market efficiency, asset returns, and the size of the risk premium in global equity markets

Journal of Econometrics 109 (2002) 195–237www.elsevier.com/locate/econbase

Market e!ciency, asset returns, and the size ofthe risk premium in global equity markets�

Ravi Bansala ; ∗, Christian Lundbladb

aFuqua School of Business, Duke University, Durham, NC 27708, USAbKelley School of Business, Indiana University, Bloomington, IN 47405, USA

Received 18 January 2002; accepted 24 January 2002

Abstract

An important economic insight is that observed equity prices must equal the present value ofthe cash 2ows associated with the equity claim. An implication of this insight is that presentvalues of cash 2ows must also quantitatively justify the observed volatility and cross-correlationsof asset returns. In this paper, we show that parametric economic models for present values canindeed account for the observed high ex post return volatility and cross-correlation observedacross 7ve major equity markets—the U.S., the U.K., France, Germany, and Japan. We presentevidence that cash 2ow growth rates contain a small predictable long-run component; this feature,in conjunction with time-varying systematic risk, can justify key empirical characteristics ofobserved equity prices. Our model also has direct implications for the level of equity prices andspeci7c versions of the model can, in many cases, capture observed price levels. Our evidencesuggests that the ex ante risk premium on the global market portfolio has dropped considerably—we show that this fall in the risk premium is related to a decline in the conditional variance ofglobal real cash 2ow growth rates. c© 2002 Elsevier Science B.V. All rights reserved.

JEL classi)cation: F3; G0; C1; C5

Keywords: Asset volatility; Correlation; Cash 2ows; Risk premia; Fundamental values

1. Introduction

An important economic insight is that observed equity prices should equal the presentvalue of the cash 2ows associated with the ownership of the equity claim. The work

� An earlier version of this paper was titled, “Market e!ciency, fundamental values, andasset returns in global equity markets”. All data employed in this paper are available atwww.kelley.iu.edu/clundbla/research.htm.

∗ Corresponding author. Tel.: +1-919-660-7758; fax: +1-919-660-8038.E-mail addresses: [email protected] (R. Bansal), [email protected] (C. Lundblad).

0304-4076/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S 0304 -4076(02)00067 -2

http://www.kelley.iu.edu/clundbla/research.htm

196 R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) 195–237

of Shiller (1981), LeRoy and Porter (1981), West (1988), and Campbell and Shiller(1987, 1988a, b), however, poses a challenge to this insight. These authors documentthe “volatility puzzle”—quantitatively, equity prices are far too volatile to be justi7edas present values of fundamental cash 2ows. This result underscores the key featureof the data that cash 2ow volatility is quite small relative to equity price volatility. Inaddition to implications for volatility, present values also restrict cross-correlations ofasset returns. In the data, the average cross-correlation in ex post returns is about sixtimes larger than that for the cash 2ow growth rates. This feature poses an additionalquantitative challenge to present values, and is labeled the “correlation puzzle”. Presentvalues of the cash 2ows are determined by the time-series dynamics of the expectedcash 2ow growth rates and the cost of capital (i.e., ex ante rate of return). In thispaper, we show that a parsimonious time-series model for cash 2ow growth rates andthe cost of capital goes a long way in explaining the observed equity market volatilityand return cross-correlations.The main insights that this paper provides can best be understood by 7rst considering

the role of the cash 2ow dynamics, followed by that of 2uctuations in the cost of capital.In the data, real growth rates have near zero autocorrelation, hence, it is common toassume that cash 2ow growth rates are i.i.d. In addition to this assumption, if cost ofcapital is constant, then news regarding cash 2ow growth rates is entirely transitory anddoes not alter future expected growth rates. Consequently, dividend yields are constantand ex post continuous return volatility equals the growth rate volatility. However,as cash 2ow growth rate volatility is smaller than return volatility, this leads to thevolatility puzzle discussed above.In sharp contrast, Barsky and DeLong (1993), argue that cash 2ow growth rates

can be modeled as an integrated process (more precisely, an ARIMA(0,1,1) process).It is important to note that in 7nite samples, the Barsky and DeLong process forgrowth rates cannot easily be distinguished from an i.i.d. process (see Shephard andHarvey, 1980), but the economic implications for asset prices are dramatically diLerent.Expected growth rates in this speci7cation contain a unit root, and consequently, newsregarding growth rates have large eLects on dividend yields as they permanently alterfuture expected growth rates. 1 Campbell et al. (1997) argue that Barsky and DeLong(1993) do not provide any direct empirical support for their growth rate dynamics—further, it is not clear if an integrated growth rate process is economically plausible.In this paper, unlike Barsky and DeLong (1993), we provide empirical evidence thatgrowth rates are well modeled as a stationary (i.e., no unit root) ARIMA(1,0,1) process.As cash 2ow growth rates contain a small predictable (and persistent) component,growth rate news leads to volatile changes in dividend yields and ex post returns. Thisstructure helps address the “volatility puzzle” and the “correlation puzzle” discussedabove.With constant cost of capital for each economy, the ex post return cross-correlations

across economies will be solely determined by the cash 2ow growth correlations. How-ever, this is unlikely to justify return cross-correlation, as growth rate correlations

1 At a 7rm level, it is well documented that cash 2ow news leads to signi7cant price reaction (see Eastonand Zmijewski, 1989).

R. Bansal, C. Lundblad / Journal of Econometrics 109 (2002) 195–237 197

across economies are quite small. One factor that may account for high return corre-lation is 2uctuations in global risk premia—a source of common 2uctuations in assetprices. This view is also consistent with Ammer and Mei (1996), who document thatmuch of the asset return covariation between national stock markets is related to newsabout future risk premia. Indeed, relying on a simple CAPM-GARCH speci7cation, asin Bollerslev et al. (1988), we show that 2uctuating global risk premia in conjunc-tion with the assumed cash 2ow dynamics can reproduce the observed ex post returncross-correlations and asset return volatility. Further, we show that the persistent com-ponent in cash 2ows is also needed for duplicating asset return cross-correlations—in the absence of this, asset price 2uctuations are dominated by common cost ofcapital 2uctuations, and hence asset returns are, counter-factually, almost perfectlycorrelated.The asset valuation model that we develop also provides insights regarding two ad-

ditional issues. First, authors, such as Ammer and Mei (1996) use cross-correlationsin cash 2ow news and expected returns to measure economic and 7nancial integra-tion, respectively, across markets. However, they do not provide any economic mech-anism to link these two measures of integration—in this paper, we do provide such amechanism and show that if there is little economic integration, then 7nancial in-tegration will be small as well. Second, Longin and Solnik (1995) show that animportant feature of global equity market data is that periods of increased marketuncertainty are also associated with a rise in the conditional correlation of returns—ourmodel, which incorporates time-varying volatility, reproduces this feature of the dataas well.Relying on the assumed cash 2ow growth rate dynamics and the speci7cation for

2uctuating global risk premia, our valuation model can account for about 70–80% ofthe volatility of asset prices (change in dividend yield or returns) and cross-correlationsin asset returns. The more standard vector autoregression (VAR) methods of model-ing cash 2ow growth rates and expected rates of returns to compute present values(as in Campbell and Shiller, 1988a) lead to asset values which have very low vari-ability (about 40% of that in the data) and very high (with many in excess of 0.9)cross-correlation in asset returns. In 7nite samples, this approach fails to capture thepersistent component in cash 2ow growth rates which leads to large asset return vari-ability, and hence also aLects asset return cross-correlations. Despite the ability of themodel to explain these particularly challenging features of the observed data, the levelof fundamental values implied by the model in particular time periods, especially forJapan (in the mid-1980s) and for the U.S. (in 1994–1998), are far from the observedequity prices. For other countries, such as France and U.K., the model matches theobserved equity prices quite well.Partly motivated by the failure to match the observed equity prices in speci7c

time periods for Japan and the U.S., we develop and estimate a model in which thetime-varying world market volatility process is assumed to be latent (see Taylor, 1986;Hansen and Hodrick, 1983). Using the valuation restrictions, we show that this latentvolatility can be recovered from the observed world equity market prices and the ex-pected cash 2ow growth of this benchmark asset. We 7nd that modeling the systematicrisk in this manner provides a signi7cant improvement over the GARCH speci7cation.


The latent volatility model matches the observed equity prices quite well, and capturesan economically signi7cant portion of the volatility (about 80%). Additionally, it jus-ti7es almost all of the observed cross-correlation, other than for Japan. In contrast tothe GARCH speci7cation, this model suggests that the aggregate risk premium in theglobal economy has fallen signi7cantly in the last decade to about 2%. This diLerencehas important eLects on measured fundamental values. We also show that much of thefall in the latent systematic risk can be attributed to a fall in the conditional worldmarket cash 2ow variance. In parallel and independent work, Fama and French (2000)“back out” the risk premia from the U.S. equity index values, and also argue that themarket risk premium has fallen.In independent papers, Dumas et al. (2000) and Chue (2000) focus on the cross-

correlation among equity returns. However, they do not focus on the joint implicationsfor return volatilities, cross-covariances, the cross-section of equity premia, and thelevel of equity prices. As they assume diLerent cash 2ow dynamics, their results andconclusions diLer from those in this paper (and that in Barsky and DeLong, 1993). Forexample, unlike the results in this paper, they can only account for a small fraction ofthe observed return volatility.The paper is organized as follows. Section 2 discusses the data used in the paper.

Section 3 provides the general present value relations used in the paper, discusses ourcash 2ow model and the evidence supporting it, and lays down the speci7c fundamentalrestrictions implied by the model. Section 4 discusses the estimation methodology, andprovides the empirical 7ndings and diagnostics. Section 5 provides evidence on thevaluation implications of our model, and Section 6 presents evidence on the size ofthe equity premium. Finally, Section 7 provides concluding comments.

2. Data description

We collect monthly data, taken from Datastream, on market prices for 7ve de-veloped equity markets: France, Germany, Japan, the U.K., the U.S., and the WorldMarket Index. 2 For each of these market indices, we also collect data on the dividendyield, earnings yield, and total returns denominated in local currencies. From these,we construct dividend and earnings growth rates; note that as in Fama and French(1995), the measure of earnings is net of depreciation. To measure various quantitiesin real terms we also collect from International Financial Statistics (IFS) a seasonallyadjusted CPI index for each country. The sample period for all the data we collect isfrom January 1973 to December 1998.The total return is de7ned as follows:

1 + Ri; t+1 =Pi; t+1

Pi; t(1 + DYi; t+1): (1)

2 The Datastream World Market Index return has a correlation of 0.99 with the Morgan–Stanley worldindex (MSCI) return.


Here Pi; t is the equity price and DYi; t represents the dividend yield, the asset’s cur-rent dividend payment divided by its current price. It is well known there are strongseasonals in the raw dividend series, thus we follow the convention in the literatureby measuring the dividend yield as the average of dividends paid on the index overthe previous year divided by the current price level, DYt = ( 1

12

∑11j=0 Dt−j)=Pt ; this is

similar to the approach taken in Heaton (1993), Bollerslev and Hodrick (1995) andHodrick (1992). For reasons of seasonality, the only data reported in Datastream forthe dividend yield and the earnings yield are constructed using the lagged 12-monthmoving average. Further, we remove additional seasonality from both dividend andearnings growth rates as a 12th order autoregressive seasonal; our empirical resultsare not sensitive to this additional step. Note that the dividend (earnings) series isconstructed using the observed equity market capitalization and the dividend yield(earnings yield) series. These valuation ratios form the focus of our computations fordetermining present values.The continuous growth rate for the cash 2ow (i.e., dividends or earnings) is the

log of the gross growth rate of the cash 2ow under consideration. When convertingnominal variables to real, we simply subtract from the relevant variable in countryi the seasonally adjusted CPI in2ation rate (i.e., log of gross in2ation) in country i.Throughout we determine the present value implications for the various markets inreal terms. We also construct a real interest rate series by using the one-month Eu-rodollar rate, and subtracting from it a measure of expected in2ation in the U.S., takenhere to be the in2ation expectation implied by an ARIMA(1,0,1) model on the in2a-tion series. 3 In Table 1, we report summary statistics for the total returns, log pricedividend ratios, log price earnings ratios, dividend growth rates, and earnings growthrates. An important feature of the data is the volatility of dividend (earnings) growthrates are on average only about 5% (10%) of the volatility of the log price dividend(earnings) ratio. In Fig. 1, we present the average (across countries) empirical autocor-relation functions for dividend and earnings growth rates; this average is representativeof the autocorrelation function for individual countries. Importantly, we observe theaverage 7rst-order autocorrelations for either the dividend or earnings growth ratesare roughly 0.06, suggesting a very low level of persistence in the observed growthrates themselves. In Table 2, we report the cross-correlations of returns, dividend andearnings growth rates among the various markets in our menu. It is evident from thetables that the correlation across the various equity markets of the 7rst diLerence ofthe log dividend yield (or the log earnings yield) is on average about six times theaverage correlation in either observed dividend or earnings growth rates across thediLerent markets. Also, note that ex post real equity return cross-correlations are ofsimilar magnitude as observed for the 7rst diLerence of the valuation ratios (the earn-ings yield and the dividend yield). All data employed in this paper are available atwww.kelley.iu.edu/clundbla/research.htm.

3 Alternative methods for constructing the ex ante real rate make very little diLerence to our empiricalresults. We have also constructed the ex ante real interest rate using diLerent methods such as removing thetrailing 12-month in2ation rate. The implied real interest rate series is very similar to the one backed outusing the ARIMA(1,0,1) series.

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Table 1Summary statistics

ln(Pt=Dt) Rln(Pt=Dt) ln(Pt=Et) Rln(Pt=Et) ri; t ln(Dt+1=Dt) ln(Et+1=Et)

Mean Std Mean Std Mean Std Mean Std Mean Std Mean Std Mean Std

FR 5.766 0.293 0.0012 0.065 5.786 0.346 − 0.001 0.073 0.0093 0.062 0.0093 0.023 0.0105 0.044(0.293) (0.074) (0.004) (0.003) (0.171) (0.134) (0.004) (0.003) (0.004) (0.003) (0.002) (0.003) (0.003) (0.005)

BD 6.080 0.211 0.0031 0.050 6.102 0.329 0.0004 0.065 0.0073 0.048 0.0042 0.017 0.0070 0.043(0.165) (0.056) (0.003) (0.003) (0.133) (0.179) (0.004) (0.003) (0.003) (0.003) (0.001) (0.002) (0.003) (0.005)

JP 6.915 0.452 0.0020 0.051 6.929 0.528 0.0018 0.056 0.0035 0.052 0.0017 0.013 0.0012 0.026(0.301) (0.041) (0.003) (0.003) (0.387) (0.080) (0.003) (0.003) (0.003) (0.003) (0.001) (0.002) (0.002) (0.003)

U.K. 5.578 0.297 0.0011 0.063 5.585 0.236 0.0002 0.069 0.0089 0.061 0.0057 0.013 0.0059 0.027(0.242) (0.118) (0.003) (0.006) (0.090) (0.137) (0.004) (0.006) (0.004) (0.005) (0.001) (0.002) (0.002) (0.003)

U.S. 5.805 0.358 0.0020 0.045 5.809 0.329 0.0009 0.049 0.0075 0.044 0.0035 0.006 0.0047 0.020(0.422) (0.118) (0.003) (0.003) (0.119) (0.204) (0.003) (0.003) (0.003) (0.003) (0.001) (0.000) (0.001) (0.001)

WD 6.017 0.342 0.0022 0.041 5.899 0.281 0.0008 0.043 0.0065 0.042 0.0052 0.013 0.0060 0.020(0.159) (0.058) (0.002) (0.003) (0.111) (0.063) (0.002) (0.003) (0.003) (0.002) (0.001) (0.001) (0.001) (0.001)

All data are reported at the monthly frequency. ln(Pt=Dt) and ln(Pt=Et) are the log price dividend and log price earnings ratios, respectively,where for the latter we scale the log price earnings ratio by the average payout ratio (D=E). R indicates the 7rst diLerence of the variable under consider-ation. ri; t is the continuous real rate of return, and ln(Dt+1=Dt) and ln(Et+1=Et) are the continuous real dividend and earnings growth rates, respectively. Theworld return, rm; t is the Datastream World Equity Index; its correlation with the commonly employed MSCI World index is 0.990. All standard errors presentedin parentheses are obtained by using a GMM-VARHAC procedure.


0.20

0.15

0.10

0.05

0.00121110987654321 121110987654321

0.20

0.15

0.10

0.05

0.00

Average observedTheoretical ARMA: �=0.973, �=0.932

Average observedTheoretical ARMA: �=0.950, �=0.890

Autocorrelation Function (Dividend Growth) Autocorrelation Function (Earnings Growth)

Fig. 1. Cash 2ow autocorrelation.

3. Present value and asset prices

It is well recognized that the fundamental value of the asset is the present value ofthe cash 2ow associated with the asset. The present value is determined by the expectedgrowth rate dynamics and the ex ante rate of return on the asset—this arithmetic iscaptured by the approach pursued in Campbell and Shiller (1988a). They show thatthe log of the ex post total return, that is ri; t+1, can be approximated as

ri; t+1 = gi; t+1 + �i;0 + �i;1zi; t+1 − zi; t ; (2)

where gi; t is the continuous growth rate of dividends and zi; t is the log price dividendratio. �i;0 and �i;1 are constants related to the Taylor-series approximation. 4 Based onthe above approximation they derive the result that

zt ≡ pt − dt =�0

1− �1+ Et

∞∑

j=0

�j1[gt+1+j − rt+1+j]

; (3)

where pt and dt are log equity price and log dividends. The above equation showsthat the key determinant of the asset valuation, zi; t , is the dynamics of the expectedcash 2ow growth rates and the ex ante rate of return on equity.

4 �i;1=1=(1+exp(di − pi)) and �i;0=−log(�i;1)−(1−�i;1)(di − pi), where (di − pi) is the steady-state(or mean) logged dividend yield. In practice, we use the approximation parameter values implied by theaverage dividend (earnings) yield observed in each market.


Table 2Observed correlations

FR BD JP U.K. U.S. WD

Correlations: ri; tFR 1.000 0.542 0.334 0.533 0.520 0.575

(0.053) (0.054) (0.058) (0.057) (0.057)BD 1.000 0.327 0.449 0.446 0.493

(0.072) (0.066) (0.075) (0.070)JP 1.000 0.337 0.342 0.677

(0.055) (0.063) (0.053)U.K. 1.000 0.608 0.614

(0.049) (0.099)U.S. 1.000 0.793

(0.049)WD 1.000

Correlations: Rln(Pt=Dt)FR 1.000 0.496 0.273 0.503 0.488 0.553

(0.054) (0.058) (0.059) (0.056) (0.062)BD 1.000 0.302 0.439 0.422 0.541

(0.070) (0.062) (0.073) (0.071)JP 1.000 0.318 0.325 0.685

(0.056) (0.061) (0.056)U.K. 1.000 0.589 0.677

(0.050) (0.050)U.S. 1.000 0.836

(0.050)WD 1.000

Correlations: Rln(Pt=Et)FR 1.000 0.444 0.182 0.476 0.434 0.478

(0.050) (0.060) (0.053) (0.059) (0.061)BD 1.000 0.285 0.371 0.341 0.422

(0.065) (0.058) (0.061) (0.065)JP 1.000 0.259 0.254 0.591

(0.060) (0.066) (0.059)U.K. 1.000 0.547 0.628

(0.056) (0.051)U.S. 1.000 0.847

(0.048)WD 1.000

Correlations: ln(Dt+1=Dt)FR 1.000 0.117 0.031 0.020 0.084 0.219

(0.065) (0.058) (0.057) (0.049) (0.062)BD 1.000 − 0.164 0.102 0.023 0.140

(0.160) (0.052) (0.051) (0.038)JP 1.000 0.101 − 0.016 0.043

(0.050) (0.057) (0.048)U.K. 1.000 0.115 0.170

(0.051) (0.040)U.S. 1.000 0.302

(0.045)WD 1.000


Table 2 (continued).


Correlations: ln(Et+1=Et)FR 1.000 0.129 0.088 0.082 0.004 0.183

(0.062) (0.060) (0.036) (0.056) (0.060)BD 1.000 0.024 0.101 0.014 0.201

(0.061) (0.056) (0.072) (0.070)JP 1.000 0.130 0.031 0.259

(0.038) (0.039) (0.047)U.K. 1.000 0.096 0.282

(0.047) (0.051)U.S. 1.000 0.633

(0.089)WD 1.000

This table presents cross-correlations of monthly returns, the 7rst diLerence of the two valuation ratios, andthe continuous growth rate of cash 2ows (dividends and earnings). Returns and growth rates are continuousand real. VARHAC adjusted standard errors are provided in parentheses.

To derive fundamental values of assets, we 7rst model cash 2ow growth ratesand then proceed to model the cost of capital. Barsky and DeLong (1993) posit anARIMA(0,1,1) process for dividend growth rates, that is, growth rates contain a unitroot. In Timmermann (1993, 1996), learning about the cash 2ow growth rate pro-cess leads to time variation in expected growth rates. Donaldson and Kamstra (1996)provide a model for the univariate dynamics of the cash 2ow growth de2ated by adiscount factor, and use boot-strapping procedures to solve for fundamental presentvalues. Campbell and Shiller (1988a, b) posit VAR dynamics for growth rates andex post rates of returns and then test for certain internal consistency restrictions thatfollow from (3); we discuss this in greater detail below. Further, all these studies focuson a single equity claim, and consequently do not address the issues associated withthe cross-correlation puzzle, which is an important focus of this paper.

3.1. Cash 7ow dynamics

In this section, we provide the description of the assumed time-series model for thegrowth rate of cash 2ows, which we demonstrate can have large eLects on impliedequity prices, volatility, and asset betas. Statistically, we posit that cash 2ow growthrates are described by an ARIMA(1,0,1) process, an assumption for which we willprovide empirical support. We also show that this process for growth rates is equivalentto a decomposition of cash 2ow levels into exponentially smoothed stochastic trend andautoregressive cyclical components. 5

5 Alternatively, following Kasa (1992), one could model cash 2ows as determined by a common trend.This speci7cation is closely related to the world business cycle described by Dumas et al. (2000). To explorethis speci7cation, we perform cointegration tests, but we do not 7nd evidence for a single common trend(world business cycle) in either the dividends or earnings series. For dividends, these cointegration resultsare broadly similar to Kasa (1992).


Let gi; t be the real growth rate of cash 2ows for equity claim i. We assume that theprocess for gi; t satis7es

gi; t = (1− !i)�i + �igi; t−1 + �i; t − !i�i; t−1: (4)

The variable that aLects present values is the conditional mean of the growth rate. Forthe ARIMA(1,0,1) process the conditional mean xi; t equals ((�i − !i)=(1 − !iL))gi; t ,where L is the standard lag operator. Consequently, the ARIMA(1,0,1) process can bemore conveniently written as

gi; t = �i + xi; t−1 + �i; t : (5)

It is assumed that g is stationary, and hence � and ! (where country subscript i issuppressed) are ¡ 1 in absolute value. Using Eq. (5), it follows that xi; t itself followsan AR(1) process,

xi; t = (�i − !i)�i + �ixi; t−1 + (�i − !i)�i; t : (6)

The parameter � determines the degree of persistence, and ! is the smoothing parameterthat aLects the construction of xi; t . It is also worth noting two special cases that theARIMA(1,0,1) representation accommodates. If �=!, the conditional mean of g is aconstant, and in fact g can be viewed as an i.i.d. process. On the other hand, if !=0,g follows a standard AR(1) process.To develop intuition regarding the implications of an ARIMA(1,0,1) process for cash

2ow expectations, and hence equity prices, consider an agent’s revision in expectedgrowth rates (for horizon n¿ 1) in response to growth rate news at date t:

Et[gt+n]− Et−1[Et[gt+n]] = �n−1(�− !)�t : (7)

Economically, Eq. (7) implies that rational agents may signi7cantly revise their long-runexpected growth rates so long as � − ! �=0. In the extreme case when � = ! (i.i.d.case), there is no revision in the expected growth rate at all. Also, the “permanence”of the expectation revision is determined by �. If � = 1, the revision in expectationsis identical across all horizons (as in Barsky and DeLong, 1993). When �¡ 1, therevision is larger for shorter horizons, and almost zero for very long horizons. Forthe case where the diLerence between � and ! is small and positive and � is large,growth rate news leads to small, but near permanent, revisions in agent’s expectationsof future growth rates.To understand the pricing implications of the expected cash 2ow growth rate process

in Eq. (6), consider the implications for the log price dividend ratio, z, in Eq. (3)(assuming that the cost of capital is constant for now):

zt = Uz +1

1− �1�(xt − Ux); (8)

where Uz and Ux refer to unconditional means. The volatility of the price dividend ra-tio is clearly increasing in the persistence of the expected growth rate; when � ap-proaches one (as in Barsky and DeLong, 1993), the price dividend ratio becomesextremely volatile. Further, the reaction of the price dividend ratio to growth rate


Table 3ARIMA(1,0,1) estimation results for dividend and earnings growth

Parameter estimatesgi; t = (1− !i)�i + �igi; t−1 + �i; t − !i�i; t−1

Dividends Earnings

�i !i R2 LR-test �i !i R2 LR-test

FR 0.939 0.882 0.027 9:56† 0.895 0.838 0.016 2:25(0.061) (0.080) (0.119) (0.141)

BD 0.906 0.760 0.108 34:70† 0.938 0.889 0.019 5:07?

(0.053) (0.082) (0.070) (0.087)JP 0.973 0.928 0.036 7:94?? 0.953 0.854 0.095 23:16†

(0.028) (0.047) (0.036) (0.060)U.K. 0.944 0.843 0.086 25:04† 0.844 0.658 0.107 31:82†

(0.038) (0.063) (0.074) (0.105)U.S. 0.952 0.864 0.075 18:02† 0.913 0.823 0.045 12:50†

(0.043) (0.067) (0.066) (0.093)

The ARIMA(1,0,1) speci7cation is estimated by maximum likelihood, using the normal distribution func-tion. Standard errors are provided in parentheses. LR-test refers to the Andrews and Ploberger (1996)likelihood ratio test of the null hypothesis that the � and ! are equal. They show that the test statistic istwo times the diLerence between the unconstrained (ARIMA) and constrained (i.i.d.) log likelihood values.A † above indicates a rejection of the null at the 0.01 level, ?? indicates a rejection at the 0.05 level, and? indicates a rejection at the 0.10 level. The critical values for these tests are obtained from Andrews andPloberger (1996). For a sample size of 250 observations, the 0.01 level critical value is 9.23, 0.05 criticalvalue is 6.13, and 0.10 critical value is 4.74; for 500 observations: 0.01 level critical value is 9.46, 0.05critical value is 6.09, and 0.10 critical value is 4.75. Intercepts are not reported.

news is

zt − Et−1[zt] = (�− !)�t

∞∑n=1

�n1�

n−1 =(�− !)�t

1− �1�: (9)

Again, when � is close to one and larger than !, the impact of cash 2ow news on theinnovation to the price dividend ratio can be very large even though the ex post cash2ow process seems very close to an i.i.d. process in a 7nite sample. In contrast, in thei.i.d. case (� = !), growth rate news has no impact on the dividend yield, a featurewhich seems counter-intuitive and empirically implausible (see Easton and Zmijewski,1989). Collectively, this suggests that the explanation of the volatility puzzle for assetprices is intimately related to large price elasticity with respect to cash 2ows—forparameter estimates presented below, this quantity is well in excess of one. In Section4.3.3, we show that the standard VAR approach (as used in Campbell and Shiller,1988a), will fail to detect, in 7nite samples, the persistent component of cash 2owswhich is important for understanding asset price volatility.

3.1.1. Empirical evidence regarding the cash 7ow dynamicsIn Table 3, we present evidence for an ARIMA(1,0,1) process for both of the two

alternative measures of cash 2ow growth rates, dividends and earnings, for the diLerent


equity markets under consideration. The ARIMA model is estimated for each countryusing maximum likelihood, assuming the normal distribution function. Our results showthat both the AR and MA parameters are highly signi7cant. The magnitude of theAR coe!cient ranges across countries from 0.844 (U.K. earnings) to 0.973 (Japandividends) and the MA coe!cient ranges from 0.658 (U.K. earnings) to 0.928 (Japandividends). In general, the AR coe!cient is fairly large, and, in all cases, exceedsthe MA coe!cient, re2ecting the fact that the observed dividend or earnings growthautocorrelations are positive and fairly small (the 7rst-order autocorrelation is about6%, on average). Fig. 1 shows that the average autocorrelation function observed inthe data is fairly close to that implied by the estimated ARIMA(1,0,1) speci7cationfor cash 2ow growth. Also, note that the R2 in all cases is fairly small (about 5%).The magnitudes of the estimates of � suggest that cash 2ow news aLects long-runexpectations of cash 2ow growth.As stated earlier, when �=!, the ARIMA(1,0,1) process collapses to an i.i.d. pro-

cess. Hence, we need to test the hypothesis that � is statistically diLerent from !. Thetest of this hypothesis is non-standard—as, under the i.i.d. null, the parameters of theARIMA(1,0,1) speci7cation are separately identi7ed only under the alternative. Fortu-nately, Andrews and Ploberger (1996) provide a likelihood ratio based test statistic for�=!, where they show that this test statistic (referred to as the LR-test) reduces to twotimes the diLerence between the unconstrained (ARIMA) and constrained (i.i.d.) loglikelihood values. Additionally, they also provide the distribution for this test statisticand the associated critical values; for convenience, these are also reported in Table 3.In Table 3, we report the Andrews and Ploberger (1996) LR-test to evaluate the

hypothesis that dividend (or earnings) growth rates are i.i.d. (equal roots). In almostall cases, the rejection of the null hypothesis is particularly strong, and hence appears toconstitute rather sharp evidence against the null of equal roots. Second, the hypothesisthat !i = 0 is sharply rejected in all cases. This implies that the AR(1) speci7cationfor cash 2ows is not supported in the data. 6

Persistence in expected growth rates is intimately related to shocks to the trendgrowth rate on the economy. One way to see this relationship is to rely directly onan extensively used alternative to decompose the level of the cash 2ow series intotrend and cyclical components: the Hodrick–Prescott (HP) 7lter (see Hodrick andPrescott, 1997). For comparison, the trend components extracted from the HP andARIMA(1,0,1) 7lters are plotted in Fig. 2 for the U.S. First, for both dividends andearnings, it is evident from the 7gure that the diLerences in implied trend compo-nents across the two 7lters are small. Further, Table 4 shows that the relative vari-ances of the trend-component growth rates are small in size, but extremely persistent(with an AR(1) coe!cient of about 0.98, on average). This evidence suggests that the

6 In our estimation of the full model described below, to maintain parsimony in the number of estimatedparameters, we restrict � and ! to be same across all equity markets. Note, in almost all cases univariateGMM estimates of the ARIMA(1,0,1) speci7cation for cash 2ow are similar to the likelihood-based estimatesprovided in Table 3. Across all 7ve countries, pooled GMM estimates for � and ! for dividend growth are0.973 (S.E. 0.027) and 0.932 (S.E. 0.037), respectively, and are 0.950 (S.E. 0.041) and 0.890 (S.E. 0.113),respectively, for earnings growth. The R2 is about 3%.


1.18

1.14

1.10

1.06

1.02

0.981970 1974 1978 1982 1986 1990 1994 1998 2002

United States

HP DividendHP Earnings

ARMA Dividends, � = 0.932,ARMA Earnings, � = 0.890

Fig. 2. Cash 2ow trends: Hodrick and Prescott (1997) and ARIMA.

persistent component captured by the ARIMA(1,0,1) process is closely linked to astochastic trend in the overall economy.A common issue that needs to be addressed in the context of equity valuation is

the appropriate choice for the cash 2ow. It is well recognized that neither the mea-sured dividends series nor the earnings series is perfect for valuation. In our data, thestochastic trends in dividends and earnings are comparable for France, Germany, andthe U.K. However, there are important periods over which trends in dividends andearnings diLer for the U.S. and Japan (see Fig. 2, for the U.S. example). For instance,the recent rise in U.S. equity prices is somewhat better mirrored in the earnings trend.The main results of our paper are driven by the persistence of shocks to the growthrate of the trend component of the cash 2ow, which should be far less susceptible tomismeasurement and the choice between dividends and earnings. When using earnings,our economic assumption is that the “trend” for earnings is identical to that of the “truedividends”. For these reasons, we employ both as measures of cash 2ows. Further, itwill be shown below that our results, in terms of the implications for volatility andcross-correlations across markets, are not very diLerent whether we use the measureddividends or earnings.

3.2. Economic models and fundamental values

We employ the world capital asset pricing model (CAPM), where we assume themarket portfolio is the world market equity index. We will show that this modelreasonably describes excess returns for the menu of global equity indices under


Table 4Trend growth rates: autocorrelations and relative volatilities

Dividends Earnings

HP ARIMA HP ARIMA

FR�1 0.9936 0.9538 0.9967 0.8990rel-vol 0.0467 0.0589 0.0270 0.0623

BD�1 0.9932 0.9696 0.9947 0.9089rel-vol 0.0921 0.0924 0.0340 0.0734

JP�1 0.9907 0.9657 0.9934 0.9376rel-vol 0.0755 0.0805 0.0716 0.1080

U.K.�1 0.9918 0.9776 0.9979 0.9453rel-vol 0.1284 0.1305 0.0461 0.1276

U.S.�1 0.9865 0.9548 0.9960 0.9322rel-vol 0.0903 0.0861 0.0534 0.0925

�1 is the 7rst-order autocorrelation of the trend growth rate. Note that rel-vol is equal to �2(gtr)=�2(g),the volatility of the trend growth rate relative to the volatility of the cash 2ow growth rate itself. The trendgrowth rates are constructed as the logged 7rst diLerence of the respective trend level. HP refers to thetwo-sided Hodrick and Prescott (1997) Filter, for which we choose the HP smoothing parameter to be 14400, which is the recommended value for the monthly frequency. The weights used in the construction ofthe ARIMA growth rates are !=0:929 for dividends and !=0:890 for earnings (the corresponding pooledMA parameter estimates presented in footnote 6).

consideration. 7 A fairly direct alternative to using the CAPM would be to rely onthe general equilibrium dynamic market based model discussed in Campbell (1996);however, this would signi7cantly increase the number of parameters to estimate. Fur-thermore, given that Campbell (1996) shows that the 7rst-order eLects in determiningrisk premia are associated with market risk and the evidence in support of the CAPM

7 There are many alternative models that one could use to model the expected risk premia for internationalequity returns. For example, Adler and Dumas (1983), Dumas and Solnik (1995), and DeSantis and Gerard(1998) argue that exchange rate risks may contribute to the risk premium for equity returns; however, foreignexchange risk may have second-order eLects relative to the market (see Ng, 2001). Bekaert and Hodrick(1992) consider a latent factor model, and Bansal et al. (1993) consider a non-linear APT model for jointlyexplaining equity and bond returns. As discussed later in the paper, when parsimony is highly valued, thestatic CAPM, at least for the 7ve equity returns under consideration, seems to be an adequate model for therisk premia in our exercise. However, we also explore a two-factor world CAPM with exchange rate risk(not reported). While we 7nd evidence in favor of a time-varying price of foreign exchange risk, the generalimplications for asset volatility and correlations are almost identical to those presented for the one-factormodel (available upon request).


that we provide below, the CAPM seems to be a reasonable model for capturing thecost of capital given that parsimony is highly valued.Assuming log normality of returns, conditional risk premia are determined cross-

sectionally as follows:

Et[ri; t+1] = rf; t + �i��2m; t − 1

2 [�2i �

2m; t + �2

i ]; (10)

where non-systematic return volatility is constant. 8 By assumption, the world investoris marginal to all the developed equity markets we consider, and the price of risk isthe (scaled) variance of the world market return. Note that rf; t is the real risk-freerate and �2

m; t is the conditional variance of the world market portfolio. The beta of theasset is �i, and �2

i is the conditional variance of the non-systematic part of the assetreturn. The market price of risk is governed by the parameter �.To solve for fundamental values, we assume the following dynamic processes un-

derlying the cash 2ow growth rates, market volatility, and the risk-free rate. First, asdescribed above, we assume the that the growth rate is an ARMA(1,1), from which itfollows that

gm; t+1 = �m + xm; t + �m; t+1; (11)

where xm; t is the expected world cash 2ow growth rate, as de7ned in Eq. (5). Analo-gously, we also assume that each country’s cash 2ow growth rate is

gi; t+1 = �i + xi; t + �i; t+1;

�i; t+1 = !i�m; t+1 + vi; t+1; (12)

where xi; t is country i’s expected cash 2ow growth rate, as stated in Eq. (5), and !iis a “cash 2ow beta” describing the relationship between country i’s cash 2ow growthand that of the world. Next, we model the time-varying world market return volatilityusing a GARCH(1,1)-M model for the world market return (see Bollerslev et al.,1988). Using the fact that �m = 1;

Rm; t+1 − Rf; t = ��2m; t + m; t+1; (13)

�2m; t = &+ ($+ %)�2

m; t−1 + %( 2m; t − �2m; t−1); (14)

where m; t+1 is distributed with mean zero and variance �2m; t . This implies that the

time-varying price of risk follows a 7rst-order autoregressive process, with the autore-gressive parameter value equal to (% + $): Further if (% + $) is large (but ¡ 1), thesystematic time-varying price of risk will be persistent and have signi7cant impact onpresent values and on asset cross-correlation. Finally, we assume the real risk-free rateevolves according to an AR(1) process as follows:

rf; t+1 = &0 + &1rf; t + rf; t+1: (15)

8 Note in Eq. (10), the component 12 [�

2i �

2m;t+�2 i ] is due to the assumption of log normality (see Campbell,

1996), or in continuous time is an Ito adjustment for log returns.


Given the preceding state variable dynamics in Eqs. (11) and (13)–(15), the presentvalue implications of the model can be evaluated. We conjecture a solution for zi; t , thelog price dividend (earnings) ratio, as follows:

zi; t = Ai;0 + Ai;1xi; t + Ai;2�2m; t + Ai;3rf; t ; (16)

where Ai;0; : : : ; Ai;3 are yet to be determined. Exploiting the processes for the statevariables, including the ARIMA process for the cash 2ow growth rate, and matching thecoe!cients, leads to the following solution for the unknown coe!cients (see AppendixA):

Ai;1 =1

(1− �i;1�i); Ai;2 =

[��i − 12�

2i ]

(�i;1(%+ $)− 1)and Ai;3 =

1(�i;1&1 − 1)

: (17)

Additionally, the fundamental betas, �i, are also endogenously determined by the un-derlying parameters (for details see Appendix A). Assuming all AR(1) coe!cients arepositive and ¡ 1, it follows that Ai;2 and Ai;3 are negative, whereas Ai;1 is positive.A rise in the cost of capital lowers the present value, and a rise in expected growthrates raises it. Further, as noted earlier, persistent changes in these variables have amuch greater impact; this is now more clearly understood, as values close to one forthe AR(1) coe!cients within the autoregressive structure will yield large values forthe coe!cients of the solution. We have an analytical expression for the fundamentalvalues, zi; t , in terms of the real risk-free rate, the market price of risk, and the expectedcash 2ow growth rate.

3.2.1. Fundamental market betas and measures of integrationThe fundamental market beta of an asset is endogenously determined. The critical

input is !i, which is the cash 2ow beta describing the relationship between a givencountry’s cash 2ow growth and that of the aggregate market. To see this more clearly,suppose that �m;1 and �i;1, the approximation parameters, are equal, and ignore theeLect of the risk-free rate on the fundamental market beta (for the complete expressionfor the fundamental asset beta see Eq. (A.13) in Appendix A). 9 In this case, thefundamental return beta satis7es

�i =CmCi!i�2(�m)C2

m�2(�m)=

Ci!iCm

; (18)

where Ci ≡ {(1 + �i;1(�− !)Ai;1)}, which is the return elasticity with respect to cash2ow news. This can be seen by examining the one step ahead innovation in the return:

ri; t+1 − Et[ri; t+1]

= {(1 + �i;1(�− !)Ai;1)}�i; t+1 + �i;1Ai;2 �; t+1 + �i;1Ai;3 rf; t+1: (19)

News regarding returns is composed of cash 2ow news, �i; t , news regarding the marketrisk premium, �; t , and that of the risk-free rate, rf; t . The key components of the

9 In practice, these assumptions are actually reasonable. �i;1s are very close to one another. Additionally,�2( rf ) is extremely small, and its inclusion matters little quantitatively.


fundamental market beta are the cash 2ow beta (i.e., !) and the asset return elasticitywith respect to cash 2ow news Ci. Our approach to deriving the restrictions for thefundamental market beta is identical to Campbell and Mei (1993)—however, unliketheir paper we will directly use observed cash 2ow information to empirically restrictthe fundamental market beta of the asset. The fundamental asset values and returns re-ported in the paper are equity prices and returns derived using the fundamental solutionfor zi; t , and the complete solution for the fundamental beta.Ammer and Mei (1996) decompose the asset return news into components related

to future cash 2ow news and cost of capital news. They measure economic integrationby the cross-correlation among the cash 2ow news components, and 7nancial integra-tion as the cross-correlation in news regarding cost of capital components. They donot provide any mechanism to connect these two measures of integration, and pointout that 7nancial integration can generate return correlation through correlation amongthe equity premium components, despite economic segmentation (i.e., near-zero cor-relation among the cash 2ow news components). Our fundamental valuation methoddirectly addresses this important issue. If a country’s cash 2ow news has close tozero correlation with aggregate market cash 2ow news (i.e., if !i ≈ 0), then the im-plied fundamental market betas will also be close to zero (see Eq. (18)) and ex postreturns across markets will have little correlation. Stated diLerently, if there is littleeconomic integration among countries, then our fundamental ex post returns will implylittle 7nancial integration among them as well. Hence, our approach provides a directeconomic link between these measures of integration.

3.2.2. Conditional second momentsTime-varying market risk in our model implies that conditional cross-correlation

across equity returns can also be time varying so that the conditional cross-correlationbetween two positive beta assets will typically rise as the market volatility increases.The conditional cross-correlation, under the assumption that non-systematic risk is ho-moskedastic, is given by the expression

(�i�j�2m; t)√

�2i �

2m; t + �2

i

√�2j �

2m; t + �2

j

: (20)

As mentioned earlier, Longin and Solnik (1995), amongst others, have documented thatduring periods of high uncertainty, conditional correlations across markets are high.The above expression allows us to quantitatively measure the degree to which ourfundamental valuation model can duplicate this important empirical feature as well.Note that if the fundamental beta of one of the assets is zero, then the conditionalcorrelation of asset returns is zero as well.Finally, to further develop the intuition regarding the fundamental sources of risk

underlying our model, we provide a tight link between the conditional volatility on theaggregate world cash 2ow process and the conditional volatility of the world marketportfolio. This link also allows us to connect the world market’s cash 2ow volatilityto the market risk premium, and interpret movements in the market risk premium inSection 6. Given the fundamental solution, consider the innovation in the market return,


ignoring the risk-free rate contribution for exposition:

rm; t+1 − Et[rm; t+1] = {(1 + �m;1(�m − !m)Am;1)}�m; t+1 + �m;1Am;2 �; t+1: (21)

If we assume that the shocks to volatility, �; t+1, are homoskedastic and uncorrelatedwith cash 2ow news, then the conditional variance of the market return can be ex-pressed as follows:

�2m; t = (�m;1Am;2)2�2

� + {(1 + �m;1(�m − !m)Am;1)}2�2�m;t

= (�m;1Am;2)2�2� +

{(1 +

�m;1(�m − !m)1− �m;1�m

)}2

�2�m;t : (22)

Conditional volatility of the world market portfolio is simply a magni7ed version ofthe world market cash 2ow conditional volatility, �2

�m; t. This provides a fundamental

justi7cation for the statistical process assumed in Campbell and Hentschel (1992),linking excess returns to dividend volatility. Global cash 7ow uncertainty determines thevolatility of the market portfolio, and hence, return cross-correlations as well. Therefore,the above equation, in conjuction with Eq. (19), suggests that ex post returns in eacheconomy are in2uenced by news regarding cash 2ows, �i; t+1, and changing economicuncertainty in global output (i.e., cash 2ows), �2

�m; t.

3.3. Latent stochastic volatility

An important input into the model, particularly in determining the equity price level,is the market price of risk, for which the GARCH model considered above is one par-ticular speci7cation. In this section, we present an alternative latent stochastic volatilityspeci7cation for the market price of risk (see Taylor, 1986). In this model, we con-tinue to assume that the conditional market CAPM model determines the ex ante costof capital, but the time-varying volatility of the world market portfolio is latent. Ex-ploiting the valuation restrictions for the world market portfolio, we can extract thislatent volatility as a linear function of the observed world price dividend ratio andexpected cash 2ow growth rates. The time-varying cost of capital for each asset un-der consideration is now determined by its market beta and the extracted time-varyingmarket volatility.We 7rst assume that the latent market volatility, as in the case with the GARCH(1,1)

speci7cation, follows an AR(1) process

�2m; t+1 = )0 + )1�2

m; t + �; t+1: (23)

The world cash 2ow dynamics are identical to what we have assumed thus far, anARIMA(1,0,1) process. Given this, the solution for the price dividend ratio for theworld market portfolio is

zm; t = Am;0 + Am;1xm; t + Am;2�2m; t + Am;3rf; t ; (24)

where zm; t is the log price dividend ratio for the world market, xm; t is the expectedgrowth rate of earnings for the world, and ��2

m; t is the market price of risk, which islatent and is not directly observable. Exploiting the fundamental solution for the world


market portfolio, it follows that the latent volatility can be extracted by inverting thevaluation restriction

�2m; t =

1Am;2

(zm; t − Am;0 − Am;1xm; t − Am;3rf; t): (25)

Exploiting Eq. (25) for measuring the latent volatility, we can rewrite the fundamentalsolution for all other assets as follows:

zi; t = Ai;0 + Ai;1xi; t + Ai;2�2m; t + Ai;3rf; t : (26)

The economic intuition in this model speci7cation is identical to the GARCH case, withone diLerence. The procedure discussed above for extracting latent volatility ensuresthat the fundamental value for the world market portfolio equals its observed price inthe data.In the latent stochastic volatility model we are exhausting the world market portfo-

lio’s present value restrictions to extract the latent volatility process, hence the presentvalue implication for the world market portfolio will be exactly satis7ed. The funda-mental values for all other assets, as in the case with the GARCH speci7cation, aredetermined by the market volatility and cash 2ow dynamics. This relative valuationapproach is similar to Bossaerts and Green (1989), where they construct a model forrelative asset valuation. Fundamental betas are endogenously determined in exactly thesame manner as before.

4. Econometric method and empirical results

4.1. GMM estimation

The varied set of economic and statistical restrictions presented above naturally mapsinto the generalized method of moments (GMM) framework for estimation (Hansen,1982). The precise orthogonality conditions we exploit and the construction of therobust VARHAC (denHaan and Levin, 1996) weighting matrix are detailed in Ap-pendix A. For the unrestricted model we consider, we estimate return �is, cash 2ow!′is (betas), the autoregressive coe!cient for the risk-free rate, &1, and the parametersassociated with the GARCH-M process for the market return, = (�; %; $)′. Addition-ally, we estimate the parameters associated with the ARIMA cash 2ow dynamics. Tokeep the number of estimated parameters manageable, we restrict the cash 2ow pa-rameters, � and !, to be identical across markets. This restriction is not rejected inthe data for either dividends or earnings based upon the GMM test of overidentifyingrestrictions associated with the pooled results presented in Table 3. We also estimatethe ARIMA parameters for the world cash 2ow growth rates. Hence, the parametervector is ( ′; !′; &1; �; !; �m; !m)′. In sum, the stochastic processes we estimate are asfollows:

Ri; t+1 − Rf; t = ,i + �i[Rm;t+1 − Rf; t] + i; t+1;

Rm; t+1 − Rf; t = ��2m; t + m; t+1;


�2m; t = &+ ($+ %)�2

m; t−1 + %( 2m; t − �2m; t−1);

rf; t+1 = &0 + &1rf; t + rf; t+1;

gi; t+1 = �i + xi; t + �i; t+1;

�i; t+1 = !i�m; t+1 + vi; t+1;

Rzi; t+1 = Ai;1(�− 1)xi; t + Ai;2(%+ $− 1)�2m; t + Ai;3(&1 − 1)rf; t + ei; t+1; (27)

where ei; t+1 is the forecast error associated with the 7rst diLerence of the log pricedividend (earnings) ratio implied by the equilibrium solution, and the fundamentalbeta is a restricted function of the estimated parameters. In the last equation, as theAi’s are also restricted by the estimated parameters, these forecast errors representimportant present value restrictions the model must confront in the data. 10 There isevery reason to believe that equity prices contain valuable information regarding cash2ow dynamics. Consequently, in addition to the information provided by the cash 2owgrowth rates, our estimation strategy exploits the information contained in zi; t+1 toidentify the parameters of the cash 2ow process. There are 18 parameters to estimate,with 41 moment conditions. GMM distribution theory (Hansen, 1982) provides anasymptotic jointly normal distribution for the parameter estimates. Note that in theinterests of parsimony, as in Campbell and Shiller (1988b), we de-mean the growthrate variables to reduce the number of parameters (intercepts) to be estimated.For the latent stochastic volatility model, we employ the analogous set of restrictions

implied by this structure using GMM, given the extracted market volatility from Eq.(25) and the de7nitions of the solution coe!cients. The precise orthogonality conditionswe exploit are also detailed in Appendix A. There are 40 moment conditions and 17parameters. We document the parameter estimates and present value implications next.

4.2. CAPM-GARCH: parameter estimates

First, in Table 5 we provide evidence that the CAPM adequately captures the riskpremia of the international assets under consideration; a standard intercept test that the,s for each index are jointly equal to zero is not rejected with a .25 =3:48. In Table 6,we present the unrestricted GMM parameter estimates for the world CAPM-GARCHmodel. In general, the parameter estimates are precise. As the cost of capital parameterestimates do not substantially diLer, we focus on the estimates obtained when earningsare employed as the cash 2ow measure. The AR(1) parameter (standard error) onthe real risk-free rate is 0.950 (0.016). The market volatility GARCH parameters areprecisely estimated with %=0:048 (0.005) and $=0:941 (0.006), which together implythat the AR(1) parameter (i.e., % + $) on the market volatility is 0.989, suggesting ahigh degree of persistence in volatility. The risk aversion parameter, �, is imprecisely

10 Note in estimation, we impose the fundamental valuation restriction on the 7rst diLerence of the log pricedividend (earnings) ratio (i.e., Rzi; t+1) as this quantity is far less persistent relative to its level. In practice,the qualitative implications of using the 7rst diLerence relative to the level of zi; t+1 for the estimates, andhence our results, are small.


Table 5CAPM tests

Ri; t+1 − Rf; t = ,i + �i(Rm;t+1 − Rf; t) + i; t+1

,i �i

FR 0.0033 0.836(0.0030) (0.079)

BD 0.0022 0.563(0.0024) (0.080)

JP − 0.0028 0.832(0.0023) (0.112)

U.K. 0.0026 0.904(0.0030) (0.190)

U.S. 0.0018 0.849(0.0017) (0.070)

.2 tests

�i = 1 56.6751dof 5P-value 0.0000�i = � 23.0583dof 4P-value 0.0001,i = 0 3.4813dof 5P-value 0.6262

This table presents GMM estimates of the standard CAPM return betas. Ri; t is the real monthly totalreturn on the Datastream Total Market Equity Index for country i. Rm;t is the monthly total return on theDatastream Total Market World Index. Rf; t is the real one-month interest rate. VARHAC adjusted standarderrors are provided in parentheses. dof denotes degrees of freedom.

estimated at 3.49 (2.27). 11 The unconditional standard deviation of the market priceof systematic risk ��2

m; t is only 0.0014, which is clearly very small.The parameters governing the cash 2ow dynamics are also precisely estimated. The

autoregressive parameter, �, for dividend and earnings growth is 0.974 (0.010) and0.968 (0.009), respectively, and the moving average parameter, !, is 0.930 (0.017)and 0.930 (0.013), respectively. The large autoregressive parameter suggests that thecomponent determining the expected cash 2ow growth rate is very persistent for bothdividends and earnings. Note that these estimates do not diLer signi7cantly from thosediscussed earlier in Section 3.1.1, where we only used information contained in cash2ows to estimate these parameters. However, it is important to note that equity pricesprovide additional information regarding the process governing cash 2ow growth rates.Consequently, any distinction in the parameter values re2ects the additional informa-tion that equity prices bear on the estimation through the fundamental present valuerestrictions.

11 Lundblad (2000) demonstrates the imprecision with which � is estimated in a univariate GARCH-Mframework.


Table 6GMM parameter estimates

CAPM-GARCH CAPM-GARCH CAPM-latent volatilitywith dividends with earnings with earnings

&1 0.951 0.950 0.951(0.014) (0.016) (0.017)

% 0.049 0.048 —(0.004) (0.005)

$ 0.939 0.941 —(0.005) (0.006)

)1 — — 0.994(0.006)

� 3.200 3.490 3.500(2.850) (2.271) (2.541)

� 0.974 0.968 0.967(0.010) (0.009) (0.008)

! 0.930 0.930 0.932(0.017) (0.013) (0.012)

�m 0.972 0.935 0.934(0.038) (0.029) (0.033)

!m 0.948 0.895 0.894(0.082) (0.043) (0.038)

!FR 0.482 0.508 0.440(0.125) (0.134) (0.151)

!BD 0.223 0.508 0.438(0.062) (0.142) (0.179)

!JP 0.047 0.284 0.205(0.037) (0.047) (0.056)

!U:K: 0.194 0.418 0.343(0.038) (0.061) (0.075)

!U:S: 0.148 0.720 0.728(0.020) (0.050) (0.058)

Tests

JT = .223 51.80 49.45 46.95P-value 0.004 0.007 0.014

This table presents the unrestricted GMM estimates associated with each of the speci7cations considered.VARHAC standard errors are provided in the parenthesis. The orthogonality conditions we exploit aredetailed in Eq. (A.16) for the CAPM-GARCH speci7cation, and Eq. (A.21) for the CAPM-latent volatilityspeci7cation. JT refers to the GMM test of overidentifying restrictions. Note that in the interests of parsimony,as in Campbell and Shiller (1988b), we de-mean the growth variables to reduce the number of parameters(intercepts) to be estimated. &1 is the AR(1) parameter for the real risk-free rate process:

rf; t+1 = &0 + &1rf; t + rf; t+1:

(%; $; �) are the GARCH-M parameters for the world market return process:

Rm;t+1 − Rf; t = ��2m;t + m; t+1;

�2m;t = & + % 2m;t + $�2m;t−1:

)1 is the AR(1) parameter on the latent stochastic volatility process:

�2m;t+1 = )0 + )1�2m;t + �; t+1:

� and ! are the ARIMA(1,0,1) parameters for the cash 2ow growth rates (�m and !m for the world). !iis the projection coe!cient for the cash 2ow innovation in country i on the innovation in the cash 2owassociated with the world market portfolio,

�i; t+1 = !i�m; t+1 + vi; t+1:


Using the GMM test of overidentifying restrictions, the world CAPM-GARCH modelis statistically rejected with a .223 of 51.80 for dividends and 49.45 for earnings. How-ever, our diagnostics reported in Table 9 suggest the model is capable of explainingmuch of the observed asset volatility and cross-correlation. Additionally, given theoverrejection associated with the GMM test documented in Ferson and Foerster (1994)and Hansen et al. (1996), caution is required. Table 7 presents the Ai’s implied by ourestimation. While the solution coe!cients on the expected cash-2ow growth rate andthe real risk-free rate are signi7cant, the coe!cient on the market volatility is not dueto the documented imprecision associated with the estimation of �.In Table 8, we report fundamental return betas implied by the estimated parame-

ters, �, as determined in Eq. (18). We perform parameter restriction tests that theestimated fundamental betas are jointly equivalent to the traditional betas. In all cases,the hypothesis of equivalence is rejected with a P-value of 0.000. In this sense, thefundamental valuation restrictions, primarily for the betas are rejected. However, whilethe fundamental market betas are smaller than the traditional, we are able to explain onaverage roughly 60% (90%) of the empirically observed market betas with dividends(earnings). In a related context, Campbell and Mei (1993) decompose the asset returnprocess itself to explain the components of the estimated beta, backing out that partof the beta which can be attributed to cash 2ow news. However, they do not directlyemploy cash 2ow information. To the best of our knowledge, this is the 7rst directempirical evidence based on observed cash 2ow information regarding asset betas. Fur-ther, note that mismeasurement of cash 2ows can signi7cantly aLect fundamental betaestimates, while having secondary eLects on the constructed fundamental value. As canbe seen in the present value solution in Eq. (3.2), the persistent component in cash2ows is a critical input into the fundamental value, and is far less likely to be sensitiveto cash 2ow mismeasurement. 12

The expected cash 2ow growth rate, the market price of risk, and the risk-freerate (the relevant state variables) are quite persistent. Consequently, the fundamentalsolution for zi is also quite persistent. The 7rst-order autocorrelation coe!cient forthe fundamental (implied) zi is close to 0.99, as is also the case for its counterpartin the data. Given the persistence in the observed (and fundamental) level of zi, wepresent our diagnostics using the 7rst diLerence of the fundamental log price dividend(earnings) ratio. However, the message from our diagnostics is the same when focusingon the level of the price dividend (earnings) ratio, its 7rst diLerence, or ex post returns.

4.3. Diagnostics

4.3.1. Volatility and correlationIn Table 9, we present the implications of our fundamental valuation model for the

volatility and cross-correlations of Rzi; t , focusing 7rst on the model’s implications for

12 For issues regarding measurement of aggregate cash 2ows see Ackert and Smith (1993) and Campbelland Shiller (1998). Additionally, Campbell (1991), Campbell and Mei (1993), and Campbell and Ammer(1993), also focus on deriving implications for betas, but do not directly employ the cash 2ow data.


Table 7Solution coe!cients

Ai;1 Ai;2 Ai;3

(A) CAPM-GARCH with dividendsFR 34.318 −146:717 −19:034

(12.522) (92.585) (5.135)BD 35.351 −113:504 −19:340

(13.299) (89.893) (5.307)JP 36.923 −97:579 −19:789

(14.526) (93.952) (5.563)U.K. 33.642 −78:589 −18:829

(12.026) (73.633) (5.023)U.S. 34.456 −80:783 −19:075

(12.624) (77.650) (5.158)WD 32.792 −220:141 −19:262

(11.179) (130.124) (5.182)

(B) CAPM-GARCH with earningsFR 28.350 −150:987 −18:979

(6.950) (72.884) (4.987)BD 29.220 −183:278 −19:358

(7.391) (84.075) (5.194)JP 30.285 −157:359 −19:812

(7.950) (87.021) (5.447)U.K. 27.826 −116:652 −18:747

(6.691) (64.627) (4.863)U.S. 28.556 −209:097 −19:070

(7.053) (83.939) (5.036)WD 14.835 −239:967 −19:260

(2.015) (150.154) (5.846)

(C) CAPM-latent volatility with earningsFR 28.349 −256:262 −32:101

(8.277) (108.918) (12.661)BD 29.221 −239:471 −33:226

(8.802) (118.506) (13.579)JP 30.287 −407:676 −34:617

(9.467) (150.450) (14.758)U.K. 27.826 −222:356 −31:428

(7.968) (101.378) (12.128)U.S. 28.556 −316:470 −32:367

(8.340) (110.546) (12.875)WD 14.836 −408:920 −19:259

(2.353) (170.615) (6.403)

The fundamental valuation solution (zi; t =Ai;0 +Ai;1xi; t +Ai;2�2m;t +Ai;3rf; t) coe!cients are Ai;1 = 1=(1−�i;1�) (on expected cash 2ow growth rates), Ai;2=[��i− 1

2�2i ]=(�i;1(%+$)−1) (on the market volatility), and

Ai;3=1=(�i;1&1−1) (on the real risk-free rate). For the latent volatility model, Ai;2=[��i− 12�

2i ]=(�i;1()1)−1).

Fundamental betas are imposed. The approximation parameter, is taken to be that implied by the averagedividend yield in each market: �1 = (0:9966; 0:9977; 0:9989; 0:9959; 0:9969) for FR, BD, JP, U.K., and U.S.,respectively VARHAC standard errors are provided in the parenthesis, computed using the delta method.


Table 8Fundamental �s

Standard CAPM �-Div �-Earn �-latent

FR 0.836 0.696 0.662 0.541(0.079) (0.090) (0.093) (0.091)

BD 0.563 0.508 0.747 0.408(0.080) (0.074) (0.095) (0.086)

JP 0.832 0.399 0.586 0.948(0.112) (0.049) (0.044) (0.046)

U.K. 0.904 0.388 0.534 0.733(0.190) (0.073) (0.057) (0.070)

U.S. 0.849 0.380 0.901 0.951(0.070) (0.053) (0.106) (0.121)

This table presents the fundamental betas implied by the covariances of the cash 2ow growth rate innova-tions with the innovations in the market cash 2ow growth rate. Standard CAPM denotes the estimated returnbetas from Table 5 for comparison, with the associated VARHAC adjusted standard errors. Note, they are notdirectly comparable, since the standard betas are estimated using the returns themselves. The fundamentalmodel-implied �-Div corresponds to the CAPM-GARCH with dividends as the cash 2ow; �-Earn corre-sponds to the CAPM-GARCH with earnings as the cash 2ow; and �-latent corresponds to the CAPM-latentvolatility with earnings as the cash 2ow. Standard errors on the fundamental betas are computed by theDelta method.

asset volatility. The range for Std(Rzi; t) observed in the data (using dividends) is from0.065 (France) to 0.045 (U.S.) (see Table 1). At the estimated parameter values, theCAPM-GARCH model with dividends explains about 65% of this observed volatility(see Table 9, Panel A). Alternatively, when earnings are used as a measure of cash2ow, the range for Std(Rzi; t) observed in the data is from 0.073 (France) to 0.049(U.S.), and the model can explain about 85% of this observed volatility (see Table9, Panel B). Asset prices are volatile in our setup due to the presence of a persistentcomponent in cash 2ow growth rates, which as discussed earlier leads to high returnelasticity with respect to cash 2ow news.Second, turning to the model’s implications for cross-correlation, Table 2 (Panel B)

presents the observed correlations in the data for Rzi; t . These correlations range from0.845 (between the U.S. and the world) to 0.332 (between Germany and the U.K.) fordividends, and the comparable range using earnings is from 0.902 (between the U.S.and the world) to 0.319 (between France and Germany). Table 9 (Panels A and B) pro-vides the correlation evidence from the perspective of the world CAPM-GARCH model.It is evident that the model generates considerable cross-correlation across asset mar-kets, despite the low cross-correlation in cash 2ow growth rates. We will subsequentlydemonstrate that this feature of the model is due to time variation in systematic risk.Further, parameter-implied fundamental asset betas greatly exceed the smaller cash 2owbetas (see Eq. (18) for the source of this magni7cation). Many of the model-impliedcorrelations are within two standard errors of the correlations observed in the data (seefor example, between the U.S. and the other economies, excluding Japan). However,some of the correlations, particularly related to Japan, are too high in the model relativeto the data.


Table 9Diagnostics: fundamental volatility and correlations Rln(Pt=Dt) (Rln(Pt=Et))

Std % Exp Corr


(A) CAPM-GARCH with dividendsFR 0.052 80 1.000 0.386 0.399 0.332 0.494 0.579

(0.004) (0.035) (0.031) (0.033) (0.026) (0.037)BD 0.038 76 1.000 0.333 0.408 0.514 0.586

(0.003) (0.026) (0.027) (0.020) (0.035)JP 0.031 61 1.000 0.501 0.655 0.674

(0.002) (0.020) (0.013) (0.030)U.K. 0.027 43 1.000 0.600 0.603

(0.002) (0.015) (0.033)U.S. 0.022 49 1.000 0.845

(0.001) (0.022)WD 0.040 98 1.000

(0.004)

(B) CAPM-GARCH with earningsFR 0.062 85 1.000 0.319 0.381 0.345 0.385 0.486

(0.004) (0.045) (0.040) (0.039) (0.042) (0.043)BD 0.065 99 1.000 0.353 0.373 0.423 0.529

(0.004) (0.039) (0.038) (0.038) (0.039)JP 0.041 73 1.000 0.490 0.619 0.731

(0.003) (0.028) (0.025) (0.024)U.K. 0.038 55 1.000 0.539 0.635

(0.003) (0.027) (0.030)U.S. 0.044 90 1.000 0.902

(0.003) (0.008)WD 0.044 102 1.000

(0.002)

(C) CAPM-latent volatility with earningsFR 0.063 86 1.000 0.280 0.430 0.350 0.370 0.495

(0.005) (0.044) (0.034) (0.032) (0.034) (0.033)BD 0.062 95 1.000 0.302 0.309 0.286 0.400

(0.004) (0.043) (0.041) (0.044) (0.044)JP 0.053 95 1.000 0.502 0.665 0.845

(0.003) (0.009) (0.005) (0.004)U.K. 0.037 54 1.000 0.471 0.585

(0.003) (0.010) (0.011)U.S. 0.036 73 1.000 0.843

(0.003) (0.009)WD 0.043 99 1.000

(0.002)

(D) Nested speci)cations

Std

CC-Div CC-Earn AR(1)-Div AR(1)-Earn

FR 0.043 0.055 0.026 0.025BD 0.031 0.056 0.027 0.032


Table 9 (continued).

Std

CC-Div CC-Earn AR(1)-Div AR(1)-Earn

JP 0.021 0.028 0.039 0.036U.K. 0.021 0.029 0.020 0.021U.S. 0.009 0.011 0.021 0.030

This table presents a diagnostic evaluation of the volatility and correlations of the 7rst diLerence in thefundamental log price dividend (earnings) ratios implied by the estimated economic model, relating to thede7nition of cash 2ows as either dividends or earnings, when the fundamental betas are imposed. “% Exp”refers to the percentage of the observed standard deviation presented in Table 1 explained by its theoreticalcounterpart. We also present several nested models: (1) “CC-Div” or “CC-Earn” refers to the case wherethe market price of risk and the real risk-free rate are constant, but cash 2ow growth follows the estimatedARIMA(1,0,1) process; (2) “AR(1)” refers to the case where the market price of risk is a GARCH(1,1)process, the real risk-free rate and cash 2ow growth follow an AR(1) process. For the AR(1) cash 2owmodel, we restrict the AR(1) parameter to be the same across countries, in a fashion analogous to therestriction we make on the ARIMA(1,0,1) processes in the more general model. The AR(1) parameter fordividend (earnings) growth is 0.0612 (0.0518), consistent with the autocorrelation function presented inFig. 1. Standard errors are computed by simulation.

4.3.2. Cash 7ow persistence and return volatility and cross-correlationsIn order to explore the separate eLects of the various sources of time variation in

the model, we consider two restricted alternatives. In the 7rst, the cost of capitalis assumed constant, and equal to the time-series average return (as in Barsky andDeLong, 1993); however, the cash 2ow growth rate follows the assumed ARIMAprocess. The evidence in Table 9 (Panel D, columns CC-Div and CC-Earn) shows thatthis speci7cation is still capable of explaining 50% of the observed volatility of Rzi; t .However, this speci7cation also implies that the cross-correlations in Rzi; t are equalto the cash 2ow growth rate cross-correlations across economies (see Table 2, PanelC), implying asset cross-correlation is counter-factually low relative to the observeddata. This evidence suggests that time variation in systematic risk is an importantcomponent in asset market cross-correlation, but the assumed cash 2ow growth rateprocess is alone capable of generating a signi7cant degree of asset volatility.In the second experiment, we consider an AR(1) speci7cation for cash 2ow growth

rates, but allow for time-varying (GARCH) systematic risk. The estimated AR(1) co-e!cients are 0.061 for dividends and 0.052 for earnings. With this speci7cation, themodel can only explain roughly 30% (see Table 9, Panel D, columns AR-Div andAR-Earn) of the observed price volatility, nearly all of which is due to time variationin the systematic risk. Since the estimated cash 2ow autoregressive parameters are sosmall, the implied price response to cash 2ow news is marginal (see Eq. (8) when� is small). Additionally, the model-implied cross-correlations are close to 0.99, en-tirely inconsistent with the correlations observed in the data. This evidence documentsthe pitfalls of relying on a low-order autoregressive speci7cation for cash-2ow growthrates. The asset return cross-correlations are very high as asset prices, in the absenceof expected growth rate eLects, are almost entirely driven by the common sources ofsystematic risk.


1974 1978 1982 1986 1990 1994 1998

US−FRUS−BDUS−JPUS−UK

0.0

0.2

0.4

0.6

0.8

1.0

NBER trough; NBER peak

Time−Varying Correlations with the US: GARCH

Fig. 3. Time-varying asset return correlations with the U.S.

To explore this further, we take a simple variance decomposition of the unexpectedchange in the fundamental price dividend ratio (implied by our full model) to determinedirectly what each sources of time variation is contributing to the total asset variance.At our parameter estimates, 46% of the variance of fundamental price shocks areattributed to cash 2ow shocks, 43% are attributed to risk premium shocks, and 5% areattributed to risk-free rate shocks, on average, across the 7ve countries we consider(this evidence is representative for all 7ve countries). Clearly, the cash 2ow shocksare important for justifying price volatility.As discussed earlier, our model also permits time-varying conditional correlations

in asset returns. Conditional correlations among equity market returns implied by thefundamental solution and estimated parameters for each country relative to the U.S. arepresented in Fig. 3. Longin and Solnik (1995) document that correlations rise duringperiods of high volatility. As can be seen (see Eq. (20)), the conditional correlations aredriven entirely by the time-variation in the price of risk, and indeed rise during periodsof elevated economic uncertainty. Interestingly, as can be see in the 7gure, correlationsalso tend to rise during documented economic downturns. This is an important featureof the data which our model facilitates.

4.3.3. Interpreting the evidenceTo explore the interactions between the growth rate dynamics and the cost of capital

in capturing asset volatility and cross-correlations, we conduct an additional MonteCarlo experiment summarized in Table 10. Cash 2ow growth rates for two countries are


Table 10Monte Carlo

Panel A: Simulated cash 7ow data ARIMA(1,0,1) population values� ! C.F. C.F. Asset Asset

Vol Corr Vol Corr0.968 0.930 0.025 0.129 0.065 0.350

Panel B: ARIMA and AR implications for asset volatility and correlationARIMA(1,0,1) AR(1)

� ! Asset Asset � Asset AssetVol Corr Vol Corr

Mean 0.921 0.871 0.065 0.341 0.090 0.025 0.962Med 0.975 0.921 0.049 0.287 0.085 0.025 0.973Std Error 0.001 0.001 0.0004 0.002 0.001 0.000 0.0003

In Panel A, we provide the parameter values used in simulating an ARIMA(1,0,1) process. These valuescoincide with those reported for our joint estimation in Table 6. Random variables are simulated from thenormal random number generator in the GAUSS statistical package. The standard deviations (C.F. Vol)for the simulated cash 2ow shocks are equal to the average values observed in the data, 0.025, and thecorrelations (C.F. Corr) across the two shocks are constructed to match that average observed in the data,0.129. Given the fundamental solution for the log price dividend ratio, zt , in Eq. (18), this structure impliesthat the population standard deviation (Asset Vol) of R(zt) is 0.065 and the correlation (Asset Corr) acrossthe two assets is 0.350. In Panel B, we report estimates of an ARIMA(1,0,1) process and an AR(1) processfor each country using the simulated growth rates based on Panel A; we take the observed cost of capitaldata and estimated processes as given above (again, see Table 6). In our Monte Carlo experiment, we relyon the same number of times series observations as are observed in the data sample. For each case, giventhe estimates, we compute the implications for asset volatility (Asset Vol) and correlation (Asset Corr)for the change in the price dividend ratio. Mean and Med are the average and median value, respectively,for the relevant quantity over the Monte Carlo repetitions; the procedure is conducted 20,000 times. Thistable also provides summary statistics on the AR(1) or ARIMA(1,0,1) estimates, as well as their respectiveimplications for asset volatility and correlations. Monte Carlo standard errors are also supplied (StdError).

simulated from an ARIMA(1,0,1) process at our estimated parameter values, obtainedfrom the joint estimation (see Table 6). The standard deviations and correlations forthe simulated cash 2ow shocks are equal to the average value observed in the data(see Panel A in Table 10 for details). Given the fundamental solution for the logprice dividend ratio, zi; t , in Eq. (18), this structure implies that the population standarddeviation of R(zi; t) is 0.065 and the correlation across the two assets is 0.350. Inaddition, the market volatility that aLects the valuations is taken to be exactly as inTable 6.Given these simulated growth rates, we estimate an ARIMA(1,0,1) process and an

AR(1) process for each. For each case, given the time-series speci7cation estimates, wecompute the implications for the volatility and cross-correlations of R(zi; t). This proce-dure is conducted 20,000 times. We provide summary statistics on the ARIMA(1,0,1)or AR(1) estimates, as well as their respective implications for volatility and correla-tions, in Table 10 (Panel B). On average, the estimated ARIMA process does extremelywell in matching observed volatilities and correlations. In sharp contrast, the estimatedAR(1) implies that less than half of the observed volatility is explained, on average,


and correlations are near 1, as the expected growth rates are estimated as if there is nopersistence, even though the data are known (in the population) to contain a highly per-sistent component. Indeed, it appears that time variation in systematic risk alone cannotexplain asset volatility. Further, a blend of the persistent cost of capital and cash 2owshocks are required to justify asset cross-correlation; in the absence of the persistentcomponent in cash 2ows, asset price 2uctuations are dominated by common cost ofcapital 2uctuations, and hence asset returns are, counter-factually, almost perfectly cor-related. Collectively, these diagnostics suggest that our stochastic trend decompositionfor cash 2ows in conjunction with time variation in systematic risk can justify, at leastat an economic level, much of the volatility and cross-correlations observed in equitymarkets.To explore these issues further, we also estimate, as in Campbell and Shiller (1988a),

a standard VAR. In particular, we estimate a unrestricted VAR(3) on returns, growthrates, the risk-free rate, and dividend yields for the 7ve countries plus the world marketportfolio. 13 For the sake of brevity, we do not provide a detailed table regarding theresults. However, the main evidence can be summarized as follows. Consistent with theMonte Carlo evidence, the VAR speci7cation fails to capture the persistent componentin growth rates, and hence cannot reproduce the magnitude of the observed variancesand cross-correlations in asset returns. This VAR setup can explain only 45% of theobserved asset volatility, on average, across the countries under consideration. Further,implied asset cross-correlations are counter-factually very high.Also, long-run growth rate predictability can be limited in 7nite samples, even from

the perspective of our speci7cation. To explore this issue, we conduct a simple MonteCarlo exercise, simulating an ARIMA(1,0,1) process for growth rates exactly as abovefor one country. We determine the fundamental price dividend ratio, zt , from ouranalytical solution, and regress the cash 2ow growth rate, gt , on the fundamental pricedividend ratio, zt , lagged either 1 or 2 years. This is conducted 1000 times. The averagerobust t-ratio on the regression coe!cients (g on appropriately lagged z) is 1.01 and0.59, respectively, which are broadly consistent with the results observed across the7ve countries we consider at the one and two year horizons. Further, the average R2’sof the regressions are 1.1% and 0.7%, respectively, in the simulation; for the observeddata, the low degree of predictability is also comparable in the observed data. In all,this evidence suggests that the price dividend ratio, even from the perspective of ourmodel, is a bad predictor of future cash 2ow growth. This is so because of two reasons:7rst, the predictive component in growth rates is very small (small R2), and second, thecost of capital variations also aLect the dividend yields. Consistent with this evidence,Fama and French (2000) 7nd limited cash 2ow predictability at longer horizons.In sum, if cash 2ow growth rates follow an ARIMA(1,0,1) process, ex post growth

rate autocorrelations may be very small, in which case the econometrician will havedi!culty distinguishing this process from an i.i.d. in 7nite samples. An estimation ofthe growth rate dynamics by projection on a 7nite distributed lag (as is usually donewhen using VARs in 7nite samples) will miss the small persistent component in growthrates which is critical for understanding asset price reaction to growth rate news. In

13 The results are not very sensitive to the choice of the lag length in the VAR.


sharp contrast, the ARIMA(1,0,1) speci7cation for growth rates can be consistent withboth the small observed 7rst-order autocorrelation coe!cient and a signi7cant impact ofgrowth rate news on present values. This explains why the results based on the AR(1)or the more general VAR structure may fail to detect, in 7nite samples, the importanceof expected growth rates for understanding asset return volatility and cross-correlations.

5. Valuation and equity premium

5.1. Valuation

In Figs. 4 and 5, we compare CAPM-GARCH fundamental price levels impliedby the GMM parameter estimates with the observed, for the case where we employearnings as the proxy for cash 2ows and fundamental betas are imposed. Generally,fundamental prices associated with this model closely track observed equity prices.We 7nd that model-implied fundamental prices more closely match the observed whenearnings are used as the underlying cash 2ows. In particular, when dividends are em-ployed, there is a substantial diLerence between the observed and fundamental pricelevels for Japan in the 1980s and the U.S. in 1994–1998. This diLerence is less promi-nent when earnings are employed; however, as can be seen, the model still cannot fullycapture the price appreciation observed in either market during this period. In Fig. 6,we present the price of risk implied by the model; at the end of our sample, the priceof risk increases substantially due to the higher volatility in the market observed duringthis period. This feature of the GARCH speci7cation hinders the ability of the modelto capture the price appreciation observed during this period. This observation suggeststhe possibility that the symmetric GARCH process may not adequately capture thetrue market volatility process. For this reason, we investigate an alternative (latent)stochastic volatility speci7cation.

5.2. Latent stochastic volatility: parameter estimates

Before discussing the valuation implications of the latent factor model, in Table6, we present estimated parameters for the latent volatility model. 14 The estimatedrisk-free rate dynamics and the cash 2ow betas are very similar to those documentedearlier. As before, they are estimated with great precision. Additionally, the AR(1)parameter for the latent volatility process ()1) is precisely estimated at 0.994 (0.006),suggesting that the volatility process is extremely persistent. This is similar to thepersistence documented earlier for the GARCH process for the market volatility, wherethe comparable number is 0.989. The standard deviation of the market price of risk inthe latent risk model is 0.0027, which as in the case of the CAPM-GARCH model isquite small. Further, as before, the parameters of the cash 2ow process (i.e., � and !)are precisely estimated, and the point estimates are very similar to those estimated in the

14 To avoid clutter we report evidence based on earnings as the proxy for cash 2ows in this alternativeframework—our results with dividend growth rates are not signi7cantly diLerent from those with earnings.


200019981996199419921990198819861984198219801978197619741972

20.2

19.8

19.4

19.0

18.8

18.2

17.8

17.4

ObservedFundamental: ARMA EarningsConstant Cost of Capital

200019981996199419921990198819861984198219801978197619741972

200019981996199419921990198819861984198219801978197619741972



10.4

10.8

11.2

11.6

12.0

12.4

12.8

13.2

13.6

14.0

13.5

13.0

12.5

12.0

11.5

11.0

10.5

10.0

9.5

France: GARCH Fundamental vs. Observed Price

Germany: GARCH Fundamental vs. Observed Price

Japan: GARCH Fundamental vs. Observed Price

Fig. 4. Fundamental asset prices: CAPM-GARCH.

CAPM-GARCH speci7cation. The risk-aversion parameter � is estimated impreciselyat 3:51 (2.54).We present diagnostics for this model using the 7rst diLerence of the fundamental

log price earnings ratio (see Table 9, Panel C). We 7nd that this model explains about84% of the observed volatility; moreover, almost all the cross-correlations are veryclose to those observed in the data, save Japan. In fact, many of the correlations arewithin one standard error of the comparable quantity observed in the data. In all, it


200019981996199419921990198819861984198219801978197619741972

200019981996199419921990198819861984198219801978197619741972

20001998199619941992199019881986198419821980197819761974197210.4

10.8

11.2

11.6

12.0

12.4

12.8

13.2

13.6

14.0

16.2

15.8

15.4

15.0

14.8

14.2

13.8

13.4

16.8

16.4

16.0

15.6

15.2

14.8

14.4

14.0

UK: GARCH Fundamental vs. Observed Price

US: GARCH Fundamental vs. Observed Price

World: GARCH Fundamental vs. Observed Price




Fig. 5. Fundamental asset prices: CAPM-GARCH.

seems that the latent stochastic volatility model does better than the CAPM-GARCHmodel along various dimensions. As can be observed in Figs. 7 and 8, the fundamentalvalues associated with the latent volatility model are capable of matching the observedquite closely; in particular, this model can facilitate the degree of price appreciationobserved in Japan in the 1980s and in the U.S. more recently. In the next section, weprovide some reasons for the somewhat better performance of this speci7cation.


200019981996199419921990198819861984198219801978197619741972

200019981996199419921990198819861984198219801978197619741972

14

10

6

2

−2

18

16

14

12

10

8

6

4

2

Latent Price of Risk

Garch−M: Market Price of Risk



Fig. 6. Price of risk.

6. The size of the global equity premium

The latent world market equity premium for the time period 1990–1998 is about2.5% as presented in Fig. 6, which is well below its historical average of about 5.0%for our sample. The comparable equity premium in the GARCH speci7cation is about4.25%, and rises sharply in 1998, due to increased market volatility. Independently,Fama and French (2000) also argue that the late 1990s implied risk premium is muchlower than ex post return data suggest. From the perspective of the latent volatility


200019981996199419921990198819861984198219801978197619741972

200019981996199419921990198819861984198219801978197619741972

200019981996199419921990198819861984198219801978197619741972

20.2

19.8

19.4

19.0

18.8

18.2

17.8

17.4

10.4

10.8

11.211.6

12.0

12.4

12.8

13.2

13.6

14.0

10.0

13.5

13.0

12.5

12.0

11.5

11.0

10.5

10.0

9.5

France: Fundamental vs. Observed Price

Germany: Fundamental vs. Observed Price

Japan: Fundamental vs. Observed Price

Fundamental: Latent Stochastic Volatility

Observed


Observed


Observed

Fig. 7. Fundamental asset prices: latent stochastic volatility model.

model, the decline in the world market equity premium is important for the rise infundamental values. Indeed, this feature facilitates the price appreciation we observeover the latter period, highlighting the limitations of the symmetric GARCH speci7ca-tion. In Eq. (23), we show that the mapping between world market return volatility andthe volatility of the cash 2ows of the world market portfolio. This mapping providesa third speci7cation for the price of risk—one based on world cash 2ow volatility asa measure of ex ante market return volatility. Given the equilibrium link between the


200019981996199419921990198819861984198219801978197619741972

200019981996199419921990198819861984198219801978197619741972

10.4

10.8

11.2

11.6

12.0

12.4

12.8

13.2

13.6

14.0

16.2

15.8

15.4

15.0

14.6

14.2

13.8

13.4

UK: Fundamental vs. Observed Price

US: Fundamental vs. Observed Price


Observed


Observed

Fig. 8. Fundamental asset prices: latent stochastic volatility model.

conditional world market volatility and world cash 2ow growth volatility, we also esti-mate a GARCH(1,1) model for the world earnings growth rate. In that case, %=0:051(0.034) and $=0:911 (0.045), suggesting persistent world cash 2ow conditional volatil-ity. 15 Fig. 9 shows that over the last 25 years, the volatility of world cash 2ows hasfallen. Hence, in equilibrium, so has the world market equity premium. From this per-spective, the decline in the world market risk premium is entirely due to a fall in thevariance of output growth. The cash 2ow implied world market risk premium in 1975was around 10% per annum, and in 1998, the same measure had fallen to around 2%.

7. Conclusions

We explore the extent to which parametric asset pricing models and the “e!cientmarket hypothesis” (i.e., asset prices should equal the present value of cash 2ows) canjustify the observed asset volatility and cross-correlation across 7ve major internationalmarkets. The idea that asset prices should equal present values of cash 2ows is fun-

15 See Bollerslev and Hodrick (1995) for additional evidence on cash 2ow conditional volatility. GARCHestimates for world dividend growth are comparable.


1974 1978 19981994199019861982

10

9

8

7

6

5

4

3

2

Fundamental Market Price of Risk (based on cash flow volatility)

Fig. 9. Price of risk.

damental to 7nance theory, and hence its quantitative implications are of considerableimportance in understanding the behavior of equity prices.The 7ve markets used in the study are the U.S., U.K., Germany, France, and Japan.

We 7nd that tractable asset pricing models which incorporate time-varying systematicrisk in conjuction with a stochastic trend decomposition for cash 2ows can explainmuch of the observed asset volatility and cross-correlation. Our empirical evidencesuggests that cash 2ows news aLects investors’ long-run growth expectations, and henceyields large eLects on fundamental present values. In economic terms, this process, forwhich we provide empirical evidence, increases the equity price elasticity with respectto cash 2ows, but also magni7es the exposure to systematic risk, hence helping tojustify the asset return cross-correlations as well. Our model also captures the observedfeature that conditional return correlations across markets increase during periods ofelevated uncertainty and economic downturns.Our analysis also has implications for the risk premium on the world market port-

folio. In particular, we show that the risk premium on the world market portfolio hasfallen considerably, approaching 2% per annum in 1998. Further, we link this declineto the fall in the volatility of world real growth rates over the last 27 years. In all,considering the implications for asset volatility, cross-correlation and risk premia, ourevidence suggests that the “e!cient market hypothesis” captures, at least in an eco-nomic sense, many of the important aspects of observed equity prices in global capitalmarkets.


Acknowledgements

We have bene7ted from conversations with Tim Bollerslev, Ron Gallant, GeorgeTauchen, and seminar participants of the 2000 WFA, London Business School,Columbia University, Duke University, Indiana University, Stockholm School of Eco-nomics, the 2001 Triangle Econometrics Conference, and Wharton.

Appendix A.

A.1. Equilibrium solution

The CAPM implies the following for risk premia cross-sectionally:

Et[ri; t+1] = rf; t + �i��2m; t − 1

2 [�2i �

2m; t + �2

i ]: (A.1)

Given the preceding statistical assumptions, the present value implications of the CAPM-GARCH model can be evaluated. First, substitute the Campbell–Shiller approximation

ri; t+1 = gi; t+1 + �i;0 + �i;1zi; t+1 − zi; t (A.2)

into Eq. (A.1) as follows:

Et[�i;0 + �i;1zi; t+1 − zi; t + gi; t+1]− rf; t + 12�

2 i = [��i − 1

2�2i ]�

2m; t : (A.3)

Conjecture the following solution for zi; t , the log price dividend (earnings) ratio:

zi; t = Ai;0 + Ai;1xi; t + Ai;2�2m; t + Ai;3rf; t : (A.4)

By substitution, this implies the following:

�i;0 + �i;1Et[Ai;0 + Ai;1xi; t+1 + Ai;2�2m; t+1 + Ai;3rf; t+1] + Et[gi; t+1]

−Ai;0 − Ai;1xt − Ai;2�2m; t − Ai;3rf; t =− 1

2�2 i + rf; t + [��i − 1

2�2i ]�

2m; t : (A.5)

Exploiting the processes for the state variables, including the ARIMA process for thecash 2ow growth rate, and matching the coe!cients, leads to the following solutionfor the unknown coe!cients:

Ai;1 =1

(1− �i;1�i); Ai;2 =

[��i − 12�

2i ]

(�i;1(%+ $)− 1)and Ai;3 =

1(�i;1&1 − 1)

; (A.6)

Ai;0 =1

(1− �i;1){�i;0 + �i;1[Ai;3&0 + Ai;2&+ Ai;1�i(�i − !i)] + �i + 1

2�2 i}: (A.7)

Using the present value restrictions, it is then possible to derive the fundamental returnbeta of an asset in Eq. (A.1). The de7nition of beta, as used by Campbell and Mei(1993):

�i =Cov(rm; t+1 − Et[rm; t+1]; ri; t+1 − Et[ri; t+1])

Var(rm; t+1 − Et[rm; t+1]): (A.8)


The computations for deriving the fundamental betas follow. De7ne the innovation involatility implied by the GARCH-M process for the world as follows:

�; t+1 = �2m; t+1 − &− (%+ $)�2

m; t : (A.9)

Given the innovations in cash 2ow growth rates, �i; t , market volatility, �; t , and thereal risk-free rate, rf; t , the one step ahead innovation in the return can be expressedas follows:

ri; t+1 − Et[ri; t+1] = Ci�i; t+1 + �i;1Ai;2 �; t+1 + �i;1Ai;3 rf; t+1 (A.10)

and Ci, the dividend elasticity of the equity price, is as follows (1 + �i;1(�− !)Ai;1).The same is true for the market itself, i.e., i = m. By substitution, Eq. (A.8) can berewritten as follows:

Cov(Cm�m; t+1 + �m;1Am;2 �; t+1 + �m;1Am;3 rf; t+1;

Ci�i; t+1 + �i;1Ai;2 �; t+1 + �i;1Ai;3 rf; t+1)

=�i Var(Cm�m; t+1 + �m;1Am;2 �; t+1 + �m;1Am;3 rf; t+1): (A.11)

Consider the projection of the cash 2ow innovation on the innovation in the marketcash 2ow implied by the model:

�i; t+1 = !i�m; t+1 + vi; t+1; (A.12)

where the projection coe!cient, !i, represents the beta of the growth rate of cash2ows with respect to the market cash 2ow innovations. Note that in Eq. (A.6) thecoe!cient Ai;2 contains �i itself. Ignoring the Jensen’s inequality adjustment (whichis quantitatively small) and assuming innovations the state variables are uncorrelated,one can solve for the fundamental beta:

�i =CmCi!i�2(�m) + �m;1�i;1Am;3Ai;3�2( rf)

C2m�

2(�m)− [�m;1�i;1Am;2( ��i; 1(%+$)−1 )− (�m;1Am;2)2]�2( �)

+(�m;1Am;3)2�2( rf):

(A.13)

A.2. GMM estimation

Let the system of economic and statistical restrictions be described in the followingway ut(�) where the parameter vector is de7ned as �. The orthogonality conditionswe exploit in our GMM estimation can be described in three categories. The 7rst setcorresponds to the cost of capital restrictions, i.e., the price of risk and risk-free rate. i; t is the return innovation, and rf; t is the innovation in the risk-free rate:

E[ i; t+1(Rm;t+1 − Rf; t)] = 0;

E[st+1] = 0;

E[ rf; t+1rf; t] = 0; (A.14)


st+1=9f( m; t+1| ; �2m; t)=9 is the score vector of the normal likelihood function, f(·|·),

associated with the GARCH-M process for the world market return. 16 The second setcorresponds to the cash 2ow restrictions. �i; t is the innovation in the cash 2ow growthrate:

E[�i; t+1xi; t] = 0; E[�i; t+1�i; t] = 0;

E[�m; t+1xm; t] = 0; E[�m; t+1�m; t] = 0;

E[�m; t(�i; t − !i�m; t)] = 0: (A.15)

The third and 7nal set of orthogonality conditions overidentify the model by evaluatingthe present-value restrictions

E[ei; t+1xi; t] = 0; E[ei; t+1�2m; t] = 0; E[ei; t+1rf; t] = 0; (A.16)

where the forecast error associated with the 7rst diLerence of the log price dividend(earnings) ratio:

ei; t+1 = Rzi; t+1 − Ai;1(�− 1)xi; t − Ai;2(%+ $− 1)�2m; t − Ai;3(&1 − 1)rf; t : (A.17)

In total, with L countries, we have L + 3 + 1 + (L + 1)2 + L + 3(L) = 7L + 6 = 41moment conditions.The optimal choice for the GMM weighting matrix is WT = (ST )−1, where ST is a

consistent estimate of the variance covariance matrix of the full set of sample moments.Additionally,

Var(�T ) =1T(D′(ST )−1;D)−1; (A.18)

where D is a matrix of partial derivatives of the sample moments with respect to theparameters. We obtain the standard Hansen’s JT test of overidentifying restrictions:

TJT ∼ .2q−p; (A.19)

where JT is the objective function. q−p, the number of degrees of freedom, is the num-ber of moments less the number of estimated parameters associated with the estimation.We have 23 overidentifying restrictions. Additionally, we perform parameter restrictiontests that the fundamental betas implied by the estimated parameters are jointly equal tothe unrestricted estimated betas. The test is the diLerence in the objective functions forthe unrestricted (in which we estimate the betas in the traditional return-based manner)and a restricted (in which we employ the fundamental betas implied by the estimatedcoe!cients) estimations, holding the weighting matrix 7xed across the two to that ofthe unrestricted model. The test statistic is a .2 with 7ve degrees of freedom.

Additionally, to account for serial correlation and heteroskedasticity in the errorstructure, we construct ST above using the VARHAC variance covariance estimationmethodology of denHaan and Levin (1996). The advantage of the VARHAC procedurein our application is in its ability to allow for a varying autoregressive structure acrossthe moments used for estimation. The VARHAC procedure 7rst estimates a VAR

16 We thank Tim Bollerslev for suggesting this approach.


representation for the errors, ut(�̂), and then constructs the spectral density at frequencyzero implied by the VAR structure. First, we choose the lag length for the VAR on thesample moments by minimizing the Bayesian information criterion. Then, we calculatethe spectral density of the pre-whitened VAR residuals. Finally, using this and theVAR coe!cient structure, we construct the VARHAC estimate of the spectral densityof the sample moments.For the latent stochastic volatility model, we employ the set of restrictions implied

by this structure using GMM, given the extracted market volatility from Eq. (25) andthe de7nitions of the solution coe!cients. 17 The projection error associated with the7rst diLerence of the log price earnings ratio is as follows:

ei; t+1 = Rzi; t+1 − Ai;1(�− 1)xi; t − Ai;2()1 − 1)�2m; t − Ai;3(&1 − 1)rf; t : (A.20)

The set of orthogonality conditions we exploit are as follows:

E[ m; t+1] = 0;

E[ i; t+1(Rm;t+1 − Rf; t)] = 0;

E[ �; t+1�2m; t] = 0;

E[ rf; t+1rf; t] = 0;

E[�i; t+1xi; t] = 0; E[�i; t+1�i; t] = 0;

E[�m; t+1xm; t] = 0; E[�m; t+1�m; t] = 0;

E[�m; t+1(�i; t+1 − !i�m; t+1)] = 0;

E[ei; t+1xi; t] = 0; E[ei; t+1�2m; t] = 0; E[ei; t+1rf; t] = 0: (A.21)

There are 40 moments and 17 parameters.

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