Can habit formation under complete market integration explain the cross-section of international equity risk premia?

Review of Financial Economics 22 (2013) 61–67

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Can habit formation under complete market integration explain the cross-section of international equity risk premia? Benjamin R. Auer ⁎ University of Leipzig, Department of Finance, Grimmaische Str. 12, 04109 Leipzig, Germany

a r t i c l e

i n f o

Article history: Received 20 June 2012 Received in revised form 16 November 2012 Accepted 22 November 2012 Available online 9 February 2013 Keywords: Bivariate GARCH CCAPM Market integration Conditional covariance Habit formation

a b s t r a c t In this article, we analyse whether the prominent habit formation model of Campbell and Cochrane (1999) can explain the cross-section of the G7 equity risk premia when formulated under the assumption of complete capital market integration. We test the conditional covariance representation of the model using a combined GARCH and GMM approach in the spirit of Bali (2008) and find that in comparison to the CAPM and the standard power utility CCAPM the habit model has superior explanatory power. It explains more than 90% of the cross-sectional variation in risk premia. Overall, our findings suggest that global consumption-based recession indicators and not returns of reference portfolios are key risk factors driving equity risk premia. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Early empirical studies found that the consumption-based asset pricing model (CCAPM), developed by Breeden (1979), is unable to account for almost any observed aspect of financial market outcomes, especially the cross-sectional variation of equity returns and risk premia. These findings propelled the widespread belief (summarized by Campbell, 2003) that the CCAPM has serious limitations as a viable model of risk. At first sight, the natural response is to dismiss the consumption-based framework altogether and instead use portfoliobased models (e. g. Fama & French, 1993) to correct for risk, to digest anomalies, and to provide cost of capital estimates. However, as emphasised by Lettau and Ludvigson (2001) and Cochrane (2005), within the rational equilibrium paradigm of finance, the CCAPM measuring systematic risk by an asset's covariance with the marginal utility of consumption has a degree of theoretical purity that is unmatched by other asset pricing models. Furthermore, there is really no alternative to the CCAPM, since other models can almost always be expressed as either special cases of, or approximations for the CCAPM. Initial empirical investigations of the CCAPM focused on the representative agent formulation of the model with time-separable power utility (standard CCAPM). However, the last two decades have brought forth an explosion of research (summarized by Ludvigson, 2011) in the field of consumption-based asset pricing where researchers have altered the standard specification to account for new preference orderings based on habits or recursive utility, or new market structures based

⁎ Tel.: +49 341 97 33 672; fax: +49 341 9733 679. E-mail address: [email protected]. 1058-3300/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.rfe.2012.11.001

on heterogeneity, incomplete markets, or limited stock market participation. In a recent paper, Chen and Ludvigson (2009) argue that within the consumption-based framework, habit formation models are the most promising and successful in describing stock market behaviour. The most prominent habit formation model is the one developed by Campbell and Cochrane (1999). It has heavily attracted the attention of recent empirical research (see e.g. Engsted & Møller, 2010; Engsted, Hyde, & Møller, 2010; Hyde & Sherif, 2010; Møller, 2009) because it can explain a large number of stylised facts on the US stock market. The model implies that individuals slowly develop habits for high or low consumption such that risk aversion becomes time-varying and countercyclical. When consumption is well above habit in cyclical upswings, risk aversion is low leading to low expected returns and high asset prices. In contrast, when consumption is close to habit, risk aversion is high leading to high expected returns and low asset prices. In this article we investigate the performance of this prominent habit formation model compared to two benchmarks: the CAPM of Sharpe (1964) and the classic time-separable power utility model. We contribute to the literature in several ways: First, except for the recent studies of Sarkissian (2003), Chue (2005), Li and Zhong (2005, 2009) and Darrat, Li, and Park (2011), empirical research on consumptionbased asset pricing models is mostly confined to a domestic setting, although theory suggests that the models are equally applicable across countries. Modelling the stochastic discount factor (SDF) of the models only with domestic data implies that individual capital markets are segmented in the sense that assets used in empirical tests are assumed to be held primarily by consumers in the country under consideration. We assume completely integrated capital markets implying that the SDF for asset returns in each country should be related to aggregate

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B.R. Auer / Review of Financial Economics 22 (2013) 61–67

world factors rather than country-specific variables. Set up in this kind of world specification, we use the above mentioned models to explain the cross-section of equity risk premia in the G7 countries. Second, to make our results directly comparable to prior empirical evidence concerning portfolio-based factor models, we follow Yogo (2006) and Darrat, Li, and Park (2011) and concentrate on estimating the covariance representation of the linear factor models that can be derived from the nonlinear Euler equation moments of the consumptionbased models. Third, our econometric methodology differs from the classic approaches used by Yogo (2006) and Darrat, Li, and Park (2011). Following Bali (2008) and Darrat, Li, and Benkato (2011), we use a two-stage estimation method. In the first stage, we extract the conditional covariances between risk premia and SDF factors from bivariate GARCH models. In the second stage, we examine the relation between risk premia and the estimated conditional covariances in a classic GMM system. With this approach, we are able to analyse the conditional rather than the unconditional implications of the models. The article is organised as follows: Sections 2 and 3 briefly present the models and the implemented estimation methodology. Section 4 delivers a description of the data. Section 5 reports the empirical results. Finally, Section 6 offers some concluding remarks. 2. Models Consumption-based asset pricing models imply that, although expected risk premia can vary across time and assets, expected discounted risk premia should always be the same for every asset. More formally, they state that h i e Et Mtþ1 Ri;tþ1 ¼ 0;

ð1Þ

e where Ri,t+1 is the real risk premium on any traded asset i held from time t to time t+1 and Mt+1 is the SDF. Under the assumption of complete market integration, the Euler Eq. (1) arises from the first-order condition for optimal consumption choice of a representative world market investor and the SDF takes the form Mt+1 =δwU′(Cw,t+1, Xw,t+1)/U′(Cw,t, Xw,t), where utility U(.) is defined over real per capita world consumption Cw,t and possibly other arguments Xw,t, U′(.) is the partial derivative of U(.) with respect to consumption and 0b δw b 1 is the investor's time discount factor. This means that Mt+1 is equal to the intertemporal marginal rate of substitution in consumption. The standard CCAPM, first tested by Hansen and Singleton (1982), assumes a period utility function defined only over real consumption.   1−γ This function U C w;t ¼ ðC w;t w −1Þ=ð1−γw Þ leads to the SDF

!−γ w C w;tþ1 C w;t ≈υ0 þ υ1 Δcw;tþ1 ;

M tþ1 ¼ δw

ð2Þ

where γw ≥ 0 (γw ≠ 1) is the investor's constant degree of relative risk aversion and Δcw,t + 1 = ln(Cw,t + 1) − ln(Cw,t). 1 The utility function in the habit formation model of Campbell and    1−γw Cochrane (1999) is specified as U C w;t ; X w;t ¼ ½ C w;t −X w;t −1= ð1−γw Þ, where Xw,t is an external habit level of consumption. Rather than specifying a process for the habit, Campbell and Cochrane (1999) introduce a surplus consumption ratio Sw,t = (Cw,t − Xw,t)/Cw,t and assure Cw,t > Xw,t by modelling the log surplus consumption ratio sw,t =ln(Sw,t) using the autoregressive process sw;tþ1 ¼ ð1−ϕw Þs w þ   ϕw sw;t þ λ sw;t tþ1 , where 0b ϕw b 1 is the habit persistence parameter, 1 For details concerning the linear approximations in Eqs. (2) and (3) see Lettau and Ludvigson (2001), Li and Zhong (2005) and Ludvigson (2011). Note that the linear form in Eq. (3) represents a scaled factor model that results from approximating the time-varying parameters of the linearised SDF by linear functions of the surplus consumption ratio.

s w is the steady state level of sw,t, and λ(sw,t) is the sensitivity function that determines how innovations in consumption growth t+1 influence sw,t. The consumption growth process is given by Δcw,t+1 =g+t+1 and t+1 ∼niid(0,σc2), where g and σc2 are the mean and variance of the consumption growth rate, respectively. The sensitivity function is specified    1=2 −1  as λ sw;t ¼ S w 1−2 sw;t −s w −1 for sw,t ≤sw,max and 0 otherwise,  2 1=2 2 where S w ¼ σ c γ w =ð1−ϕw Þ , sw;max ¼ s w þ 0:5ð1−S w Þ and s w ¼ lnðS w Þ. This specification leads to a SDF of the form !−γ w C w;tþ1 Sw;tþ1 C w;t Sw;t ≈ ξ0 þ ξ1 sw;t þ ξ2 Δcw;tþ1 þ ξ3 sw;t Δcw;tþ1 ;

M tþ1 ¼ δw

ð3Þ

where relative risk aversion is no longer determined by γw alone but is given by γw/Sw,t. In addition to these models, we also analyse the empirical performance of the CAPM. Following Darrat, Li, and Park (2011), in our application the SDF of this model is a linear function e

Mtþ1 ¼ ψ0 þ ψ1 Rw;tþ1

ð4Þ

e of the risk premium Rw,t + 1 on a world market portfolio usually approximated on the basis of a broad international stock market index. 2

3. Methodology Smith and Wickens (2002) and Yogo (2006) show that with a SDF being an exact or approximate linear function of F risk factors fj,t + 1, Eq. (1) can be formulated as F h i X   e e Et Ri;tþ1 ¼ bj Covt Ri;tþ1 ; f j;tþ1 :

ð5Þ

j¼1

This covariance representation says that in equilibrium, variation in expected returns over time must reflect variation in the quantity of risk over time, measured by the conditional covariance of risk premia with risk factors. Therefore, the expected risk premium of an asset i is equal e to the factor risk price bj times the quantity of risk Covt(Ri,t+1 ,fj,t+1), summed over all factors j = 1,…,F. The premium is high when the covariances of an asset's risk premia with the factors are high, provided that bj >0. Our models can be easily mapped into this form because for the standard CCAPM we have f1,t+1 = Δcw,t+1, the habit formation model yields f1,t +1 = sw,t, f2,t+1 = Δcw,t+1 and f3,t +1 = sw,tΔcw,t+1, and the e CAPM specifies f1,t+1 = Rw,t+1 . To estimate and test the representation (5), we use a slightly modified version of the approach of Bali (2008) and Darrat, Li, and Benkato (2011): In a first stage, we extract the conditional covariances between risk premia and factors from bivariate GARCH models. 3 The bivariate e GARCH(1,1) model for a risk premium Ri,t +1 and a factor fj,t+1 is speci4 fied as follows : e

i

Ri;tþ1 ¼ μ þ i;tþ1

ð6Þ

2 Cuthbertson and Nitzsche (2004, chpt. 13) and Cochrane (2005, chpt. 9) show how the CAPM can be derived as a special case of the CCAPM. 3 As an alternative, one could follow Yogo (2006) and use the instrumental variables approach of Campbell (1987) and Harvey (1989). However, as this approach is sensitive to the choice of instruments, we concentrate on the GARCH approach. 4 Note that we also estimated bivariate GARCH(1,1) models with autoregressive components in the mean equations. However, these alternative specifications did not alter our main conclusions.

B.R. Auer / Review of Financial Economics 22 (2013) 61–67 j

f j;tþ1 ¼ μ þ j;tþ1

ð7Þ

2

i

i 2

i

2

ð8Þ

2

j

j 2

j

2

ð9Þ

ij

ij

σ i;tþ1 ¼ ϕ0 þ ϕ1 i;t þ ϕ2 σ i;t σ j;tþ1 ¼ ϕ0 þ ϕ1 j;t þ ϕ2 σ j;t ij

σ ij;tþ1 ¼ ϕ0 þ ϕ1 i;t j;t þ ϕ2 σ ij;t :

ð10Þ

It is estimated using the method of maximum likelihood. Denoting the parameters in the GARCH model and the number of observations for each series with ω and T, respectively, we can write the loglikelihood function as

63

different from zero. In our application, the degree of over identification relevant for this test is N⋅ K − F. As a second measure we use the mean T absolute pricing error MAE= T−1ΣiN=1Σt=1 |eit|, where |eit| is the absolute pricing error for asset i at time t. Finally, we also construct the R2 statistic of Campbell and Vuolteenaho (2004) being defined as one minus the ratio of the mean squared pricing error to the variance of average risk premia. Because the weighting matrix in the second GMM stage is model-specific, we cannot use the J-test to directly compare models. We therefore calculate MAE and R2 on the basis of first stage estimates where the weighting for the moments are the same for all models and use these two metrics for model comparison.

4. Data T h i 1X ′ −1 lnð2πÞ þ lnjV t j þ ε t V t εt ; LðωÞ ¼ − 2 t¼1

ð11Þ

where " εt ¼

Rei;t −μ i f j;t −μ

# ð12Þ

j

and " Vt ¼

2

σ i;t

σ ij;t

σ ij;t

σ 2j;t

# :

ð13Þ

After the estimation, the conditional covariance σij,t + 1 of the risk e premium Ri,t + 1 with the factor fj,t + 1 can be easily extracted. Estimation and extraction is performed for all combinations of i and j. As in Li (2009), in the second stage the conditional covariances serve as explanatory variables in a system of equations that we estimate with GMM. Simultaneously considering i = 1,…,N assets this system arises from the restrictions E

hn o i e Ri;tþ1 −b1 σ i1;tþ1 −…−bF σ iF;tþ1 ⊗zt ¼ 0:

ð14Þ

The system is built on the idea that the forecast errors of the covariance representation should be unconditionally and conditionally mean zero. The conditionality is modelled by the (K × 1) vector of instruments zt.5 We consider both the time series and the cross-sectional dimension and restrict the parameters b1,…,bF to be the same of all assets because Bali (2008) notes that by so doing, we not only guarantee the crosssectional consistency of the estimated intertemporal relation, but also gain statistical power by pooling multiple time series together for a joint estimation with common slope coefficients. Furthermore, following Yogo (2006), the abnormal returns are restricted to zero for all assets as implied by the theory. The system is estimated using two-step GMM, and the identity matrix is used as the weighting matrix in the first stage estimation. In the second stage, we use the statistically optimal weighting matrix (the inverse of the covariance matrix of the sample moments) estimated with a Newey–West correction.6 As in Yogo (2006) and Darrat, Li, and Park (2011), we employ three different metrics to measure model performance: The first measure is the J-test proposed by Hansen (1982). This test allows us to investigate whether the moment conditions of a given model are significantly 5 As the GMM estimator is a generalisation of the instrumental variables estimator, it can handle the problem that generated explanatory variables are used in the estimations. 6 Unlike many recent studies, we do not restrict our attention to identity weighting estimation only. Cochrane (1996) and Parker and Julliard (2005) emphasise that when first and second stage GMM estimates do not differ much it is preferable to use the second stage estimates because they are efficient. This is the case in our application. Furthermore, using efficient weighting we can make our statistical conclusions invariant to the choice of test assets (see Ludvigson, 2011).

The models are evaluated on the basis of international quarterly data from the first quarter of 1970 to the second quarter of 2010. Our focus is on the cross-sectional variation of the risk premia in the G7 countries Canada, France, Germany, Italy, Japan, UK and USA.7 We calculate the returns of representative stock market investments in those countries on the basis of Morgan Stanley Capital International (MSCI) stock market indices that are adjusted for dividend reinvestment and represent value-weighted portfolios of large firms traded in the national equity markets. Following Harvey (1991b), Li and Zhong (2005) and Darrat, Li, and Park (2011), the associated risk premia are obtained as the returns on the MSCI country indices in US dollars less the returns on the risk-free asset as approximated by the three-month US Treasury bill rate sourced from Datastream. Since data on household consumption of non-durables and services are unavailable for most countries, we follow Engsted et al. (2010) and use household (private) total consumption in each country instead. To calculate time series of national real per capita consumption growth, we use seasonally adjusted aggregate consumption data, population data and national consumer price indices from the IMF International Financial Statistics.8 To arrive at the per capita consumption, the aggregate consumption for each country is divided by the quarterly population estimates. These estimates are obtained by linearly interpolating annual population figures. From this data, real consumption growth is calculated using the national consumer price indices. Following Sarkissian (2003), we construct the world per capita consumption growth as the gross domestic product (GDP) weighted average of the countries' real per capita consumption growth rates. The weighting for each quarter is based on GDP in US dollars at the beginning of the quarter. Since the seasonally adjusted GDP data from the IMF International Financial Statistics are in local currencies, we convert the data to US dollars using foreign exchange data from Reuters. 9 The world surplus consumption ratio is calculated using the formulas outlined in Section 2. Following Li (2001) and Li and Zhong (2005), we set ϕw = 0.95 and γ w ¼ 2. 10 g and σc are estimated from the data. The starting value for sw,t is set to s w . 7 We also extended our analysis by additionally considering Australia, Austria, Belgium, Denmark, Hong Kong, Netherlands, Norway, Spain, Sweden and Switzerland. As this extended sample does not alter our main conclusions, we do not report the corresponding results here. They are available from the author upon request. 8 As some of the IMF series are not seasonally adjusted we apply the X-12-ARIMA adjustment method. 9 Harvey (1991a) emphasises that national consumption data should not be expressed in US dollars because then the time series properties of the changes in consumption would be dominated by the changes in exchange rates. Therefore, meaningful aggregate consumption growth can only be constructed by the aggregation of unitless consumption growth rates rather than consumption levels. Exchange rate fluctuations affect the world consumption growth only indirectly through the US dollar denominated GDP. 10 We also tested other values for ϕw and γ w that are commonly used in empirical applications as well as values resulting from the iterative procedure suggested by Engsted and Møller (2010). We find that our results prove to be robust because other parameter values mainly affect the level and not the characteristic development of the ratio over time.

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B.R. Auer / Review of Financial Economics 22 (2013) 61–67

Table 1 reports the summary statistics for the main variables in our dataset. Panel A shows means and standard deviations of the equity risk premia and the consumption growth rates in the G7 countries. The highest average risk premium over the sample is 1.66% per quarter in Japan, the lowest positive is 0.18% per quarter in UK. Consistent with the findings of Harvey (1991b), the risk premium in the US is lower than in most other markets and exhibits the lowest standard deviation. We also confirm the negative risk premium in Italy found by Darrat, Li, and Park (2011), indicating that returns on the market index in US dollars in Italy are less on average than the returns on the US Treasury bills. As far as the consumption growth rates are concerned, their means and standard deviations do not differ much across countries. Panel B of Table 1 describes the main explanatory variables of our asset pricing models. These variables are the world consumption growth, the world surplus consumption ratio and the risk premium of the world market portfolio approximated by the risk premium on the MSCI world stock market index. Real world consumption growth over the sample period is on average 0.42% per quarter and in comparison to the national growth rates less volatile. Consistent with the findings of Li and Zhong (2005) we find an average world surplus consumption ratio of 4.67%. The risk premium on the world market portfolio shows a negative sign. Fig. 1 plots the time series of the real world consumption growth and the corresponding world surplus consumption ratio. As we can see, the explanatory variables of our consumption-based models are recession indicators because the major spikes in the series accompany periods of world recession such as 1973–1975 and 1980–1981. Even the beginning of the recent financial crisis is reflected by the series. Before turning to the estimation results a few last data issues have to be mentioned: First, based on prior research (see Bekaert & Harvey, 1995; Darrat, Li, & Park, 2011; Li & Zhong, 2005) our instrumental variables in the GMM tests are the lagged world consumption growth, the lagged US consumption-wealth ratio, and the lagged US term spread. The data for the US consumption-wealth ratio developed and first used as an instrumental variable by Lettau and Ludvigson (2001) come from Martin Lettau's website. The US term spread is the 10-year US government bond yield minus the US 3-month Treasury bill rate. Second, Darrat, Li, and Park (2011) show that world consumption variables led by one quarter are more highly correlated with risk premia on country indices than contemporaneous ones because unlike the smoothed consumption data, stock returns quickly

Fig. 1. Consumption-based variables. Notes: The figure shows the time series of the real world consumption growth rate and the corresponding world surplus consumption ratio from 1970:Q1 to 2010:Q2. Low (high) values of the measures indicate business cycle downswings (upswings).

respond to news released in the market. Thus, we use consumption variables led by one quarter in our estimations. 11

Table 1 Summary statistics. Panel A: Country-specific data Equity risk premia

Canada France Germany Italy Japan UK USA

Consumption growth

Mean

SD

Mean

SD

0.86 0.90 1.33 −0.44 1.66 0.18 0.44

10.31 12.58 11.86 14.11 12.31 11.54 8.47

0.44 0.44 0.43 0.57 0.45 0.53 0.39

0.90 0.89 1.11 0.92 1.36 1.20 0.80

Panel B: World variables

Δcw,t Sw,t e Rw,t

Mean

SD

0.42 4.67 −1.03

0.69 1.78 8.56

Notes: The table reports means and standard deviations of the main variables used in our estimations. The sample spans from 1970:Q1 to 2010:Q2. Panel A describes the quarterly risk premia on the MSCI country indices and of the real per capita consumption growth in the G7 countries. Panel B shows summary statistics of the world consumption growth rate Δcw,t, the world surplus consumption ratio Sw,t and the risk premium on the MSCI world stock market index. All numbers are given in percent.

5. Empirical results Table 2 reports the main results of our estimations, namely the estimated factor risk prices for the CAPM, the standard CCAPM and the habit formation CCAPM as well as our measures for model evaluation. A first look reveals that the J-test fails to reject any model at conventional significance levels. For the world CAPM in which the excess return on the MSCI world index is the only pricing factor, we find a positive factor risk price. This is consistent with the theory because assets having a positive correlation with the market portfolio should have a positive risk premium. However, in accordance with the evidence presented by Li and Zhong (2005), we also find that the factor risk price is insignificant. The mean absolute pricing error of the model is 0.54% per quarter and the associated R 2 statistic is close to zero. As Yogo (2006) notes, this indicates the failure of the CAPM since the model has less explanatory power than a simple model, which predicts that the average 11 This is equal to using the beginning-of-period timing convention for consumption (see Campbell, Lo, & MacKinlay, 1997, chpt. 8).

B.R. Auer / Review of Financial Economics 22 (2013) 61–67 Table 2 Estimation results. Factor risk prices e Rw,t+1

CAPM Standard CCAPM Habit CCAPM (a) Habit CCAPM (b) Habit CCAPM (c)

Measures for model evaluation MAE(%)

R2(%)

J-Test

0.5406

00.08

0.4036

55.28

0.1073

95.59

28.7194 [0.3746] 25.1163 [0.5679] 20.1162 [0.7407]

4.5439 0.2981a (0.3213) (4.0033)

0.1028

95.49

20.2550 [0.7793]

0.3246a (5.0302)

0.1332

94.74

20.5232 [0.8081]

Δcw,t+2

sw,t+1

sw,t+1 Δcw,t+2

0.6305 (1.0560) a

35.5032 (2.6308) 2.7958 0.3069a 4.4065 (0.1837) (3.8730) (0.1887)

Notes: This table shows the estimated factor risk prices for the models described in Section 2. Estimations are performed for the cross-section of the G7 risk premia using the approach outlined in Section 3. Following Cochrane (1996) we use four Newey–West lags in the second step of the GMM estimation. Instruments are lagged values of the world consumption growth rate, the US term spread and the US consumption-wealth ratio. t-Statistics are given in round brackets, p-values in square brackets. a Denotes significance at the 1% level.

returns across markets are just a constant. This failure may arise from the fact that the model neglects important sources of risk (see Karolyi & Stulz, 2003) or that a broad stock market index is not an adequate proxy for the world market portfolio also containing human capital (see Roll, 1977). When we turn to the power utility CCAPM, we see that the factor risk price associated with the world consumption growth factor is positive and highly significant. This is consistent with the findings of Darrat, Li, and Park (2011) and model theory because assets with a positive covariance with consumption growth must have high expected risk premia. Such assets tend to have low returns when investors have low consumption and therefore high marginal utility. They are risky in that they fail to deliver wealth precisely when wealth is most valuable to investors. Investors therefore demand a large risk premium to hold them (see Campbell, 2003). The mean absolute pricing error of the power utility model is lower than the one found for the CAPM. Furthermore, we can observe a substantially higher R 2 of 55.28% so that more than half of the cross-sectional variation in risk premia can be explained by the model. This superior explanatory power can also be seen in Fig. 2. This figure plots the average realized risk premia against the premia predicted by the models based on first-stage GMM estimates. The plots clearly show that the world CCAPM fits the data better than does the world CAPM. The results for the world habit model are most promising. Despite the fact that the world consumption growth factor and the interaction term between world consumption growth and world surplus consumption growth are not significant, we find lower mean absolute pricing errors and a higher R 2 than with the standard CCAPM. The model can explain a striking 95.59% of the cross-sectional variation in risk premia. 12 Fig. 2 visualizes the almost perfect forecasts of the model. As Table 2 shows, even after exclusion of the insignificant variables R 2 stays above 90% and the factor risk price for the surplus consumption ratio remains significant at a 1% level. Consistent with the

12 We also considered the model extension proposed by Wachter (2006) and performed estimations with different values of its interest rate parameter κ that enters in the denominator of S w as −ðκ=γ w Þ. Møller (2009) and Engsted et al. (2010) find that this parameter is usually insignificant in their estimations and summarise studies that give mixed evidence on the cyclicality of the risk-free rate. It is therefore not surprising that our main conclusions are not altered.

65

theory, the risk price is positive because an asset that is positively correlated with the surplus consumption ratio (high in booms, low in recessions) commands a higher risk premium as it fails to deliver wealth when it is most valuable to the investor (when the surplus consumption ratio is low). 13 Even though the full model specification is not supported by the data (not all factor risk prices are significant), our results underscore the importance of covariance with the world surplus consumption factor in determining expected risk premia. Looking at this evidence, the question arises whether our results represent a rejection of the efficient market hypothesis (EMH). Recall that a capital market is said to be efficient if prices always fully reflect available information (see Fama, 1970). 14 Thus, in an efficient market, world consumption growth and other related factors should be completely reflected by the world market portfolio. The market risk premium in the CAPM should fully capture variations in these macroeconomic factors, and thus the CAPM should show some validity. Our results lend support to the CCAPM but reject the CAPM. However, this cannot be interpreted as a rejection of the EMH because, as mentioned above, a stock market index cannot be regarded as an adequate proxy for the world market portfolio. An adequate proxy may reflect variations in consumption factors and provide some CAPM validity. A closely related reason for the differences in the empirical validity of the CAPM and the CCAPM can be found in their theoretical relation. The CAPM can be derived from the CCAPM if the return on the world market portfolio Rw,t+1 is perfectly negatively conditionally correlated with the marginal utility of world consumption U′(Cw,t+1). However, this strong assumption makes the models unlikely in general to be empirically consistent with each other. This can be seen best if utility is quadratic. In this case, marginal utility is linear in consumption, and the assumption then implies perfect correlation between Cw,t+1 and Rw,t +1, but this is unlikely to hold in the data (see Cuthbertson & Nitzsche, 2004, chpt. 13). 6. Summary and conclusions In this article, we analysed whether the CAPM, the standard power utility CCAPM and the habit formation CCAPM of Campbell and Cochrane (1999) can explain the cross-sectional variation of equity risk premia in the G7 countries in an international (complete capital market integration) setting. Instead of testing the unconditional implications of the models, we implemented a combined GARCH and GMM procedure to estimate the conditional covariance representation of the models. So in contrast to the majority of empirical investigations concerning consumption-based asset pricing models, we turned our backs to classic instrumental variable test approaches and a modelling of stochastic discount factors based on national data. Our results show that as in prior research the CAPM completely fails to explain the cross-section of risk premia. The consumption-based models can drastically reduce the pricing errors of the portfolio-based model. The power utility CCAPM can account for more than 50%, and the habit formation model for more than 90% of the variation in risk premia. We are able to provide the important evidence that consumption-based world market factors (the world consumption growth rate and the world surplus consumption ratio) being global recession indicators and not the returns of a global reference portfolio are key risk factors driving international equity risk premia. 13 Darrat, Li, and Park (2011) find a negative risk price and conclude that “such potently negative risk price is economically plausible since the surplus consumption ratio is countercyclical (high during recessions and low during economic booms)”. This is misleading because theory and Fig. 1 say that the ratio is procyclical. Therefore, the evidence presented by Darrat, Li, and Park (2011) is actually evidence against the empirical validity of the habit model. 14 Note that efficient markets and the random walk hypothesis are two distinct ideas (neither one necessary nor sufficient for the other) being equivalent only under very special circumstances like risk neutrality (see Lo & MacKinlay, 1999).

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Fig. 2. Realized risk premia vs. fitted risk premia. Notes: This figure plots are realized average risk premia against fitted risk premia (based on first-stage GMM estimates) on the MSCI indices of the G7 countries. The sample period is 1970:Q1–2010:Q2. Vertical distances to the 45° line can be interpreted as pricing errors. Risk premia above (below) the line are underestimated (overestimated) by the corresponding model.

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