Global Long-Run Risk in Durable Consumption and Asset Pricing

Nicole Branger∗

Michael Semenischev‡

∗

Finance Center M¨ unster, Westf¨alische Wilhelms-Universit¨at M¨ unster, Universit¨atsstr. 14-16, 48143 M¨ unster, Germany. E-mail: [email protected]., Phone: +49 251 83-22854 ‡

Finance Center M¨ unster, Westf¨alische Wilhelms-Universit¨at M¨ unster, Universit¨atsstr. 14-16, 48143 M¨ unster, Germany. E-mail: [email protected]., Phone: +49 251 83-22854

Global Long-Run Risk in Durable Consumption and Asset Pricing First version: January 15, 2014 This version: March 10, 2014

Abstract Consumption based asset pricing models explain prices by the properties of aggregate consumption. There is ample evidence that durable and nondurable consumption have significantly different properties with far-reaching implications for asset pricing. This also holds true in an international context. Looking at post-war data from the US and the UK, we report that the cross-country correlation of consumption growth is larger for durable than for nondurable consumption growth. Moreover the price-dividend ratios strongly predict future durable consumption growth rates while there is little evidence for nondurable consumption in the two countries. We analyze the role of durable consumption for national and international asset pricing in a long-run risk model. The model that captures best the stylized facts in the data features a joint long-run risk component in durable consumption levels in the two countries while nondurable consumption levels follow random walks. On national grounds, the incorporation of durable consumption improves the predictability pattern in consumption growth rates while basic asset pricing moments are matched too. In international asset pricing, the variability of the exchange rate is moderate due to durable consumption. Furthermore the joint long-run risk factor drives the large cross-country correlations of durable consumption growth, price-dividend ratios and return correlations. Keywords: Durable consumption, long-run risk, international asset pricing JEL: F31, G12, G15

1

Introduction

Consumption of nondurable goods differ from durable goods in the response to economic shocks. The former is rather insensitive and stable, whereas the latter is pro-cyclical, i.e. it disproportionately rises during economic booms and disproportionately shrinks during economic busts. Recent general equilibrium consumption based asset pricing models, e.g. Eraker, Shaliastovich, and Wang (2013) and Yang (2011), account for this difference and show its implication for asset prices. While previous work has a national focus, we contribute to the literature by proposing a two-country two-good long-run risk general equilibrium model that analyzes national and international asset pricing moments within a consistent framework. The proposed model is motivated by several empirical facts. Focusing on postwar US and UK data, we examine the differences of durable and nondurable consumption in an international context. We show that the findings in Yang (2011) for the US transfer to the UK. Durable consumption is highly autocorrelated and predictable by the price-dividend ratio in the two countries. To put it differently, the growth rate of durable consumption contains a persistent and predictable component. In the US (UK), the R2 statistic ranges from 13% (32%) at 1-year horizon to 20% (40%) at 5-year horizon. Furthermore, since the price-dividend ratios in the two countries are highly correlated with a coefficient equal to 88%, we conclude that the time-varying persistent growth rates comove strongly in the two countries. Moreover, the durable consumption time series are significantly positively correlated with a coefficient equal to 43% and have similar statistical moments and. On the contrary, nondurable consumption is neither (strongly) predictable nor are the moments of the time series very similar across the two countries. The correlation coefficient is lower and equals to 33%. Overall, the empirical findings suggest that durable consumption is exposed to (global) long-run risk. In the general model setup the representative investor entertains preferences 1

based on Epstein and Zin (1989) and draws a nonseparable utility flow over durable and nondurable consumption. The two good preference specification nests the well understood single perishable good setup and thus the role of durable consumption is straightforward to assess. The economy is based on the long-run risk dynamics introduced by Bansal and Yaron (2004). In order to understand the role of durable consumption for national and international asset pricing, we consider four model calibrations. The parameter sets are selected along following reasoning. First, we select parameter sets that reconcile the results of closely related models in the literature. Since the consistent pricing of national and international assets is of particular interest of this study, we choose the parameterization of Bansal and Yaron (2004) and Colacito and Croce (2011). Both studies consider a single perishable good setup. While the former explains national asset pricing, the latter focuses on international moments. We extend the analysis of both models to the complementary markets. The third and forth parameter sets open the durable consumption channel. The third parameterization features longrun risk exposure in both consumption goods. Finally, motivated by the empirical findings, we specify that (global) long-run risk impacts durable consumption while nondurable consumption follows a random walk. We calibrate the models to the post-war data sample and evaluate the performance in a simulation study. In the first step, basic moments of the economy and several national asset pricing moments are compared with the data. In the next step, international moments, e.g. real exchange rate or return correlations, are analyzed. Notably, besides setting cross-country shock correlations we do not recalibrate the models in the international setting. The model that captures the stylized facts in the data most satisfactory features a (joint) long-run risk component in durable consumption while nondurable consumption levels follow random walks, i.e. the forth parameterization. The simu-

2

lation results suggest that durable consumption risk is important for national and international asset pricing, in particular within a consistent pricing framework. It turns out that the exposure of durable consumption to long-run risk has farreaching consequences. In particular durable consumption growth is highly persistent due to the long-run risk component, which results in a robust predictability pattern by the price-dividend ratio as in the data. However, as common in the long-run risk setup, the predictability of (dividend growth) excess stock returns is still (over) underestimated. Evidently, modeling nondurable consumption levels as a random walk results in lack of predictability, which is confirmed by the post-war data sample. Moreover as a result of the high sensitivity to long-lasting economic growth shocks the model generates a sizeable equity premium. In the international setting, the model generates a smooth real exchange rate and reconciles return and cash-flow correlations that are in line with the data. It turns out that nonseparable utility over the two goods and the joint time-varying expected growth rate in durable consumption growth are driving the results. Assuming that nondurable and durable goods are to a certain degree complements, the nonseparable utility specification decreases the pricing kernel’s sensitivity to volatile short-run shocks in nondurable consumption as a result the volatility of the real exchange rate decreases. The global long-run risk causes that the price-dividend ratios are highly correlated and therefore the cross-country returns too, as observed in the data. The parametrization of Bansal and Yaron (2004) performs relatively poor in international asset pricing. The volatility of the exchange rate is more than twice of the sample average because of the high sensitivity of the pricing kernel to nondurable short-run innovations. Furthermore the cross-country correlations are underestimated. Colacito and Croce (2011) fix the shortcomings by decreasing the risk aversion coefficient of the investor, i.e. decreasing the pricing kernel’s sensitivity to shocks, and increasing the AR(1) coefficient of the long-run risk process. However

3

due to the extremely high persistency in the expected growth rate, the model significantly overestimates the predictability of cash-flow growth by the price-dividend ratio. Finally the simulation study shows that the third model specification, i.e. durable and nondurable consumption growth are exposed to long-run risk, is inferior for two reasons. First, the predictability of future nondurable consumption growth is clearly overestimated. Second, the model generates excess volatility in the real exchange rate. However all model specifications all models (over) underestimate the (risk-free rate) equity premium due to the relatively low consumption volatility in the post-war data. This paper is related to studies that consider multiple consumption goods. Studies on nonseparable utility over durable and nondurable consumption goods include Ogaki and Reinhart (1998), Dunn and Singleton (1986), Okubo (2011) and Pakos (2011) who focus on the estimation the intra- and intertemporal elasticity of substitution. Yogo (2006) estimates expected returns in the cross-section using the recursive pricing kernel with the two goods. Marquez and Nieteo (2011) and Guo and Smith (2012) apply the estimation to non-US data. Lustig and Verdelhan (2007) and DeSantis and Fornari (2008) extend the analysis to currency returns. In a theoretical context, Branger, Dumitrescu, Ivanova, and Schlag (2011) study the implications of relative share movements of two general consumption goods for equilibrium returns and volatility dynamics within a recursive preference framework. Gomes, Kogan, and Yogo (2009) introduces a production economy setting with durable and nondurable goods and address the equilibrium pricing implications in the cross-section of stock returns. Our economic model follows the prominent long-run risk literature. Closely related work is for instance Bansal and Yaron (2004), who study a real economy with a single nondurable good. Colacito and Croce (2011) extend the model to a single

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good-two country real economy setup. Bansal and Shaliastovich (2013) consider a single good-two country nominal framework. Yang (2011) studies a two-good real economy setup, where nondurable consumption levels follows a random walk and durable consumption has a long-run risk component. Eraker, Shaliastovich, and Wang (2013) setup a nominal two good economy with separate processes in expected growth for both goods and the inflation rate. The rest of the paper is organized as follows. In Section 2, we discuss the data and the empirical findings, which motivate our general equilibrium model. Section 3 presents the model setup and describes the equilibrium solutions. Section 4 presents the simulation setup and results. Section 5 concludes.

2 2.1

Empirical Motivation Data

We collect quarterly data on nominal consumption expenditure on nondurable goods and services, on durable goods, the corresponding price levels and population data for the US and the UK. The US seasonally adjusted consumption data is obtained from the national income and product accounts (NIPA) tables of the Bureau of Economic Analysis (BEA) covering the sample from 1966.Q2 to 2011.Q4. The BEA classifies consumption goods as durable that have an average life of at least 3 years. UK seasonally adjusted data is collected from the Office for National Statistics (ONS) covering the same sample period. In contrast to BEA, ONS distinguishes between expenditures on nondurable, semi-durable and durable goods. The ONS defines semi-durable goods that are used continuously over a period of a year, but have a shorter lifespan and are cheaper than durable goods, e.g clothing. To en-

5

sure a coherent consumption good specification, we interpret semi-durable goods as nondurable goods. The long-lasting property of durable goods poses the challenge to measure the service flow of durable goods within given period. We adopt the convention in the literature and assume that the service flow of durable goods is proportional to the stock of durable consumer goods in the economy. The BEA reports yearend data on the net stock of consumer durable goods. Following Yogo (2006) we compute quarterly durable stock data using the implied depreciation rate and quarterly expenditure data. The implicit depreciation rate in the US is 5.2% per quarter. Unfortunately, the ONS does not release corresponding data, so that we need to estimate the consumer durable stock in the economy. We rely on the methodology proposed by BEA (2003) and use Williams (1998)’ yearend durable stock estimate of 1965 as our starting value. Analogously to the US timeseries, we then infer quarterly consumer durable stock data in the UK. The implied depreciation rate is approximately 3.9% per quarter. All nominal series are deflated by the aggregate price level for all consumption goods and divided by the total population to obtain real per-capita data. Monthly financial data, i.e. value-weighted market returns, risk-free rates, dividend yields and exchange rates are obtained from the website of Global Financial Data and cover the sample from 1966.Q2 to 2012.Q4. We approximate the market return by the total return index of the S&P 500 and the FTSE All Share Index. The dividend yields are based on the same indices. We back out dividends by multiplying the dividend yield with the price index in the respective country. The US and UK risk-free rates are approximated by yields on the 3-month T-bills. We use the country-specific consumer price index to deflate the nominal returns.

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2.2

Descriptive Statistics

Table 1 summarizes basic descriptive statistics for durable consumption, nondurable consumption and dividend growth in the US and the UK. Figure 1 displays the times series. The trajectories of durable consumption is displayed in the upper panel. The middle panel shows nondurable consumption and the lower panel illustrates the trajectories of dividend growth in the two countries. The descriptive statistics are strikingly similar for durable consumption in the two countries. On average, it increases by 4.10% (3.51%) in the US (UK) p.a. with a standard deviation of 1.08% (1.08%). The US time series is slightly left-skewed (-0.29), whereas the UK time series exhibits positive skewness (0.50). The US (UK) kurtosis coefficient is 2.9 (3.8). Notable, durable growth rates in both countries are highly autocorrelated. The autocorrelation coefficients stay positive and significant up to eight quarters, as shown by the upper panel of Figure 2. The average annualized growth rate for nondurable consumption is 2.17% in the US and 2.09% in the UK. The US time series has an annualized standard deviation of 0.91%. In contrast, nondurable growth in the UK is almost as twice as volatile with a standard deviation of 1.69%. It is apparent that our post war data sample is overall less volatile than pre-war data samples, e.g. used in Bansal, Kiku, and Yaron (2007). US nondurable growth is slightly left-skewed (-0.66) and nondurable growth UK is almost symmetrically distributed (0.06). Both time series have a similar kurtosis coefficient approximately equaling to 4.4. The middle panel of Figure 2 yields a striking picuter. While the nondurable growth rate in the US is significantly autocorrelated up to four quarters, there is no significant persistency in the UK time series. The insignificant autocorrelation of nondurable consumption growth in the UK is consistent with findings in Guo and Smith (2012). Basic statistics for dividend growth differ strongly in both countries. Real div7

idends grow on average by 0.99% in the US and by 0.46% in the UK p.a. Their annualized standard deviations are 3.88% respectively 8.83%. The US time series is slightly right-skewed (0.08). In contrast, the UK time series is left-skewed with a coefficient equal to -0.68. The kurtosis coefficients are equal to 9.58 (US) and 4.85 (UK). Similar to nondurable consumption growth, dividends in the US are significantly autocorrelated up to four quarters, while dividends in UK are uncorrelated. The lower panel of Figure 2 illustrates this findings Table 2 presents the correlation matrix of our macroeconomic dataset. Growth rates of durable and nondurable consumption are positively correlated in both countries. The US coefficient equals to 46% and the UK coefficient is 42%. US-dividend growth rates are positively correlated with US durable (25%) and nondurable consumption growth rates (15%). On contrary, UK dividend growth rates show no significant contemporaneous correlation with consumption growth rates. The crosscountry correlation coefficient for nondurable consumption is 33%, which is in line with Colacito and Croce (2011). Dividends are mildly positively correlated across both countries with a coefficient of 21%. Most interestingly, durable consumption growth exhibits a significant positive cross-country correlation coefficient, which is equal to 43%. Overall, the descriptive statistics of our macroeconomic dataset reveal interesting findings. While nondurable consumption and dividend growth show significant differences in basic statistics, durable consumption shows striking similarities in the US and UK. The findings are emphasized by the correlation matrix reported in Table 2. Next, we analyze the predictability of consumption and dividend growth.

2.3

Predictability of Consumption and Dividend Growth

In order to study the long-run properties of our dataset, we follow the long-run risk literature and regress the average cumulative future growth rate on the current log 8

price-dividend ratio: h

1X i ∆gt,t+j = ai (h) + b(h)i pdit + t,t+h , h j=1

i = US, UK

(1)

i where ∆gt,t+j subsumes average future cumulative growth rates over h-quarters of

durable consumption, nondurable consumption and dividend growth in country i. Table 3 reports the regression results. High current level in the price-dividend ratio predicts high future real growth of durable consumption in the US. The slope coefficient is significantly positive and the explanatory power increases with the horizon. The R2 value reach 20% at a 5-year horizon. In addition, we extend the analysis to the UK. The price-dividend ratio predicts strongly future durable growth rates as well. The slope coefficient is significantly positive, twice as large as the US coefficients, and the R2 value reaches 44% (41%) at three (five) year horizon. In the US, we find no evidence for predictability in nondurable consumption growth by the price-dividend ratio. The results are in line with Beeler and Campbell (2012)1 . In the UK, nondurable consumption growth exhibits a little degree of predictability. However, the slope coefficients are significant only up to two years and R2 values decrease from 8% at one year to 4% at three years. Similar to nondurable consumption, we find no significant evidence for predictability in dividend growth in the US. Interestingly, the price-dividend ratio predicts negative dividend growth at five years in the UK. The slope coefficient is -0.0074 and R2 -statistics equals to 4.6%. The UK results are in line with findings in Engsted and Pedersen (2010). We confirm the results of earlier studies on the predictability of durable consumption, nondurable consumption and dividend growth. Moreover, we provide 1

Beeler and Campbell (2012) use quarterly data from 1947.Q2 to 2008.Q4.

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novel evidence, that the predictability pattern found in the US is evident, in particular for durable consumption, in the UK too. To sum up, we find hat durable consumption share similar descriptive statistics, are positively contemporaneous correlated and, importantly, contain a highly persistent and predictable component in the US and the UK. Since the correlation coefficient of the price-dividend ratios equals to 88%, the results suggest that the two durable consumption time series are exposed jointly to a global long-run risk component. In contrast, we find no (weak) evidence for a (global) long-run risk factor in nondurable consumption. This empirical facts motivate our general equilibrium model to analyze national and international asset pricing in a coherent framework. The model is presented in the next section.

3 3.1

Model Setup Preferences

We analyze an economy with two countries that are denoted as home and foreign. For notational convenience, we describe the preferences, the economy dynamics and the equilibrium solution only for the home country. Identical assumptions hold for the foreign country. The foreign country is indexed by an asterisk. We assume an infinite, discrete-time, endowment economy where the representative investor gains utility from nondurable consumption C and a service flow proportional to the stock of durable consumer goods. The stock of durable consumer goods evolves according to St = (1 − δ)St−1 + Et ,

(2)

where Et denotes the expenditures of durable goods at time t and δ is the constant depreciation rate. 10

In each period, the investor aggregates C and S to a nonseparable utility flow according to the constant elasticity function (CES) h i 1 1− 1 1− 1 1− 1 u(C, S) = (1 − α)C , + αS

(3)

where α ∈ [0, 1] indicates the relative importance of durable consumption and  ∈ [0, ∞) denotes the intratemporal elasticity of substitution between both goods. A large value implies that the two goods are easily to substitute, while a small value indicates that the goods are complements. Setting  = 1, e.g. done by Yang (2011), the CES aggregator function in (3) collapses to the Cobb-Douglas form. Moreover, setting α = 0, the investor derives utility just from nondurable consumption C. The inter-period utility function is given by the recursive form introduced by Epstein and Zin (1989)  1−1 1  1− 1 ψ 1 ψ 1− ψ , Ut = (1 − β)ut (Ct , St ) + β (Et (Ut+1 )) 1−γ

(4)

where Ut is the life-time utility function, ut is defined in (3), β is the subjective time discount factor, ψ is the elasticity of intertemporal substitution and γ is the relative risk aversion coefficient. Recursive preferences disentangle aversion to fluctuations in time, captured by ψ, and aversion to fluctuations across states, captured by γ. Note, if γ =

1 , ψ

the preferences in (4) collapse to the standard CRRA expected

utility function. In the case γ >

1 , ψ

the investor has preference for early resolution

of uncertainty, which we assume in the following. As Yogo (2006) shows, the pricing kernel valued in units of nondurable consumption, is given by Mt+1 = β θ=

1−γ 1 1− ψ

θ



Ct+1 Ct

− ψθ 

Zt+1 Zt



θ 1− 1 

( ψ1 − 1 )

θ−1 Rg,t+1 .

(5)

, Rg,t+1 is the return on total wealth and Zt denotes the relative share of

nondurable consumption and evolves accordingly to Zt =

Ct , Ct + Qt St 11

where Qt is the user cost of durable goods defined by the marginal rate of substitution between the two goods uc,t α Qt = = us,t 1−α



St Ct

− 1 .

Like in a single perishable good economy, the pricing kernel is determined by the marginal utility of nondurable consumption. Assets which have low (high) returns when marginal utility is high, have high (low) expected returns. However, adding durable consumption, introduces composition risk captured by the relative share process Zt . Assuming preference for early resolution of uncertainty, i.e. θ < 0, the effect of variations in the share process depends on the relative magnitude of  and ψ. Consider 0 < 1ψ < , then for a given nondurable consumption level, the stochastic discount factor decreases in durable consumption. Intuitively, a high  indicates that both goods are substitutes and thus low nondurable consumption is offset by high durable consumption. On the other hand, if 0 <  < 1 < ψ the stochastic discount factor increases in durable consumption. A low  indicates complementarity between the two goods and thus low nondurable consumption cannot be offset by high durable consumption.

3.2

The Economy

Motivated by the empirical findings in Section 2, the economy is characterized by Yt ∈ R5 and evolves accordingly Yt+1 = µ + F Yt + Gt t+1 ,

(6)

where µ ∈ R5 is the vector of unconditional means, F ∈ R5×5 is the persistency matrix which governs the conditional expected growth rate of the economy, G ∈ R5×5 is the volatility matrix and  ∈ R5 is the vector of innovations.

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Yt+1

More precisely, we specify    ∆st+1 µs        ∆ct+1   µc       =  xt+1  µ= 0     2    σt+1   (1 − ν)¯ σ2    ∆dt+1 µd

          



0 0 φs    0 0 φc   F = 0 0 ρ    0 0 0  0 0 φd

0 0 0



  0 0 0    0 0 0 .   ν 0 0   0 0 0

The first element of the economy vector, ∆st+1 , is the growth rate of durable consumption. The second element, ∆ct+1 denotes the growth rate of nondurable consumption. xt is the persistent long-run risk component in the economy. The 2 fourth element, σt+1 , is the time-varying variance in the economy. The dividend

growth rate is denoted by ∆dt+1 . The persistency of the long-run risk component is determined by ρ. As it is common in the literature, we assume 0  ρ < 1. The parameter ν is the persistency of the variance. The parameters φs , φc and φd are the loadings of the cash flow dynamics on the long-run risk component. Following Drechsel and Yaron (2011), we impose an affine structure on the covariance variance matrix G0t Gt Gt G0t = h + Hσ σt2 ,

(7)

where h ∈ R5×5 and Hσ ∈ R5×5 are symmetric and positive definite. Moreover we set
h = diag [0, 0, 0, ϕ2σ , 0] ,

Hσ = diag [ϕ2s , 1, ϕ2x , 0, ϕ2d ]




In fact, it is the specification introduced by Bansal and Yaron (2004), extend by the durable stock in the economy. The vector of innovations is t+1 = [s,t+1 , c,t+1 , x,t+1 , σ,t+1 , d,t+1 ]0 ∼ N(0, 1).

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3.3

The Equilibrium

3.3.1

Consumption Claim

We solve for the equilibrium of the economy following Drechsel and Yaron (2011). In equilibrium, each asset j has to satisfy the Euler condition Et [exp(mt+1 + rj,t+1 )] = 1.

(8)

The logarithm of the stochastic discount factor in (5) is   θ 1 1 θ − ∆zt+1 + (θ − 1)rg,t+1 . mt+1 = θ log β − ∆ct+1 + ψ 1 − 1/ ψ 

(9)

Following Eraker, Shaliastovich, and Wang (2013), we log-linearize the relative share process: ∆zt+1

  1 ≈χ 1− (ic − is )0 Yt+1 , 

where ∆st+1 = i0s Yt+1 with is = [1, 0, . . . , 0]0 and ∆ct+1 = ic Yt+1 with ic = [0, 1, . . . , 0]0 . The parameter χ ∈ (0, 1) is equal to the average expenditure on durable consumption goods (expressed in nondurable goods) in the economy, χ =

¯ QS ¯ C ¯. QS+

The log-

linearization enables us to obtain analytical solutions for the asset pricing variables of interest. However, it shuts of the non-linear risk channel due to changes in the composition, which is analyzed e.g. in Cochrane, Longstaff, and Santa-Clara (2008) and Piazzesi, Schneider, and Tuzel (2007).2 If  > 1, a positive shock to nondurable consumption growth increases the relative share of nondurable consumption of the investor. On the other hand, if  < 1, the investor will decrease her share of nondurable consumption, since both 2

If the relative share expenditure on durable goods is stable in time, the log-linearization is

justifiable. Eraker, Shaliastovich, and Wang (2013) show that the share is relatively constant in the US data. Hamilton and Morris (2002) come to the same conclusion for the UK. The relative share expenditure on durable goods is approximately 15%.

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goods are complements. Setting  = 1 the relative share is insensitive to consumption shocks. Notably, setting χ = 0 implies α = 0. Then, the intra-period utility function in (3) collapses to u(Ct , St ) = Ct . In turn, the lift-time utility function in (4) collapses to the well-known single good recursive form. For the return on wealth, we rely on the approximation in Campbell and Shiller (1988) rg,t+1 ≈ κ0 + κ1 υt+1 − υt + ((1 − χ)ic + χis )0 Yt+1 , where υt denotes the log wealth to consumption ratio at time t. As usual, we conjecture that υ is affine in the state variables υt = A0 + A01 Yt , where A01 = [0, 0, Ax , Aσ , 0]. It is well-known that the assumption of preferences for early resolution of uncertainty (γ > /1 ψ) implies a dominance of the substitution effect over the wealth effect. Thus, the wealth to consumption ratio increases in the conditional expected growth rate, i.e. Ax > 0. Moreover, an increase in the economic uncertainty results in an decrease of the wealth to consumption ratio, i.e. Aσ < 0. Relying on the approximation of the return on wealth, we solve for the innovation of the pricing kernel at time t + 1:3 mt+1 − Et (mt+1 ) = −Λ0 Gt t+1 where Λ ∈ R5 contains the market prices of risk. Decomposing the vector of market 3

A detailed derivation can be found in Appendix A.2.

15

prices of risk yields 

γχ − θχ1/    γ(1 − χ) + θχ1/   Λ =  (1 − θ)κ1 Ax    (1 − θ)κ1 Aσ  0

      .    

(10)

The first and second element of Λ contain the market prices of short-run consumption risk. The prices are determined by two factors. The first factor is the risk aversion coefficient times the average expenditure share for the respective good of the investor. This factor is in line with a single good economy.4 In addition, the market prices of short-run consumption risk is impacted by the intratemporal elasticity of substitution of the two goods. The higher the complementary of the two goods, the stronger its impact. In fact, under certain parameter sets the market price of risk for short-run nondurable consumption is negative. Intuitively, the investor dislikes a ceteris paribus increase in nondurable consumption due to the strong complementary of both goods. The third and forth elements are the market prices of expected growth risk and volatility risk. Since θ < 0, it is straightforward to verify that the representative investor appreciates an increase in the conditional expected growth rate (ceteris paribus, the pricing kernel decreases) and dislikes an increase in the uncertainty/variance (ceteris paribus, the pricing kernel increases).

3.3.2

Risk-free Rate

Once the pricing kernel is known, we can compute the risk-free rate. It turns out that the risk-free rate is affine in the state variables rf,t = r0 + r10 Yt , 4

Setting the average expenditure for durable goods to zero, i.e. χ = 0, verifies the conclusion.

16

where r0 ∈ R, and r1 ∈ R5 are given in Appendix A.2. An increase in the consumption volatility decreases the risk-free rate due to a higher precautionary savings motive of the investor. The reaction is in line with a single good economy. However, the risk-free rate’s sensitivity to positive shocks in the expected growth rate is ambiguous. Modeling nondurable consumption as a random walk, Yang (2011) shows that if ψ > , the effect of intratemporal smoothing dominates that of intertemporal smoothing, resulting in a counter-cyclical behavior of the risk-free rate. In contrast to a single good economy, a positive shock in the expected growth rate decreases the risk-free rate. Modeling long-run risk in both consumption processes, the sensitivity will depend on their relative exposure. Assuming identical exposure, the risk-free rate increases given a positive shock in the expected growth rate due to the intertemporal smoothing motive.

3.3.3

Dividend Claim

To price a dividend claim, we proceed analogously as for the consumption claim. We log-linearize the return rm,t+1 ≈ κm,0 + κm,1 ωt+1 − ωt + i0d Yt+1 , where id = [0, 0, . . . , 1]0 and ∆dt+1 = i0d Yt+1 . Next, we assume that the log price to dividend ratio ω is affine in the state variables ωt = B0 + B10 Yt , where B0 ∈ R and B1 = [0, 0, 0, Bx , Bσ , 0]. Since sgn(B1 ) = sgn(A1 ), the pricedividend ratio rises if the expected growth rate increases and decreases if economic uncertainty rises. Once the equilibrium solution is known, we can compute the con-

17

ditional equity premium5 Et [rm,t+1 − rf,t ] + 0.5V art (rm,t+1 ) = −Cov t (mt+1 , rm,t+1 ) = Λ0 Gt G0t Γ. 3.3.4

Real Exchange Rate

In order to analyze international asset pricing moments within a two country economy we allow the shocks to the economies to be cross-country correlated Cov(t+1 , ∗t+1 ) = Σ, where Σ ∈ R5×5 denotes the (co)variance matrix of the shocks. To facilitate the analysis, we specify Σ = diag[ρs , ρc , ρx , ρσ , ρd ]. As Backus, Foresi, and Telmer (2001) show, under complete and frictionless markets, the log expected real spot exchange growth rate (home currency per unit of foreign currency) is the difference of the expected log pricing kernels. Et (∆et+1 ) = Et (m∗t+1 − mt+1 )

(11)

= (1 − θ)A0 (Yt∗ − Yt ) − Λ0 F (Yt∗ − Yt ). The expected change in exchange rate is determined by differences in the state variables of the two economies. For instance, given an ceteris paribus increase in the economic uncertainty of the foreign country, the foreign currency is expected to appreciates. Intuitively, from the perspective of the US-investor, since the foreign economy is riskier, the investor demands a higher premium for holding foreign assets. If the state variables in both countries strongly comove, the expected spot exchange growth rate is negligible. The innovation in the exchange rate is given by ∆et+1 − Et [∆et+1 ] = −Λ0 (G∗t ∗t+1 − Gt t+1 ). 5

For a detailed derivation refer to Appendix A.3.

18

(12)

Thus, movements in the real exchange rate are determined by differences in innovations weighted by the corresponding market price of risk. Again, if the state variables in both countries are sufficiently high correlated, real exchange rate is driven by short-run consumption innovations.

4 4.1

Results Simulation Setup

We rely on the simulation study approach outlined in Beeler and Campbell (2012) to compute the moments and statistics of interest. We run 10,000 simulations and choose a sample period equal in length to our empirical sample, i.e. 45 years. In the tables we report the means and standard deviation of the moments of interest from 10,000 simulation runs. Moreover, concerning the predictability results, we report the proportion of those samples below that of the empirical moment. As Beeler and Campbell (2012) note, this fraction can be interpreted as a p-value for a one-sided test of the considered model applying to the particular moment. For notational convenience, we abbreviate the the four model specification. More specifically, the model of Bansal and Yaron (2004) is denoted by “BY”, Colacito and Croce (2011) is abbreviated by “CC”. The third model specification is denoted by “DurNon” and the final model where solely durable consumption is exposed to long-run risk is represented by “Dur”.

4.2

Calibration

We calibrate the models to the quarterly data set covering the period from 1966.Q2 to 2011.Q4 We set the parameters so that average statistics of consumption and

19

dividend data of the US and the UK are matched. Table 4 summarizes the preference and endowment parameters. Note that all parameters are displayed in monthly terms. Nondurable consumption has a monthly growth rate of 0.18% while durable consumption grows on average with 0.32% per month. The mean dividend growth rate is approximately 0.1%. We set the unconditional volatility to 0.5% in order to match the annualized volatility of nondurable consumption growth. We appropriately scale the volatilities of the durable and dividend growth dynamics to the empirical values . Further, we impose the same volatility process for all models under consideration. We rely on the specification in “BY”. The persistency of volatility, ν, is set to 0.987 and the standard deviation of volatility innovations is set to 2.3×10−6 . The persistency implies a half-life of volatility innovations of roughly 52 months. “BY” set the monthly autocorrelation coefficient of the stochastic growth rate to 0.979, implying a half-live of 32 months. On the other hand, “CC” choose a higher persistency factor. It is set to 0.987, which implies a half-live of 52 months. In the third and forth model specification, we choose the moderate persistency factor of 0.979. In the third model, “DurNon” we set the loadings to the long-run risk factor to unity. In the last model specification, “Dur”, we allow only for long-run risk exposure in durable consumption growth. Motivated by the strong evidence for long-run risk in durable consumption growth, we set φc = 0 and φs = 1.25. We calibrate the cross-country correlations in order to match the empirical correlation of the consumption and dividend growth rates. We justify our choice of parameters in Section 4.4 in more details. We use common preference parameters for the simulation study. First, we set the subjective discount factor uniformly to 0.999. In the “BY” parameterization and “CC” parameterization, our choice for the relative risk aversion and IES is in line 20

with the original work. Specifically, in “BY”, the risk aversion is at 10 and the IES at 1.5. In “CC”, we set the risk aversion at 4.25 and the IES at 2. For both durable consumption models, we set the risk aversion at 10 and the IES at 1.75. In both two-good models the intratemporal elasticity of substitution coefficient, , is 0.80. The value is in line with empirical estimates in Eraker, Shaliastovich, and Wang (2013) and Guo and Smith (2012). Moreover, we set the average expenditure for durable goods at 15%, which approximately corresponds to the empirical value in the US and the UK.

4.3

National Moments

First, we evaluate the models along the basic properties of the consumption and dividend dynamics and along national asset pricing moments. Table 5 summarizes the results. Both durable consumption models, “DurNon” respectively “Dur”, match the mean growth rate and standard deviation of durable consumption. Discrepancies in both models arises due to the different long run risk exposure. Higher long-run risk exposure in the “Dur” model results in higher autocorrelation coefficients, which are closer with empirically values. Turning our attention to nondurable consumption, it is apparent that all models match the mean growth rate and standard deviation well. Again, discrepancies in the long run risk exposure imply differences in the autocorrelation of nondurable consumption. While the first three models generate a positive first order autocorrelation of approximately 0.34, the last model generates a lower coefficient of 0.20. The empirical results are ambiguous. The US time series exhibits positive autocorrelation. In contrast, the UK time series shows no autocorrelation. All four models match the average empirical moments of the dividend growth rates in the US and the UK well. 21

Next, we compare the moments of the equity premium, risk-free rate and the price-dividend ratio. All models underestimate the equity premium due to the low post war consumption volatility. The “BY” model generates an equity premium of 2.1% with a standard deviation of 12%. The “CC” model produces an equity premium of 2.4% (despite of a risk aversion at 4.25) with a standard deviation of 25% due to the higher persistency in the stochastic growth rate. Since the “DurNon” model incorporates long run risk in the two goods, it generates the highest excess return of 3.73% with a annualized volatility of 15%, which is approximately in line with the US equity premium. The “Dur” model underestimates the equity premium, i.e. 2.7%, albeit has a reasonable standard deviation of 17%. All models successfully generate an insignificant autocorrelation in excess stock returns. All models have difficulties to match the level and standard deviation of the risk-free rate. The risk-free rate is overestimated by approximately 1% and the volatility is clearly to low. Notably, the “Dur” model generates the lowest volatility of 0.1%. The low volatility is due to the opposed effect of the intra- and intertemporal smoothing motive on the risk-free rate. When the expected growth rate is high in the economy, the intratemporal smoothing motive drives the investor to save for the future, while the intertemporal smoothing motive will cause the investor to borrow for the future. As the result, the risk-free rate’s sensitivity to innovations in the growth rate decreases. The offsetting effect is most pronounced if nondurable consumption levels follow a random walk. All models overstate the autocorrelation of the risk-free rate. All models match the level of the price-dividend ratio well. However, except for the “CC” model, the models understate the standard deviation.6 The “BY” model underestimates the volatility most severely. While the “DurNon” and “Dur” 6

Among others, Beeler and Campbell (2012) point the difficulty of the single good long run risk

model setup to produce a reasonable volatility of the price- dividend ratio. Thus, incorporating durable consumption does not resolve the drawback.

22

model generate a standard deviation of approximately 20%. The high volatility of the price-dividend ratio (40%) in the “CC” model is due to the high persistency of the long-run risk factor. The price-dividend ratio’s sensitivity to shocks in the timevarying growth component rises in the persistency factor. The strongly pronounced autocorrelation in the price-dividend ratio is matched well by all models. Next, we evaluate the model’s predictive power for cash-flow growth rates and excess stock returns.7 We regress the cash flow growth rates and excess returns on lagged values of the price to dividend ratio as suggested by Equation (1). We choose predictive horizons over 1, 3 and 5 years. Table 6 summarizes the results. The Table reports the regression coefficients, the corresponding t-statistics and R2 -statistics. As noted earlier, the Table provides the proportion of those simulation samples below that of the empirical result, i.e. the average of the US and UK value. To begin with, we discuss the predicability of durable consumption growth. In Section 2 we report strong predictive power of the price-dividend ratio for durable consumption growth. Both durable consumption models, “NonDur” and “Dur”, are capable to produce the predictability pattern observed in the data. Albeit, both models overstate the predictability at 1-year horizon. The model endogenous R2 statistics are between 43% (“DurNon”) and 61% (“Dur”), while the R2 statistic is between 13% (US) and 32% (UK) in the data. The model performance increases with the horizon. The regression coefficients are somewhat higher, but the t-statistic and R2 s are in the range of their empirical values. The “Dur” model predictive power exceeds the predict power of the “NonDur” model as a result of the higher exposure to the stochastic growth rate. While durable consumption growth is strongly predictability, we find little evidence (no evidence at longer horizons) for nondurable consumption growth in 7

Among those who investigate the predictive power of the price to dividend ratio for stock excess

returns are Fama and French (1988), Hodrick (1992), Campbell and Yogo (2006) and Kellard, Nankervis, and Papadimitriou (2010).

23

our post war data sample. However, both single good models, i.e. the “BY” model and “CC” model, generate a strong predictability pattern. At the 1-year horizon, the R2 -statistics are between 23% (“BY”) and 39% (“CC”), whereas the R2 s in the data are up to a maximum of 7%. The discrepancy between both models and the data are even stronger pronounced at longer horizon. We find no evidence for predictability neither in the US data nor in the UK data. However, in the regressions of the “BY” (“CC”) model the R2 is 21% (36%). While the very high persistency of the growth rate in the “CC” model helps to match national asset pricing moments despite a relative low risk aversion, it causes a clear overestimation of the predictability of consumption growth. Based on the simulation results, the model is rejected at 1% significance level. Since the “DurNon” model incorporates long run risk exposure in nondurable consumption growth, it is not surprising that it shares same drawbacks. On the contrary, we cannot rejected the “Dur” model at conventional significance levels. Thus, at least for our post war quarterly data set, we conclude that nondurable consumption levels follow a random walk as modeled in the “Dur” model. Next we turn our attention to the predictability of dividend growth and excess stock returns. Table 7 displays the results. Our empirical regression results are in line with Engsted and Pedersen (2010), who analyze the predictability of dividend growth and excess returns for the US, UK and other countries. We find no evidence of predictable dividend growth in the data. In contrast, according to the models under considerations, forecasting regressions of the dividend growth have significant positive coefficients and the R2 s are between 17% (“BY”) and 42% (“CC”) at a 1-year horizon and between 16% (“BY”) and 37% (“CC”) at a 5year horizon. It is worth noting that although all models overstate the predictability of dividend growth, the “CC” model generates a predictive power of the pricedividend ratio that is almost twice that of the other models. Again, this result is due to the high persistency of the predictive component in the cash flow dynamics.

24

Overall all models are at odds with the data. This far-reaching discrepancy is known and remains a challenge for further research effort in asset pricing. Next we investigate the predictability of stock excess returns. Before turning the attention to the simulation results, it is noteworthy that, in the data the price to dividend ratio has clearly more predictive power in the UK than in the US. R2 values range from 20% at a 1-year horizon to 51% at a 5-year horizon. In comparison, the R2 statistic in the US does not exceed 15% at a 5-year horizon.8 All models reconcile the basic predictability pattern of stock returns. A high price-dividend ratio predicts lower expected returns in the future. Moreover, as in the data the regression’s R2 values rise in the horizon. However, all models clearly understate the predictive significance of the price-dividend ratio. In the “BY” model, which has the lowest predictive power, the R2 values range from 1.7% at a 1-year horizon to 8.3% at a 5-year horizon. The “CC” model has the highest predictability among the compared models. As a result of the high persistency, the price -dividend ratio reacts stronger to innovations in the stochastic growth rate. Albeit, the model sill understate the predictability of stock returns. The R2 values range from 2.1% at a 1-year horizon to 11.2% at a 5-year horizon. The “DurNon” model produces slightly lower regressions results. More surprisingly, although only durable consumption growth is exposed to long run risk in the “Dur” model, it produces comparable results as the “NonDur” model. At a 1-year horizon the R2 is 1.9%, whereas at a 5-year horizon the R2 increases to 9.3%.9 To sum up, the considered models share commonalities but also differ in major aspects. The basic properties of consumption and dividend dynamics are matched 8

Since we impose a symmetric calibration, we cannot shed light on the strong discrepancy in

the predictability of stock returns. It is left for future work. 9 We acknowledge that in the “Dur” model, we set the long run risk exposure to φs = 1.25. However the result is still somehow surprising, since the average expenditure share for durable goods in the economy is set to 15%.

25

well. However, as a result of the low volatility in the economy, all models (over-) underestimate the (risk-free rate) equity premium. The level of the price-dividend ratio is matched. On the contrary, the variability of the price to dividend ratio, except for the “CC” model, is underestimated. The main differences lie in the predictability of consumption and dividend growth rates. As shown earlier, we find no evidence for predictability in nondurable consumption growth, whereas the price to dividend ratio significantly predicts durable consumption growth. At odds with the data, except for the “Dur” model, the models generate strong predictability in nondurable consumption growth. In this respect, the “CC” model is most clearly rejected. Overall, as a result of the low volatility in the consumption dynamics in the post-war dynamics, the models under considerations underestimate the equity premium, and overestimate the risk-free rate. However, introducing durable consumption enhances the pricing performances, since the “BY” model performs worst. Differences in the performance of the “CC” model is caused by the high persistency of the expected growth rate. The model endogenous predictive power of the pricedividend ratio of each specification is at odds with empirical patterns. However, the “Dur” model generates the most convincing results, since it matches the lack of predictability in nondurable consumption growth.

4.4

International Moments

As Colacito and Croce (2011) show, the correlation between the domestic and foreign pricing kernel has to be sufficiently high in order to generate a smooth exchange rate. In other words, the correlation between the state variables has to be sufficiently high. To achieve this, we assume that innovations to the expected growth rate are identical across the two countries, i.e. we set ρx = 1. Setting the correlation coefficient to unity, reflects that the long-term prospects of the two countries are similar. However, similar to Bansal and Shaliastovich (2013), we assume less than perfect correlated 26

volatility innovations. More precisely, we set ρσ = 0.7. Nondurable consumption shocks are correlated by ρc = 0.3. The correlation of dividend innovations, ρd , is 0.15. Furthermore, we assume uncorrelated durable consumption shocks, i.e. ρs = 0. This parameterization choice allows us to highlight the role of long run risk in durable consumption for international asset pricing. Table 8 summarizes the empirical and theoretical results of international key moments. To begin with, we turn our attention to the foreign exchange market. All considered models have an average exchange growth equal to zero due to the symmetric calibration, which is very close to the sample average of 0.59%. Moreover all models replicate the lack of autocorrelation of the exchange rate. However, the models differ significantly with respect to the volatility of the exchange rate. As Equation (12) shows, the variability in the exchange rate is determined by innovations to the consumption dynamics and state variables weighted by the corresponding market price of risk. since the state variables are highly correlated the variability of the spot exchange growth rate is strongly determined by short-run innovations in the consumption growth rates times the corresponding market prices of risk.In the single good economy, the impact of short run shocks to nondurable consumption on the standard deviation is determined by the risk aversion of the investor. In the “BY” model the risk aversion is at 10. As a result, the BY model has a standard deviation of 24%, which is more than twice that of the sample average of 10%. The “CC” model generates an annualized standard deviation of 11%, which is the result of the relative low risk aversion. The risk aversion at 4.25 decreases the pricing kernel’s sensitivity to short run shocks in the consumption growth rate and therefore the variability in the exchange rate. The “DurNon” model generates a standard deviation of 18%, although the risk aversion is at 10. The reduction in the volatility is due to durable consumption risk 27

channel. Once the investor considers durable goods, she “splits” her overall risk aversion according to the preference towards the two goods. Since durable consumption is less volatile than nondurable consumption the overall dispersion in the pricing kernel decreases as a result. However, the standard deviation is still at odds with the data. Interestingly, the “Dur” model has a standard deviation of 14%, which is not too far away from the sample average. As previously mentioned, incorporating durable goods implies that the investor not only cares about the intertemporal distribution of risk, but also about the intratemporal allocation of the two goods. Assuming complementarity between the two goods, i.e. 0 <  < 1, both motives interact in opposition to each another. As a result, the marginal utilities’ sensitivity to changes in the state variables decreases and simultaneously the standard deviation of the exchange rate too. The offsetting effect is most pronounced if nondurable consumption levels follows a random walk. Hence, the “Dur” model has a lower standard deviation than the “DurNon” model. Next, we turn the attention to cross-country correlations of cash flow growth rates and asset pricing moments. Since we impose an exogenous correlation structure, the correlation coefficients of consumption and dividend growth rates are matched quite well. Noteworthy, the correlation rises in the persistency of the stochastic growth rate underlying both economies. Due to the relative low volatility and the exposure to long-run risk of durable consumption growth, the “Nondur” and “Dur” models generate a sizeable positive correlation although the correlation of short run shocks is set to zero. Backus and Smith (1993) report the lack of correlation between consumption differentials and exchange rate movements. We confirm the findings for the considered data sample. Neither the durable consumption differentials (12%) nor the nondurable consumption differentials (6%) is significantly correlated with exchange

28

rate movements. Neither of the considered model resolves the Backus and Smith (1993) puzzle. The single good models produce an excess correlation of approximately 80%. The two good models lessen the correlation to 50% - 60%.10 In the data the price-dividend ratios have a correlation coefficient of 88%. The correlation coefficient in the “BY” model is 83%, which is the lowest of all models. The “CC” model has a correlation of 95%, while the durable consumption models are closest to the data. Recalling the return approximation in Campbell and Shiller (1988), it is evident that the cross-country correlation of returns is either due to correlated dividend growth rates or to price to dividend ratios. The data suggest that the return correlation is driven by the latter. Since all models generate highly correlated price-dividend ratios, the returns are highly correlated too. However, it is surprising how a small (over) underestimation of the correlation of the price-dividend ratios is intensified in return correlation. The “BY” models generates a return correlation of 50%, while the CC model has a coefficient equal to 87%. Consequently, the durable consumption models are the closest to the data, with a return correlation of approximately 70%. All models generate excess correlation of the risk-free rates. Considering Equation (11), it is evident that the risk-free rate is a linear function of the state variables in the economy. Since we assume highly correlated state variables (in order to a ensure a smooth exchange rate), the excess correlation is consequent. The challenge to match the correlation of the risk-free rates and the standard deviation simultaneously is left for future work. 10

Colacito and Croce (2013) solve the Backus and Smith (1993) anomaly with recursive prefer-

ences and highly correlated long run risk by introducing an endogenous risk-sharing scheme. Due to the importance of durable goods in international trades, e.g. Engel and Wang (2011), it would be interesting to study the role of durable consumption within Colacito and Croce (2013) risk-sharing scheme.

29

It is natural to study the international investment opportunity set by analyzing foreign returns expressed in domestic units. More specifically, for an (UK) US UK US-investor the foreign return is computed by (rt+1 + ∆et+1 ) rt+1 − ∆et+1 . Overall,

since all models understate the equity premium, the foreign Sharpe ratio is underestimated as well. However, incorporating durable consumption brings the results closer to the average sample. While the single good models have a foreign Sharpe ratio of 8%, the two good models produce a ratio approximately of 14%. Discrepancies in the pricing performance of the models are particulary highlighted in the correlation of foreign returns expressed in domestic units and domestic returns. In the data, the correlation coefficients are between 63% (UK) and 69% (US). In the “BY” model, the return correlation coefficient is equal to 27% for two reasons. First the “BY” model already has a relatively low correlation of returns expressed in local units (50%). Second, the high variability of the exchange rate lowers the correlation coefficient further. On the contrary, in the “CC” model the correlation is equal to 85% due to the excess return correlation and the low standard deviation of the exchange rate. Analogously to the “BY” model, the “DurNon” model creates an insufficient correlation between the foreign (expressed in domestic) and domestic returns. Although the return (expressed in local units) correlation is in line with the data, the volatility of the exchange rate is too high. As a result, the correlation is equal 53%, which is lower than the sample average. Solely the “Dur” model manages to match the sample average closely. The correlation coefficient equals to 63%. Finally, the lack of correlation between returns and movements in the exchange rates is captured by all models under consideration. Summing up, the models differ significantly with respect to international key moments. Based on the supportive simulation results, we conclude that a joint expected growth rate in durable consumption levels and non-separable utility over the two goods are necessary to match salient moments in international asset pricing.

30

The “Dur” model features the necessary model ingredients.

5

Conclusion

In the US and UK data, durable consumption growth is positively contemporaneous correlated, highly persistent and predictable by the price-dividend ratio. Whereas nondurable consumption growth has lower the contemporaneous correlation coefficient and there is little evidence for persistency and predictability. In other words, durable consumption is exposed to long-run risk. Moreover since the price-dividend ratios of the US and UK are highly correlated in the data, we conclude that the time-varying expected growth rate is similar in the two countries. Motivated by the empirical facts, the role of durable consumption for national and international asset pricing is studied in a (two-good) consumption-based general equilibrium model. In the model, the representative investor entertains Epstein and Zin (1989)’ preferences and the underlying economy is exposed to long-run risks and stochastic economic uncertainty. In order to emphasize the role of durable consumption we study a single-nondurable good and a two-good economy each with two parameter sets. In the single good economy, we rely on the parametrization in Bansal and Yaron (2004) and Colacito and Croce (2011). The first parameter set in the two good economy models long-run risk in both consumption goods, while the second set ensures that only durable consumption is exposed to long-run risk. The pricing performance is assessed along national key moments, predictability of cash flow growth rates and excess stock returns and international asset pricing moments. The simulation results reveal several interesting finding. While the parameterization in Bansal and Yaron (2004) successfully matches national moments, it fails in international asset pricing. In particular, the model generates excess volatility in the exchange rate due to the high sensitivity of the pricing kernel to short-run 31

consumption shocks. The parameterization in Colacito and Croce (2011) is characterized by a low risk aversion coefficient and a extremely high persistency factor in the long-run risk process. The low risk aversion coefficient ensures a smooth exchange rate while the high persistency factor generates a sizeable equity premium. However, the extremely high persistency factor generates counterfactual high predictability of consumption and dividend growth rates by the price-dividend ratio. It turns out, that a consistent pricing of national international assets pose contradictive challenges to the single-good economy setup. The incorporation of durable goods partly resolves the contradictive challenges. The simulation results suggest that the two good economy that features long-run risk in durable consumption growth while nondurable consumption levels follow a random walk is the most satisfactory. The model acceptably matches national asset pricing moments, reconciles the predictability pattern in consumption growth growth rates, generates a relatively smooth exchange rate and reproduces the large cross-country correlations of returns and price-dividend ratios. The next step in this line of research is to address the role of durable goods for international asset pricing within a framework which explicitly allows for trade in consumption goods. We leave this challenge for future research.

32

A A.1

Solving for the Equilibrium Consumption Claim

The derivation follows Drechsel and Yaron (2011) and relies on the standard log linearization of the return on wealth and the relative share process of nondurable consumption. Substituting the affine guess for the wealth to consumption ratio (10) into the return on wealth (10) and exploiting the Euler condition (8) yields   Et exp θ log β − θ(1/ψ(1 − χ) + χ1/)∆ct+1 + θχ(1/ − 1/ψ)∆st+1 +  0 0 θ(κ0 + κ1 (A0 + A1 Yt+1 ) − A0 − A1 Yt + (1 − χ)∆ct+1 + χ∆st+1 ) =1 Defining is = (1, 0, . . . , 0)0 and ic = (0, 1, . . . , 0)0 and rearranging yields   Et exp θ log β + θκ0 + θ(κ1 − 1)A0 − θA01 Yt + 0



(((1 − γ)(1 − χ) − θχ1/)ic + ((1 − γ)χ + θχ1/)is + θκ1 A1 ) Yt+1

=1

In order to evaluate the expectation, we establish the functional relationship for u ∈ Rn : Et [exp(u0 Yt+1 |Yt )] = exp(f(u) + g(u)0 Yt ) 1 f(u) = µ0 u + u0 hu 2 1 g(u) = F 0 u + [u0 Hi u]i∈{1...n} . 2

(A.13)

Thus, the Euler conditions holds, if and only if the system of n + 1 is solved θ log β + θκ0 + θ(κ1 − 1)A0 + f(u) = 0 g(u) − θA1 = 0 where u = ((1 − γ)(1 − χ) − θχ1/)ic + ((1 − γ)χ + θχ1/)is + θκ1 A1 . The log linearization constants are given by exp(¯ υ) 1 + exp(¯ υ) = ln (1 + exp(¯ υ )) − κ1 υ¯,

κ1 = κ0

where the unconditional mean of the log wealth consumption ratio is υ¯ = A0 + A01 E(Y ). 33

Rearranging the terms yields the identities A0 + A01 E(Y ) = ln κ1 − ln(1 − κ1 ) κ0 = −κ1 ln κ1 − (1 − κ1 ) ln(1 − κ1 ), which implies κ0 + (κ1 − 1)A0 = − ln κ1 + (1 − κ1 )A01 E(Y ). Since κ0 and A0 are given in terms of κ1 and A1 , we solve for κ1 and A and exploit the identity for κ0 and A0 .

A.2

Pricing Kernel, Market Prices of Risk and the Risk-free Rate

Once the equilibrium solution is known, we can solve for the pricing kernel. The next-period log pricing kernel (in terms of the state variables) is given by mt+1 = m0 + (1 − θ)A01 Yt − Λ0 Yt+1 where m0 = θ log β + (θ − 1)(κ0 + (κ1 − 1)A0 ) and Λ ∈ R5 = (γ(1 − χ) + θχ1/)ic + (γχ − θχ1/)is + (1 − θ)κ1 A1 contains the market prices of risk. In order to derive the risk-free rate, we set rj,t+1 = rf,t in the Euler condition (8) and evaluate the conditional expectation with the functional relationship in (A.13). Thus the log risk-free rate is rt,f = −θ log β − (θ − 1)(κ0 + (κ1 − 1)A0 ) − f(−Λ) − (g(−Λ) + (1 − θ)A1 )0 Yt 1 = −θ log β − (θ − 1)(κ0 + (κ1 − 1)A0 ) + µ0 Λ − Λ0 hΛ  0 2 1 0 − [Λ Hi Λ]i∈{1...n} + F 0 (−Λ) + (1 − θ)A1 Yt 2 = r0 + r10 Yt

A.3

Dividend Claim

As for the return on wealth, we rely on the log-linearization of the market return rm,t+1 ≈ κm,0 + κm,1 ωt+1 − ωt + ∆dt+1 and conjecture that the log price to dividend ratio is affine in the state variables υd,t = B0 + B10 Yt . 34

The solution for the coefficients is computed analogously to the consumption claim - the market return is set into the Euler condition (8) and the affine guess is substituted for the log price to dividend ratio. Again, defining the selection vector id = (0, 0, . . . , 1)0 and Γ = κ1,m B1 + id and rearranging, yields   Et exp θ log β + (θ − 1)(κ0 + (κ1 − 1)A0 ) + κ0,d + (κ1,d − 1)B0 −  0 0 ((θ − 1)A1 + B1 ) Yt + (Γ − Λ) Yt+1 = 1. As for the consumption claim, the equilibrium solution is computed by solving the following n + 1 system θ log β + (θ − 1)(κ0 + (κ1 − 1)A0 ) + κ0,d + (κ1,d − 1)B0 + f(Γ − Λ) = 0 g(Γ − Λ) + (1 − θ)A1 − B1 = 0. Note, using the identity κm,0 + (κm,1 − 1)B0 = − log κm,0 + (1 − κm,1 )B10 E(Y ) we first solve for κm,1 and B1 and then compute κm,0 and B0 . Once the equilibrium solution is known, we can compute the market return by rm,t+1 = κ0 + (κm,1 − 1)B0 + (Γ0 F − B10 )Yt + Γ0 Gt t+1 . where Γ = [κm B1 + id ] ∈ R5 denotes the exposure of the market return to the risk factors in the economy.

35

References Backus, D., S. Foresi, and C.I. Telmer, 2001, Affine Term Structure Models and the Forward Premium Anomaly, Journal of Finance 56, 279 – 304. Backus, D., and G. Smith, 1993, Consumption an Real Exchange Rates in Dynamic Exchange Economies with Nontraded Goods, Journal of International Economics 35, 297–316. Bansal, R., D. Kiku, and A. Yaron, 2007, Risks for the Long-Run: Estimation and Inference, Working Paper. Bansal, R., and I. Shaliastovich, 2013, A Long Run Risks Explanation of Predictability Puzzles in Bond and Currency Markets, Review of Financial Studies 26, 1–33. Bansal, R., and A. Yaron, 2004, Risks for the Long Run: A potential Resolution of Asset Pricing Puzzles, Journal of Finance 59, 1481–1509. BEA, 2003, Fixed Assets and Consumer Durable Goods in the United States, 192597, Government Printing Office. Beeler, J., and J.Y. Campbell, 2012, The Long-Run Risks Model and Aggregate Asset Prices: An Empirical Assessment, Critical Finance Review 1, 141–182. Branger, N., I. Dumitrescu, V. Ivanova, and C. Schlag, 2011, Two Trees the EZ Way, Working Paper. Campbell, J.Y., and R.J. Shiller, 1988, The Dividend-Price Ratio and Expetation of Future Dividends and Discount Factors, Review of Financial Studies 1, 195–228. Campbell, J.Y., and M. Yogo, 2006, Efficient Test for Stock Return Predictability, Journal of Financial Economics 81, 27–60. Cochrane, J., F. Longstaff, and P. Santa-Clara, 2008, Two Trees, Review of Financial Studies 21, 347–385. Colacito, R., and M. Croce, 2011, Risks for the Long Run and the Real Exchange Rate, Journal of Politcal Economy 119, 153–182. Colacito, R., and M. Croce, 2013, International Asset Pricing with Recursive Preferences, Journal of Finance 68, 2651–2686. DeSantis, R.A., and F. Fornari, 2008, Does Business Cylce Risk Account for Systematic Returns from Currency Positioning? The International Perspective, Working Paper.

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Drechsel, I., and A. Yaron, 2011, Whats Vol Got to Do with It, Review of Financial Studies 24, 1–45. Dunn, K., and K.J. Singleton, 1986, Modeling Term Strucutre of Interest Rates under Nonseparable Utility and Durability of Goods, Journal of Financial Economics 17, 27–55. Engel, C., and J. Wang, 2011, International Trade in Durable Goods: Understanding Volatility, Cyclicality and Elasticities, Journal of International Economics 83, 37– 52. Engsted, T., and T.Q. Pedersen, 2010, The Dividend-Price Ratio does Predict Dividend Growth: International Evidence, Journal of Empirical Finance 17, 585–605. Epstein, L.G., and S.E. Zin, 1989, Substiution, Risk Aversion, and the Intertempora Behavior of Consumption and Asset Returns: A Theoretical Framework, Econometrica 57, 937–969. Eraker, B., I. Shaliastovich, and W. Wang, 2013, Durable Goods, Inflation Risk and the Equilibrium Term Structure, Working Paper. Fama, E.F, and K.R. French, 1988, Dividend Yields and Expected Stock Returns, Journal of Financial Economics 22, 3–26. Gomes, J.F., L. Kogan, and M. Yogo, 2009, Durability of Output and Expected Stock Returns, Journal of Political Economy 117, 941–986. Guo, B., and P.N. Smith, 2012, Durable Consumption, Long-Run Risk and the Equity Premium, Working Paper. Hamilton, B., and B. Morris, 2002, Durables and the Recent Strength of Household Spending, Mimeo, Bank of England. Hodrick, R.J., 1992, Dividend Yields and Expected Stock Returns: Alternative Procedures for Inference and Measurement, Review of Financial Studies 5, 357–386. Kellard, N.M., J.C. Nankervis, and F.I. Papadimitriou, 2010, Predicting the Equity Premium with Dividend Ratios: Reconciling the Evidence, Journal of Empirical Finance 17, 539–551. Lustig, H., and A. Verdelhan, 2007, The Cross-Section of Foreign Currency Risk Premia and Consumption Growth Risk, American Economic Review 97, 89–117. Marquez, E., and B. Nieteo, 2011, Further International Evidence on Durable Consumption Growth and Long-Run Consumption Risk, Quantitative Finance 11, 195–217. 37

Ogaki, M., and C.M. Reinhart, 1998, Measuring Intertemporal Substitution: The Role of Durable Goods, Journal of Political Economy 106, 1078 – 1098. Okubo, M., 2011, The Intertemporal Elasticity of Substitution: An Analysis Based on Japanese Data, Economica 78, 367–390. Pakos, M., 2011, Estimating Intertemporal and Intratemporal Substitutions When Both Income and Substitutions Effects are Present: The Role of Durable Goods, Journal of Business and Economic Statistics 3, 439–454. Piazzesi, M., M. Schneider, and S. Tuzel, 2007, Housing, Consumption and Asset Pricing, Journal of Financial Economics 83, 531–569. Williams, G., 1998, The Stock of Consumer Durables in the United Kingdom: New Estimates 1948-95, Review of Income and Wealth 44, 417–435. Yang, W., 2011, Long-Run Risk in Durable Consumption, Journal of Financial Economics 102, 45–61. Yogo, M., 2006, A Consumption-Based Explanation of Expected Stock Returns, Journal of Finance 61, 539–580.

38

Statistics Mean Std.-Div. Skewness Kurtosis AC(1) AC(3)

Dur. US UK 0.0410 0.0351 0.0108 0.0108 -0.2875 0.5062 2.9217 3.8201 0.7553 0.5903 0.5606 0.4608

Nondur. US UK 0.0217 0.0209 0.0091 0.0169 -0.6601 0.0667 4.6158 4.2465 0.5332 0.0190 0.3871 0.0498

Div. US UK 0.0099 0.0046 0.0388 0.0883 0.0875 -0.6793 9.5830 4.8598 0.5246 -0.1643 0.3321 0.2071

Table 1: Descriptive Statistics The table reports mean growth rate, volatility, skewness, kurtosis and autocorrelation coefficients of real durable consumption, nondurable consumption and dividend growth. Mean growth rates and volatilities are annualized. Descriptive statistics for consumption growth are based on quarterly sample period from 1966.Q2 to 2011.Q4. Descriptive statistics of dividend growth are based on the sample period from 1966.Q2 to 2012.Q4.

(1) Dur.us (2) Dur.uk

(1) 1.0000

(2)

0.4341

1.0000

(3)

(4)

(5)

(6)

(0.0000)

(3) Nond.us (4) Nond.uk (5) Div.us (6) Div.uk

0.4639

0.1529

(0.0000)

(0.0405)

1.0000

0.3869

0.4262

0.3331

(0.0000)

(0.0000)

(0.0000)

0.2569

0.0590

0.1576

0.0854

(0.0005)

(0.4316)

(0.0346)

(0.2545)

0.0909

0.0616

0.0990

0.1159

0.2167

(0.2250)

(0.4113)

(0.1863)

(0.1212)

(0.0035)

1.0000 1.0000 1.0000

Table 2: Correlation Matrix The table reports the correlation matrix of durable consumption, nondurable consumption and dividend growth in the US and UK and corresponding p-values in parentheses.

39

1 yr Dur. slope t-stat R2 Nond. slope t-stat R2 Div. slope t-stat R2

US 2 yr

3 yr

5 yr

1 yr

UK 2 yr

3 yr

5 yr

0.0041

0.0036

0.0033

0.0033

0.0084

0.0082

0.0078

0.0063

(2.5462)

(2.1273)

(1.9783)

(2.3419)

(6.5011)

(7.9256)

(8.6698)

(6.1208)

0.1383

0.1313

0.1438

0.2052

0.3303

0.3957

0.4430

0.4114

-0.0001

-0.0006

-0.0007

-0.0006

0.0045

0.0026

0.0011

0.0004

(-0.0823)

(-0.4492)

(-0.6121)

(-0.7284)

(2.4311)

(1.4198)

(0.5950)

(0.2415)

0.0002

0.0090

0.0197

0.0272

0.0850

0.0390

0.0096

0.0019

0.0045

0.0033

0.0025

0.0033

0.0029

-0.0026

-0.0066

-0.0074

(0.8068)

(0.6641)

(0.5385)

(1.0316)

(0.3193)

(-0.3668)

(-1.1086)

(-1.3959)

0.0355

0.0226

0.0144

0.0477

0.0026

0.0030

0.0237

0.0458

Table 3: Cash Flow Predictability: Price to Dividend Ratio The table reports predictive regressions for the average cumulative durable, nondurable consumption and dividend growth rate on the forecasting variable price to dividend ratio in the respective country. Corresponding t-statistics are reported in parentheses. Standard errors are Newey-West adjusted using ten lags.

40

Parameter

Symbol

Mean Nondurable Growth Nondurable Volatility Multiple Mean Dividend Growth Dividend Volatility Multiple Mean Volatility Persistency of Volatility Volatility of Volatility Correlation Nondurable Correlation LRR Correlation Volatility Correlation Dividend Nondurable LRR Exposure Mean Durable Growth Durable Volatility Multiple Durable LRR Exposure Correlation Durable Dividend LRR Exposure LRR Persistency LRR Volatility Multiple

µc ϕc µd ϕd σ ¯2 ν ϕσ ρc ρx ρσ ρd φc µs ϕs φs ρs φd ρ ϕx

1.0000 3.0000 0.9790 0.0440

Calibration CC DurNon Economy 0.0018 1.0000 0.0008 5.0000 0.0050 0.9870 2.30e-06 0.3000 1.0000 0.7000 0.1500 1.0000 1.0000 0.0032 0.6000 1.0000 0.0000 5.0000 4.0000 0.9870 0.9790 0.0480 0.0480

Relative Risk Aversion Intertemporal ES Subjective Discount Facotr Average Expenditure Durable Intratemporal ES

γ ψ β χ 

10 1.5 0.9990 -

Preference 4.25 10 2.0 1.75 0.9990 0.9990 0.1500 0.8000

BY

Table 4: Model Parameter The Table displays the endowment process parameterns and the preference parameters of the representative investor. ”BY” denotes the specification in Bansal and Yaron (2004). ”CC” represents the specification in Colacito and Croce (2011). The model specification ”DurNon” models long run risk in both consumption goods, while the model specification ”Dur” incorporates long run risk in durable consumption growth. All parameters are given at monthly frequency.

41

Dur

0.0000 0.0032 0.5000 1.2500 0.0000 4.0000 0.9790 0.0480 10 1.75 0.9990 0.1500 0.8000

0.0410 0.0108 0.7553 0.5606 0.0217 0.0091 0.5332 0.3871 0.0099 0.0388 0.5246 0.3321 0.0380 0.1683 0.0879 -0.0372 0.0089 0.0169 0.3065 0.2986 3.5723 0.4207 0.9867 0.9396

0.0351 0.0108 0.5903 0.4608 0.0209 0.0169 0.0190 0.0498 0.0046 0.0883 -0.1643 0.2071 0.0438 0.2003 0.1044 0.0268 0.0118 0.0259 0.4506 0.3140 3.2057 0.2863 0.9530 0.8031

Data

CC Mean SD 0.0211 0.0093 0.0166 0.0024 0.3608 0.1002 0.1727 0.1202 0.0088 0.0466 0.0831 0.0122 0.3626 0.0982 0.1739 0.1187 0.0239 0.0382 0.2581 0.0344 -0.0054 0.0810 -0.0056 0.0789 0.0177 0.0046 0.0040 0.0011 0.9476 0.0240 0.8163 0.0795 3.4207 0.2351 0.4043 0.1121 0.9493 0.0232 0.8165 0.0795

Table 5: National Moments

BY Mean SD 0.0212 0.0059 0.0158 0.0022 0.3023 0.0855 0.1012 0.1008 0.0098 0.0207 0.0756 0.0102 0.2420 0.0790 0.0361 0.0882 0.0209 0.0177 0.1183 0.0153 -0.0036 0.0786 -0.0065 0.0789 0.0214 0.0037 0.0040 0.0010 0.9331 0.0262 0.7679 0.0852 3.4270 0.0547 0.1244 0.0279 0.9088 0.0327 0.6888 0.1062

DurNon Mean SD 0.0381 0.0059 0.0109 0.0017 0.4489 0.0995 0.2571 0.1239 0.0212 0.0064 0.0160 0.0022 0.3181 0.0871 0.1167 0.1045 0.0095 0.0267 0.0777 0.0107 0.2808 0.0840 0.0790 0.0965 0.0373 0.0223 0.1521 0.0189 -0.0065 0.0794 -0.0056 0.0779 0.0178 0.0034 0.0038 0.0009 0.9330 0.0261 0.7683 0.0845 3.0877 0.0848 0.1858 0.0440 0.9274 0.0275 0.7440 0.0922

Dur Mean SD 0.0379 0.0073 0.0108 0.0019 0.5902 0.0989 0.4076 0.1334 0.0213 0.0027 0.0147 0.0020 0.2009 0.0755 -0.0081 0.0824 0.0095 0.0264 0.0777 0.0106 0.2787 0.0834 0.0781 0.0958 0.0270 0.0254 0.1726 0.0223 -0.0071 0.0811 -0.0077 0.0785 0.0195 0.0010 0.0010 0.0002 0.9361 0.0254 0.7784 0.0827 3.2912 0.0973 0.2149 0.0510 0.9294 0.0272 0.7500 0.0904

The Table displays empirical moments from the quarterly data sample and theoretical moments for the considered models. ”BY” denotes the specification in Bansal and Yaron (2004). ”CC” represents the specification in Colacito and Croce (2011). The model specification ”DurNon” models long run risk exposure in both consumption goods, while the model specification ”Dur” incorporates long run risk in durable consumption growth. The growth rates, standard deviations and returns are annualized.

E(∆s) σ(∆s) AC1(∆s) AC3(∆s) E(∆c) σ(∆c) AC1(∆c) AC3(∆c) E(∆d) σ(∆d) AC1(∆d) AC3(∆d) E(rm ) σ(rm ) AC1(rm ) AC3(rm ) E(rf ) σ(rf ) AC1(rf ) AC3(rf ) E(pd) σ(pd) AC1(pd) AC3(pd)

Statistics

0.0041 2.5462 0.1333 0.0033 1.9783 0.1387 0.0033 2.3419 0.2002 -0.0001 -0.0823 -0.0055 -0.0007 -0.6121 0.0138 -0.0006 -0.7284 0.0211

BY Mean (< x ˆ)% 0.0189 0.0067 5.2389 0.0286 0.2300 0.0555 0.0137 0.0154 4.3035 0.0138 0.2631 0.0440 0.0099 0.0445 3.3387 0.0328 0.2150 0.0893

CC Mean (< x ˆ)% 0.0087 0.0024 8.2390 0.0039 0.3939 0.0073 0.0068 0.0029 6.7540 0.0026 0.4416 0.0082 0.0053 0.0093 5.1410 0.0079 0.3675 0.0255

DurNon Mean (< x ˆ)% 0.0140 0.0100 8.2827 0.1484 0.4369 0.0995 0.0101 0.0887 5.8743 0.4994 0.3854 0.3149 0.0073 0.2298 4.1718 0.5824 0.2823 0.5717 0.0140 0.0066 5.7044 0.0193 0.2638 0.0358 0.0101 0.0125 4.5836 0.0120 0.2904 0.0354 0.0074 0.0414 3.5083 0.0321 0.2351 0.0808

Table 6: Predictive Regression: Consumption Growth

0.0084 6.5011 0.3265 0.0078 8.6698 0.4397 0.0063 6.1208 0.4077 0.0045 2.4311 0.0798 0.0011 0.5950 0.0036 0.0004 0.2415 -0.0044

Data

Dur Mean (< x ˆ)% 0.0159 0.0005 12.1350 0.0221 0.6111 0.0072 0.0115 0.0314 7.3816 0.3034 0.4712 0.1527 0.0084 0.1478 4.9090 0.4558 0.3285 0.4598 0.0000 0.7573 -0.0138 0.7892 0.0191 0.8090 0.0000 0.5232 -0.0130 0.4981 0.0513 0.3729 0.0000 0.4782 0.0065 0.4412 0.0730 0.3090

The Table displays predictive regressions of consumption growth from the quarterly data sample and the simulation study. The Table covers regression coefficients, the corresponding t-statistics and R2 -statistics. The percentile statistics indicates the proportion of those samples below that of the empirical result. ”BY” denotes the specification in Bansal and Yaron (2004). ”CC” represents the specification in Colacito and Croce (2011). The model specification ”DurNon” models long run risk exposure in both consumption goods, while the model specification ”Dur” incorporates long run risk in durable consumption growth. Standard errors are Newey-West adjusted using ten lags.

β1 (∆s) tstat1 (∆s) R12 (∆s) β3 (∆s) tstat3 (∆s) R32 (∆s) β5 (∆s) tstat5 (∆s) R52 (∆s) β1 (∆c) tstat1 (∆c) R12 (∆c) β3 (∆c) tstat3 (∆c) R32 (∆c) β5 (∆c) tstat5 (∆c) R52 (∆c)

Statistics

0.0029 0.3193 -0.0029 -0.0066 -1.1086 0.0181 -0.0074 -1.3959 0.0401 -0.0755 -4.4938 0.2023 -0.0569 -6.6215 0.4284 -0.0420 -7.4152 0.5120

BY Mean (< x ˆ)% 0.0733 0.0046 4.9091 0.0086 0.1734 0.0303 0.0451 0.0303 3.4686 0.0242 0.1862 0.1015 0.0316 0.0729 2.6689 0.0710 0.1620 0.2782 -0.0177 0.2053 -0.6672 0.0810 0.0171 0.9806 -0.0183 0.2743 -1.0969 0.0621 0.0557 0.9699 -0.0167 0.3383 -1.3318 0.0424 0.0838 0.9602

CC Mean (< x ˆ)% 0.0453 0.0002 9.2330 0.0002 0.4228 0.0003 0.0345 0.0007 7.0655 0.0007 0.4509 0.0073 0.0267 0.0055 5.2114 0.0071 0.3694 0.0534 -0.0193 0.1583 -1.0029 0.1163 0.0216 0.9687 -0.0188 0.2216 -1.4992 0.0860 0.0720 0.9458 -0.0173 0.2934 -1.8804 0.0731 0.1123 0.9185

DurNon Mean (< x ˆ)% 0.0633 0.0024 5.9097 0.0044 0.2455 0.0126 0.0420 0.0137 4.3169 0.0122 0.2566 0.0558 0.0299 0.0443 3.2592 0.0456 0.2101 0.1914 -0.0222 0.2225 -0.9882 0.1156 0.0219 0.9683 -0.0214 0.2874 -1.4389 0.0862 0.0699 0.9488 -0.0191 0.3573 -1.7295 0.0603 0.1037 0.9344

Table 7: Predictive Regression: Dividend Growth and Excess Market Return

0.0045 0.8068 0.0302 0.0025 0.5385 0.0087 0.0033 1.0316 0.0420 -0.0155 -1.2033 0.0243 -0.0149 -1.7623 0.0798 -0.0138 -3.4324 0.1365

Data

Dur Mean (< x ˆ)% 0.0560 0.0019 6.1332 0.0034 0.2564 0.0106 0.0380 0.0082 4.5842 0.0067 0.2767 0.0421 0.0273 0.0313 3.4840 0.0332 0.2269 0.1657 -0.0197 0.1911 -0.8564 0.0957 0.0196 0.9755 -0.0187 0.2474 -1.2892 0.0733 0.0625 0.9615 -0.0169 0.3119 -1.5953 0.0538 0.0932 0.9475

The Table displays predictive regressions of dividend growth and of market excess return from the quarterly data sample and the simulation study. The Table covers regression coefficients, the corresponding t-statistics and R2 -statistics. The percentile statistics indicates the proportion of those samples below that of the empirical result. ”BY” denotes the specification in Bansal and Yaron (2004). ”CC” represents the specification in Colacito and Croce (2011). The model specification ”DurNon” models long run risk exposure in both consumption goods, while the model specification ”Dur” incorporates long run risk in durable consumption growth. Standard errors are Newey-West adjusted using ten lags.

β1 (∆d) tstat1 (∆d) R12 (∆d) β3 (∆d) tstat3 (∆d) R32 (∆d) β5 (∆d) tstat5 (∆d) R52 (∆d) β1 (rm ) tstat1 (rm ) R12 (rm ) β3 (rm ) tstat3 (rm ) R32 (rm ) β5 (rm ) tstat5 (rm ) R52 (rm )

Statistics

0.1768 0.6940 0.1623

0.0790 0.2721 0.0509

0.1432 0.0927 0.1123

BY Mean SD -0.0001 0.0341 0.2434 0.0232 -0.0082 0.0771 0.3852 0.0792 0.1907 0.0826 0.8561 0.0388 -0.0209 0.0761 0.5078 0.0587 0.8387 0.0677 0.9571 0.0354 0.0876 0.8508 0.0639

0.1470 0.0251 0.1953

CC Mean SD -0.0002 0.0163 0.1110 0.0107 -0.0067 0.0787 0.4364 0.0901 0.3236 0.1021 0.7972 0.0638 -0.0194 0.0732 0.8759 0.0223 0.9596 0.0319 0.9537 0.0419

Table 8: International Moments

0.2184 0.6355 -0.0326

Data US UK 0.0059 0.1010 0.1332 0.4341 0.3331 0.2204 0.1270 0.0610 -0.0271 0.7055 0.8815 0.5053 0.1668 0.5315 0.1013

0.1383 0.0873 0.1587

DurNon Mean SD -0.0001 0.0239 0.1861 0.0156 -0.0115 0.0772 0.3284 0.1146 0.3992 0.0807 0.2349 0.0865 0.4323 0.0527 0.5097 0.0549 -0.0464 0.0704 0.6760 0.0436 0.8872 0.0594 0.9567 0.0352

0.1249 0.6397 0.0698

0.1466 0.0632 0.1637

Dur Mean SD -0.0001 0.0203 0.1467 0.0135 -0.0094 0.0774 0.5120 0.1147 0.2915 0.0774 0.2357 0.0855 0.4565 0.0480 0.6489 0.0507 -0.0300 0.0734 0.7492 0.0360 0.9177 0.0507 0.8654 0.0898

The Table displays empirical international moments from the quarterly data sample and theoretical moments for the considered models. ”BY” denotes the specification in Bansal and Yaron (2004). ”CC” represents the specification in Colacito and Croce (2011). The model specification ”DurNon” models long run risk exposure in both consumption goods, while the model specification ”Dur” incorporates long run risk in durable consumption growth. The growth rates, standard deviations and returns are annualized.

+ ∆e) Corr(rm , ∆e)

∗ Corr(rm , rm

E(rm +∆e) σ(rm +∆e)

E(∆e) σ(∆e) AC1(∆e) Corr(∆sU S , ∆sU K ) Corr(∆cU S , ∆cU K ) Corr(∆dU S , ∆dU K ) Corr(∆e, ∆sU S − ∆sU K ) Corr(∆e, ∆cU S − ∆cU K ) Corr(∆e, pdU S − pdU K ) U S , rU K ) Corr(rm m Corr(pdU S , pdU K ) Corr(rfU S , rfU K )

Statistics

Figure 1: Growth Rates Durable consumption, nondurable consumption and dividend growth rate for the US and UK. The shaded areas indicate US-recessions defined by the NBER. The consumption growth figures are based on quarterly observations from 1966.Q2 - 2011.Q1. The dividend growth figure is based on quarterly observations from 1966.Q2 - 2012.Q4.

46

Autocorrelation: Durable Consumption Growth 1 US UK Bounds

0.5

0 0

1

2

3

4

5

6

7

8

9

10

8

9

10

8

9

10

Autocorrelation: Nondurable Consumption Growth 1

0.5

0 0

1

2

3

4

5

6

7

Autocorrelation: Dividend Growth 1

0.5

0 0

1

2

3

4

5

6

7

Figure 2: Autocorrelation of Consumption Growth Autocorrelation function of durable, nondurable consumption and dividend growth rate in the US and UK and corresponding confidence bounds.

47