205、bout 50basis points, an increase in the earnings yield increases international risk premiums by about 65basis points, whereas an increase in the US short rate of 1% decreases the equity premium byalmost 3.5%. It may be that US factors dominate global discount rates and that this leads to theobserved

206、 pattern, but it seems hard to come up with an international asset-pricing model thatwould explain these predictability patterns.Such an international model would also have to capture predictability patterns in exchangerate returns. We measure the exchange rate return for a US based investor, using

207、the logarithmicexchange rate change (in dollars per foreign currency) plus the foreign-US interest rate differen-tial. Thisreturnisthe topicofthe vastliteratureon theUnbiasednessHypothesisin internationalfinance. If no instruments predict this return, the interest differential or forward premium is

208、anunbiased predictor of future exchange rate changes. Whereas earlier work finds very strong re-jections of this hypothesis, recent tests yield weaker results (see Bekaert and Hodrick (2001).In Table (11) we report regressions of foreign exchange returns on the US instruments and theinstruments of t

209、he currencys country. Since France and Germany now share a common cur-rency as of 1 January, 1999, and were included in the EMS during the 1990s we include onlythe US Dollar-Deutsch Mark exchange rate.In the Unbiasedness literature it is customary to regress foreign exchange returns or ex-change rat

210、e changes onto the forward premium or interest differential, an exercise which typi-cally results in strong negative slope coefficients. In our framework, this would correspond tofindingnegativecoefficients on the USinterest rateand positiveones on the foreigninterest rate.This pattern is only valid

211、 for the pound; in the other countries the US interest rate enters witha positive sign and is not statistically significantly different from zero. In the UK both interestvariables are significantly different from zero, whereas the only other significant interest ratecoefficient is the Japanese inter

212、est rate for the yen equation. Perhaps surprisingly, some of theyield variables do seem to have predictive power for foreign exchange returns, but no clear pat-tern emerges. For example, the US earnings yield is only significant in the pound equation, andthen only forf, whereas the dividend yield is

213、 significant for the Deutsche Mark equationat bothandf. Local instruments are also significant. For example, UK dividendyields predict pound foreign exchange returns, and German dividend yields predict Mark for-eign exchange rates. Perhaps such complex patterns are not surprising in a globally integ

214、ratedworld. An international pricing model will typically require the exchange rate change to be the8We pool UK, French and German returns and exclude Japan because of its smaller sample size.24difference of the two pricing kernels in the two countries, so that factors driving equity pricesin both c

215、ountries may affect exchange rates as well.To obtain more powerful tests, we also conduct a number of joint tests in the bottom panelof Table (11) across horizons, 12 and 60 months. For all of our tests, we find strongrejections of the null of no predictability for the Japanese foreign exchange retu

216、rns, but theseare hard to interpret because of the collinearity problems that plague the covariance matrix inthis case. Therefore we focus our discussion on the pound and Mark returns. First, we contrastthe predictive power of local versus US instruments. We fail to find significant rejections of th

217、enull of no predictability in both cases, but local instruments seem to have more predictivepowerthan US instruments. Second, we contrast the predictive power of the interest rate instruments,with the predictive power of the yield variables. Surprisingly, the yield variables are significantat the 5%

218、 level in the pound return regression and at the 1% level in the Mark return regression,but the interest rate variables are not significant. Whereas the literature has typically focused oninterest rate instruments to predict returns, they do not appear strong predictors, relativeto yieldvariables. T

219、hird, joint tests strongly reject the null of no predictability in both regressions. Weconclude that for foreign exchange predictability returns, there is strong predictability by yieldvariables but not by interest rates, the instruments used in the standard literature.7ConclusionsThe predictable co

220、mponentsin equity returns uncoveredin empiricalwork overthe last 20 yearshavehad a dramatic effect on finance research. Theoretical research on equilibrium models usesthe predictability evidence as a stylized fact to be matched. The partial equilibrium dynamic as-set allocation literature investigat

221、es the impact of the predictability on hedging demands. Muchof the focus has been on the predictive prowess of the dividend yield, especially at long hori-zons, but Lamont (1998) shows that earnings yields have independent predictive power. In thisarticle, we pose the question whether this predictab

222、ility is real. After carefully accounting forsmall sample properties of standard tests, our answer is surprising but important. We show thatthe standard predictability patterns are not statistically significant, not robust across countriesor sample periods and fail to conform to the economic interpr

223、etation given by Lamont (1998).Moreover, there is no evidence of long-horizon predictability in any of the 5 countries we ex-amine. In this sense, the predictability that has been the focus of most recent finance researchis simply not there.Nevertheless, we do find that stock returns are predictable

224、, calling for a re-focus of thepredictability debate in three directions. First, our results suggest that predictability is mainlya short-horizon, not a long-horizon, phenomenon. Second, the strongest predictability comes25fromthe shortrateand notfrom yieldvariableswithpricein the denominator. The r

225、esultthattheshort rate predicts equity returns goes back to at least Fama and Schwert (1977), but somehowrecent research has failed to address what might account for this predictability and has mostlyfocused on dividendyield predictability. Third, thereare tantalizing cross-country predictabilitypat

226、terns that appear stronger than domestic predictability patterns. The emergence of a globallyintegrated capital market over the last 20 years should refocus research towards determinants ofglobal discount rates.We hope that our results will, in the short run, affect the asset allocation literature,

227、whichoften has taken predictability of the dividend yield variable as given, and in the longer run,will stimulate research on theoretical models that might explain the predictability patterns wedemonstrate, particularly short rate predictability at short horizons and across countries.26AppendixAProo

228、f of Proposition 2.1From the Dividend Discount Model we can write:?(A-1)where?and? ?.We claim that:?!#?$%?&)(Z*+-,*.0/!)1(A-2)which we show by induction.The initial conditions are given by:?!#&2?!#4365!3879;:.&)3./!379:.:?!#:.=?+)A.B+?/!+=3&$(Z7+-,7.0/!(A-3)where(Z7N+A.B+:+79+A.+Aand,7C3?+?.%+)ATo p

229、rove the recursive relation, assume this relation holds for*. Then using iterative expectations:?!#?%?&2?D!+E?+?2?!4365!379F:.$(Z*+-,*./3)3./!G379:.+-,*+7.0/!(A-4)where(Z*+72(Z*D+H+-,*.B+I,*5.:+79H+-,*.H+-,*and,*+7J3+?.+I,*5$KHence the price-earnings ratio is given by:?#?$?L&?)(Z*+-,*./(A-5)27BProof

230、 of Observation 2.1We would like pricing kernel variability with earnings growthand log payout to be the same if were to use dividendgrowth. DenoteM83:.:+:.=. In particular, we would like covFN13Min a system with statevariables5;N.as covOG +13Min our system with state variables5. We note thatcov$OG+

231、13MC3+A.:KDenotingNas the 22 covariance matrix under the dividend growth system, and:Nas the 21 vector of pricesof risk under this system, thencovN13M3C3.PST4U+:;RQR+Q RVST4U+:;T4UQRVST)U+QT4UW:NPQPXSRZY+:NRYQRY(B-1)where:P:R:T)U.from our three-factor earnings growth and log payout system and:N:NP:N

232、RY.from adividend growth system. Note thatQPXSRYcov51OG+QPXST)U+Q PXSRandQNcovOG +X1O +3WQT4U+9QRVST)U+QRHence we can re-write equation (B-1) as:;RXQR+:;R+:FT4UQ RVST4U+:FT4UQT4U:NRYQR+Q RVST4U+QT4U(B-2)This relation is satisfied if:;R:FT4UW:NRZY.CProof of Corollary 2.1Lemma C.1 Under the restrictio

233、n for?in equation (9) the coefficients,*A_*.Proof: From the initial condition for,7in equation (7) we have:,737+?+73?3?=3?A+73?A7AWe can verify that,*)Ain the recursive relation in equation (6) by direct substitution.Using equation (1) we can write:?#a&7?#5+A.b/+?)(Z*+A.0/dc+Ie?(C-1)noting that,*A_*

234、from Lemma C.1. The termcis given by:c%+$A.B+H+)A.?/+79%+)A.+)A%+$A.B+?+)A.0/+79hH+)A.H+$A5(C-2)28where we substitute?+-,*5.?+$A.N%+)A.?andeis given by:e?$(Z*+A.B+79+A.+A+A.?/?$(Z*+73%+$A.+.A/+$(Z7+.A/E./3H+$A.)(Z7D+.A/E./3%+)A.:?E./3+A.:3c(C-3)Hence the expected simple total return is:?#a&2fE5(C-4)

235、wheref43H+)A.5%+)A.B+79%+)A.)3%+)A.)(Z773$(Z75(C-8)DProof of Corollary 2.2The conditional squared simple total return is given by:?#a&7?5+A.i/+?$(Z*+A.b/7?$j+9lk+Im(D-1)29wherej?#9+A./&c+A.?$(Z*+9H+)A.0/&h?j(D-3)whereh?is given by equation (C-8) andmn?$(Z*+A.b/h?j(D-4)Hence the conditional expectati

236、on of the squared simple total return is:?#a&7+9h?+h?Ej7+h?Ec%+$A.%+)A?#o&E+A.+A(D-5)where the last expression results from substituting in the conditional simple total return given in equation (C-7).We can write the conditional variance of the simple total return as:varaA2?#a&3?#a&?#a&+A.+A537Cf#+A

237、.+A537&95(D-6)wheref)3p+$A.:. This is also the conditional variance of the simple risk premium. Note that?+)A.p+$Ais the conditional variance of dividend growth but expressed in earnings growth and logpayout ratios.To look at unconditional variances we use the formula:vara?q#vara&+var?aThe first ter

238、m is given by:?G#vara&2f#5%+)A.H+$A537&9hBP+hQP(D-7)wherehBPandhQPare the unconditional mean and variance of5respectively. The second term is given by:var$?a3Cfvar55Cf9hBP+hQPXhQP37(D-8)Combining equations (D-7) and (D-8) we have:vara32fE9hBP+hQP#XhQP+H+)A.9hBP+hQPXhQP37.30ETesting Predictability Ac

239、ross HorizonsThe moment conditions for the system in equation (20) are:?1)rZ?stvu?rtxw.rtZy?ztxw?.ztZyn?Czstq|(E-1)where7., a7vector andztztwKKKztZy.From standard GMM;3n1Xwithe!k!e!,e!|?C.5andk!?1)rZ?str.?st?z?stz?st|.(E-2)The Hodrick (1992) estimatekDofk!is given by:k7m.m(E-3)wheremis amatrixmmtwKK

240、KXmtZywheremtis given bymt.t1KKK.t, andt,7, is:tt?!?1(E-4)since under the null of no predictability the one-step ahead errors?zare uncorrelated andzt+-EEEZ+t. Denoting/.1KKK., an estimate ofe!is given by:e7|/./(E-5)To test the hypothesisvwe use the Newey (1985)htest:.#.&ranki(E-6)witheke.FTestingPre

241、dictabilityPoolingCross-Sectional InformationLet the dimension ofbe37so there will be a total ofregressors, including the constant terms?for eachofcountries. In equation (21) denote the free parametersHKKK h., and the unrestricted parametersstacked by each equation$.KKK H. We can estimate the system

242、 in equation (21) subject to therestriction thatv, where C is a3737matrix of the form:36KKK36KKK.KKK36(F-1)whereis a377vector of zeros,is a3737matrix of zeros, andis a37rankidentity matrix.DenotetNtKKKt.7?N7l?7ztNztKKKzt.7/.$K(F-2)31Then the system can be written as:t/.+zt(F-3)subjecttov. Towriteinc

243、ompactnotationleta.tKKK.t.,/.KKK/.,z.tKKKz.t.Then the compact system can be written as:a/+subject tov(F-4)A consistent estimateofis given by:U)i3/./.#/./.&vU)i(F-5)withU$i/./.a. This gives us an estimateof.The moment conditions of the system in equation (F-3) are:?C$rt2?1/zt(F-6)By standard GMMhas d

244、istribution;3?1.!k!(F-7)with.!?rt.(F-8)andk!2?1$rtr.t(F-9)The Hodrick (1992) estimatekofk!is given by:k7?tpp.(F-10)where(7) ispt?!/?K(F-11)Under the null hypothesis of no predictabilityzt+2KKKtwhereare the 1-step ahead seriallyuncorrelated errors. This is the SUR equivalent of the Hodrick (1992) est

245、imate for univariate OLS regressions.An estimateof!is given by:.7?!rt.1(F-12)KKKh.with3rt.77.7KKK(F-13)32The estimatehas distribution;3?1#.k&$K(F-14)There are+37free parameters inwithmoment conditions. This gives3+37over-identifying restrictions. The Hansen (1982)J-test of over-identifying restricti

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265、ces from Statistics Based on Multiyear Asset Re-turns,” Journal of Financial Economics, 25, 323-348.42 Shiller, R. J., and A. E. Beltratti, 1992, “Stock Prices and Bond Yields: Can Their Comovements Be Ex-plained in Terms of Present Value Models?” Journal of Monetary Economics, 30, 25-46.43 Stambaug

266、h, R. F., 1999, “Predictive Regressions,” Journal of Financial Economics, 54, 375-421.44 Valkanov, R., 2000, “Long-Horizon Regressions: Theoretical Results and Applications to the Expected Re-turns/Dividend Yields and Fisher Effect Relations,” working paper.35Table 1: Sample Moments of Returns and I

267、nstrumentsUS&-5E-mean0.0766-3.3632-2.62710.07690.0689-0.7364stdev0.14770.40590.39150.03400.08190.1596-0.00800.98190.98390.96990.09400.9871UK&-5mean0.0765-3.0780-2.46010.1027stdev0.18480.25560.36080.0347-0.01540.96010.96900.9521France&-5mean0.0689-3.2192-2.82190.0948stdev0.20880.40850.61710.04730.079

268、50.98030.94730.8464Germany&-5mean0.0690-3.3512-2.74350.0576stdev0.18800.33240.47570.02450.05820.97940.98120.9806Japan&-5mean0.03580.01060.02840.0454stdev0.19110.00510.01740.03020.02750.98080.98110.9678Summary statistics of monthly excess returns and instruments. In the tablerepresents excessreturns,

269、AElog dividend yields,-log earnings yields,5short rates. Excess returns and shortratesare continuouslycompoundedmonthlyreturns. Theequityreturnsare fromMSCIand theshortrates are EURO 1 month rates. The excess return for a country is equal to the equity return in localcurrency less the EURO 1 month r

270、ate for that country. The mean and standard deviation ofand5are obtained by multiplying the sample mean and standard deviation by 12 and79respectively.denotes the sample autocorrelation. Dividend (earnings) yields are from MSCI which use the sumof dividends (earnings) over the past twelve months. Th

271、e table reports continuously compoundedgrowthin earnings ratesEwhich is the growthin earnings summed overthe past year (1KKK37x7)to the past year beginning one month earlier (371KKK379) for the US. The log payout ratio Efor the US is the difference ofA-andE. The sample period is from Feb 1975 to Dec

272、 1999 forthe US, UK, France and Germany and from Jan 1978 to Dec 1999 for Japan. Earnings yields forJapan are negativeduring 1999, so we report levels, instead of logs, for dividend and earnings yieldsfor Japan.36Table 2: Null Model VAR EstimationConstant and Companion FormconstCompanion form?Null M

273、odelB550.00020.97110.00000.0000(0.0001)(0.0140)0.0216-0.2894-0.00780.0079(0.0012)-0.02221.28940.00780.9921(0.0012)(0.0900)(0.0105)(0.0010)Alternative Present Value ModelB550.00020.97110.00000.0000(0.0001)(0.0140)0.0125-0.24520.27390.0096(0.0037)(0.0597)(0.1669)(0.0036)-0.01040.05840.02040.9867(0.002

274、4)(0.9690)(0.2237)(0.0044)Conditional Volatilities and CorrelationsVolatilityConditional CorrelationsQ550.00071.0000(0.0000)0.06470.10381.0000(0.0137)(0.0318)0.0351-0.1758-0.84961.0000(0.0129)(0.4035)(0.1874)We estimate the monthly VAR/2B+?/+=,=1of/V5., where5is the continuously-compounded risk-free

275、 rate,is monthly (unobserved) earnings growth andis the monthly (unobserved) log payout ratio. We set?E?A, where subscripts denotematrix elements (row and column). Estimation of the Alternative VAR proceeds in two steps. First,we estimate the equation for5on US EURO 1 month rates. Second, holding th

276、ese parameters asfixed, we estimate the remaining parameters inB,?, andusing Simulated Method of Moments.We match the first and second moments of MSCI data on log earnings growth and log payout ratiowhich use summed earnings and dividends over the past year in their construction. We also usethe cros

277、s-moment of current (annual) growth and (annual) payout with lagged one-year growth andpayout. The Alternative Model is exactly identified. To estimate the Null Model we hold fixed thecovariance matrix-&-5-AEEmean0.1877-0.09130.1603-0.08290.1175-0.0604stdev0.37670.22070.27580.21080.16490.1607Panel D

278、: Trivariate Regression Coefficientshorizon779(?AEE5AEE5AEE5mean0.2411-0.10930.00030.1983-0.0954-0.00100.1266-0.0624-0.0004stdev0.39320.23860.13570.28280.22280.11990.16490.16130.0784The table reports the mean and standard deviation of the small sample distribution of the slopecoefficients in the reg

279、ression:t+.+=tStwheret79x+2EEE+tis the cumulated and annualized-period ahead return,is the log dividend yield alone in PanelA, the log earnings yield alone in Panel B, the log dividend yield and log earnings yield togetherin Panel C, and the log dividend yield, log earnings yield, and the short rate

280、 in Panel D. The smallsample distribution is based on 5000 replications of a sample size of 299 observations using theConstant Expected Return Null Model in Table (2) as the DGP.38Table 4: Size Properties of T-Statistics from the Constant Expected Return Null ModelDividend Regression(779(?Nominal si

281、ze0.1000.0500.1000.0500.1000.050OLS0.1050.0520.4710.4020.6110.541Robust Hansen-Hodrick0.1080.0550.2100.1440.4700.391Hodrick 1B0.1080.0550.1060.0520.1440.076Earnings Regression(779(?Nominal size0.1000.0500.1000.0500.1000.050OLS0.1060.0520.6010.5390.7860.747Robust Hansen-Hodrick0.1110.0550.1900.1240.4

282、750.403Hodrick0.1110.0550.0960.0440.0790.036Lamont Regression(779(?Nominal size0.1000.0500.1000.0500.1000.050Dividend coefficients onlyOLS0.1330.0690.5830.5140.7860.722Robust Hansen-Hodrick0.1370.0710.2540.1870.5290.457Hodrick0.1370.0710.1450.0810.1390.077Earnings coefficients onlyOLS0.1360.0750.661

283、0.6000.8350.800Robust Hansen-Hodrick0.1380.0780.2230.1520.4970.426Hodrick0.1380.0780.1270.0660.0980.051Extended Lamont Regression(779(?Dividend coefficients onlyNominal size0.1000.0500.1000.0500.1000.050Dividend coefficients onlyOLS0.1570.0860.6130.5490.7880.744Robust Hansen-Hodrick0.1620.0890.2970.

284、2230.5610.486Hodrick0.1620.0890.1700.0990.1450.079Earnings coefficients onlyOLS0.1460.0840.6680.6100.8260.801Robust Hansen-Hodrick0.1490.0860.2490.1750.5240.448Hodrick0.1490.0860.1310.0750.0910.047Short Rate coefficients onlyOLS0.1230.0640.6430.5780.7940.761Robust Hansen-Hodrick0.1270.0680.2260.1520

285、.4860.407Hodrick0.1270.0680.1190.0600.0890.048The table lists nominal versus empirical size properties of the OLS, Robust Hansen-Hodrick andHodrick t-statistics. We simulate 5000 samples of length 299 from the Constant Expected ReturnNull Model in Table (2), calculate the t-statistics for each metho

286、d and record the percentage ofobservations greater than the nominal critical values under the null hypothesis of no predictability.39Table 5: Predictability of US Excess ReturnsUnivariate RegressionsLamont RegressionTrivariate Regression&-onlyEonly&-AE5-5Full Sample 1975:02 - 1999:121-0.0820-0.0415-

287、0.29940.2445-0.38960.5747-3.7341(-1.0571)(-0.5301)(-1.6837)*(1.3720)(-2.1524)*(2.5620)*(-2.7539)*12-0.1063-0.0673-0.25810.1635-0.27100.2433-0.9604(-1.1315)(-0.8086)(-1.3550)(1.0190)(-1.4465)(1.3738)(-0.8302)60-0.0942-0.08590.2280-0.23940.1815-0.26160.6659(-0.5249)(-0.9866)(0.2890)(0.5094)(0.2168)(-0

288、.5890)(0.7111)Restricted Sample I 1975:02 - 1994:1210.09570.05520.2272-0.09750.54940.0518-4.5453(0.6615)(0.5674)(0.5705)(-0.3725)(1.3249)(0.1939)(-2.9723)*120.04800.02170.1859-0.10090.2837-0.0598-1.3252(0.3260)(0.2324)(0.4980)(-0.4547)(0.7318)(-0.2634)(-1.0870)600.04150.01410.2574-0.19170.1991-0.204

289、5-0.6636(0.2662)(0.0957)(0.7205)(-0.5266)(0.6007)(-0.5679)(1.2746)Restricted Sample II 1975:02 - 1991:1210.10500.07600.1417-0.03060.54710.0315-4.5079(0.4763)(0.4094)(0.2553)(-0.0633)(0.9757)(0.0656)(-2.9744)*120.08740.07000.07470.01110.19600.0281-1.3268(0.3904)(0.3559)(0.1629)(0.0256)(0.4201)(0.0654

290、)(-1.0924)600.08730.03200.4615-0.33060.3669-0.32100.5557(0.3317)(0.1363)(1.0938)(-0.7625)(0.9128)(-0.7834)(0.5873)We estimate regressionsof the formt+.+=tStwheret79xV+EEE?+tis the cumulated and annualized-period ahead return, with instruments. The first two columnscontain the coefficients for univar

291、iate regressions (AE, the log dividend yield, and5-, the log earnings yield). Columns 3 and 4 contain the coefficients for the Lamont regression&-E), and the last three columns give the coefficients for the trivariate regressionA-5, where5is the annualized 1-month short rate. T-statistics are in par

292、entheses andcalculated using Hodrick standard errors. T-statistics significant at 95% (99%) are denoted with *(*).40Table 6: Predictability Across Horizons in 5 CountriesUSUKFranceGermanyJapan?Univariate Regressions&-0.85560.50140.62070.58210.08185-0.58690.0404*0.87320.88030.0572Lamont Regression -A

293、Eand-&-only0.61450.61290.54640.13940.94965-only0.37230.81160.70300.18810.7433JointAEand-0.14950.15020.82710.24530.0101*Trivariate Regression -AE,E,5&-only0.70160.54740.88570.35230.96915-only0.08490.25830.65190.29660.7915JointAEand-0.0097*0.0111*0.78970.40820.0010*5only0.0399*0.10630.38020.12790.9318

294、JointAE,-,50.0105*0.0312*0.72600.09840.0018*The table lists p-values of joint tests across horizonse71791. P-values less than 5% (1%) areasterixed * (*).&-denotes the log dividend yield,Ethe log earnings yield and5the shortrate. Japans (marked by?) regressions are performed in levels, not logs.Table

295、 7: Pooled-Country Estimations Excluding JapanLamont SystemA-and-(7(79n&-0.02340.03520.1084(-0.2803)(0.3594)(1.2303)-0.0046-0.0590-0.0593(0.0859)(-1.1053)(-1.4923)hTest p-valuesJ-test0.26050.53430.1667&10.95920.51670.2253Trivariate SystemAE,E,5779(2AE0.07190.07260.0860(0.7884)(0.7119)(1.0057)-0.0236

296、-0.0521-0.0613(0.4626)(-1.0446)(-1.7079)5-1.7334-0.6554-0.3357(-2.5247)*(-1.1852)(0.8094)hp-valuesJ-test0.10010.39600.1397&1515(0.08510.48490.1451A10.49510.56530.16135(0.0116*0.23590.4183The predictability regressiont+.+=tStwheret79l+nEEEV+tisthe cumulated and annualized-period ahead return with ins

297、truments, is estimated jointly acrosscountries, constraining the coefficients to be the same across countries (see Appendix F for details).Japan is excluded from estimation because of a shorter sample and negative earnings yields during1999. The constants in the regressions are allowed to differ acr

298、oss countries (and are not reportedin the table). We report the Lamont system whereAE-and a tri-variate system with&-E5, where&-andEdenote the log dividend and earnings yields respectively,and5is the annualized one-month short rate. T-statistics of the predictability coefficients are inparentheses a

299、nd are based on Hodrick (1992) standard errors. In thetests, the J-test is a test ofthe over-identifying restrictions and the other tests refer to joint test of coefficients equalling zero.T-statistics significant at levels greater than than 95% are asterixed *.41Table 8: Cross-Country Predictabilit

300、y of Equity ReturnsUS&-US5-US5LocalAELocal-Local5Local Market is UK1-0.48060.1215-3.23670.52820.5427-2.0319(-1.1352)(0.2201)(-1.5434)(1.0346)(1.0204)(-1.1247)12-0.34860.2964-0.48310.4699-0.1061-0.8179(-1.1527)(0.8534)(-0.2761)(1.2299)(-0.4139)(-0.6138)600.0467-0.00240.66020.3559-0.1419-0.6771(0.0532

301、)(-0.0052)(0.4802)(1.1660)(-1.2958)(-0.4557)Local Market is FR1-0.60610.6967-3.05570.0512-0.01880.1898(-1.8558)*(1.7115)(-1.1329)(0.2383)(-0.2786)(0.1090)12-0.36390.3448-1.39040.0991-0.06220.3813(-1.1643)(1.2332)(-0.7624)(0.5041)(-1.0004)(0.4261)600.0527-0.15380.64490.1095-0.02920.3064(0.0330)(-0.22

302、38)(0.3249)(0.4004)(-0.3712)(0.7332)Local Market is GER1-1.08310.8229-4.16650.61420.00120.0589(-2.6604)*(1.8206)(-1.7929)(1.4097)(0.0081)(0.0205)12-0.60140.4281-1.45250.5002-0.1343-1.1546(-1.5260)(1.3244)(-0.9198)(1.2788)(-1.0086)(-0.4912)600.1639-0.15310.17160.2981-0.17290.8250(0.1153)(-0.2313)(0.0

303、982)(0.8964)(-1.7115)(0.5810)Local Market is JAP1-6.99483.6414-1.99650.17940.0011-0.4109(-0.5427)(0.6959)(-0.9995)(1.0262)(0.0145)(-0.2348)127.0783-1.3938-1.51910.1658-0.0166-0.4580(0.5893)(-0.2835)(-0.8545)(0.8256)(-0.2091)(-0.3125)6016.8843-5.0788-0.14810.05680.01730.0583(0.4863)(-0.6249)(-0.1294)

304、(0.2003)(0.4359)(0.0589)We regress local excess equity returnstof horizononto US instruments and local instruments.Hodrick standard errors are used to calculate t-statistics given in parantheses. T-statistics significantat the 95% (99%) level are asterixed * (*). The regressions for Japan are run in

305、 levels forAEand5-(not logs).42Table 9: Cross-Country Predictability of Equity Returns(?TestsBaseLocalBase PredLocal PredUK0.16390.1761USFR0.30160.9773GR0.0498*0.2441JP0.78700.5249US0.86560.0213*UKFR0.51970.9311GR0.63850.6695JP0.41370.4154US0.52990.0306*FRUK0.38260.1797GR0.99290.6612JP0.63550.1822US

306、0.49250.0491*GRUK0.29930.1587FR0.46660.9912JP0.0250*0.6342US0.59050.0013*JPUK0.49890.1619FR0.91830.2911GR0.93220.6202We regress localexcess equity returns on base country instruments and local instruments (A-,Eand annualized short rates5). For example, the top entry regresses UK equity excess return

307、s on USinstruments and UK instruments; the second entry regresses French equity excess returns on USinstruments and French instruments. We reporthp-value tests for the hypothesis of base countrypredictability (“Base Pred”) and local country predictability (“Local Pred”) at a one-month horizon.P-valu

308、es less than 5% are asterixed *, those less than 1% are asterixed *. Regressions for Japan(indicated by?) are run in levels, not logs.43Table 10: US Predictability of Foreign Equity ReturnsCoefficientshPredictability TestsUSAEUSEUS5&-,5-5AE,E,5UK on US instruments1-0.29540.6406-4.34670.11070.0354*0.

309、1867(-1.2797)(1.8791)(-2.1040)*12-0.10800.2140-1.19070.64000.46510.8233(-0.5372)(0.8931)(-0.7305)600.2872-0.17200.22190.89470.80360.8584(0.4010)(-0.4460)(0.2487)FR on US instruments1-0.57160.7090-3.05220.06790.17890.1179(-2.2632)*(2.1677)*(-1.3422)12-0.30330.3647-1.46700.42440.40040.5711(-1.1262)(1.

310、2971)(-0.8410)600.3506-0.24210.38160.81330.75470.8258(0.3277)(-0.4238)(0.3124)GR on US instruments1-0.62580.7185-2.51470.0279*0.07080.0308*(-2.4711)*(2.6343)*(-1.8071)*12-0.39030.4392-1.18230.13660.26070.1989(-1.4450)(1.9760)*(-1.1247)600.3340-0.32681.27530.14310.34020.0910(0.2691)(-0.4968)(0.9537)J

311、P on US instruments1-0.29840.5406-2.50620.25030.16670.4201(-1.0893)(1.5631)(-1.3828)120.04690.2005-1.84740.36470.23390.5681(0.1673)(0.6744)(-1.1903)600.9097-0.4119-0.19680.35250.82540.3183(0.7357)(-0.5853)(-0.2207)Pooled Estimation (excluding JP) on US instruments1-0.47060.6607-3.41190.0061*0.0032*0

312、.0036*(-2.8419)*(3.1855)*(-2.9440)*12-0.26820.3153-1.20010.13410.16830.1690(-1.6076)(2.0033)*(-1.3776)600.2883-0.25060.63610.32420.47080.3839(0.3771)(-0.6151)(0.7212)We regress UK, FR, GR and JP excess equity returnstfor horizonon US instruments only(US&-,-and annualized short rates5). The last pane

313、l presents the estimations of predictabil-ity coefficients constraining the coefficients to be the same across countries, excluding Japan. Theconstants in the regressions are allowed to differ across countries (and are not reported in the ta-ble). T-statistics of the predictability coefficients are

314、in parenthesis and are calculated using Hodrick(1992) standard errors. In thetests, we perform a joint test of coefficients equalling zero. For thejoint estimation thep-values for a J-test of over-identification are 0.2876, 0.0940 and 0.6197 for(7, 12 and 60 respectively. P-values less than 5% (1%)

315、are asterixed * (*).44Table 11: Cross-Country Predictability of Exchange Rate ReturnsUSAEUSEUS5LocalA-LocalELocal5US-UK Exchange Rate Return1-0.16180.0846-2.3076-0.55440.00141.9074(0.8279)(0.2819)(-2.2235)*(-2.3974)*(0.0061)(1.7577)*12-0.11890.5339-2.1985-0.2221-0.36070.9876(-0.6261)(1.9995)*(-2.552

316、0)*(-1.0779)(-2.0673)*(1.2725)60-0.02020.0551-0.29270.0408-0.16380.0085(-0.0463)(0.2400)(-0.3894)(0.2360)(-2.4828)*(0.0105)US-GR Exchange Rate Return10.6462-0.10470.0707-0.83420.0790-0.0619(3.2389)*(-0.4852)(0.0514)(-3.8443)*(0.9796)(-0.0385)120.51110.0105-0.8178-0.72060.10110.0671(2.5128)*(0.0607)(

317、-0.8983)(-3.5886)*(1.2660)(0.0505)600.2615-0.13370.4888-0.25700.0459-0.0697(0.5061)(-0.5518)(0.6322)(-1.9243)(0.9329)(-0.1090)US-JPExchange Rate Return14.7153-6.04830.00120.2085-0.06663.7250(0.5470)(-1.6499)(0.0007)(1.5573)(-1.2474)(-2.2651)*12-0.4377-1.7852-1.4859-0.01360.02561.6005(-0.0519)(-0.604

318、5)(-1.1226)(-1.1096)(0.4801)(1.5244)6011.5440-3.61620.2162-0.13450.02270.1839(0.4527)(-0.6076)(0.2755)(-0.7003)(0.9389)(0.2619)hTests Joint Across Horizons7, 12, 60USLocalShort RatesYieldsAll VariablesUS-UK0.18760.06150.28690.0465*0.0000*US-GR0.29820.14490.52860.0051*0.0000*US-JP0.01620.0002*0.0049*

319、0.0047*0.0000*We regress exchange rate returnst(expressed in dollars per foreign currency) onto US instru-ments and local (foreign country) instruments. We investigate horizonse7, 12 and 60. Hodrickstandard errors are used to calculate t-statistics given in parentheses. T-statistics significant at t

320、he95% (99%) level are asterixed * (*). The regressions for Japan are run in levels forA-andE(not logs). The bottom panel reports p-values oftests joint across horizons7, 12, 60 for ex-change rate predictability. The column labeled “US” tests joint predictability of US dividend yields&-, earnings yie

321、lds5-and US short rates5. The column labeled “Local” tests joint predictabil-ity of foreign countryA-,-and5. The column labeled “Short Rate” tests for joint US andforeign country5predictability. The column labeled “Yields” tests for joint US and foreign country&-and-predictability. Finally, the colu

322、mn labeled “All Variables” tests joint predictability ofall US and foreign country instruments. P-values less than 5% (1%) are asterixed * (*).4565432100.0200.020.040.060.080.10.12Risk Premium AnnualizedBetaRisk PremiumModel log risk premium Model simple risk premium Empirical log risk premium Empir

323、ical simple risk premiumAnnualized simple risk premiums (dashed line) and log risk premiums (solid line). The simple risk premiumsare calculated using Corollary 2.1 and the log risk premiums are produced by simulation. The simple (log)risk premium in the sample used for calibration is 0.0880 (0.0766

324、). A price of risk vector:!:;R :;T4Uwith:R:T4Uh:of -4.655 matches both the empirical simple risk premium and the empirical log risk premium.Figure 1: Risk Premium from the Null Present Value Model46510152025303540455055600.140.130.120.110.10.090.08Horizon (months)CoefficientDividend Coefficients US5

325、101520253035404550556010987654321Horizon (months)TstatDividend Significance US TstatisticsOLS Robust HansenHodrickHodrick 510152025303540455055600.090.080.070.060.050.04Horizon (months)CoefficientEarnings Coefficients US51015202530354045505560987654321Horizon (months)TstatEarnings Significance US Ts

326、tatisticsOLS Robust HansenHodrickHodrick 510152025303540455055600.250.20.150.10.0500.050.10.150.2Horizon (months)CoefficientLamont Coefficients USDividendEarnings5101520253035404550556032101Horizon (months)TstatDividend Significance US TstatisticsRobust HansenHodrickHodrick 5101520253035404550556021

327、01Horizon (months)TstatEarnings Significance US TstatisticsRobust HansenHodrickHodrick The left column shows coefficientsin the regression?Clwhere;p?lllFD is the cumulated and annualized-period ahead return, with instrumentsg2(log dividendyields) in the top row,g?$(log earnings yields) in the middle

328、 row and2$in the bottomrow. The right column shows t-statistics from these regressions. Horizonsare on the x-axis.Figure 2: Coefficients and Standard Errors from US Regressions4701020304050600.160.140.120.10.080.06CoefficientDividend Coefficients US01020304050601.210.80.60.4Horizon (months)Hodrick T

329、statDividend Significance US Tstatistics01020304050600.050.10.150.20.25CoefficientDividend Coefficients UK01020304050600.20.40.60.811.21.41.6Horizon (months)Hodrick TstatDividend Significance UK Tstatistics01020304050600.040.0200.020.040.060.08CoefficientDividend Coefficients FR01020304050600.40.200

330、.20.40.60.81Horizon (months)Hodrick TstatDividend Significance FR Tstatistics01020304050600.10.0500.050.1CoefficientDividend Coefficients GR01020304050600.80.60.40.200.20.40.6Horizon (months)Hodrick TstatDividend Significance GR Tstatistics01020304050601011121314151617CoefficientDividend Coefficient

331、s JP01020304050601.61.822.22.42.6Horizon (months)Hodrick TstatDividend Significance JP TstatisticsDividend coefficients and t-statistics calculed using Hodrick standard errors for the regression;lwhere?2xxF;Fis the cumulated and annualized-period ahead return,with instrumentsH. Horizonsare on the x-

332、axis. For Japan, the dividend yield is in levels but forall other countriesXrepresents the log dividend yield.Figure 3: Dividend Regressions in 5 Countries4801020304050600.10.090.080.070.060.050.040.03CoefficientEarnings Coefficients US01020304050601.210.80.60.4Horizon (months)Hodrick TstatEarnings

333、Significance US Tstatistics010203040506000.050.10.150.2CoefficientEarnings Coefficients UK010203040506000.511.5Horizon (months)Hodrick TstatEarnings Significance UK Tstatistics01020304050600.10.080.060.040.020CoefficientEarnings Coefficients FR010203040506021.510.50Horizon (months)Hodrick TstatEarni

334、ngs Significance FR Tstatistics01020304050600.080.070.060.050.040.030.020.01CoefficientEarnings Coefficients GR01020304050601.41.210.80.60.40.20Horizon (months)Hodrick TstatEarnings Significance GR Tstatistics01020304050601234567CoefficientEarnings Coefficients JP01020304050600.511.522.5Horizon (mon

335、ths)Hodrick TstatEarnings Significance JP TstatisticsEarnings coefficients and t-statistics calculated using Hodrick standard errors for the regression;where;xllF;is the cumulated and annualized-period aheadreturn, with instruments. Horizonsare on the x-axis. For Japan, the earnings yield is in leve

336、lsbut for all other countries$represents the log earnings yield.Figure 4: Earnings Regressions in 5 Countries49510152025303540455055600.250.20.150.10.0500.050.10.150.2CoefficientLamont Coefficients USDividendEarnings510152025303540455055600.0500.050.10.150.20.25CoefficientLamont Coefficients UKDivid

337、endEarnings510152025303540455055600.10.080.060.040.0200.020.040.060.080.1CoefficientLamont Coefficients FRDividendEarnings510152025303540455055600.20.100.10.20.3CoefficientLamont Coefficients GRDividendEarnings5101520253035404550556005101520CoefficientLamont Coefficients JPDividendEarningsDividend a

338、nd earnings coefficientsfromtheregression2g;where;p?xllF-is the cumulated and annualized-period ahead return, with instruments)X$.Horizonsare on the x-axis. For Japan dividend and earnings yields in levels are used, all other countriesuse log dividend and earnings yields.Figure 5: Lamont Regressions

339、 Coefficients in 5 Countries500102030405060432101CoefficientShort Rate Coefficient from Trivariate Regression US010203040506032101Horizon (months)Hodrick TstatShort Rate Significance in Trivariate Regression US Tstatistics010203040506032.521.510.50CoefficientShort Rate Coefficient from Trivariate Re

340、gression UK01020304050601.81.61.41.210.80.60.4Horizon (months)Hodrick TstatShort Rate Significance in Trivariate Regression UK Tstatistics010203040506021.510.500.51CoefficientShort Rate Coefficient from Trivariate Regression FR01020304050601.510.500.51Horizon (months)Hodrick TstatShort Rate Signific

341、ance in Trivariate Regression FR Tstatistics0102030405060321012CoefficientShort Rate Coefficient from Trivariate Regression GR010203040506021.510.500.511.5Horizon (months)Hodrick TstatShort Rate Significance in Trivariate Regression GR Tstatistics010203040506021.510.50CoefficientShort Rate Coefficie

342、nt from Trivariate Regression JP01020304050601.510.50Horizon (months)Hodrick TstatShort Rate Significance in Trivariate Regression JP TstatisticsShort rate coefficients and t-statistics constructed using Hodrick standard errors from the trivariate regression;p2IW;lwhere;?llx;is the cumulated and ann

343、ualized-periodahead return, with instruments$?. The short rateis annualized. We report only thecoefficient on. Horizonsare on the x-axis. For Japan dividend yieldsand earnings yields$arein levels, for the other countries these are log dividend and earning yields respectively.Figure 6: Short Rate Coefficients in Trivariate Regressions in 5 Countries51