Asset Pricing with Durable Goods and Non-Homothetic Preferences

Transcription

1 Asset Pricing with Durable Goods and Non-Homothetic Preferences Michal Pakoš First Draft: Sep 5th This Draft: April 5th, 3 Abstract The purpose of this paper is to understand the pattern of the demand for the durable goods over time and how that relates to and helps explain asset pricing puzzles. The conclusion of the previous literature is that the relative demand changed a lot in response to a pure substitution effect, that the marginal rate of substitution is not volatile enough to pass Hansen-Jagannathan bounds and that there is not enough cross-sectional variation in the covariances between the excess returns on assets and the discount factor. I show how a correct modelling of income vs. substitution effects changes all that. The substitutability between non-durable consumption and the services flow from the stock of durables is very low, in stark contrast to results of the previous literature, which magnifies risk premia. It was the falling price of durables that raised the real income, which lead to a rise in the consumption of durables - there is an important non-homotheticity. The resulting marginal rate of substitution passes Hansen-Jagannathan bounds and also helps account for the cross-section of expected returns on 5 Fama-French portfolios. I interpret the stochastic discount factor of Fama and French (99,993) as a projection of the marginal rate of substitution of the C-CAPM with durables on the closed subspace of returns. I also show that the market price of risk varies and that the co-integrating residual tracks the longer horizon returns on 5 Fama-French portfolios with R s as high as 44%. Working Paper, Graduate School of Business, University of Chicago. This is a substantionally revised version of my paper submitted at the end of spring quarter to satisfy finance requirements at GSB, U of Chicago. I am grateful to my advisors John Cochrane and John Heaton for their valuable comments. I have also benefitted from helpful comments of George Constantinides, Max Gillman, Lubos Pastor, Ruy Ribeiro, Tano Santos and Brown Bag seminar participants at GSB, University of Chicago. Remaining errors are mine. mpakos@gsb.uchicago.edu

2 Introduction This paper proposes a novel mechanism to explain the quantitative asset pricing puzzles. It is built around the blend of two ideas not considered in the asset pricing literature before, namely, non-homotheticity and low substitutability between non-durable consumption and another good, or possibly a multiplicity of goods. It offers a very simple and intuitive alternative to the various generalizations of the canonical C-CAPM, which ranges from relaxing the timeseparability of the preferences (Constantinides (99), Heaton (993, 995), Campbell and Cochrane (999)), expected utility assumption (Kreps and Porteus (97x), Epstein and Zin (9xx)), market incompleteness (Constantinides and Duffie (996)), contract enforceability (Alvarez and Jermann (), Lustig ()), prospect theory (Barberis, Huang and Santos ()), among others. In this paper I develop this idea by considering the other good to be the flow of services from the stock of durables. Preferences that exhibit low substitutability between non-durables and services flow are Leontief preferences. By their nature they translate the small nondurable consumption risk into large risk premia. Intuitively, in recessions the non-durable consumption and durables investment both fall, but the stock of durables rises. As a result, the nondurable consumption and the flow of services depart from the optimal proportion which is very costly for the consumer. An influential study by Lucas (987) of the welfare cost of the variability of non-durable consumption estimated the cost to be very small. Consumers are not afraid of recessions because their non-durable consumption falls and not surprisingly in the canonical consumption-based CAPM they demand low risk premia on assets. It is the close-to-perfect complementarity between non-durable consumption and the flow of services and the distortion in the optimal consumption basket in recessions what consumers are afraid of. Therefore, stocks that pay off badly in recessions have to offer higher equilibrium risk premia. This mechanism also generates a time-varying market price of risk. I show that the cointegrating residual from the stochastic cointegrating regression forecasts returns on Fama-French portfolios (especially small stocks). Non-homotheticity explains why the ratio of flow of services over nondurables increases over time. The old view has been that consumers optimally substituted to ever-cheaper durables as their relative price was falling. The new view is that the substitution effect (in the sense of Hicks) is very small. Consumers are buying more durables because the fall in the relative price of durables increased their real income. This result is very natural as durables do not have a substitute and thus the Hicksian price effects are small. Most studies have imposed See Ait-Sahalia, Parker and Yogo (3) for an exciting exception

3 homotheticity. Models with homothetic preferences neglect income effects and counterfactually ascribe all changes in the relative demand to a pure substitution effect. Furthermore, because non-durables and services flow are held in nearly fixed proportions, one may eliminate the services flow from the preferences. This is essentially what most researchers have been doing implicitly when they wrote down a preferences specification defined over nondurables only. I conjecture that this near-observational equivalence (durables are still in the budget constraint) explains how macroeconomics could have gone on for years getting the right quantities and still miss the asset prices due to the low volatility of the Lucas-Breeden stochastic discount factor. The ambition of this paper is also to try to unify the equity premium puzzle literature and the cross-section of expected returns literature. Traditionally, models that claim to have explained the equity premium puzzle do not provide evidence as to how their discount factor would price any subset of portfolios. The cross-section of expected returns literature cannot generate quantitatively high risk premia in the time-series. The Lucas-Breeden stochastic discount factor of the canonical C-CAPM is not volatile enough to get inside Hansen-Jagannathan bounds. The cross-sectional variation in covariances between the discount factor and the returns on Fama-French portfolios is also tiny, evidenced by the basically flat plot of the fitted vs. the realized average returns. I show that the C- CAPM with durables is capable to explain the equity premium puzzle and the risk-free rate puzzle with a low γ. The marginal rate of substitution is volatile enough to get inside the Hansen-Jagannathan bounds. In addition, although the J T statistics rejects the model in quarterly data, the plot of average vs. the actual returns is significantly better compared to the canonical C-CAPM. Nearly all studies that claim to have explained the equity premium puzzle do not impose upon themselves the additional restriction that their discount factor actually price some subset of portfolios in addition to the risk-free rate and the value-weighted market return. The results in annual data are also interesting. The discount factor gets inside HJ bounds. The plot of fitted vs. average returns is better than any in the literature. In fact, R s of that plot is about 9%. However, it is still rejected statistically because it is estimated very precisely. In contrast, the pricing errors of the canonical C-CAPM are very noisy, they and their standard errors are an order of the magnitude larger than for CCAPM with durables. I differ from the cross-section of expected returns literature in that my factor is the marginal Some imposed subsistence level but it is not clear a priori how general non-homotheticity such an assumption allows for.

4 rate of substitution - I have a one factor model capable to account for the cross-sectional variation in expected returns on Fama-French portfolios. I do not allow for arbitrary loadings on the discount factor. The only free parameters are preference parameters. In contrast, papers such as Fama and French (99,993), Lettau and Ludvigson (), Lustig and Van Nieuwerburgh(), Piazzesi, Scheider and Tuzel (3), Santos and Veronesi (), and other, are multifactor (often 3 or 4) models, where the factor is not the marginal rate of substitution. I interpret Fama-French (99, 993) linear stochastic discount factor as a projection of the marginal rate of substitution onto the closed subspace of returns. I argue that CCAPM with durables offers a macroeconomic rationalization of the exciting comovement of stock returns unveiled by Fama and French (99, 993). Previous studies such as Eichenbaum and Hansen (99), Ogaki and Reinhart (998) already introduced the services flow from durables. Their agenda however was not asset pricing. Rather, they were interested in estimating the elasticity of substitution between non-durable consumption and the services flow and worked with homothetic preferences (CES aggregator). Pakoš (3) argues that homotheticity biases the estimate of the elasticity of substitution up. Dunn and Singleton (986) assume that the preferences over nondurables and durables are Cobb-Douglas and investigate the term-structure implications of the durability. However, in addition to homotheticity they restrict the elasticity of substitution to be one. Another related paper is Heaton(995) who uses one good, called services which is a weighted average of past nondurable consumption and is a high-tech treatment of durability (at short horizons many goods are durable) vs. habit persistence. Other related recent papers are Piazzesi, Schneider and Tuzel (PST) (3) and Lustig and Van Nieuwerburgh() who instead of durables use housing. They impose homotheticity but as I show later on, their demand equations are misspecified - income effects are crucial for a correct modelling of housing. Recall from the Slutsky equation that the income effects are proportional to the expenditure shares which is largest for housing, and moreover, housing has no substitutes. This is reflected in a biased estimate of the elasticity of substitution (PST estimate it to be.7). Furthermore, PST do not test the Euler equations. They linearize their stochastic discount factor. It seems that they actually argue that elasticity of substitution close to one, Cobb-Douglas preferences, is capable to explain quantitative asset pricing puzzles, the result with which I disagree. Lustig and Van Nieuwerburgh set up a heterogenous-agent economy, but do not solve it and instead postulate a linear discount factor. None of these two papers tests neither the restrictions on the coefficients in their discount factor nor perform a GRS test (or J T test) whether the pricing errors are jointly significantly different from zero. 3

5 Preferences and Asset Prices. Preferences The preferences of the representative consumer are defined over nondurable consumption c t and the imputed services flow s t { } U ({c t, s t }) = E β t u (Ω(c t, s t )) where u (Ω) is an iso-elastic felicity function defined over the consumption index t= u (Ω t ) = γ Ω(c t, s t ) γ () The consumption index Ω(c t, s t ) is a generalized-elasticity of substitution (GES) sub-utility function (Pakoš (3)) Ω(c t, s t ) = { } (a c t ) θ + (( a) st ) η θ θ θ, (θ, η, a) R+ (, ) (3) and displays non-homotheticity. The canonical case of iso-elastic felicity function defined over CES sub-utility index is subsumed as a special case 3. () I assume that the consumer produces the flow of services by a time- and state-independent linear household production function s t = k d t (4) I normalize k = and substitute the services flow s t with the stock of durables 4 d t. I then write the consumption index as Ω(c t, d t ). The elasticity of substitution is defined as a percentage change in the relative Hicksian demand in response to a percentage change in the relative price, ES = log(c t /d t ) log q t and the expenditure elasticity is defined as a percentage change in the Marshallian demand in response to a percentage change in expenditures, η = log c t log e t and η = log d t log e t, where e t is the withinperiod expenditure on the consumption goods. In a related paper, Pakoš (3) interprets the parameter θ as the elasticity of substitution 5 and the ratio η as the ratio of expenditure 3 For example, Dunn and Singleton (986) assume that the consumption index Ω(c t, s t ) is Cobb-Douglas and their implied θ = and η =. Eichenbaum and Hansen (99) and Ogaki and Reinhart (998) relax the restriction θ = but still keep the homotheticity assumption η =. However, that eliminates the income effect from the demand equation and biases the estimate of θ as we ascribe all changes in the relative demand to the substitution effect. 4 I occasionally refer to d t as the flow of services instead of saying that the flow of services is a linear function of d t. 5 This is not exactly correct. See the discussion that follows. 4

6 Figure : Indifference Curves and the Income Expansion Path: Prais-Houthakker (955) model 9 8 Services Flow From Durables Nondurables Flow NOTE - The graph is plotted for the parameter η =.5. elasticities between nondurable consumption c t and durables d t. The special case of θ delivers the non-homothetic case of Leontief sub-utility function, so-called Prais-Houthakker model, first proposed by Prais-Houthakker (955) Ω(c t, s t ) = min {a c t, ( a) η s η t } (5) with the income expansion path s t = s t (c t ) defined implicitly by a c t = ( a) η s η t (6) I plot the indifference curves and the income expansion path in Figure. The preference specification has the feature that both goods are normal and services flow is a luxury good (i.e. income elasticity η is greater than one) and non-durable consumption is a necessity good with income elasticity η less than one. Consumers cannot substitute from non-durables to services flow (in the sense of Hicks) but as their real income rises they choose to consume more services flow from the stock of durables. This view is consistent with the empirical results discussed in subsequent sections. 5

7 . Homotheticity vs. Non-Homotheticity At first sight it may seem that allowing for non-homotheticity is not important and just introduces another parameter. In this section I argue how non-homotheticity leads to a substantial reinterpretation of the results in Eichenbaum and Hansen (99), Ogaki and Reinhart (998), and partly Piazzesi, Schneider and Tuzel (3) and Lustig and Van Niewerburgh (). There is also huge empirical evidence against homotheticity - the income elasticities vary across categories of goods and they probably also depend on income and prices themselves as suggested by their time variation. Houthakker (957) and Houthakker and Taylor (97), and Ogaki (99), using cross-sectional and time-series data, respectively, find empirical support for the Engel s law that the budget share of food declines with the level of wealth. Costa () estimates the income elasticities for food at home.47 in 96-94,.6 in Those for total food are.65 in and.68 in and in Those for recreation are.37 in 97-94,.4 in 97-35, and.8 in Homotheticity of the felicity function u(c t, d t ) eliminates scale effects in that the relative demand c t /d t depends only on the relative price and hence the Engel curves are straight lines. It is an analog to the constant-returns-to-scale (CRS) production function, often used in the theory of the firm. In the context of the parametrization proposed in this paper, it corresponds to the case η =. Eliminating income effects results in ascribing all changes in the relative demand to the pure substitution effect, which biases the estimate of the elasticity of substitution (Pakoš (3)). Slutsky equation decomposes the price effect into a substitution effect and an income effect and it implies for example that the relative demand for durables depends on the relative durables price and the real expenditure (see Deaton and Muellbauer (98), Pakoš (3) and the discussion at the end of the section dealing with the intratemporal first-order condition). Formally d log (c t /d t ) = θ d log q t + (η η ) d log ê t (7) where θ is the elasticity of substitution, and η and η are income elasticities. Setting η = η = yields d log (c t /d t ) = θ d log q t (8) It is an empirical fact that the ratio of durables to nondurables has been increasing steadily and the relative price has been decreasing. Is it because consumers substituted to ever-cheaper durables or because declining relative price increased real income? Homothetic preferences do not even allow you to ask this question. By writing down a homothetic felicity function (i.e. constant-elasticity of substitution) the answer is immediately that it was due to pure substitution. This is often claimed to be confirmed by estimating a big elasticity of substitution. The 6

8 argument is however misleading. The relative demand may change in response only either due to income effect or substitution effect. If we don t allow for non-homotheticity, by construction it can t be income effect. But the relative demand changed a lot and not surprisingly we find a large elasticity. But this is not a proof that the relative demand changed in response to substitution effect. In fact, substitution effect is about compensated changes in demand. It answers the question - what would a consumer do if the relative price changed, holding real income constant - it is about moving along an indifference curve. The language is often misused in practice with people arguing that relative price changed and consumers substituted away - but was it a substitution in the sense of Hicks or was it a response to a change in real income. In fact, in what sense may a consumer substitute from apples to CDs? Even by introspection it is not so unreasonable to think that the Hicksian price effects are very small - people want to buy durables and non-durable in nearly fixed proportions (think of the example of food and refrigerator). They certainly may substitute within a category (i.e. food or refrigerator). Allowing for non-homotheticity (i.e. η ) is an important generalization, not yet considered in the asset pricing literature (see Ait-Sahalia, Parker and Yogo (3) for an exciting exception). The most often used approach to impose non-homotheticity in the macroeconomics has been to consider subsistence levels. The approach advocated in this paper has the advantage that it allows to specify expenditure elasticities explicitly rather than implicitly through subsistence levels. It is also not clear how general non-homotheticity subsistence levels actually allow for. The specification in this paper is very general, the only restriction is that the ratio of expenditure elasticities be kept constant and it is likely to be found useful in future research in macroeconomics in general. In this sense, the paper makes a methodological contribution of finding a particularly convenient mathematical form of non-homothetic felicity function with easily interpretable parameters. It turns out that the answer to the question why the ratio of durables has been changing steadily is absolutely crucial for asset pricing. Introducing income effects produces an unbiased estimate of the elasticity of substitution, we no longer force it to pick the effect of missing income effects. Low substitutability and thus high complementarity between nondurables and durables as reflected in a small elasticity of substitution - the felicity function close to Leontief - has the potential to explain quantitative asset pricing puzzles. The feature of the Leontief preferences is that consumers want to keep durables and non-durables in fixed proportion, holding real income constant. I show that this mechanism enables to translate a small variation in non-durable consumption into a large change in consumer s well-being and thus amplify the risk premia. This also shows why linearization (or log-linearization) of the model may not be such a good idea. Leontief preferences are not differentiable - they have a kink - and hence 7

9 the parameters estimated from the linearized version are nearly impossible to interpret..3 Consumer s Optimization Problem I allow the consumer to trade in i I types of Lucas (978) trees, with the number of each type normalized to one, at price p it. The trees yield each period dividends div it. The consumer s problem is { } max E β t u (Ω(c t, d t )) (9) subject to the budget constraint t= c t + q t x t + i I p it a it+ = i I a it (p it + div it ) + w t () where w t is the labor income (I do not model the labor-leisure choice) and the law of motion for the stock of durables d t+ = ( δ) d t + x t () with δ denoting the depreciation rate of durables. The proposed model of consumer durables is neoclassical in that I do not impose non-negativity constraints on durables investment, so-called irreversible investment. It is an empirical fact that the per-capita durables purchases are always positive and therefore that constraint is never binding in equilibrium. I also do not introduce adjustment costs, gestation lags, transaction costs etc. In equilibrium, it must be true that a it = () and that the demand for the consumption goods c t and x t is such that the goods markets clear..4 Intertemporal First-Order Condition and Asset Prices The marginal utility of non-durable consumption is u c (c t, d t ) = a θ θ c θ t { } (a c t ) θ + (( a) dt ) η θ γ θ θ (3) The intertemporal marginal rate of substitution (or stochastic discount factor (SDF)) is M t+ = β u c(c t+, d t+ ) u c (c t, d t ) (4) 8

10 The first-order condition for the optimal portfolio choice is given by the dynamic Euler equation where R it+ = (p it+ + div it+ ) /p it is an asset s gross return. = E t {M t+ R it+ } (5) Fama and French (99, 993) propose an empirically determined SDF of the form M F F t+ = b + b R mkt,t+ + b SMB t+ + b 3 HML t+ (6) to understand the risk premia on assets. I interpret their SDF as a projection of M t+ onto the closed subspace of returns R t+, namely, M F F t+ = proj(m t+ R t+ ) (7) which follows from = E t {M t+ R it+ } = E t { proj(mt+ R t+ ) R it+ } (8).5 Intratemporal First-Order Condition One has to be a bit careful to derive the first-order condition for the optimal choice between the durables and non-durables. Specifically, I distinguish two cases. Firstly, if the elasticity of substitution is non-zero (really significantly different from zero from the statistical point of view), then it must be true that nondurables c t and the services d t satisfy { ( )} ud (c t+, d t+ ) q t = E t M t+ u c (c t+, d t+ ) + ( δ)q t+ Intuitively, in perfect rental market the value of durables q t is determined as any other asset value by simply discounting the next-period payoff with the marginal rate of substitution M t+. The payoff consists of two terms. The first one is the dividend u d (c t+, d t+ ), expressed in terms of utils, or, u d (c t+, d t+ )/u c (c t+, d t+ ) units of nondurable consumption. The second is the next-period market value of the depreciated durable ( δ)q t+. equation q t u c (c t, d t ) u d (c t, d t ) {[ = β E t ( δ) q ] } t+ u c (c t+, d t+ ) ud (c t+, d t+ ) + u d (c t, d t ) u d (c t, d t ) (9) Reshuffling the previous () and computing the derivatives yields η θ θ ( a) η θ a θ ( s η ) θ t c t {[ = β E t ( δ) q ] } t+ u c (c t+, d t+ ) ud (c t+, d t+ ) + u d (c t, d t ) u d (c t, d t ) () 9

11 It is plausible to assume that the right-hand side is stationary and thus this equation implies that if the series log q t, log c t and log d t are co-integrated, then we can take logs (the model is actually log-linear) on the left and estimate the preference parameters θ and η super-consistently by running a regression in levels (Ogaki(99), Ogaki and Reinhart (998)) log c t = constant + θ log q t + η log d t + ɛ t () I interpret this equation as a conditional Marshallian demand function c t = c t (q t, d t ). This specification differs substantially from Ogaki and Reinhart (998). They impose the restriction that η = and rewrite the previous regression as log (c t /d t ) = constant + θ log q t + ɛ t (3) However, such a regression is misspecified because it neglects income effects and yields a biased estimate of the elasticity of substitution θ >. The same problem is apparent in Piazzesi, Schneider and Tuzel (3) who also impose homotheticity to estimate the demand for housing (relative to nondurable consumption) and obtain θ > as well. Neglecting income effects is especially perilous for goods with large expenditure shares 6 and no substitutes, i.e. housing, durables etc. Secondly, if the elasticity of substitution θ is zero, then the felicity function u(c t, d t ) is not differentiable and we have to use a different argument. Specifically, as θ then the felicity function converges to a CRRA utility defined over the Leontief consumption index, namely, u(c t, d t ) = γ (min {a c t, ( a) η s η t }) γ (4) From the non-satiation 7 of u(c t, d t ) and the properties of the Leontief consumption index we obtain that in the optimum, a c t = ( a) η s η t (5) In reality, the world is stochastic and hence c t and d t may depart for a while after a shock. The previous equation implies that c t and d t are cointegrated and we can estimate η by running a regression (in levels) log c t = constant + η log d t + ɛ t (6) Notice that this is basically the previous specification obtained by imposing the restriction θ =. 6 The Slutsky equation says that the income effects are proportional to the expenditure shares. 7 That is, u c(c t, d t) > and u d (c t, d t) >.

12 Pakoš (3) interprets θ as the elasticity of substitution and η as the ratio of income elasticities. The intuition for this interpretation is simple. For simplicity and without loss of generality, assume that the setting is deterministic. If we think of consumers as renting durables in a perfect rental market and paying the user cost of capital, then the preferences are weakly separable. Weak separability is necessary and sufficient for the second-stage of two-stage budgeting to hold (Deaton and Muellbauer (98)). We can therefore think of the previous equation as a conditional Marshallian demand function. More specifically, the second-stage of two-stage budgeting program yields that the Marshallian demands for the non-durables and services flow satisfy after log-differentiating (Pakoš (3)) d log c t = ε d log q t + η d log ê t d log d t = ε d log q t + η d log ê t where ε ij denote Hicksian price elasticities and η i the income elasticities, ê t is the real expenditure on both consumption goods. Eliminating the real expenditure yields d log c t = (ε ε η) d log q t + η d log d t where I denoted η = η / η. For example, η < means that services flow has income elasticity greater than one and non-durable consumption less than one 8. This is intuitively plausible and is also consistent with the Engel s law. The elasticity of substitution ES between nondurables and services flow is defined as ES = ε ε. We see that, up to the multiple η, ES = θ. Although they are not exactly equal, I interpret θ as a yardstick of substitutability between the goods. For θ we get that ES is very close to θ and they are exactly equal for θ =, in which case there is no substitutability between the goods. One may integrate the previous equation - this is perfectly consistent with the model which is log-linear (see the intratemporal condition above) - and hence the parameters are constant, log c t = constant + (ε ε η) log q t + η log d t 8 Recall the restriction that the average income elasticity must be one. Intuitively, a percent increase in expenditures must be matched by a percent increase in real consumption.

13 3 Empirical Section Figure portrays 4 macroeconomic series - durables purchases, nondurables, durables stock and the ratio of the stock of durables over nondurables, all quarterly. There is a clear secular rise in the consumption of services flow (from durables) over non-durables. The interpretation advanced and empirically supported in this paper is that the rising real income in the U.S. economy enabled consumers to buy more durables. Furthermore, although durables purchases and nondurables are procyclical the durables purchases are always positive. This provides empirical support for not imposing non-negativity constraints on the durables investment x t in the theoretical section. 3. Testing the Euler Equations: Generalized Method of Moments Approach I firstly estimate the parameter vector ( θ, η, γ, β, a ) R 5 + off the asset prices by applying Generalized Method of Moments to the dynamic Euler equation. I perform a first-stage GMM with the weighting matrix W = inv(diag(cov(r))). I estimate the spectral density matrix Ŝ using the Bartlett weight (with 7 lags, sample size T = 5). I plot the realized vs. the fitted average returns by using the formula E T (R) = cov T (m, R) E T (m) (7) The results for quarterly data are reported in Table. As a benchmark, I firstly estimate the canonical C-CAPM (Breeden (979), Lucas (978)). It is well-known that the Lucas-Breeden stochastic discount factor (c t+ /c t ) γ does not do a very good job pricing risky assets (Hansen and Singleton (99, 993), Mehra and Prescott (985) and others). My results are consistent with this literature. The parameter vector (γ, β) is outside the parameter space R+. I statistically reject the model. I plot the fitted vs. realized returns in Figure 8. The plot is flat - there is practically no cross-sectional variation in covariances of the nondurable consumption growth with excess return and the model predicts tiny (i.e. ɛ) risk premia. The volatility of Lucas-Breeden discount factor is small and it does not pass into Hansen-Jagannathan bounds (Figure 5). The results for the C-CAPM with durables are more encouraging. The parameter vector lies in the parameter space. I estimate the elasticity of substitution θ =.3 with std(θ) =.3 and therefore cannot reject the hypothesis that there is zero substitutability between nondurables and the services flow - case of Leontief preferences. This is already a novel result (see Eichenbaum and Hansen (99), Ogaki and Reinhart (998) for estimates with homothetic preferences and Pakoš (3) for how non-homotheticity overturns their result). It illuminates

14 the importance of allowing for non-homotheticity - income effects are crucial and substitution effects are negligible. In fact, I estimate the ratio of income elasticities η =.5339 (std(η) =.9), which implies that the income elasticity of durables is greater than nondurables. On average, the income elasticity must be one and thus we obtain the very intuitive result that durables are luxury goods, with income elasticity greater than one, and nondurables necessary goods, with income elasticity positive but less than one. This is also consistent with the Engel s law (Ogaki (99)). Furthermore, I estimate γ =.53 with quite a precision, its standard error is std(γ) =.9. This contrasts the estimate in annual data which is very noisy (see Table and the section below for more). Although γ is not a yardstick of risk-aversion, I conjecture (but not prove, to be done) that the value function will inherit the concavity of the felicity function and thus W J W W J W γ (8) The intuition for this is as follows. The estimate ˆθ is not significantly different from zero and thus the consumption index Ω is Leontief. In addition, I provide additional evidence based on the intratemporal first-order condition in favor of this. Therefore, from the theoretical section we obtain that Ω(c t, d t ) = min {a c t, ( a) η d η t } (9) In deterministic setup, we get that a c t = ( a) η d η t. In reality, the world is stochastic and hence there may be some error ɛ t. Eliminating durables d t from the preferences yields and the felicity function becomes Ω(c t, d t ) Ω(c t ) = a c t + ɛ t (3) u (Ω(c t )) = γ Ω(c t) γ = γ (a c t + ɛ t ) γ (3) This already looks like a standard model with CRRA preferences, in which case there are already results that the value function will inherit the concavity of the felicity function under certain conditions. But the durables still remain in the budget constraint, that s why the conjecture. The ambition is to show that the coefficient of the relative risk aversion defined in terms of atemporal gamble as a concavity of the value function will be tightly related to the estimated γ and hence small. This with the additional result that the intertemporal marginal rate of substitution M t+ gets inside Hansen-Jagannathan bounds suggests that the proposed generalization of the canonical C-CAPM may be capable to explain the equity premium and risk-free rate puzzles. Interestingly, whereas most studies claim to have explained the equity premium puzzle by calibrating their model and getting inside the Hansen-Jagannathan (HJ) 3

15 bounds, I get inside HJ bounds (Figure 6) with the estimated parameter vector. And the HJ bounds are constructed using 5FF portfolios and hence are tighter than the ones usually considered in the literature (i.e. using only aggregate market return). The J T test of overidentifying restriction (an analogue to GRS test) rejects the hypothesis that the pricing errors are zero. However, this is still informative. The plot of the actual vs. fitted average returns on 5 Fama-French portfolios and the risk-free rate R f looks substantially better than for the canonical C-CAPM. The plot actually shows that the pricing errors are smaller and have lower standard errors than the benchmark model. There is also a statistical issue of whether the asymptotic standard errors are correct for such highly non-linear model. I emphasize that my stochastic discount factor is the marginal rate of substitution. I do not linearize it, in stark contrast to most cross-section of expected returns literature (i.e. Lustig and Van Nieuwerburgh (), Piazzesi, Schneider and Tuzel (3), Santos and Veronesi () and others) 3. Longer-Horizon Results: Annual Data In this section I estimate the canonical C-CAPM and C-CAPM with durables at the yearly frequency. Annual data have the advantage that they are not seasonally adjusted and not surprisingly the model fits a lot better. Using longer horizon returns is also consistent with findings of Marshall and Daniel(9xx), Parker and Julliard (3) and is actually the frequency I used in the original version of this paper (before Parker and Julliard came out). Because annual data leaves me with 38 observations, I estimate the model using both all 5 Fama-French portfolios and a subset of them. I cannot test the model in the first case as I have too many moments relative to the time series (6 moments vs. T=38 observations). Table reports estimated parameter vector for all 5 Fama-French portfolios and for a subset, so that I can test the model. The estimates based on all 5 FF portfolios are more precise. The estimates for the ratio of income elasticities η =.56 with std(η) =. and thus are consistent with the results from quarterly data. The same is true for θ. One parameter that is estimated differently than in quarterly data is γ which is above. However, the estimate ˆγ is very noisy and is estimated with a huge standard error. As a result, it is statistically indistinguishable from its quarterly counterpart. The model with durables is estimated very precisely. In Table 3 I report the pricing errors u T = E T (M t+ R t+ ) and their asymptotic standard errors for canonical C-CAPM and C-CAPM with durables. The pricing errors for C-CAPM with durables are an order of magnitude smaller and have an order of magnitude smaller standard errors. The fact that I 4

16 still reject the model with durables (p-value 3.5%) but not the standard C-CAPM (p-value 5%) reflects the very high precision of the pricing errors of C-CAPM with durables. The canonical model is very noisy! This is evident also from Figure 8. The implied R for the C- CAPM with durables (annualy) is above 9% and so if I follow many papers in the literature and just compare the models based on R I would reject all other models and accept this one. But, of course, R is not a test of an asset pricing model. As i quarterly data, I display the Hansen-Jagannathan bounds for both models in Figure 5. 5

17 3.3 Reconciliation with the Intratemporal First-Order Condition 3.3. Tests for Non-Stationarity I test the null hypothesis that the series log(c t ), log(d t ) and log(q t ) are difference stationary against the alternative of trend stationarity using Phillips-Perron test (I included a constant and a linear time trend). I cannot reject the hypothesis that the data may have been generated by a random walk with drift at 5% significance level. The results are summarized in the Table xx Tests for Deterministic Cointegration Under deterministic cointegration, the same vector which removes stochastic trends also has to remove deterministic trends (Engle and Granger (987)). A. Quarterly Data Following the discussion in the theoretical section, I distinguish two cases. Firstly, I consider the case when the elasticity of substitution θ is non-zero. Then the intra-temporal equation implies that log c t, log d t and log q t are co-integrated and the co-integrating vector (θ, η) can be estimated super-consistently by running the following regression (in levels) log c t = constant + θ log q t + η log d t + ɛ t (3) I obtain θ =.959, η =.599 (Table 7). I compute the Phillips-Ouliaris z ρ and z t statistics. The results are summarized in Table 5. I cannot reject the hypothesis of deterministic co-integration at % significance level (for lags q = 5, 6 in Newey-West correction), see Table 5. Secondly, if the elasticity of substitution θ is zero, then the ratio of expenditure elasticities η may be estimated super-consistently by running the following regression (in levels) log c t = constant + η log d t + ɛ t (33) I obtain η =.54 (Table 7). I cannot reject the hypothesis of deterministic co-integration at % significance level (for lags q = 4, 5, 6 in Newey-West correction), see Table 5. It seems that at the same time (log c t, log d t, log q t ), and (log c t, log d t ) are co-integrated. This is suggestive that θ and the Phillips-Ouliaris test does not have power to distinguish between the case of θ very small and θ =. This justifies why I estimated the parameter vector including θ and η off the asset prices. The conventional approach in the literature (Ogaki (99), Ogaki and Reinhart (998)) would be to estimate the parameters (θ, η) 6

18 super-consistently from the intratemporal condition, substitute them into the GMM objective function and estimate the rest of the parameter vector off the asset prices. The co-integration results however produce a bit conflicting results - should I use (ˆθ, ˆη) = (.9,.599) based on the first regression specification or set θ = and use ˆη =.54? Note however that the parameters as estimated off the asset prices in the GMM section (ˆθ, ˆη) = (.3,.5339) in quarterly data (Table ), and (ˆθ, ˆη) = (.,.5649) using all 6 assets and (ˆθ, ˆη) = (.5,.56) using the subset of assets, both in annual data (Table ). This is consistent with the case where θ is small and η ( ). Furthermore, if it is really true that (log c t, log d t, log q t ) are co-integrated then it shouldn t matter if we estimate the regression as log q t = const + const log c t + const log d t + ɛ t (34) instead and apply the Phillips-Ouliaris co-integration test to the residual ɛ t. However, as Table 5 shows, this specification accepts the null of no co-integration! I interpret these results as providing evidence in favor of the hypothesis that θ is very small (of the order.) and η is about.5. I therefore conclude that the parameter vector as estimated in the GMM section off the asset prices is consistent with the intratemporal condition. B. Annual Data As in quarterly data, I distinguish between the two cases. The estimates of the co-integrating vectors is reported in Table 7. I reject the hypothesis of deterministic co-integration at % level. This is surprising given that the series seem to be co-integrated at % level in quarterly data. Either the critical values are inappropriate as they are for sample size T = 5 whereas the actual one is T = 39. Or, the series are stochastically co-integrated with a small trend in the preference parameters, the Phillips-Ouliaris test does not have power against this alternative, and the trend becomes more important in annual data where the time variation of the series is significantly lower. 7

19 3.4 Time-Series Predictability of Long-Horizon Returns on 5 Fama-French Portfolios In this section I unveil the novel result that the co-integrating residual ɛ t from the co-integrating regression log c t = constant + η log d t + ɛ t (35) forecasts returns on 5 FF portfolios, mostly the small stocks. Technically, for each year I use the 4th quarter residual from the regression estimated in quarterly data as those are cointegrated at % level (see the discussion above also) and thus ɛ t is stationary 9 I run predictive regressions for horizons (in years) h=,,3,4,5 r t r t+h r f t+... rf t+h = α + β ˆɛ t + error t+h (36) where r denotes the log of gross return and r f is the log of risk-free rate. I report the results in the Table 8 in the Appendix. The R s in general rise with the horizon. The most predictable seem to be the smallest stocks SB-SB5 where R s reach levels above 3%. Secondly, the distressed stocks (stocks with a large book value relative to the market capitalization) SB5, SB5 and S3B5 are predictable with R s as high as 44.8%, 43.4% and 9.7% and in fact for the two smallest categories of stocks the distressed stocks SB5 and SB5 seem to be the most predictable. Interestingly, the largest stocks S5B-S5B5 are not very predictable which implies that after value-weighting the results do not contradict the predictability literature where R s for the aggregate market above 5% are rare. This result uncovers an exciting evidence that the equilibrium risk-premia on small stocks and distressed stocks are related to the distortion of the investor s consumption basket in recessions as measured by the residual ɛ t. As shown above, the data are suggestive of Leontief preferences in which case consumers want to choose a c t = ( a) η s η t but because of unanticipated shocks these two quantities depart, with ɛ t measuring by how much. It is well-known that there are econometric problems with interpreting long-horizon regressions. Firstly, the regressor ˆɛ t is persistent, but although it is predetermined it is not exogenous. Nelson and Kim (993) and Stambaugh (999) warn that this creates a small sample bias in favor of finding predictability. Secondly, Valkanov () shows that t-statistics do not converge to a well-defined distribution when the forecasting horizon is a non-trivial fraction of the sample size. I address this issue by plotting time-series of the actual 4-year long-horizon 9 If you run the regression in annual data, the results will be about the same. See also the discussion in the intratemporal first-order condition section where I discuss co-integration tests. 8

20 return and the predicted one ˆα + ˆβ ˆɛ t in addition to relying on t-statistics and R s. Note that even though we may worry that the t > test and R magnitude comparison may be inappropriate, the forecast tracks the long-horizon return rather closely, especially for the stocks SB-SB5. I interpret this result as another evidence in favor of predictability. Even though we may not believe the R s and worry that t-statistics are uninformative, it is unlikely that a spurious result would deliver such a good predictive plot. 9

21 4 Model Intuition The paper proposes a novel mechanism to generate risk premia in line with the empirical evidence. Investors care about the composition of their consumption basket and dislike when their consumption of non-durables and the services flow from durables get out of line from the optimal proportion. Specifically, non-durable consumption c t provides utility only together with the services flow. Preference specification to deliver this is Leontief preferences, which naturally generate a recession factor. This is in line with the empirical evidence found in the paper, namely, that the consumption index has the form Ω(c t, s t ) = min {a c t, ( a) η s η t } Recessions are times when both the durables investment x t and non-durables c t fall. However, investment x t does not fall to zero, that would imply zero sales for U.S. manufacturing - really a super-great depression. From the law of motion for the stock of durables d t+ = ( δ) d t + x t+ we see that as long as x t is large enough to replace depreciated stock ( δ) d t, the stock of durables d t+ increases. This is easily visible in Figure xx where I plot the non-durable consumption against the stock of durables. In recessions, c t falls and d t rises and thus they get out of line. This implies large welfare losses for the consumer. Consumers are afraid of stocks because they do badly in recessions which are times when their consumption basket is already distorted. There are two features of the model that are crucial. Firstly, it is the multiplicity of goods. Most studies focus only on non-durable consumption. However, that doesn t fluctuate much and this gives rise to the equity-premium puzzle. Secondly, consumers cannot substitute (in fact, only very little) from c t to d t. However, the relative Marshallian demand d t /c t changed a lot. This implies that the income effects are very important - preferences are non-homothetic. Studies that did introduce another good such as leisure, durables, housing etc. assumed homotheticity. I emphasize that it is the interplay between multiplicity of goods and non-homotheticity that is crucial.

22 5 Conclusion This paper uncovers a novel mechanism to explain quantitative asset pricing puzzles. I show that the substitutability between nondurables and services flow from durables is very low - consumers prefer their consumption of nondurables and the flow of services to be in nearly constant proportions. In other words, recessions are costly not because the consumption of nondurables falls per se but because such a fall distorts the consumption basket. Preferences in addition exhibit non-homotheticity - the relative demand for durables has been rising because the declining relative price of durables raised the real income. The distortion of consumption basket in recessions also implies a time-varying market price of risk. I contribute to, and unify, the equity-premium puzzle and risk-free rate literature, and the cross-section of expected returns literature. Specifically, I show that the marginal rate of substitution passes Hansen-Jagannathan bounds for both quarterly and annual data. I explain the cross-section of expected returns on 5 Fama-French portfolios with just the marginal rate of substitution, in stark contrast to most other studies that log-linearize the model and use multiple factors. The paper also contributes to the predictability literature and shows that the cointegrating residual tracks the long-horizon excess returns on 5 Fama-French portfolios. In addition, I contribute to the empirical macroeconomics literature dealing with the substitutability of goods. I show that a correct modelling of income and substitution effects leads to a substantial reinterpretation of the time pattern of consumption data in the postwar U.S. economy. Finally, the paper makes a methodological contribution to dynamic economics of finding a particularly convenient mathematical form of non-homothetic felicity function with easily interpretable parameters.

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