Does inflation explain equity risk premia?

Transcription

1 Does inflation explain equity risk premia? Paulo Maio November 2017 Abstract I derive a simple linear macro asset pricing model that contains inflation as a risk factor in addition to the standard consumption growth and market return factors. This model nests the baseline CCAPM and Epstein Zin models as special cases. Inflation arises endogenously in the pricing kernel by assuming an intra-temporal utility that depends on both real and nominal consumption, which suits an investor with partial money illusion. However, the model can not explain the cross-sectional dispersion in risk premia associated with 60 equity portfolios (related to six major CAPM anomalies). Moreover, the inflation risk price estimates, and implied preference parameter estimates, are statistically insignificant and/or economically implausible in most cases. To a large degree, these results hold when one estimates the Euler equations associated with the non-linear macro model. The evidence from this paper largely suggests that inflation does not help explaining the cross-section of stock returns. Keywords: Asset pricing; inflation risk; consumption-based asset pricing models; macro asset pricing models; cross-section of stock returns; money illusion; Euler equations; stock market anomalies JEL classification: E31; E44; G11; G12. Maio: Hanken School of Economics (Department of Finance and Statistics); paulofmaio@gmail.com. I am grateful to Kenneth French for providing stock return data. 1

2 1 Introduction There is widespread evidence showing that the Consumption-CAPM (CCAPM) of Lucas (1978) and Breeden (1979) cannot explain the cross-section of average stock returns (e.g., Breeden, Gibbons, and Litzenberger (1989), Lettau and Ludvigson (2001), Brav, Constantinidies, and Geczy (2002), Jacobs and Wang (2004), among others). In response to this failure, several studies have estimated the single-factor model by employing alternative measures of consumption growth in an attempt to improve its performance (e.g., Parker and Julliard (2005), Jagannathan and Wang (2007), Savov (2011), Kroencke (2017)). Another branch of the macro-finance literature has focused on estimating multifactor macroeconomic asset pricing models that contain other macro variables as risk factors in addition to the standard consumption (non-durables and services) growth (e.g., Epstein and Zin (1991), Lettau and Ludvigson (2001), Lustig and Van Nieuwerburgh (2005), Yogo (2006), Gomes, Kogan, and Yogo (2009), Lioui and Maio (2014), Chen and Lu (2017)). However, these studies largely ignore inflation as a risk factor and do not analyze whether it improves the performance of consumption-based models when it comes to explaining cross-sectional equity risk premia. This paper attempts at filling this gap in the empirical asset pricing literature. To achieve that purpose, I derive a three-factor macro asset pricing model that contains inflation as a risk factor: the factors in the model are the standard real consumption growth employed in the baseline CCAPM, the standard excess market return employed in the CAPM of Sharpe (1964) and Lintner (1965), and the inflation rate. The underlying framework assumes an intertemporal utility of the recursive form (Epstein and Zin (1989, 1991) and Weil (1989)), combined with an intra-temporal utility that depends on both real consumption and nominal consumption (with a Cobb-Douglas specification) as in Basak and Yan (2007). The intratemporal utility suits the case of a partially money illusioned investor, who can not totally differentiate nominal from real consumption in his consumption/asset allocation decisions. The degree of money illusion is captured by the money-illusion parameter (ε), which varies between zero and one. Hence, this specification nests the standard fully-rational case of no money illusion (ε = 0) as well as the case of complete money illusion (ε = 1). Therefore, the assumption of money-illusion allows one to derive a macro model in which inflation appears endogenously as one of the variables in the pricing kernel. There are three preference parameters in the model: the coefficient of relative risk aversion, the moneyillusion parameter, and the elasticity of intertemporal substitution, which allows for a one-toone correspondence between the factor risk prices and the underlying structural parameters. Given the assumption of a positive money-illusion parameter, the inflation risk price should be positive as long as the risk aversion parameter is above one (more risk averse than an investor with log utility). Critically, the three-factor model nests the baseline CCAPM and 2

3 the two-factor model of Epstein and Zin (1991) (which contains consumption growth and the market return as risk factors) as special cases. I estimate a linear version of the three-factor macro model (in expected return-covariance representation) by first-stage GMM with a rich cross-section of stock returns. Specifically, I use decile portfolios associated with six stock characteristics: book-to-market ratio; asset growth; price momentum; accruals; net share issues; and long-term return reversal. These portfolios are associated with some of the most important market anomalies, that is, they produce significant cross-sectional spreads in average returns and are not explained by the baseline CAPM. Moreover, these portfolios are widely used in the empirical asset pricing literature to test and evaluate linear factor models (e.g., Fama and French (2015, 2016), Hou, Xue, and Zhang (2015), among others). In the linear model, the implied preference parameters are retrieved from the factor risk price estimates and the corresponding standard errors are obtained by using the delta method. Overall, the empirical results do not provide much support for the linear three-factor macro model, and also do not suggest that the inflation factor helps pricing cross-sectional equity risk premia. Specifically, in the augmented cross-sectional test with the 60 portfolios the macro model produces a negative cross-sectional R 2, which implies that the model does even worse than a trivial model that predicts constant cross-sectional risk premia. Moreover, the model does not improve much the baseline CCAPM and Epstein Zin models when it comes to explaining the risk premia associated with these 60 portfolios. The macro model performs somewhat better when it comes to pricing some of the single groups of deciles (e.g., book-to-market or return reversal deciles). However, it is difficult to assess the contribution of the inflation factor for such fit: the inflation risk price estimates are either insignificant or (marginally in some cases) significantly negative. These negative risk price estimates imply negative estimates for the underlying money-illusion parameter, which is at odds with the assumptions and predictions of the theoretical model. These findings are robust to several robustness checks: using a shorter sample that ends in 2007; employing alternative definitions of the macro factors; estimating the model with an unrestricted zero-beta rate; and including the market factor in the menu of testing assets. The main qualitative results also hold when one estimates a restricted macro model that contains the consumption growth and inflation rate as risk factors. This two-factor model is a special case of the benchmark model by assuming that the investor has time-separable power utility (that is, the risk aversion parameter equals the inverse of the elasticity of intertemporal substitution). To complement the empirical results associated with the linear model, I estimate the Euler equations associated with the original non-linear macro model. This stems from a possible significant approximation error associated with the linear pricing equations, which may imply 3

4 that the pricing errors and implied preference parameter estimates differ substantially from those obtained in the estimation of the Euler equations (see Lettau and Ludvigson (2009)). As with the estimation of the linear macro model, I use first-stage GMM to estimate the Euler equations corresponding to the 60 excess portfolio returns. In the estimation with 60 portfolios, the performance of the three-factor model is only marginally better than that of the Epstein Zin model with an average pricing error of 0.28% (versus 0.34%). In some of the single-anomaly tests (book-to-market, asset growth, and momentum deciles) there is a substantial improvement in fit relative to the Epstein Zin model. However, although the estimates for the money-illusion parameter are positive, there is no statistical significance in most cases. Given these results, it is difficult to assess the contribution of the inflation factor for the performance of the non-linear benchmark macro model. These results are even more clear when one estimates the non-linear restricted macro model based on power utility: the estimates of the money-illusion parameter are largely insignificant in all cases, which implies that the inflation factor (from a statistical viewpoint) does not add explanatory power within the macro model. In sum, the results of this paper suggest that the standard inflation rate does not help explaining equity risk premia in the context of a structural macro asset pricing model. Hence, future extensions of the baseline CCAPM that aim in explaining the cross-section of stock returns should rely on other macro variables as risk factors. Nevertheless, this does not preclude that other risk factors associated with inflation within alternative asset pricing frameworks might help explaining cross-sectional risk premia. 1 This paper is related to the work of Liu and Zhang (2008). They estimate the five-factor model of Chen, Roll, and Ross (1986) (which contains unexpected inflation and the change in expected inflation as risk factors) on a cross-section of size, book-to-market, and momentum portfolios. My paper has two major differences relative to that study. First, I derive and estimate a structural macro model in which the factor risk prices are linked to the underlying preference parameters. Second, I use a richer cross-section of stock returns in the empirical asset pricing tests. My work is also related to Duarte and Mishara-Blomberger (2012), who estimate a simple unconditional model containing inflation as the risk factor. Among others, there are three major differences in the two studies. First, in their model the inflation risk price is not analytically linked to structural preference parameters. Second, Duarte and Mishara-Blomberger (2012) test their model by using the cross-section of individual stock returns while I use equity portfolios. These are known to have a significant dispersion in average returns, and thus provide a more powerful asset pricing test (in addition to being widely used in the empirical asset pricing literature). Third, they focus on the risk price 1 For example, Maio (2013) shows that a scaled factor linked to the lagged inflation rate helps pricing portfolios sorted on book-to-market and momentum. 4

5 estimates and do not provide goodness-of-fit measures to evaluate the statistical performance of their model for cross-sectional equity risk premia. 2 The rest of the paper is organized as follows. Section 2 presents the theoretical background. Section 3 describes the data and econometric methodology. Section 4 presents the empirical results associated with the estimation of the linear macro model. Section 5 shows the results for the estimation of the non-linear macro model. Section 6 concludes. 2 The model In this section, I derive the macroeconomic asset pricing model, which is tested in the following sections. 2.1 Euler equation The representative investor has an intertemporal utility function with the recursive form introduced by Epstein and Zin (1989, 1991) and Weil (1989), U t = {(1 δ) v 1 γ θ t + δ [ E t ( U 1 γ t+1 )] 1 } θ 1 γ θ, (1) where v denotes the intratemporal utility; γ is the relative risk aversion (RRA) coefficient; ψ is the elasticity of intertemporal substitution (EIS); and θ is an auxiliary parameter defined as θ (1 γ)ψ. This specification has the advantage of disentangling relative risk aversion ψ 1 from the elasticity of intertemporal substitution and has been increasingly popular in the macro-finance literature. 3 When θ = 1, which means that the RRA parameter is equal to the reciprocal of EIS (γ = 1 ), one gets the standard time-separable power utility specification: ψ U t = v1 γ t 1 γ. (2) Following Basak and Yan (2007), the intratemporal utility has a Cobb Douglas specification and depends on both real consumption (C) and the price level (Π), v t = C 1 ε t (C t Π t ) ε, (3) 2 In related work, Boons, Duarte, De Roon, and Szymanowska (2016) also estimate the unconditional inflation risk premium. However, they use a non-parametric approach rather than estimating the inflation risk premium within a cross-sectional test of an asset pricing model (containing inflation as a risk factor). 3 See, for example, the large long-run risks literature Bansal and Yaron (2004), Bansal, Gallant, and Tauchen (2007), Hansen, Heaton, and Li (2008), Bansal, Dittmar, and Kiku (2009), and Malloy, Moskowitz, and Vissing-Jørgensen (2009), among others. 5

6 where 0 ε 1 represents the degree of money illusion. This preference specification suits the case of a partially money illusioned investor, who can not totally differentiate nominal (CΠ) from real consumption in his consumption/asset allocation decisions. 4 A higher value of ε implies a larger impact of money illusion on utility. This specification has the attractive property of nesting the standard fully-rational case of no money illusion (ε = 0) as well as the case of complete money illusion (ε = 1). Hence, an investor with partial money illusion will represent a mixture of these two extreme cases. 5 is also consistent with bounded rationality (see Simon (1978)). This intra-period utility specification The investors s portfolio contains risky assets and the risk-free asset, with the following dynamics over time, W t+1 = R w,t+1 (W t C t ), (4) R w,t+1 = N ω i,t (R i,t+1 R f,t+1 ) + R f,t+1, (5) i=1 where W t+1 represents total real wealth at the end of period t + 1; R w,t+1 is the real gross return on total wealth; ω i is the portfolio weight associated with risky asset i; R i,t+1 is the corresponding real gross return; and R f,t+1 denotes the real gross risk-free rate from t to t + 1. It follows that the Euler equation associated with the dynamic problem presented above is given by 6 where the stochastic discount factor (SDF) is as follows: 0 = E ( Q t+1r e i,t+1), (6) Q t+1 = δ θ ( Ct+1 C t ) 1 γ θ ( Πt+1 Π t ) ε(1 γ) R θ 1 w,t+1. (7) This specification represents a three-factor asset pricing model in which the factors are the real gross consumption growth (C t+1 /C t ), gross inflation (Π t+1 /Π t ), and the real gross market return (R w,t+1 ). Hence, inflation emerges endogenously as a risk factor in the model, which helps to price equity risk premia. 4 Several empirical and experimental studies document the presence of money illusion in economic and financial decisions (e.g., Modigliani and Cohn (1979), Shafir, Diamond, and Tversky (1997), Fehr and Tyran (2001), Ritter and Warr (2002), Campbell and Vuolteenaho (2004b), among others). 5 Basak and Yan (2010) specify an alternative theoretical framework in which money illusion is modeled as a bias in the investor s beliefs. 6 A full derivation is available upon request. 6

7 2.2 Linear specification I linearize the macro model presented above. Following Yogo (2006) and Lioui and Maio (2014), I use the following general expected return-covariance representation in unconditional form, E ( Ri,t+1) e = Cov (qt+1, R i,t+1 ), (8) where q t+1 ln (Q t+1 ) denotes the log SDF. By applying the general pricing equation (8) to the SDF in Eq. (7), and by using the following definitions, c t ln (C t ), c t+1 = c t+1 c t, π t ln (Π t ), π t+1 = π t+1 π t, r w,t+1 ln(r w,t+1 ), one obtains the expected return-covariance representation of the macro model: E ( R e i,t+1) = γc Cov (R i,t+1, c t+1 ) + γ π Cov (R i,t+1, π t+1 ) + γ w Cov (R i,t+1, r w,t+1 ). (9) The covariance risk prices above are related to the preference parameters as follows, γ c θ + γ 1 = 1 γ ψ 1, γ π ε (γ 1), γ w 1 θ = γψ 1 ψ 1, (10) and given these definitions, the implied structural parameters are given by γ = γ c + γ w, ε = γπ, γ c+γ w 1 ψ = 1 γw γ c. (11) In this model, the signs of the factor risk prices are constrained by theory. Given the restriction that 0 ε 1 and by assuming that the investor is more risk averse than an investor with log utility (γ > 1), it follows that the inflation risk price should be positive. On the other hand, with γ > 1 the consumption risk price is positive if ψ < 1. Finally, the market risk price is positive if γ < 1/ψ (assuming again that ψ < 1). The macro model represents a rich specification since it nests several other models. Specifically, the two-factor model of Epstein and Zin (1991) can be obtained as a special case of 7

8 the three-factor model by setting ε = 0, which means no money illusion and implies γ π = 0: E ( R e i,t+1) = γc Cov (R i,t+1, c t+1 ) + γ w Cov (R i,t+1, r w,t+1 ). (12) In this model, the risk prices are related to the structural parameters as follows: γ c θ + γ 1, γ w 1 θ. (13) The standard CCAPM of Lucas (1978) and Breeden (1979) can be obtained by setting both ε = 0 and θ = 1, E ( Ri,t+1) e = γc Cov (R i,t+1, c t+1 ), (14) where γ c = γ. Therefore, the three-factor macro model can help improving the CCAPM or the Epstein Zin models in terms of pricing the cross-section of stock returns if the dispersion (among test assets) in the covariances (betas) associated with the inflation factor, Cov(R i,t+1, π t+1 ), matches the corresponding cross-sectional dispersion in average returns, E(Ri,t+1). e 3 Data and Econometric framework 3.1 Data and variables I use quarterly data spanning the period from 1963:IV to 2016:IV, where the starting date is constrained by the availability of data on some of the equity portfolios used in the asset pricing tests. Consumption growth ( c) corresponds to the log change in the quarterly real consumption of non-durables and services. To obtain real per capita consumption, real consumption is divided by total population (observed in the last month of each quarter). I use the end-of-period timing convention for consumption, in which asset returns between t and t + 1 are matched with the consumption growth measured over the same time interval. The inflation rate ( π) represents the log difference in the Personal Consumption Expenditures (PCE) price index. The choice of PCE instead of the Consumer price index (CPI) stems from preserving consistency with the real consumption data. Following Epstein and Zin (1991), Yogo (2006), and Lioui and Maio (2014), among others, I use the value-weighted stock market return from CRSP as the proxy for the log return on total wealth (r w ). To obtain the real market return, I deflate the quarterly nominal return (which represents the compounded monthly returns) by the quarterly inflation rate based on the CPI index (which represents the growth rate in the price index in the last month of each quarter relative to the same month in the previous quarter). 8

9 Quarterly portfolio returns represent the compounded monthly returns. To obtain excess portfolio returns, I subtract the quarterly return on the three-month Treasury bill (which is equal to T B3M/4, where T B3 is the three-month interest rate observed in the first month of a given quarter). Real portfolio returns are obtained by deflating the raw quarterly returns by the quarterly CPI inflation rate. The consumption, interest rate, and price index data are all obtained from the FRED database (St. Louis FED). The equity portfolio return data are obtained from Kenneth French s website. The testing assets represent decile portfolios associated with six stock characteristics: book-to-market ratio (BM10); asset growth (AG10); price momentum (M10); accruals (ACC10); net share issues (NSI10); and long-term return reversal (REV10). These portfolios are associated with some of the most important market anomalies, that is, they are not explained by the baseline CAPM of Sharpe (1964) and Lintner (1965). Moreover, these portfolios are widely used in the empirical asset pricing literature to test linear factor models (e.g., Fama and French (2015, 2016), Hou, Xue, and Zhang (2015), Dittmar and Lundblad (2017), among others). The value-growth anomaly arises from the evidence that value stocks (stocks with high book-to-market (BM) ratios) outperform growth stocks (low BM) (e.g. Rosenberg, Reid, and Lanstein (1985) and Fama and French (1992)). Price momentum refers to a pattern in which stocks with high prior short-term returns outperform stocks with low prior returns (Jegadeesh and Titman (1993) and Fama and French (1996)). The asset growth anomaly comes from the evidence that stocks of firms that invest more exhibit lower average returns than the stocks of firms that invest less (Titman, Wei, and Xie (2004), Cooper, Gulen, and Schill (2008), Fama and French (2008), and Lyandres, Sun, and Zhang (2008)). The accruals anomaly stems from the evidence that stocks of firms with low accruals enjoy higher average returns than stocks of firms with high accruals (Sloan (1996) and Richardson, Sloan, Soliman, and Tuna (2005)). The net share issues effect describes a cross-sectional pattern in which stocks with high net share issues earn lower returns going forward than stocks with low net share issues (Loughran and Ritter (1995), Daniel and Titman (2006), and Pontiff and Woodgate (2008)). The long-term reversal in returns anomaly (De Bondt and Thaler (1985, 1987)) represents the evidence that stocks with low returns over the last five years have higher subsequent returns, while past long-term winners have lower future returns. The descriptive statistics associated with the macro factors are presented in Table 1. Both c and π have a similar volatility, yet inflation is considerably more persistent than real consumption growth (autoregressive coefficients of 0.71 and 0.34, respectively). 7 Figure 1, which plots the time-series of both c and π, suggests that both factors tend to be 7 The persistence of inflation is consistent with previous evidence (e.g., Fuhrer and Moore (1995), Ang, Bekaert, and Wei (2007), among others). 9

10 more volatile around recession periods (including the great recession of ). Not surprisingly, r w is substantially more volatile than the other two factors in the macro model. The correlations displayed in Panel B of Table 1 indicate that the three factors are close to being uncorrelated. Table 2 shows the descriptive statistics associated with the quarterly high-minus-low spreads in returns between the last and first decile within each portfolio group. Most return spreads are statistically significant at the 5% or 1% level, the exception being the spread corresponding to REV10, which is borderline insignificant at the 5% level (t-ratio of 1.95). 8 All six anomalies are economically significant as the magnitudes of the average return spreads are above 1% per quarter in all cases. The anomaly showing the largest spread in average returns is clearly momentum with an average gap of 3.74% per quarter. The anomalies with lower magnitudes are accruals and net share issues with average gaps in returns around 1.20% (in magnitude). Momentum exhibits the highest volatility followed by long-term return reversal (both with standard deviations above 10% per quarter). At the other end of the spectrum, accruals has the least volatile return spread (5.27%). 3.2 Econometric framework The macro model is estimated by first-stage GMM (Hansen (1982), Cochrane (1996, 2005)). The first-stage estimation uses the identity matrix (equally-weighted moments). This is conceptually equivalent to an OLS cross-sectional regression of average excess returns on factor covariances (or equivalently, single-regression betas), which is widely employed in the empirical asset pricing literature (e.g., Black, Jensen, and Scholes (1972), Jagannathan and Wang (1998), Campbell and Vuolteenaho (2004a), Kan, Robotti, and Shanken (2013), among others). Using first-stage GMM enables one to evaluate whether the macro model is capable of pricing a chosen set of economically interesting assets rather than a combination (with extreme positive and negative weights) of these original assets as it is the case with the second-stage estimation (see Cochrane (2005) and Ludvigson (2013) for further discussion). The GMM system has N + 3 moment conditions, where the first N sample moments 8 I do not include portfolios associated with other traditional anomalies in the asset pricing tests since the quarterly return spreads are clearly insignificant for the sample period used in this paper. These include the size anomaly (Banz (1981)), operating profitability (Novy-Marx (2013) and Fama and French (2015)), idiosyncratic variance (Ang, Hodrick, Xing, and Zhang (2006)), and market beta (Frazzini and Pedersen (2014)). 10

11 correspond to the pricing errors for each of the N testing assets: 1 T T t=1 g T (b) Ri,t e γ c R i,t ( c t µ c ) γ π R i,t ( π t µ π ) γ w R i,t (r w,t µ w ) c t µ c = 0, π t µ π r w,t µ w i = 1,..., N. (15) In this system, the last three moment conditions capture the factor means, thus implying that the estimated risk prices account for the estimation error in the factor means (as in Cochrane (2005) (Chapter 13), Yogo (2006), Maio and Santa-Clara (2012), and Maio (2013)). There are N 3 overidentifying conditions (N + 3 moments and 6 parameters to estimate). For full details on the GMM estimation, see Maio and Santa-Clara (2012) and Maio (2013). Since the signs of the risk prices are constrained by theory, as discussed in the last section, I use one-sided p-values when evaluating the statistical significance of the risk price estimates. The standard errors associated with the implied preference parameter estimates are obtained by using the delta method. 9 By defining the first N residuals from the GMM system as the pricing errors associated with the N testing assets, α i, i = 1,..., N, the χ 2 -statistic of overidentifying restrictions is defined as ˆα Var ( ˆα) ˆα χ 2 (N 3), (16) where denotes a pseudo inverse. Although this statistic represents a formal test of the null hypothesis that the pricing errors are jointly equal to zero, it suffers from several biases. First, in many applications the covariance matrix of the pricing errors is close to singular, leading to an overstated inverse and the consequent rejection of the null hypothesis even when the pricing errors have small magnitudes. Second, in other situations the inverse of the covariance matrix is underestimated leading to a non-rejection of the null hypothesis even when the pricing errors are large in magnitude. In both situations, the miscalculation of the covariance matrix s inverse leads to a qualitative test decision that is the opposite of what the pricing errors errors magnitudes would suggest. Therefore, a more robust measure to assess the fit of the linear model is the OLS cross- 9 Full details are available upon request. 11

12 sectional coefficient of determination, Ni=1 R 2 ˆα 2 i = 1 Ni=1 R 2, (17) i where R i = 1 Tt=1 R e T i,t 1 ( Ni=1 1 ) Tt=1 R e N T i,t denotes the (cross-sectionally) demeaned (average) excess returns, and ˆα i represents the (cross-sectionally) demeaned pricing errors. R 2 measures the proportion of the cross-sectional variance of average excess returns explained by the factors associated with the model. This metric can achieve negative values if the model s fit is lower than a model containing just an intercept (see Campbell and Vuolteenaho (2004a) and Maio and Santa-Clara (2017)). This means that the model underperforms a model that predicts constant risk premia in the cross-section of testing assets. An alternative goodness-of-fit measure is the mean absolute pricing error (MAE), which takes in account only the magnitude of the pricing errors (thus, ignoring the relationship to the raw risk premia): MAE = 1 N 4 Main results N ˆα i. (18) i=1 In this section, I present the estimation and evaluation results for the linear macro model presented in Section Benchmark model I estimate the macro model on each of the six groups of decile portfolios described in the last section: BM10, AG10, M10, ACC10, NSI10, and REV10. To assess the fit of the model for the joint six anomalies, I also conduct an augmented test including simultaneously the 60 portfolios. As a reference point, I present the estimation results associated with the baseline CCAPM, which are presented in Table 3. In the augmented cross-sectional test with 60 portfolios, the R 2 is negative ( 41%), which indicates that the performance of the CCAPM is worse than a trivial model containing just an intercept. The explanatory ratios in the single-anomaly tests are also negative in most cases, which shows that the CCAPM cannot price the risk premia associated with each of these portfolio sorts. The exception is in the estimation with BM10, in which one obtains a positive R 2 (24%), although this level of fit is modest in comparison to other studies that test macro models with portfolios sorted on BM. 10 It is interesting to 10 For example, Yogo (2006) and Lioui and Maio (2014) obtain R 2 above 80% when estimating their macro 12

13 see that the single-factor model is not rejected by the χ 2 -test in most cases (p-values above 5%) despite the large magnitudes of the pricing errors (as indicated by the R 2 and MAE estimates). This arises from a problematic inversion of the covariance matrix of the pricing errors and confirms that this statistic is not a robust test to assess the validity of a given asset pricing model. Regarding the consumption risk price estimates, we can see that most estimates of the RRA coefficient are quite large (between 157 and 213), which represents the equity premium puzzle documented in Mehra and Prescott (1985) and vastly confirmed in the macro-finance literature (see Cochrane (2007)). Next, I estimate the Epstein Zin model, with the results being presented in Table 4. Overall, the two-factor model does not improve substantially the CCAPM. In the estimation with 60 portfolios, the R 2 is again negative (-30%) and the average pricing error is 0.45% per quarter (compared to 0.50% for the CCAPM). Moreover, the two-factor model is clearly rejected by the specification test (p-value of zero). Turning to the single-anomaly crosssectional tests, the explanatory ratios are either negative (estimation with AG10, M10, ACC10, and NSI10) or around zero (estimation with REV10). The exception to this pattern is in the estimation with the BM deciles as indicated by the R 2 of 66%, which represents almost three times the fit obtained for the CCAPM. However, the market risk price estimate is largely insignificant (t-ratio of 0.92). Therefore, these results suggest that the market factor adds little in terms of explaining the cross-section of average returns of the 60 portfolios. The results for the benchmark macro model are presented in Table 5. In the augmented asset pricing test, the three-factor model performs similarly to the Epstein Zin model as indicated by the R 2 and MAE estimates of 13% and 0.44%, respectively. The major improvement relative to the two-factor model occurs in the estimation with the NSI10 and REV10 portfolios as indicated by the positive explanatory ratios. However, this level of fit is relatively modest, especially when the testing assets are NSI10 (R 2 = 19%). 11 On the other hand, in the estimation with BM10 the fit is basically the same as that of the Epstein Zin model. The inflation risk price estimates are negative in all cases and there is statistical significance in the estimation with M10, ACC10, NSI10, REV10, as well as in the augmented test. Given the implied estimates of γ above one, this implies implausible negative estimates for the money-illusion parameter (ε), although only in one case (augmented test) is there statistical significance. Therefore, in the few cases where the inflation factor is priced and helps pricing cross-sectional risk premia (as in the estimation with NSI10 and REV10), it does so in a way that is inconsistent with the underlying theoretical model. We can also observe that the implied estimates of ψ are largely insignificant in all cases. models (that contain consumption growth as one of the factors) on the traditional Fama French 25 size-bm portfolios (with quarterly returns). 11 To put the results for the REV10 portfolios in perspective, Lioui and Maio (2014) obtain R 2 above 60% when estimating their macro model on portfolios sorted on size and return reversal (with quarterly returns). 13

14 Overall, the results of this subsection do not provide much support for the three-factor macro model and also do not suggest that the inflation factor helps pricing cross-sectional equity risk premia. 4.2 Restricted macro model In this subsection, I estimate a restricted version of the macro model, which is based on the standard power utility instead of the recursive utility underlying the benchmark model. Specifically, by setting θ = 1 in the linear benchmark model in Eq. (9), one obtains the following two-factor model: E ( Ri,t+1) e = γc Cov (R i,t+1, c t+1 ) + γ π Cov (R i,t+1, π t+1 ). (19) The risk price estimates are given by γ c γ, γ π ε (γ 1), (20) and the implied structural parameters can be retrieved as follows: γ = γ c, ε = γπ. (21) γ c 1 The results for the restricted model are presented in Table 6. We can see that the performance of the two-factor model is very close to that of the benchmark model. Specifically, in the estimation with 60 portfolios, the R 2 and MAE estimates are 16% and 0.43%, respectively. These estimates are also quite similar across the two models in the single-anomaly asset pricing tests. The main difference is that the two-factor model does not pass the specification test in the estimation with the 60 portfolios. Similarly to the benchmark model, the estimates for γ π are negative in all cases, although only in one case (estimation with NSI10) is there statistical significance. Hence, these results suggest that the inflation risk factor seems to be less relevant than in the benchmark model when it comes to explaining most of these anomalies. The implied estimates for ε are negative, but largely insignificant, in all cases. In sum, the evidence from this subsection suggests that the main conclusions driven for the benchmark model poor statistical performance and implausible risk price estimates also hold for the restricted model that excludes the market factor. Hence, the assumption of intertemporal recursive utility, as opposed to time-separable power utility, is not critical in driving the asset pricing implications of the macro model, and in particular, the inflation 14

15 factor. 4.3 Sensitivity analysis In this subsection, I conduct several robustness checks to the results presented above. First, I estimate the model for a sample that ends in This enables one to assess the impact of the financial crisis, and the resulting spike in stock return volatility, in the performance of the macro model. The increased stock return volatility has been more evident in some dimensions of the cross-section of stock returns. 12 The results are displayed in Table 7. Overall, the performance of the benchmark model is similar to the one registered for the full sample. The main difference is in the estimation with the NSI10 and REV10 portfolios: the fit of the three-factor models improves significantly as indicated by the R 2 of 46% and 54% for NSI10 and REV10, respectively. However, as in the benchmark case, the inflation risk price estimates are significantly negative, which implies a negative estimate of ε (although insignificant) in the estimation with REV10. In the estimation with NSI10, both the consumption risk price and implied RRA estimates are also negative (although not statistically significant). Hence, the increased fit of the model comes at the cost of implausible risk price and preference parameter estimates. Second, I use the beginning-of-period timing convention for consumption. In this case, asset returns between t and t + 1 are paired with the consumption growth and inflation rate between t + 1 and t + 2. Several studies use this alternative timing convention because the lead consumption growth series is more correlated with stock returns than the current consumption growth series, leading to a better explanation of the equity premium puzzle and cross-sectional risk premia (see Campbell (2003), Yogo (2006), and Savov (2011), among others). The results are presented in Table 8. We can see that the model s performance improves considerably when it comes to pricing the M10, NSI10, and REV10 portfolios, as indicated by the explanatory ratios around or above 65%. However, the estimated risk prices and implied structural parameter are largely inconsistent with the theoretical model. Specifically, in the estimation with NSI10 and REV10, the estimates of γ π are significantly negative, as in the benchmark case. On the other hand, in the estimation with the momentum portfolios, the market risk price is significantly negative and the consumption risk price has a very large magnitude. This implies an implied risk aversion estimate above 400 in the estimation with M10. Hence, the increased fit of the model for these three anomalies comes again at the cost of implausible risk price estimates. When it comes to pricing the BM10 portfolios, the explanatory ratio declines by more than half in comparison to the benchmark case, while the inflation risk price is positive but strongly insignificant. In the augmented test 12 For example, there was a momentum crash in 2009 (see Barroso and Santa-Clara (2015) and Daniel and Moskowitz (2016)). 15

16 with 60 portfolios, the performance of the model is even slightly lower than the benchmark model with an R 2 of 26% and an average pricing error of 0.47%. Overall, using the alternative factor definitions does not alter the main qualitative results obtained for the benchmark macro model. Third, I estimate the macro model without imposing the restriction that the zero-beta rate equals the risk-free rate. Specifically, I estimate the following specification, E ( Ri,t+1) e = γ0 + γ c Cov (R i,t+1, c t+1 ) + γ π Cov (R i,t+1, π t+1 ) + γ w Cov (R i,t+1, r w,t+1 ), (22) where γ 0 denotes the zero-beta rate in excess of the average risk-free rate (three-month T-bill rate). The estimation results are reported in Table 9. We can see that the excess zero-beta rate estimates are both statistically and economically significant in all cases, varying between 2% per quarter (estimation with BM10) and 8%(estimation with M10). This suggests that there is clear misspecification in the macro model, that is, there are missing risk factors. The inflation risk price estimates (and implied estimates of the money-illusion parameter) assume both positive and negative values, but are strongly insignificant in all cases. Further, the explanatory ratios are considerably higher than in the benchmark estimation with no intercept, varying between 57% (augmented test) and 88% (estimation with AG10). In most cases, this stems exclusively from the inclusion of the intercept in the pricing equation. Specifically, in the estimation with AG10, M10, ACC10, and the 60 portfolios the benchmark results (with constrained zero-beta rate) indicate that the macro model does not explain any cross-sectional dispersion in risk premia (as indicated by the negative R 2 estimates). Hence, the positive R 2 in the specification with an unrestricted zero-beta rate indicates that the model only picks the cross-sectional average risk premium. These results illustrate well the danger of estimating a linear factor model with an intercept and not showing the model s performance when the intercept is excluded (a common practice in the literature): in some cases (as in this study), one might erroneously conclude that the model has a large explanatory power for the cross-section of stock returns (by looking exclusively at the performance of the unrestricted specification) when it fact it only explains the cross-sectional mean (that is, the model drives the level or market factor). Fourth, the quarterly market factor is included in each cross-sectional asset pricing test. This is consistent with the prescriptions in Lewellen, Nagel, and Shanken (2010), who advocate that the traded factors in a model (the market factor in this case) are included in the set of testing assets. I use the quarterly market factor available from Kenneth French s data library. The results displayed in Table 10 are very similar to the benchmark case. Specifically, in the estimation with the 60 portfolios the R 2 and MAE estimates are 13% and 0.44%, respectively, sensibly the same as in the benchmark test. Therefore, including 16

17 the market factor in the menu of testing assets has a negligible effect in the performance of the macro model. 5 Euler equations In this section, I estimate the Euler equations associated with the original non-linear macro model derived in Section 2. From an econometric perspective, the linear specification estimated in the last section is more appropriate than the corresponding Euler equation since the GMM estimation of the linear model might have better finite sample properties. This is especially relevant when the number of testing assets is relatively large, as in the estimation with the 60 portfolio returns. More importantly, the linear specification provides a better intuition of the model s explanatory power for the cross-section of excess stock returns. However, it is possible that there is a significant approximation error associated with the linear pricing equations, which may imply that the pricing errors and implied preference parameter estimates differ substantially from those obtained in the estimation of the Euler equations (see Lettau and Ludvigson (2009) for further discussion). 5.1 Econometric framework There are three preference parameters to estimate from the data: the coefficient of relative risk aversion (γ); the money illusion parameter (ε); and the elasticity of intertemporal substitution (ψ). The objective of this analysis is to assess whether the estimation of the Euler equations associated to the macro model originates lower pricing errors and more economically plausible estimates of the preference parameters than the standard consumption and Epstein Zin models. I estimate the Euler equations corresponding to the macro asset pricing model by firststep GMM, where the identity matrix is used as the weighting matrix. The set of moment conditions corresponds to the Euler equations associated with excess stock returns: 0 = E ( Q t+1 R e t+1), (23) where 0 is a vector of zeros and R e t+1 is a vector of excess returns. 13 Thus, the number of orthogonality conditions is N, where N is the number of excess returns being priced. 14 I compute the asymptotic test of overidentifying restrictions to test the null hypothesis 13 The time discount factor, δ, is not included in the SDF since it is not identified in the moment conditions associated with excess returns. 14 I do not estimate an Euler equation for the real risk-free rate since the focus of this paper in in explaining equity risk premia. 17

18 that the orthogonality conditions are satisfied (J-statistic), g T (b) Var[g T (b)] g T (b) χ 2 (N K), (24) where b denotes the vector of parameter estimates; g T (b) is the vector of stacked moments; and Var[g T (b)] denotes a generalized inverse of the moments covariance matrix. This statistic is distributed as χ 2 (N K), where K is the number of parameters to be estimated (K = 3 in the benchmark model). The mean absolute error, MAE, is given by MAE = 1 P N g i,t (b), (25) i=1 where g i,t denotes the error associated with the ith moment condition. I use asymptotic heteroskedasticity-robust standard errors to compute both the t-statistics associated with the structural parameters and the covariance matrix of the errors from the moment conditions. Since the signs of the structural parameters are constrained by theory, I use one-sided p-values similarly to the estimation of the linear specification. The formulas for the GMM standard errors are provided in Appendix A. In the non-linear estimation of the Euler equations, I use the Newton-Raphson numerical optimization algorithm. To obtain starting values for the first-stage estimation, a discrete grid search approach is used, as in Brav, Constantinidies, and Geczy (2002). In this method, for each specified vector of values for the parameters, (γ, ε, ψ), the GMM objective function, g T g T is evaluated, and the parameter values that minimize the objective function are obtained. The range of values for γ is (0, 1,..., 400); for ε the interval is (0, 0.05,..., 1); while the corresponding interval for ψ is given by (0, 0.05,..., 2). Using this two-step approach to obtain the parameter estimates helps one in avoiding local (instead of global) optimal solutions. 5.2 Empirical results As in the estimation of the linear model in the last section, I start by estimating the Euler equations associated with the baseline consumption model, Q t+1 = ( Ct+1 C t ) γ, (26) 18

19 as well as the two-factor Epstein Zin model, Q t+1 = ( Ct+1 C t ) 1 γ θ R θ 1 w,t+1, (27) which serve as the references for the benchmark three-factor macro model. The estimation results for the consumption model are presented in Table 11. The average pricing errors are quite large in magnitude (above 1% per quarter in most cases) and clearly below the corresponding estimates associated with the linear CCAPM. Moreover, the estimates for the risk aversion coefficient vary between 72 and 80, which are substantially below the corresponding estimates in the linear model. Unlike the linear model, the nonlinear consumption model is rejected by the specification test in most cases. 15 These results suggest that the performance of linear and non-linear consumption models can differ in a non-trivial way. The results for the Epstein Zin model are displayed in Table 12. The fit of the two-factor model is substantially better than the baseline consumption model: In the estimation with the 60 portfolio the MAE is 0.34%, which is less than one third the corresponding estimate for the single-factor model. In the single-anomaly tests, the average pricing error estimates vary between 0.24% per quarter (estimation with ACC10) and 0.65% (M10), which indicates a considerable improvement relative to the baseline model. However, the estimates for ψ alternate in sign and are largely insignificant in all cases. In the estimation with BM10, AG10, and REV10, the estimates for γ are around one, although there is no statistical significance. statistically significant. In the other cases, one obtains risk aversion estimates above 80, which are The estimation results for the benchmark macro model are presented in Table 13. In the estimation with 60 portfolios, the performance of the three-factor model is only marginally better than that of the Epstein Zin model with an average pricing error of 0.28% (versus 0.34%). Yet, in some of the single-anomaly tests there is a substantial improvement in fit relative to the two-factor model: in the estimation with BM10, AG10, and M10 the MAE estimates decline by more than 20 basis points by including the inflation factor in the SDF. One also notes that the average errors are substantially smaller than the corresponding estimates for the linear three-factor model in the estimation with AG10, M10, and the 60 portfolios. Hence, the performance of the linear and non-linear specifications can vary in a significant way. Moreover, the macro model is not rejected by the specification test when the testing assets are BM10, AG10, REV10, and the 60 portfolios. However, although the estimates for ε are positive, there is no statistical significance 15 There is wide evidence showing that Euler equations associated with the baseline consumption model (based on power utility) cannot explain asset returns (e.g., Hansen and Singleton (1982), Kocherlakota and Pistaferri (2009), Lettau and Ludvigson (2009), Savov (2011), among others). 19

20 in most cases. The sole exception is in the estimation with M10, where one obtains an (implausible) estimate for the money illusion parameter above one (1.65). The estimates for ψ are also insignificant in most cases, the exceptions being the tests with M10 and NSI Given these results, it is difficult to assess the contribution of the inflation factor for the performance of the non-linear macro model. I also estimate the Euler equations associated with the restricted macro model, which is based on time-separable power utility (θ = 1): Q t+1 = ( ) γ ( Ct+1 Πt+1 C t Π t ) ε(1 γ). (28) The estimation results are shown in Table 14. The major qualitative results observed for the benchmark model also apply for the two-factor model. Specifically, the magnitudes of the average pricing errors are very similar to those in the three-factor model (actually, in some cases the MAE estimates are marginally lower in the restricted model). On the other hand, the estimates for ε are positive, but largely insignificant, in all cases. Hence, excluding the market factor from the SDF does not have a meaningful impact on the performance of the model and on the preference parameter estimates. 6 Conclusions In this paper, I derive a three-factor macro asset pricing model that contains inflation as a risk factor in addition to the standard real consumption growth and excess market return factors. The underlying framework assumes an intertemporal utility of the recursive form (Epstein and Zin (1989, 1991) and Weil (1989)), combined with an intra-temporal utility that depends on both real consumption and nominal consumption (with a Cobb-Douglas specification). The intratemporal utility suits the case of a partially money illusioned investor, who can not totally differentiate nominal from real consumption in his consumption/asset allocation decisions. The degree of money illusion is captured by the money-illusion parameter (ε), which varies between zero and one. There are three preference parameters in the model: the coefficient of relative risk aversion, the money-illusion parameter, and the elasticity of intertemporal substitution, which allows for a one-to-one correspondence between the factor risk prices and the underlying structural parameters. Critically, the three-factor model nests the baseline CCAPM and the two-factor model of Epstein and Zin (1991) as special cases. I estimate a linear version of the three-factor macro model by first-stage GMM with a rich cross-section of stock returns. Specifically, I use decile portfolios associated with six stock 16 The large standard errors of the structural parameter estimates in tests of non-linear asset pricing models is also present in previous studies (e.g., Constantinides and Ghosh (2011), Savov (2011), among others). 20