A Pure Production-Based Asset Pricing Model∗ Frederico Belo† July 15, 2008

Abstract This paper explores the implications of the producers’ first order conditions for asset pricing and provides an explanation of the cross-sectional variation in average stock returns. Theoretically, I derive a stochastic discount factor for asset returns from the equilibrium marginal rate of transformation, the rate at which a producer can transform output in one state of nature into output in another state. I propose a procedure to measure the marginal rate of transformation in the data and I show that the inferred marginal rate of transformation implies a novel macro-factor asset pricing model in which the pricing factors are a function of industry output and prices growth. Empirically, the model captures well the risk and return trade-off of several portfolio sorts, including the 25 Fama-French portfolios sorted on size and book-to-market. The returns on small stocks and value stocks have a large negative covariance with the marginal rate of transformation, which explains their high average returns relative to big stocks and growth stocks. The estimated preducers’ elasticity of substitution of output across states of nature is high, a result that is in contrast with the assumption of a zero elasticity that is implicitly made in current empirical and theoretical representations of the technology of a firm that operates in an uncertain environment.

JEL Classification: E23, E44, G12 Keywords: Production-Based Asset Pricing, Production Under Uncertainty, Macro-Factor model, Cross-Sectional Asset Pricing, Size Premium, Value Premium ∗

This paper is based on my PhD thesis at the University of Chicago. I am very grateful to the members of my dissertation committee, John Cochrane (Chair), John Heaton, Monika Piazzesi and Pietro Veronesi for their constant support and many helpful discussions. I have also benefited from comments by Andy Abel, Murray Carlson (WFA discussant), Hui Chen, George Constantinides, Bob Goldstein, João Gomes, François Gourio, Luigi Guiso, Lars Hansen, Urban Jermann, Boyan Jovanovic, Christian Julliard, Stavros Panageas, Andrew Patton, Ioanid Rosu, Nikolai Roussanov, Nick Souleles, Maria Ana Vitorino, Zhenyu Wang, Amir Yaron, Motohiro Yogo, and seminar participants at the University of Chicago, University of Pennsylvania (Wharton), Central Bank of Portugal, Imperial College, London School of Economics, London Business School, Federal Reserve Bank of New York, University of Illinois Urbana-Champaign, University of Minnesota, Universidade Nova de Lisboa and the WFA 2007. I also thank Eugene Fama and Kenneth French for making their datasets available as well as João Gomes, Leonid Kogan and Motohiro Yogo for sharing with me their data. I gratefully acknowledge the financial support from Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology). All errors are my own. † Assistant Professor, University of Minnesota, Carlson School of Management. Contact: [email protected]. Web page: http://www.tc.umn.edu/~fbelo/

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The goal of this paper is to do for the production side the exact analog of the consumption-based asset pricing paradigm. In a consumption-based model we use the consumers’ first order conditions to recover a discount factor for asset returns from the equilibrium marginal rates of substitution. The goal of a pure production-based model is to use the producers’ first order conditions to recover a discount factor from the equilibrium marginal rate of transformation, the rate at which a producer can transform output in one state of nature into output in another state. As we need a utility function to measure marginal rates of substitution from consumption data, in this approach we need to specify a production function to obtain marginal rates of transformation from production data. Alas, standard representations of the technology of a firm that operates in an uncertain environment don’t let us do that. In standard representations, there is nothing the producer can do to transform output across states of nature and thus the marginal rates of transformation are not well defined. To see this, consider a typical production function of the form Y (s) = (s)F (Kt )

(1)

where Kt is the input (chosen today), Y (s) is the output and (s) is an exogenous productivity level, which are a function of the state of nature s (tomorrow). The producer can only transform output today into output tomorrow in fixed proportions across states of nature. In order to produce more in one state (by increasing the use of the input Kt ), it must produce more in all the other states as well. Thus standard representations of the technology are Leontief across states of nature, as illustrated in the left panel of Figure 1. The bold lines in this figure represent the production possibilities frontier generated by this standard technology for a given amount of inputs Kt . By definition, the marginal rate of transformation is given by the slope of the production possibilities frontier at a given point such as A. Since there is a kink in the production possibilities frontier, the marginal rate of transformation is not well defined.1 [Insert Figure 1 here] To address this issue, I consider an alternative representation for the firm’s technology in which the producer can transform output across states of nature. The technology that I use was first proposed theoretically in Cochrane (1993) but it has not been explored empirically. In this representation, the producer has access to a standard technology such as (1) but where the input set is augmented by a constrained choice of the state-contingent productivity level (s). With this additional choice, the producer can choose to produce more in high-value states at the expense of producing less in low-value states. The production possibilities frontier across states of nature is then smooth (differentiable) and thus the marginal rates of transformation are well defined, as illustrated in the right panel of Figure 1. 1

This result also holds in more general representations of the technology in which some inputs such as capital or labor utilization are allowed to be adjusted after the state of nature is realized. Naturally, once a state of nature is realized, no transformation of output across states is possible by definition.

2

I consider the production decision problem of a producer that has access to the smooth production technology described above and chooses its inputs in order to maximize the contingent-claim value of the firm. The producer is competitive and takes as given the relative price Pt of the good produced by the firm as well as the market-determined stochastic discount factor Mt to value its cash-flows. As I show in section 2.3, the first order conditions with respect to the state-contingent productivity level, and hence for output Yt , can be written as Mt = φ

µ

Pt−1 Pt

¶µ

Yt Yt−1

¶α−1

θ−α state-by-state, t

(2)

where α, φ are parameters of the firm’s technology and θt > 0 is a state-contingent technological parameter that describes the ability of the firm to produce in each state of nature. The technology is discussed in detail in section 2.1. Equation (2) is the crucial prediction from the theoretical model that I explore in the empirical work. This equation states that in order to maximize the contingent claim value of the firm, the producer equates the stochastic discount factor Mt to the marginal rate of transformation state-by-state. Thus with this condition we can recover the stochastic discount factor in the economy from the producers’ first order conditions without any information about preferences in the same way that we recover a discount factor in the consumption-based model from the consumers’ first order conditions without any information about the technologies. This equation summarizes the goal of a pure production-based approach to asset pricing. The marginal rate of transformation in (2) depends on the unobserved state-contingent technological parameter θt and thus it cannot be taken directly to the data, an issue that lead Cochrane (1993) to abandon this approach. In this paper, I solve this identification problem. The crucial assumption that I make is the assumption that the state-contingent technological parameter θt has a factor structure and thus it can be decomposed into a small number of common (across technologies) factors. This assumption provides an additional cross-equation restriction linking the first order conditions of the producers in this economy. As I show in section 3.1, this restriction allows me to identify the unobserved state-contingent technological parameter θt from the observable relative movements in the price and output growth across technologies. In turn, this allows me to identify the marginal of transformation (2) in the data as well. In one of the empirical specifications explored in this paper I focus on the relative movements across only two technologies and I then show that the observed equilibrium marginal rate of transformation implies the asset pricing model E[Rte ] = bp Cov (∆p2t − ∆p1t , Rte ) + by Cov (∆y2t − ∆y1t , Rte ) ,

(3)

where Rte is the excess return of any tradable asset, ∆pit and ∆yit are, respectively, the growth rate of prices and output in technology i = 1, 2, and the factor risk prices bp and by are a function of the parameters of the firm’s technology. Thus I show that a novel macro-factor asset pricing model follows from a pure production-based asset pricing setup. One benefit of having a structural asset pricing model as opposed to an ad hoc asset pricing model is that the estimates of the factor risk prices in (3) have an economic interpretation. In the

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production-based model, the factor risk prices are related to the parameters of the firm’s technology, in particular to the parameter α in (2). This feature allows me to check whether the model fits the data with theoretically plausible parameter values, which is an important diagnostic in the evaluation of any asset pricing model as emphasized by Lewellen, Shanken and Nagel (2006) and many others. In the empirical section, I test the production-based model by examining if the linear asset pricing model (3) captures the observed variation in the cross-section of expected stock returns. I interpret the two production technologies to be the durable goods and the nondurable goods sectors in the US economy. As I show below, this specification is justified by the time series properties of the output growth in these two sectors. To check the robustness of the results to this specification, I also explore an empirical specification that uses information on output and price data from a larger cross section of technologies and I show that the results are similar. This result suggests that the particular choice of the two technologies is not the main determinant of the empirical performance of the model. The main empirical findings in this paper can be summarized in terms of their relevance for asset pricing and for macroeconomics. In terms of asset pricing, I find that the marginal rate of transformation captures well the risk and return trade-off of many portfolio sorts. The returns on small stocks and value stocks have a large negative covariance with the marginal rate of transformation, which explains their high average returns relative to big stocks and growth stocks. The model explains about 75% of the cross-sectional variation in the returns of the 25 Fama-French size and book-to-market portfolios (the standard benchmark portfolios used in the empirical asset pricing literature), about 68% of the cross-sectional variation in the returns of 9 risk double sorted portfolios and about 95% of the cross sectional variation in the returns of 5 industry portfolios. The model is also able to explain about 72% of the cross-sectional variation in the returns of 10 momentum portfolios, but the estimates of the factor risk prices are not consistent with those from the other test assets. In addition, the production-based model outperforms the standard consumption-based asset pricing model with power utility and compares well with the empirical Fama-French (1993) three factor model, when the 25 Fama-French portfolios sorted on size and book-to-market are used as test assets. In terms of the relevance for macroeconomics, the parameters estimates that I obtain here suggests that the producer’s elasticity of substitution of output across states of nature is high. This result is in sharp contrast with the assumption of a zero elasticity that is implicitly made in current empirical and theoretical representations of the technology of a firm that operates in an uncertain environment. Without any ability to transform output across states of nature, I estimate that the observed volatility of the productivity level in the economy should be about 140% (annually). In the data however, typical values of the volatility of the productivity level are an order of magnitude below, suggesting that firms are substantially smoothing their productivity level (and hence output) across states of nature. The paper proceeds as follows. Section 1 discusses the related literature. Section 2 discusses the 4

smooth production technology used in this paper and presents the production-based asset pricing model. Section 3 explains the procedure to measure the marginal rates of transformation in the data, discusses two alternative empirical specifications of the model, derives a linear asset pricing model and discusses the estimation methodology. Section 4 tests the production-based model on the cross-section of expected stock returns of several portfolio sorts, compares the performance of the production-based model with that from other asset pricing models and provides an economic interpretation of the results. Finally, Section 5 concludes.

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Related Literature

The central novel contribution of this paper is to estimate an asset pricing model based on operating marginal rates of transformation. Many successful macro-factor models of the form of (3) have been evaluated, including Chen Roll and Ross (1986), Li, Vassalou and Xing (2003), Cochrane (1996), and Jagannathan and Wang (1996). However, the theoretical motivation in these papers relied on consumers’ first order conditions and the estimated factor risk prices b are estimated as free parameters. One can view the contribution of this paper as providing a theory behind successful empirical work, as well as in extending that empirical work to the precise specification of (3). In turn, the strategies for identifying the state-contingent technological parameter θt in (2) in the data are the central innovation that let this empirical work go through. The work most closely related to mine is the Cochrane’s (1988, 1993) and Jermann’s (2007) work on pure production-based asset pricing as well as the work by Balvers and Huang (2006).2 Cochrane (1993) proposes the smooth production technology that I use in this paper but abandoned the approach since the evaluation of the model required the identification of the state-contingent technological parameter θt in the data. In this paper, I solve this identification problem by extending Cochrane’s framework from an economy with only one aggregate technology to an economy with an arbitrarily number of technologies (industries) and by exploring cross-equation restrictions across the producers’ first order conditions. In addition, this extension allows me to derive a novel multifactor macro asset pricing model that I test on the cross-section of average stock returns. Cochrane (1988) and Jermann (2007) uses two technologies and two states of nature to infer contingent claims prices from production decisions and are able to replicate some interesting stylized asset pricing facts such as the equity premium and the term premium. One of the key features that differentiate my analysis is that I can read the marginal rate of transformation off a single technology and for any number of states of nature. I only use multiple technologies in the empirical work for identification of the marginal rate of transformation in the data, not for spanning. In contrast, Cochrane (1988) and Jermann (2007) are limited to standard spanning results: to identify the marginal rate of transformation they require the number of technologies in the economy to be equal to the number of states (an assumption that Jermann defines as "complete technologies"). 2

Cochrane (1991, 1993 and 2005) and Jermann (2007) discusses the motivation for a production-based asset pricing approach. Cochrane (2005) provides a brief survey of this literature.

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In addition, I provide a set of moment conditions for the cross section of expected stock returns and I test the model in the data. In contrast, Cochrane (1988) and Jermann (2007) evaluate their models through simulation and calibration. Balvers and Huang (2006) derive a stochastic discount factor that is a function of production variables. They find interesting support for an empirical asset pricing model in which an aggregate productivity shock is the single asset pricing factor, and the capital stock relative to a productivity measure is a conditioning variable. Balvers and Huang model however, is not pure productionbased. First, Balvers and Huang recover a stochastic discount factor because of the ability of the consumers, not of the producers, to substitute consumption across states of nature. The neoclassical production function considered in the paper does not have any cross-sectional implications since there is nothing the producer can do to shift production across states of nature. Second, they impose restrictions on preferences by ruling out features such as durable goods, habits or preference shocks. Another important feature that differentiate Balvers and Huang from my empirical work, is the fact that the sign and magnitude of the factor risk prices in Balvers and Huang are not restricted by the theory and thus are estimated as free parameters. In addition, the empirical tests of the model focus only on the cross-section of the Fama-French 25 portfolios sorted on size and book-to-market and not on a broad set of test assets as I do here (see Lewellen, Shanken and Nagel (2006) for a general critique of this procedure). Both facts limit the economic interpretation of Balvers and Huang’s results and the scope of the evaluation of the model. My work departs from typical work in production-based asset pricing that focus on either (i) the study of the firms’ optimal investment decisions or (ii) the study of the asset pricing implications of nontrivial production functions in general equilibrium models. The first group of studies in the production-based asset pricing literature explores the fact that with linearly homogenous production functions (average q equals marginal q), the investment return should be equal to the market return on a claim to the firm’s capital stock. Cochrane (1991) finds that investment returns, a function of investment and output data, are highly correlated with stock returns. Cochrane (1996) and Li, Vassalou and Xing (2003) extend this approach to the study of the cross section of equity returns and Gomes, Yaron and Zhang (2006) incorporate costly external finance into this framework. But once again, the individual production functions used in these studies have no cross sectional asset pricing implications since there is nothing the producer can do to transform output across states of nature. Therefore, the empirical implementation of these studies is based on the hypothesis that the investment returns are factors for asset returns, but this hypothesis is not a direct prediction of these models. Interestingly, as I show in this paper (see section 2.3), investment returns and the stochastic discount factor in the economy do not coincide in general. The second group of studies in the production-based asset pricing literature focus on the endogenous link between returns and production variables. Examples of this approach include Brock (1982), Rouwenhorst (1995), Jermann (1998), Berk, Green and Naik (1999), Boldrin, Christiano and Fisher (2001), Gomes, Kogan and Zhang (2003), Gourio (2005), Gala (2005), Gomes, Kogan and Yogo (2006), Panageas and Yu (2006) and Papanikolaou (2007). These studies are also not pure production-based since the production functions in these models do not have well defined 6

marginal rates of transformation of output across states. Therefore, these models still rely on the consumer’ first order conditions to find marginal rates of substitution or a discount factor across states of nature to obtain the equilibrium conditions. Finally, this paper is also related to a vast literature on production under uncertainty. For example, Chambers and Quiggin (2000) (and references therein) argue that a state-contingent production approach is a realistic description of the production process of firms that operate in an uncertain environment. According to the authors, if the different inputs used in the production process are subject to different productivity shocks, the choice of the mix of inputs is equivalent to a state-contingent choice of output. This approach is not operational however, since we don’t observe all the different inputs used by firms in the data, but it provides theoretical support for the aggregate smooth (across states) technology that I use in this paper.

2 2.1

A Pure Production-Based Asset Pricing Model Technology

Each producer in the economy has access to a production technology that is smooth (differentiable) both across time and across states of nature. I use the analytically tractable specification of a smooth technology proposed in Cochrane (1993). In this specification, a producer produces output Yt using a standard technology of the form Yt = t F (Kt ) where F (.) is an increasing and concave function of the inputs Kt . In order to allow the producer to substitute output across states of nature, the choice set is augmented with the choice of the state-contingent productivity level t , subject to a constraint set. In defining this constraint set, Cochrane (1993) proposes a standard CES aggregator E

∙µ

t

θt

¶α ¸ 1

α

≤1

(4)

where α > 1 is a parameter and θt > 0 is a state-contingent technological parameter that describes the ability of the firm to produce in each state of nature: it is relatively easy to produce (i.e. chose high productivity levels t ) in states with high θt and vice-versa. In this specification, the curvature parameter α controls the ability of the producer to transform output across states of nature. This parameter is related to the elasticity of substitution of output across states parameter which, as standard in a CES aggregator, is defined as σ = (α − 1)−1 . The case α → ∞ corresponds to the standard representations of the technology. In this case, the firm has effectively no ability to transform output across states of nature since the chosen productivity level t must converge to θt state-by-state in order to satisfy the restriction (4). The choice t = θt is always feasible and as α decreases, it becomes easier for the firm to transform output across states.

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Thus this restriction can be interpreted as follows: nature hands the firm an underlying statecontingent productivity level θt , which the firm distorts into a new state-contingent productivity level t in order to produce more in some states of nature (high value states) at the expense of producing less in other states of nature (low value states). Appendix A-1.2 explains the derivation of the constraint set (4) from more primitive assumption about the firms’ technology. The hypothesis that producers have some control over their state-contingent productivity level, and hence its state-contingent output, is plausible. According to the evidence provided in Sheffi (2005) and, more generally, in the literature on operational risk management (e.g. Apgar (2006)), firms respond to uncertainty by adjusting their production practices. More formally, Cochrane (1993) shows that smooth production sets across states of nature can occur when one aggregates standard production functions that are not smooth (Appendix A-1.1 presents an example of this result). Ultimately however, the reasonability of the hypothesis that producers can substitute output across states of nature is an empirical question. By nesting standard representations of the technology as a special case (α → ∞), the technology that I use here allows me to formally test this hypothesis in the data.

2.2

The Producer Maximization Problem

Each producer in the economy, indexed by i = 1, ...N, has access to one technology to produce one differentiated good. Here, I consider the maximization problem of producer i. The producer is competitive and takes as given the market-determined stochastic discount factor Mt , measured in units of a numeraire good, to value the cash-flows arriving at the end of period t. I assume markets are complete, in which case the stochastic discount factor is unique and is equal to the contingentclaim price divided by the probability of the corresponding state of nature. The existence of a strictly positive stochastic discount factor is guaranteed by a well-known existence theorem if there are no arbitrage opportunities in the market (see for example, Cochrane (2001), chapter 4.2). The producer makes its production decisions with the purpose of maximizing the contingent claim value of the firm. The timing of the events is as follows. Output is realized at the end of each period. The producer then chooses the current period investment Iit−1 , the next period state-contingent productivity level it and distributes the total realized output minus investment costs as dividends Dit−1 to the owners of the firm. To derive the first order conditions, it is useful to state the problem recursively. Define the vector ˜ t , ˜θit , P˜it , ) where Kit−1 is the current period stock of state variables as xt−1 = (Kit−1 , it−1 , Pit−1 , M of capital, it−1 is the current period productivity level, Pit−1 = pit−1 /pt−1 is the current period ˜ t is the next relative price of good i with respect to the price of the numeraire good (pt−1 ), M period’s distribution of the stochastic discount factor in units of the numeraire good and ˜ θit and ˜ Pit are the next period’s distribution of the underlying productivity level and the relative price of good i respectively. Let V (xt−1 ) be the contingent claim value of the firm at the end of period t − 1 given the vector of state variables xt−1 . The Bellman equation of the firm is

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V (xt−1 ) =

max

{Iit−1 ,

{Dit−1 + Et−1 [Mt V (xt )]}

it }

subject to the constraints, Dit−1 = Pit−1 Yit−1 − Iit−1 Yit−1 =

it−1 F

1 ≥ Et−1 Kit

(Dividend)

i

(Kit−1 ) ∙µ ¶α ¸ 1

(Output)

α

it

(Productivity Level)

θit = (1 − δ i )Kit−1 + Iit−1

(Capital Stock)

for all dates t. Et−1 [.] is the expectation operator conditional on the firms’ information set at the end of period t − 1 and δ i is the depreciation rate of the capital stock in technology i. In this specification, I ignore capital adjustment costs and the choice of labor inputs since, under some assumptions, these features do not affect the equilibrium marginal rate of transformation across states of nature as I discuss below.

2.3

Recovering the Stochastic Discount Factor from the Producer’s First Order Conditions

The first order condition for the state-contingent productivity level is in Appendix A-2) it it−1

1 1−α

1 α−1

= φit−1 Mt

α α−1

θit

µ

Pit Pit−1

¶

it

is given by (all the algebra

1 α−1

,

(5)

where φit−1 is a variable pre-determined at time t that does not play any role in my empirical work.3 Intuitively, condition (5) states that the firms’ optimal choice of the productivity level (and hence output) is determined by prices and technological constraints. Since α > 1, the firm chooses a higher productivity level in states of nature in which output is more valuable, i.e. high Mt and Pit states, and in states of nature in which it is easier to produce, i.e. high θit states. We can invert the first order condition (5) to recover the stochastic discount factor from the firms’ optimal choice of the productivity level. Rearraging terms, we have Mt = φit−1

µ

Pit−1 Pit

¶µ

it it−1

¶α−1

θ−α it .

(6)

This condition states that in order to maximize the contingent claim value of the firm, the producer equates the stochastic discount factor Mt to the marginal rate of transformation state-bystate. Thus with this condition we can recover the stochastic discount factor from the producers’ 3 The mean of the stochastic discount factor E[Mt ] only affects the risk-free rate. Since I will asses the model using excess return, I ignore the predictions on E[Mt ], and hence the value of φit−1 .

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decisions without any information about preferences in the same way that we recover a discount factor in the consumption-based model from the consumers’ first order conditions without any information about the technologies. Equation (6) is the main prediction from the theoretical model that I explore in the empirical work. For empirical purposes, it is convenient to express the stochastic discount factor in terms of directly observed variables, up to the underlying productivity level θit which I discuss below. Using the fact that the observed output is given by Yit = it F i (Kit ) and that F i (Kit ) is pre-determined at time t, we can express the stochastic discount factor in (6) as ¯ Mt = φ it−1

µ

Pit−1 Pit

¶µ

Yit Yit−1

¶α−1

θ−α it

(7)

¯ where φ it−1 is again a variable pre-determined at time t. Representing the stochastic discount factor in terms of output instead of the unobserved productivity level it simplifies the empirical implementation of the model. Even though the productivity level it could be measured in the usual way as a Solow residual, this procedure would be subject to possible misspecification errors in the functional form of the production function F (.), as discussed in Burnside, Eichenbaum and Rebelo (1996), for example. The first order condition for physical investment is given by I ] = 1, Et−1 [Mt Rit

(8)

where I = (1 − δ i ) + Pit Rit

i it Fk

(Kit )

(9)

is the (stochastic) investment return. This is the standard condition that the investment return is correctly priced. According to this condition, the firm removes arbitrage opportunities from the physical investment and whatever assets the firm has access to. Interestingly, in my setup, the stochastic discount factor (7) is in general not equal to the investment return (9) a result that is in sharp contrast with the empirical implementation of the investment-based models of Cochrane (1996) and Li, Vassalou and Xing (2003) who use the investment return as a proxy for the stochastic discount factor in the economy. I abstract from capital adjustment costs since these costs only affect the investment returns and not the across-states predictions in (6) that I explore. For the same reason, I also abstract from the choice of labor inputs by the firm. If labor is included, the marginal rate of transformation is still given by (7) provided that labor inputs are chosen before the state of nature is realized. In this case, the production function with labor, F i (Kit , Lit ), is still pre-determined at time t and thus the analysis from equation (6) to equation (7) is unchanged. This analysis changes however, if the firm is allowed to adjust labor ex post in response to the realized productivity level. In this case, labor is another source of variation across states of nature and thus, to recover the marginal rate of transformation, it would be necessary to separately identify the productivity level and the

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level of labor inputs in the data. This is an interesting generalization that I don’t pursue in this paper in order to keep the model simple and transparent and to emphasize the role of the choice of the state contingent productivity level in the results. It is important to emphasize however, that allowing labor to adjust ex post is not a substitute for the mechanism to substitute output across states of nature that I consider here, since once a state of nature is realized, it is not possible to transfer output across states by definition. To measure a marginal rate of transformation across states of nature it is necessary to have a decision in which more in one state of nature costs less in another state. The mere option to adjust something ex post does not tell us anything about the rate at which a producer give up one thing in one state to get it in another.4 Substituting the discount factor (7) into the standard asset pricing equation E[Mt Rte ] = 0 for excess returns (Ret ) and some algebra, yields the standard asset pricing condition5 E[Ret ] = −

Cov(Mt , Ret ) . E[Mt ]

(10)

This equation tells us that cross sectional variation in average stock returns is explained by cross-sectional variation in the level of risk. The main proposition of the production-based model is that the risk of any traded asset in the economy can be measured by the covariance of its returns with the marginal rate of transformation. An asset is risky if it delivers low returns in states of nature in which the marginal rate of transformation is high and thus it must offer higher expected returns in equilibrium as a compensation for its level of risk.

3

The Empirical Content of the Production-Based Model

In order to take the model to the data, we need to measure the unobserved underlying productivity level θit in (7). In this section, I propose a procedure to identify this variable in the data. I then derive a production-based linear asset pricing model from the producer’s first order conditions. Finally, I discuss two alternative empirical specifications of the model and the estimation methodology.

3.1

Identification

The simplest way to solve the identification of the underlying productivity level θit problem would be to assume that this variable was constant across states of nature. In economic terms, this assumption specifies that it is not easier to produce in one state of nature relative to another 4

To show this more formally, consider a two-period setup with only ex-post adjustments in labor. The firm chooses the state-contingent labor inputs (Lt+1 ) to maximize its contingent claim value: maxE [Mt ( Lt+1

t+1 F (Lt+1 )

− wt+1 Lt+1 )]

The first order condition for the state-contingent labor is t+1 F 0 (Lt+1 ) = wt+1 state-by-state, where wt+1 is the wage rate. This condition does not provide any information about the stochastic discount factor Mt . 5 To obtain this result use the standard decomposition E [Mt Rte ] = E [Mt ] E [Rte ] +Cov(M, Re ).

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state. This assumption would make the estimation of the model straightforward since in this case, this variable could be estimated as an additional parameter. This is the approach suggested, but not empirically pursued, in Cochrane (1993). Unfortunately, this simple approach generates a marginal rate of transformation with counterfactural variation across states of nature leading to wrong predictions for equilibrium excess returns. This conclusion follows from the observation that with a constant underlying productivity level θit , the first order condition (5) imply that the firm chooses a higher productivity level, and hence a higher level of output, in states of nature with high values of the stochastic discount factor Mt . However, it is well known that states with high values of the stochastic discount factor are associated with less output (recessions), not more. Thus, in order to match the real world, it must be true that the underlying productivity level θit does vary across states of nature and it is higher in states with low values of the stochastic discount factor. This observation follows naturally from a general equilibrium argument and it is not an assumption about the stochastic discount factor: consumers who eat the output would place an higher value for the stochastic discount factor Mt in states of nature with low output. To solve the identification problem, I explore the hypothesis that the underlying productivity level is related across technologies. Accordingly, I specify a factor structure for the underlying productivity level as stated in Assumption 1. This assumption imposes a strong restriction on the model thus providing testable empirical content to it. Assumption 1 (Identification): The unobserved underlying productivity level in each technology i = 1, .., N has the following factor structure J X j ¯ λij ¯ θt αθit = j=1

j where ¯θit = log(θit ), ¯ θt is the j th common productivity factor, with j = 1, .., J, and λij is the loadings of the underlying productivity level of technology i on the common productivity factor j. The loadings for technology 1 are normalized to λ1j = 1, ∀j.

This assumption is motivated by the well documented existence of common factors in production technologies. Aggregate production possibilities are higher for one firm when they are higher for another; that is, business cycles have common components. In addition, some technologies/firms’ are more cyclically-sensitive than others and thus the loadings of each firms’ underlying productivity level on the common components, here captured by the parameter λij , may vary across firms. The assumption of a log linear factor structure with no technology specific idiosyncratic term is naturally a strong assumption but it can be justified as follows. The linear structure can be interpreted as a first order linear approximation of a non linear relationship. More importantly, the absence of a technology specific idiosyncratic term is guided by the empirical implementation of the model. In the empirical section, I identify a technology using industry level data, not firm level data, and at the industry level of aggregation it is more reasonable to assume that the technology specific idiosyncratic term is averaged out across firms. Ultimately however, whether this assumption is 12

plausible or not is an empirical question that I address in the empirical section. Technically, Assumption 1 imposes one additional restriction across the producer’s first order conditions that allows me to infer the underlying productivity level θit in the data, and hence the equilibrium marginal rate of transformation, from the relative movements of observed output and price data across the different technologies. This conclusion is formally stated in Proposition 1. Proposition 1 Under Assumption 1 and with J ≥ 1 common productivity factors, the equilibrium marginal rate of transformation can be identified from output and price data in J + 1 technologies. The marginal rate of transformation is approximately given by "

Mt ≈ κt−1 exp −

J+1 X

[bpi

i=2

(∆pit − ∆p1t ) + byi

(∆yit − ∆y1t )]

#

where ∆pit and ∆yit are, respectively, the growth rate in the price and in the output of technology i0 s good, κt−1 is a variable pre-determined at time t that does not play any role in the empirical analysis, and the factor risk prices bpi and byi are a function of the curvature parameter α and the loadings λij of the individual production technologies (see Appendix A-3 for exact formula). For the one common productivity factor case (J = 1), the two factor risk prices can be written as "

bp by

#

=

"

1/(1 − λ) (α − 1)/(λ − 1)

#

where λ21 = λ. Proof. See Appendix A-3.

3.2

A Linear Asset Pricing Model

In this paper, I focus on excess returns, which allows me to consider a simplified version of the marginal rate of transformation defined in Proposition 1. Since for a vector of excess returns (Rte ) of tradable assets any valid discount factor Mt satisfies Et−1 [Mt Rte ] = 0,

(11)

we can divide the pre-determined variable κt−1 in the equilibrium marginal rate of transformation defined in Proposition 1 into the zero on the RHS of this equation. Equivalently, we can set κt−1 = 1 without changing the pricing errors of the model. This implies that an alternative valid discount factor for excess returns from the production-based model is given by Mt∗

"

= exp −

J+1 X i=2

[bpi

(∆pit − ∆p1t ) + byi

(∆yit − ∆y1t )]

#

(12)

where the factor risk prices b0 s are specified in Proposition 1. This discount factor is proportional to the true marginal rate of transformation in the model: it measures the component of the marginal 13

rate of transformation that varies across states of nature and therefore has pricing implications for excess returns. To simplify the empirical implemention of the production-based model, I aproximate it as a linear asset pricing factor model. Examining the nonlinear model using the nonlinear marginal rate of transformation in (12) produces qualitatively similar results (results available upon request). I linearize the marginal rate of transformation (12) by a first order Taylor expansion around the unconditional mean of the factors. Normalizing the mean of the marginal rate of transformation to one (since the mean is not identified from the estimation of the model on excess returns) and substituting it into the standard pricing equation (10), yields the linear asset pricing model E[Rte ] =

J+1 X i=2

[bpi Cov (∆pit − ∆p1t , Rte ) + byi Cov (∆yit − ∆y1t , Rte )] ,

(13)

where the factor risk prices bpi and bpi are related to the parameters of the production technologies as described in Proposition 1.6 This result shows that a macro-factor asset pricing model follows from a pure production-based asset pricing setup. For the one common factor productivity factor model, the model simplifies to E[Re ] = bp Cov (∆p2t − ∆p1t , Rte ) + by Cov (∆y2t − ∆y1t , Rte )

(14)

where the factor risk prices are "

bp by

#

=

"

1/(λ − 1) (α − 1)/(1 − λ)

#

(15)

Although the exact empirical specification of the linear factor model (13) is new, its specification is closely related to other empirical popular macro-factor asset pricing models such as the Cochrane (1996) and the Li, Vassalou and Xing (2003) investment-based models. These models use the investment growth rate in several technologies as the pricing factors whereas I use output and price growth rates. To the extent that investment growth rates and output growth are highly correlated within technologies, the production-based model can thus be used to understand some of the puzzling empirical findings of these models. For example, Li, Vassalou and Xing (2006) and Cochrane (1996) find that the factor risk prices in their models have typically opposing signs across technologies. Cochrane (1996, table 9) obtains this result when domestic and non-domestic investment growth are used as pricing factors. The estimated pattern of the risk prices is not 6

To obtain this result: a first order Taylor expansion of the marginal rate of transformation (12) around the unconditional mean of the factors and normalizing its mean to one yields %J+1 & J+1 [ p [ y ∗ bi (∆pit − ∆p1t − E [∆pit − ∆p1t ]) + bi (∆yit − ∆y1t − E [∆yit − ∆y1t ]) Mt ≈ 1 − i=2

i=2

Substitute this linearlized marginal rate of transformation into equation (10) and the fact that E[Mt∗ ] = 1, yields the linear asset pricing model (13).

14

explained in these models since the factor risk prices are free parameters not restricted by theory. In contrast, in the production-based model, the factor risk prices b in (13) or (14) are related to the parameters of the technology of the firms. Thus, considering an economy with two technologies (as in Cochrane (1996)) and the asset pricing model (14) implied by the one common productivity factor specification, the opposing pattern in the sign of the investment (output) growth factors is a prediction of the production-based model: the factor risk prices for the output growth factors are by in technology 2 and -by in technology 1.

3.3

Data and Empirical Specification

I use annual data from 1932 to 2006. I identify a technology in the model as an industry in the US economy, and I use the industry level data provided in the National Income Product Accounts (NIPA) available through the Bureau of Economic Analysis (BEA) website. NIPA provides industry level data for four sectors: durable goods, nondurable goods, services and structures. Output in each technology is measured by the real gross domestic product. Data for real gross domestic product for these industries is from the NIPA table 1.2.3, lines 7, 10, 13 and 14. The price data for each industry is also from NIPA, table 1.2.4, lines 7, 10, 13 and 14. The asset market data used in the estimation of the production-based model is standard and the description of the data is provided in Appendix A-5, together with a description of the additional macro data used. The theoretical model described in the previous sections is silent about the number of common productivity factors in the economy. As such, to establish the robustness of the empirical findings, I estimate and test the production-based model under two alternative empirical specifications. In the first specification, I assume the existence of only one common productivity factor in the underlying productivity level in which case the identification of the marginal rate of transformation requires data from two technologies only. This specification is appealing since it makes the analysis particularly tractable by keeping the number of pricing factors small thus avoiding parameter proliferation. It also facilitates the economic interpretation of the estimation results as I show below. The second specification is more general and allows for the existence of an arbitrarily large number of common productivity factors in the underlying productivity level. This extension is also interesting since it allows me to incorporate in the estimation of the model information from a larger cross-section of technologies and also to make the results less sensitive to a particular choice of two technologies. The analysis of this model however, is complicated by the fact that the number of pricing factors in this specification increases with the assumed number of common productivity factors in the economy. To maintain the tractability of the empirical model in this specification, I reduce the number of pricing factors through a principal components analysis. 3.3.1

One common productivity factor specification

Under the assumption of one common productivity factor and as described in section 3.2, the production-based model can be approximated by the linear asset pricing factor model specified in equation (14). This specification thus requires price and output data for two production tech15

nologies. I interpret the first (reference) technology as the durable goods sector and the second technology and the nondurable goods sector in the US economy. Naturally, the nature of the technologies is an additional modeling choice. This specification is convenient since the time series properties of the output growth in these two sectors suggests that the application of Assumption 1 is particularly appropriate for these two sectors. This fact can be seen in Table 1 which reports the descriptive statistics of selected macroeconomic variables across the whole sample period and during expansions and recessions. I define a year as being a recession if there are at least five months in that year that are defined as being a recession by the NBER. Clearly, output growth in these two sectors has a business cycle component: not surprisingly, output growth in both sectors tends to be high in expansions and low in recessions. In addition, the two sectors seems to have different sensitivities to the business cycle: output growth in the durable goods sector is on average 12.32% higher in expansions than in recessions while the output growth in the nondurable goods sector is only 1.35% higher in expansions than in recessions. [Insert Table 1 here] 3.3.2

Multiple common productivity factors specification

Under the assumption of multiple common productivity factor, the production-based model can be approximated by the linear asset pricing factor model specified in equation (13). This specification thus requires price and output data from more than just two technologies. In particular, as specified in Proposition 1, with J > 1 common productivity factors, the marginal rate of transformation can be identified from price and output data in J + 1 technologies. With a possibly large number of common productivity factors, estimating the asset pricing model (13) is not feasible in practice. Output and price growth are highly correlated across sectors (Murphy, Shleifer and Vishny (1988)), which creates multicollinearity problems and makes inference unreliable. In addition, the number of pricing factors, and thus the number of parameters to be estimated, increases with the number of common productivity factors exacerbating the problem. For example, J = 3 common productivity factors imply six pricing factors, namely the relative growth rate of output and relative prices in the three technologies. To overcome these problems I reduce the number of pricing factors through a principal components analysis. This analysis summarizes the information contained in the cross section of relative output and relative prices growth in a small number of orthogonal variables - the principal components - that by construction retain most of the information of the original variables. This procedure also allows me to identify the components of the cross section of output and relative prices growth that are relevant for pricing. Mardia, Kent and Bibby (1979) provide a textbook treatment of principal components analysis. Appendix A-4 explain the procedure in detail. For this empirical specification, I consider all the four NIPA sectors, namely the durable goods, nondurable goods, services and structures sectors. As in the one common factor productivity specificaiton, I specify the durable goods sector as the technology 1 (reference technology). Thus, by using data from four sectors, I’m considering three common productivity factors. Naturally, 16

this analysis can be extended to any arbitrary number of common productivity factors but this extension would require the use of more disagragated sectoral data. As I show in the Appendix A-4, for this empirical specification, the marginal rate of transfomation is well approximated by Mt ≈ 1 − bp PFPCt − by OFPCt

(16)

where PFPCt (price first principal component) is the first principal component of the cross-section of the relative price growth and OFPCt (output first principal component) is the first principal component of the cross-section of relative output growth. Each of these components capture 77% (output) and 64% (price) of the cross-sectional variation of the corresponding variable. In addition, for asset pricing purposes, this approximation seems appropriate as suggested by a series of asset pricing tests which show that only the first principal component of these variables are relevant for pricing (results not reported here but available upon request). Substituting (16) into the standard pricing equation (10), yields the linear asset pricing model E[Re ] = bp Cov (PFPCt , Rte ) + by Cov (OFPCt , Rte ) .

(17)

The main limitation of this empirical specification is that, with only two pricing factors, it is not possible to identify all the underlying technological parameters (α,and the loading λ0ij s) for each technology. This fact limits the economic interpretation of the results since in this case the sign and the value of the factor risk prices b’s are not restricted by the theory.

3.4

Estimation Methodology

To estimate and test the production-based model, I use the standard moment condition g(b) = E

£¡

¢ ¤ 1 − b0 Ft Rte = 0,

(18)

where Rte is a vector of excess returns, 1 − b0 Ft is a linear aproximation of the marginal rate of transformation, b is the vector of factor risk prices that we want to estimate and Ft is the matrix with the (demeaned) factors. This condition fits naturally into a generalized method of moments (GMM) framework (Hansen and Singleton (1982); see Cochrane (2002) for a textbook treatment of GMM). To estimate the factor risk prices, define gT (b) as the sample counterpart of the population pricing errors (18) . If the model holds, then gT (b) = 0. The GMM estimates are formed by choosing the factor risk prices b that minimizes a weighted sum of the sample pricing errors min JT = gT (b)0 WT gT (b) b

(19)

where WT is the weighting matrix. I report second stage (efficient) estimates. In the first stage I use the identity matrix as the weighting matrix, WT = I, while in the second stage I use WT = S −1 where S is the Newey-West estimate of the covariance matrix of the sample pricing errors in the 17

first stage. In the estimation of this matrix, I use two period lags to account for the possibility of time aggregation in output and price data (see Hall (1988) on consumption data). For the linear factor models that I consider here, the minimization in (19) has a closed form solution. The first ˆ2 ) estimates are given by ˆ1 ) and the second stage (b stage (b ˆ1 = (d0 d)−1 d0 E[Re ] b T 0 −1 ˆ2 = (d S d)−1 d0 S −1 E[Re ] b T where d is the covariance matrix between the returns and the factors. Therefore, the first (second) stage GMM estimate that I report is simply an OLS (GLS) cross-sectional regression of the mean excess returns on the covariance between returns and the factors. In this regression, I also include an intercept term thus allowing the model to possibly missprice the risk free rate. Under the null that the model is correct, this intercept should be zero. Testing if this hypothesis is true in the data thus provides one additonal check of the model. Given the estimates of the factor risk prices, I obtain the implied technological parameters λ and α using equation (15) and the correponding standard errors by the delta-method. To test the production-based model, I use the J-test (Hansen (1982)) of overidentifying restrictions. This test is conceptally similar to the standard GRS test of alphas (Gibbons, Ross and Shanken (1989)), applied to GMM cross-sectional regressions. As a diagnostic of the fit of the model, I report two commonly used, albeit imperfect, measures: the cross-sectional R-squared (R2 ) and the mean absolute pricing error (MAE). The R2 is obtained from an OLS regression of the realized excess returns on the predicted excess returns by the model. Since I include a constant in the estimation, the R2 is bounded between 0 and 1. The MAE is obtained by first computing the pricing error of each asset i, αi = E[Rie ]observed − E[Rie ]Predicted , and then take the average across P assets of the absolute value of the pricing errors to obtain MAE= N1 N i=1 Abs(αi ), where N is the number of test assets and Abs(αi ) is the absolute value of the pricing error of asset i. It is important to emphasize that these measures are imperfect and should not be interpreted as a formal test of the model but only as an additional check. In particular, it is the well known that the R2 measure is not invariant to set of test assets used. As shown in Kandel and Stambaugh (1995) and Roll and Ross (1995), a carefull choice of test assets can produce an arbitrarily good or arbitrarily bad R2 on the plots of expected returns versus predicted excess returns. In addition, as Lewellen, Nagel and Shanken (2006) show, an high cross-sectional R2 (or low pricing errors) is easy to obtain and thus represents a low hurdle for asset pricing models, when the set of test assets has a strong common factor structure and the pricing factors have some, even if small, correlation with these common factors.

4

The Production-Based Model in Practice

In this section, I estimate and test the production-based model for the one common aggregate productivity factor and the multiple common productivity factors specifications. In the discussion 18

of the results, I focus most of my analysis of the former specification since, as discussed before, this specification allows for an economic interpretation of the sign and magnitude of parameter estimates. To establish the robustness of the empirical findings, I test the production-based model on several portfolio sorts and I compare the performance of the production-based model with the standard consumption-based model and the empirical Fama-French (1993) three factor model. Finally, I provide an economic interpretation of the results.

4.1

Fama-French 25 Portfolios Sorted on Size and Book-to-Market

I start by examining if the production-based model is able to explain the variation in the average returns of the Fama-French 25 portfolios sorted by size and book-to-market equity, which are the standard benchmark portfolios currently used in the empirical asset pricing literature. The choice of these portfolios is motivated by a large empirical literature who found that size and value premia capture a large fraction of the cross-sectional variation in expected returns. [Insert Table 2 here] The first column in Table 2, Panel A, reports the second stage GMM estimates and tests of the production-based model on these portfolios for the one common productivity factor specification. In addition, it reports the estimates of the factors risk prices and the implied parameters of the technologies in the durable and nondurable goods sectors. The production-based model captures well the cross-sectional variation in the returns of these portfolios with an high cross sectional R2 of 75.3% and low mean absolute pricing error of 1.4% (annually). Importantly, the model is comfortably not rejected by the J− test of overidentifying restrictions (p-value of 92.9% ). The estimates of the factor risk prices suggest that the relative price growth factor is the most important for pricing these assets. Finally, the implied curvature parameter has the correct theoretical sign (α > 1) and the intercept (cte ) in the cross sectional regression is not statistically significantly different from zero, which provides additional support to the model. [Insert Figure 2 here] The top-left panel in Figure 2 provides a visual description of the overall good fit of the production-based model on these portfolios. This figure plots the predicted versus realized excess returns implied by the first stage GMM estimates of the model. The straight line is the 45◦ line, along which all the assets should lie. The deviations from this line are the pricing errors which provides the economic counterpart to the statistical analysis. Here, all portfolios lie close to the 45◦ line. Interestingly, the production-based model is able to price the small-growth portfolio (portfolio 11 in the figure) which is known to be hard to price. D’Avolio (2002) and Lamont and Thaler (2003) suggests that short sale constraints are binding on a typical small-growth stock which creates limits to arbitrage which might explain why several asset pricing models cannot price this portfolio. It is thus interesting that the frictionless production-based model that I consider here can price these stocks. 19

[Insert Table 3 here] To help in the interpretation of the good fit of the production-based model on the 25 FamaFrench portfolios sorted on size and book-to-market, Table 3, Panel A reports the average annual excess returns on the 25 Fama-French and Table 3, Panel B reports the opposite of the cross sectional covariances between the fitted marginal rate of transformation and returns on these portfolios. The pattern of the covariances in Panel B is consistent with the pattern of excess returns reported in Panel A, which explains the good fit of the production-based model on these portfolios. The opposite of the covariances of the value stocks are on average almost four times higher than that of the growth stocks thus explaining the value premium. Small stocks tend to have higher opposite covariances than big stocks thus explaining the size premium. In the evaluation of any asset-pricing model it is important to understand which facts in the data are driving the results. Table 3, Panels C and D reports the covariance of the relative output growth factor and the relative price growth factor with the returns of the 25 Fama-French portfolios. In Panel C, there is some, albeit small, spread in the covariances of the relative output growth factor with the returns on these portfolios, especially along the book-to-market dimension. In Panel D, there is a very large spread in the covariances of the relative price growth factor with the returns of these portfolios both along the size and the book-to-market dimension, and the pattern of the covariances matches that of the average returns reported in Panel A. In turn, this fact helps to explain why the relative price growth factor is the most relevant pricing factor in the estimation of the production-based model on these portfolios. Taken together, these results suggests that the relative price growth factor is capturing most of the cross-sectional variation in the returns of these portfolios, with the relative output growth factor adding some explanatory power along the book-to-market dimension. The results for the multiple common productivity factor specification are reported in the first column of Table 2, Panel B. Overall, the results confirm the ability of the production-based model to capture the cross-sectional variation in the returns across these portfolios, suggesting that the results for the previous specification are not specific to a particular choice of two technologies (nondurable and durable goods sectors). The model is comfortably not rejected by the J− test of overidentifying restrictions (p-val of 96.9%), it has an high cross sectional R2 of 66.5% and low mean absolute pricing errors of 1.6% annually. Interestingly, contrary to the one common aggregate productivity factor specification, both the price and the output pricing factors are statistically significant in this specification, suggesting the the first principal component of the relative output growth factor contains some additional relevant information for pricing that is not contained in the nondurable and durable goods output growth factor.

4.2

Other Portfolios

I now examine if the production-based model is able to explain the cross sectional variation in the average returns of the following additional five sets of portfolios: (i) 9 risk double sorted portfolios, which are value weighted portfolios that are formed based on the "pre-ranking" relative price 20

growth and relative output growth betas of each individual stock; (ii) 10 momentum portfolios; (iii) the Gomes, Kogan and Yogo 5 industry portfolios; (iv) all portfolios together; and (v) all portfolios together except the momentum portfolios. Appendix A-5.2 explains the construction of these portfolios in detail. The choice of these portfolios can be justified as follows. I examine the 9 risk double sorted portfolios since sorting on pre-ranking betas provides a rigorous test for asset pricing models by creating a large spread in the post-formation betas or covariances. Table 4, Panels B and C, shows that this procedure achieves its goal. In Panel B, reading across each column, the ex-post covariance of the high pre-ranking relative price growth beta portfolio is subtantially higher than the ex-post covariance of the low pre-ranking relative price growth beta portfolio. In Panel C, reading across each row, the ex-post covariance of the high pre-ranking relative output growth beta portfolio is subtantially higher than the ex-post covariance of the low pre-ranking relative price growth beta portfolio. In addition, consistent with the hypothesis that these factors are important risk factors, this sorting procedure generates a large spread in average returns. The average excess return of the high price-low output ranking beta portfolio is 7.4% higher than the average return of the high price-low output pre-ranking beta portfolio. Finally, the relationship between average returns and the corresponding covariances with the factors is monotonic across the two dimensions. In addition to the risk sorted portfolios, I examine 10 momentum portfolios because sorting on prior return deciles not only generates an impressive spread in average returns (the difference between the average returns of the winner portfolio and the looser portfolio is 15.7% (annually) in my sample) but this large spread in returns has been difficult to rationalize by current popular asset pricing model. For example, Fama and French (1996) show that momentum is the only deviation from the CAPM that is not explained by the Fama and French (1993) three factors model. Finally, I examine the Gomes, Kogan and Yogo (2007) 5 industry portfolios since these portfolios also generate some spread in the average returns across industries and because the sorting is based on industry classification which is close in spirit to the production-based approach. [Insert Table 4 here] Expanding the set of test assets allows me to address Daniel and Titman (2005) and Lewellen, Nagel and Shanken (2006) critiques. These authors criticize the procedure of focusing exclusively on the 25 Fama-French portfolios in testing asset pricing models. Since the 25 Fama-French portfolios have a strong factor structure (the average R2 of the Fama and French (1993) three factors explains more than 90% of the time-series variation in the returns of these portfolios), these authors argue that any factor that is marginally correlated with HML and SMB will appear successful in explaining the cross section of the 25 Fama-French portfolios. Studying these sets of portfolios addresses this concern by relaxing the tight factor structure of the 25 Fama-French portfolios. Columns two to four in Table 2, Panel A, reports the second stage GMM estimates and tests of the production-based model for each one of these portfolios under the one common productivity factor specification. Column five and six report the estimation results when all assets are considered and when all assets minus the 10 momentum portfolios are considered. Across all test assets, the 21

model is not rejected by the J− test of overidentifying restrictions. In addition, the estimation produces high cross sectional R2 with low mean absolute pricing errors that ranges from 0.19% (annually) for the industry portfolios to 1.6% for the momentum portfolios and are only slightly higher when all the assets (including and excluding the momentum portfolios) are considered jointly. The good fit of the production-based model across these portfolios can also be seen in Figure 2 which plots the predicted versus realized excess returns implied by the first stage GMM estimates on these portfolios. Most of the test assets lie along the 45◦ line, thus generating low pricing errors. [Insert Figure 2 here] In the evaluation of the performance of any asset pricing model, it is also important to study the consistency of the estimates of the factor risk prices (sign and magnitude) across portfolios. Under the null that the model is correct, the estimates of the factor risk prices should be identical. The results reported in Table 2, Panel A reveal that the estimates of the factor risk prices are indeed consistent across all the test assets except across the 10 momentum portfolios. In particular, for the 10 momentum portfolios, the estimated factor risk price for the relative output growth is positive and statistically significant. In turn, this fact implies that the point estimate of the curvature parameter α for these portfolios does not satisfy the theoretically restriction α > 1, implying a negative elasticity of substitution of output across states. The fact that the 10 momentum portfolios requires estimates of the factor risk prices that are different from the other test assets can also be seen by comparing the estimation of the model including all test assets (column six) with the estimation of the model that includes all test assets excluding the 10 momentum portfolios (last column). Excluding the 10 momentum portfolios greatly increases the cross sectional R2 and reduces the pricing errors. The results for the multiple common productivity factor specification reported in Table 2, Panel B are largely consistent with the analysis for the one common productivity factor specification, and thus are ommited here for brevety. The results from this section allows us to conclude that the production-based model is able to explain the cross-sectional variation in the average returns of these portfolios but the model has some difficulties when the 10 momentum portfolios are used as test assets. Despite this negative fact, the overall results presented here are supportive of the production-based model.

4.3

Comparison With Other Asset Pricing Models

In order to evaluate the production-based model, it is also important to compare it to close competitors rather than simply reject or fail to reject it on the basis of statistical tests. In fact, it is not hard to statistically reject any of the current popular models if one uses a sufficiently rich set of test assets or a data sample covering a long period. In this section, I compare the productionbased model with the Lucas (1978) and Breeden (1979) standard consumption-based model with power utility and the empirical Fama-French (1993) three factor model. I include the standard consumption-based model since it is a natural theoretical benchmark for the production-based

22

model. I also include the Fama-French three factor model since this model has been very successful in pricing several portfolio sorts thus also providing an interesting benchmark. Appendix A-6 provides a complete description of these two models. In what follows, I only discuss the results for the one common productivity factor specification since the qualitative conclusions are similar for the multiple common productivity factors specification. [Insert Table 5 here] Table 5 presents the second stage GMM estimates and tests of the three asset pricing models using the 25 Fama-French portfolios sorted on size and book-to-market as test assets. None of the three models is rejected in this data. More interestingly, the performance of the production-based model is comparable to that of the successfull empirical Fama-French three factor model both in terms of cross sectional R2 and mean absolute pricing erros. In addtion, the performance of the production-based model compares favorably to the standard consumption-based asset pricing model. The results for the consumption-based model confirm the difficulty of this model in explaining the cross sectional variation in the returns of these portfolios. The annual mean absolute pricing error of the consumption-based model is 2.25% which is substantially higher than the 1.39% obtained in the production-based model. In addition, the consumption-based model requires an implausibly high coefficient of relative risk aversion of 41.28 to price these portfolios, a re-statement of the equity premium puzzle using cross sectional data.

4.4

Economic Interpretation of the Results

One benefit of having a structural asset pricing model as opposed to an ad hoc asset pricing model is that the estimates of the factor risk prices can be linked to interpretable deep parameters which, in the case of the production-based model, are parameters related to the firm’s technology. In this section, I discuss the economic interpretation of the estimates and results. In this discussion, I focus on the results from the estimation of the production-based model on the Fama-French 25 portfolios sorted on size and book-to-market under the one common productivity factor specification, reported in the first column of Panel A in Table 2. The estimates of the technological parameters (α and λ) reveal interesting information about the characteristics of the technologies in the nondurable and the durable goods sectors. First, the parameter that controls the sensitivity of the underlying productivity level in the nondurable goods sector to the common productivity factor (parameter λ) is positive but smaller than one. Thus, according to this estimate, the underlying productivity level in the two sectors are positively correlated but the underlying productivity level in the nondurable goods sector is less sensitive to the common productivity factor. To the extent that the common productivity factor is closely related to the business cycle, this fact might explain why the output growth in the nondurable goods sector is less cyclical than the output growth in the durable goods sector as reported in Table 1.

23

Second, the point estimate of the curvature parameter α is small and close to one, which is the minimum admissible value for this parameter. Recall that this curvature parameter is related to the elasticity of substitution of output across states, defined as σ = (α − 1)−1 , which is the productionbased analogue of the coefficient of relative risk aversion in the standard consumption-based model. Given the small point estimate of the curvature parameter α, the elasticity of substitution of output in the two sectors is estimated to be σ ˆ = 33.3. Unfortunately, since this parameter is new in the literature, there is no benchmark to compare this value with. In order to interpret the economic magnitude of the estimate of the elasticity of substition of output across states, I examine the properties of the implied time series of the common productivity level and relate it to well documented time series properties of firm specific realized productivity levels or Solow residuals. As discussed in section 2.1, the specification of the technology used in this paper implies that both the underlying and the realized productivity levels should be equalized state by state when the firm does not have any ability to substitute output across state of nature, which corresponds to the case where the elasticity of substitution of output across states is zero (σ = 0). Examining the difference between the properties of the two series provides a measure of how much substitution of output across states firms are doing. As shown in Appendix A-3, equation (22), the log common productivity level in can be inferred from price and output data in the two sectors from £ ¤ ¯θt = (λ − 1)−1 γ NDt−1 − γ Dt−1 − (λ − 1)−1 [∆pDt − ∆pNDt + (α − 1) (∆yNDt − ∆yDt )] .

(20)

where γ NDt−1 and γ Dt−1 are pre-determined variables at time t. The difficulty with this equation is that these variables, in general, vary over time. In order to study the properties of the estimated underlying productivity level, I thus make the additional assumption that the states of nature are independent and identically distributed (i.i.d.) in which case γ NDt−1 and γ Dt−1 are constants over time. Note that this assumption is not required for any of the asset pricing tests reported in the previous sections, and it is only necessary here in order to interpret the results. Finally, because the mean of the underlying productivity level is not pinned down in the estimation of the model on excess returns, I focus my analysis on the innovations (i.e. demeaned values) of the underlying productivity factor. Figure 3, Panel A plots the time series of the innovations in this variable. The shaded bars are NBER recession years. Interestingly, the plot reveals that the common productivity factor tends to be particularly low in recessions and high in expansions. The mean innovation in the log common productivity factor is −0.47 in recessions and 0.17 in expansions, which suggests that recession periods corresponds to the realization of states of nature in which it is difficult to produce. Interestingly, the standard deviation of the innovations in the log common productivity factor is 140%. For comparison, typical estimates of firm level volatilities of the realized productivity level reported in Cooper, Russell, and John Haltiwanger (2005) are around 22%, and estimates at the industry or at the aggregate level are even lower. Thus, these results suggest that the firms have a large ability to substitute output across states of nature. In turn, this allows us to conclude that the estimated elasticity of substitution of σ ˆ = 33.3 is an economically large number, a result that

24

is in sharp contrast with the assumption of a zero elasticity of substitution that is implicitly made in current empirical and theoretical representations of the technology of a firm that operates in an uncertain environment. [Insert Figure 3 here] It is also interesting to examine the time series of the innovations in the fitted marginal rate of transformation. This time series is interesting since it provides information about the realized time series of the contingent-claim prices in the US economy. The estimated time series of the innovations in the log marginal rate is plotted in Figure 3, Panel B. The estimation of this series follows directly from equation (23) in Appendix A-3 assuming, as before, that the states of nature are i.i.d. As expected, state-contingent claim prices, as measured by the marginal rate of transformation, tend to be high during recessions. The mean log innovation in the marginal rate of transformation is 0.69 in recessions and −0.24 in expansions. Interestingly however, the estimated innovations in the marginal rate of transformation reveal recession states that are not captured by the NBERdesignated business cycle recessions dates. For example, in 1976 we observe large innovations in the marginal rate of transformation and hence high contingent claim prices, but there isn’t a single month in this year that is classified by the NBER as a recession. In addition, not all NBER recessions were equally important. According to the production-based model, the recessions in 1938 and 1949 were particularly severe since they correspond to the realization of states of nature with very high state-contingent claim prices. In short, the estimated marginal rate of transformation captures information about recessions that transcends the NBER-designated business cycle recessions dates. Figure 3 also shows that the innovations in the marginal rate of transformation and the innovations in the common productivity factor are almost the mirror image of each other. The correlation between the two innovations is −0.97 and the volatility of the innovations in the common productivity factor (140%) is approximately equal to the volatility in the innovations in the marginal rate of transformation (148%). This result is not surprising given the relatively low volatility of output and price growth in the two sectors when compared with the required volatility of any valid discount factor that prices assets in the US economy. To understand this fact, recall that the marginal rate of transformation in the sector 1 (here, the durable goods sector) in an i.i.d. world can be written as µ ¶ µ ¶ PDt −1 YDt α−1 −α ¯ θDt , (21) Mt = φ PDt−1 YDt−1 which follows from equation (7). From Table 1, the standard deviation of annual output growth in the durable goods sector is approximately 12%. The standard deviation in the relative price growth is around 2.5%.7 In addition, the Sharpe ratio in the US economy in the post-war period is approximately 0.4 which implies that the standard deviation of the discount factor must be at 7

Note that, in this equation, PD t is the relative price of the durable good, with respect to the price of the numeraire good. Here, I consider the numeraire good to be the aggregate consumption basket in which case the CPI for all goods is used.

25

least 40% on annual data.8 To be consistent with these values, and given the low point estimates of the curvature parameter α, equation (21) immediately implies that we need a volatile common productivity factor that is highly negatively correlated with the stochastic discount factor in order to make the model consistent with the observed relatively low volatility of output and relative price growth.

5

Conclusion

I recover a stochastic discount factor for asset returns from the equilibrium marginal rate of transformation, inferred from the producer’s first order conditions and I propose an empirical strategy for measuring the marginal rate of transformation in the data. The central insight of the pure production-based model is that the marginal rate of transformation of output across states of nature is an appropriate measure of risk in the economy. The measured marginal rate of transformation implies a novel macro-factor asset pricing model in which the pricing factors are relative movements in output and price growth across industries (technologies) in the economy. I test the model in the data and the results provide empirical support for a production-based approach to asset pricing. The marginal rate of transformation captures well the risk and return trade-off of many portfolio sorts, including the size and the value premia. The production-based model developed here can be extended in many directions not only to improve its empirical performance, but also to address questions in both the asset pricing and in the macroeconomics literature. The production-based model has some difficulties in pricing the portfolios sorted on momentum. One possible way to address this issue is to explore more complex specifications of the basic smooth production function considered in this paper. For example, incorporating production side features such as externalities, learning by doing, gestation lags, multiple production inputs such as labor, into the smooth production function considered here might generate measured marginal rates of transformations with better pricing properties. In addition, this extension would allow us to understand which properties of the production functions are more relevant for pricing thus helping us to distinguish between alternative specifications (and propose new specifications) of currently used production functions. This extension is the exact analog of the several successful extensions of the standard consumption-based model with power utility. Consumption side features such as habit formation, long-run risk or multiple consumption goods considerably improved the empirical performance of the consumption-based approach to asset pricing as well as our understanding about utility functions. 8

This analysis follows from the basic pricing equation for excess returns (Re ) 0 = E [MRe ] = E [M] E [Re ] + ρ [M, Re ] σ [M] σ [Re ]

we have σ [M] = −

E[M] E [Re ] . ρ [M, Re ] σ [Re ]

The Sharpe ratio in the US postwar data is about E [Re ] /σ [Re ] = 0.4 annualy. Thus, even if the discount factor and returns are perfectly correlated (ρ [M, Re ] = 1) we need σ[M] = 40% annually.

26

By linking the stochastic discount factor to marginal rates of transformation and thus to production data, the model developed here also provides a new framework for examining classic issues in the macro-international finance literature. For example, empirical and theoretial work on risksharing, gains of financial integration and real exchange rates typically relies on the consumer’s first order conditions and consumption data. Exploring these issues from the perspective of a production-based approach and with production data may provide new insights to this literature. In this paper, I intentionally didn’t study the implications of the production-based model for the risk-free rate and the term-structure. Pricing these assets requires additional assumptions on the joint distribution of the forcing processes in the model (the stochastic discount factor Mt and the underlying state contingent productivity level θt ) to solve for the actual level of the marginal rate of transformation in Proposition 1. By focusing on excess returns, I avoided any distributional assumptions on my empirical work, helping to make the results robust to possible mispecifications of the distribution of these variables. It is well known however, that many asset pricing models fail to simultaneously fit the equity premium or the cross sectional variation in expected returns and the level and time series properties of the risk free rate. It is thus an interesting question for future research to examine if the production-based model can simultaneously match both facts. Another interesting extension is the study of the implications of the results presented here for consumption-based asset pricing. In a world with complete markets and no frictions, with perfect consumption and output data and with perfect knowledge of the production and utility functions, the marginal rate of substitution inferred from the consumers first order conditions should be exactly equal to the marginal rate of transformation inferred from the producers’ first order condition. By recovering a stochastic discount factor from the producers first order conditions without any information about preferences we can thus use the production-based model to discriminate between different popular specifications of utility functions. The opposite is true as well. Finally, the economic interpretation of the parameters estimates that I obtain here suggests that the producer’s elasticity of substitution of output across states of nature is quite high, a result that is in sharp contrast with the assumption of a zero elasticity of substitution that is implicitly made in current empirical and theoretical representations of the technology of a firm that operates in an uncertain environment. Future research can examine the robustness of this conclusion by exploring (i) a setup with variable labor as well as labor and capital adjustment costs that I ignore here; and (ii) incorporate the smooth production function studied in this paper in a fully specified general equilibrium model and verify if we can generate artificial time series that simultaneously match the asset pricing and business cycle (consumption and sectoral output, investment and good prices) facts in the data.

27

APPENDIX A-1

Smooth production sets

In this section I show that a smooth (differentiable) production possibilities frontier can be justified by an aggregation result of individual production functions that are not smooth and I explain h³ ´α ithe < derivation of the restriction set for the choice of the state-contingent productivity level E θtt 1 used in the paper. This section is based on Cochrane (1993) and the reader is referred to that paper for a more detailed analysis of the smooth production function presented here.

A-1.1

Aggregation

A smooth production possibilities frontier across states of nature can occur when one aggregates standard production functions which are not smooth. This is analogous to the standard result that an aggregate of Leontief production functions can produce a smooth production function such as a Cobb-Douglas. As a simple example to illustrate this claim, consider a two-state world, in which a farmer can plant in two fields (technologies). Let the technology of field i have the following standard form p (1) yi (s) = i (s) ki s = wet, dry and i = 1, 2 where yi (s) is the output in field i in state s, and ki is the number of seeds planted in field i. In addition, consider the following simple structure for the shocks in each field 1 (s)

=

½

1 if s = wet and 0.5 if s = dry

½ 0.5 if s = wet 2 (s) = 1 if s = dry

so that field one is relatively more productive if the weather is wet and field two is relatively more productive if the weather is dry. The left panel in Figure 4 plots the production possibilities frontier in each one of this standard technologies for the case ki = 1 in each technology. Total output is Y (s) = y1 (s) + y2 (s) and the number of seeds available to the farmer are constrained to be K = k1 + k2 We only observe the aggregates K and Y (s) but we know that the farmer can vary the amount of seeds in each of the two fields. This structure implies that aggregate production possibilities frontier that relates the total inputs of the firm (K) to its total output (Y ) across states is a smooth set. This is illustrated in the right panel in Figure 4 which plots the production possibilities frontier across states, when K = 1 and as we vary the amount of seeds in each of the two fields subject to the constraints k1 + k2 ≤ 1, k1 , k2 ≥ 0. As the figure shows, even though the individual production technologies have kinks, the aggregate technology is smooth. The farmer can shape the risk exposure of his total output to weather by varying the amount planted in each of the two fields.

28

[Insert Figure 4 here]

A-1.2

A tractable representation of a smooth technology

Motivated by the analysis of the previous section, Cochrane (1993) posits an aggregate, smooth production set with an analytically tractable functional form. Cochrane (1993) models directly the aggregate production function of the firm and not the individual technologies yi (s) in (1), since these individual technologies are in general not observable to economists. Output across states of nature st is assumed to satisfy the restriction,9 " X

#1

α

[a(st )Y (st )]α

st

≤ F (Kt ).

(2)

Thus the smooth production set for output across states of nature is directly specified as a CES transformation curve. Here, α > 1 is a parameter, a(st ) is a state-continget technological parameters and F (.) is the (certain) production function which is increasing and concave in the input Kt (here, capital). The restriction α > 1 guarantees that the set of feasible outputs in each state lies along a strictly concave transformation curve defined by (2). Therefore, in order to increase output in one state of nature the producer must decrease output in the other states of nature and at an increasing rate. This sensible property of the production function captures diminishing returns to scale in the production of output in each state of nature. In a continuous state-space, the transformation curve (2) can be expressed as ∙Z

α

dM (ω)y(ω)

¸1

α

≤ F (Kt )

In this representation, a(st ) or dM are not necessarily a probability measure. Since it is convenient to use a probability measure in order to take this technology to the data, Cochrane (1993) uses the Radon-Nikodym derivative and express the previous transformation curve as ∙Z

α

d Pr(ω) (y(ω)/θ (ω))

¸1

α

≤ F (Kt )

or Et−1

∙µ

Yt θt

¶α ¸ 1

α

≤ F (Kt )

(3)

where the expectation is conditional on the information set in period t − 1. Since F (Kt ) is predetermined at time t, we can finally express the previous production function as 9

Feenstra (2003) proposes a similar transformation curve. However, instead of choosing the output across states, he considers the choice of different output varieties.

29

Yt = t F (Kt ) ∙µ

Et−1

t

θt

¶α ¸ 1

(4)

α

≤1

(5)

where I divided both sides of (3) by F (Kt ) to obtain (5). This is the representation of the technology that I use in the paper.

A-2

Solving the producer’s maximization problem

˜ t, ˜ Define the vector of state variables as xt−1 = (Kit−1 , it−1 , Pit−1 , M θit , P˜it , ) where Kit−1 is the firm i’s current period stock of capital, it−1 is the current period productivity level, Pit−1 = pit−1 /pt−1 is the current period relative price of firm i0 s good with respect to price of the numeraire good ˜ t is the next period distribution of the stochastic discount factor in units of the numeraire (pt−1 ), M good and ˜θit and P˜it are the next period distribution of the underlying productivity level and relative price of good i respectively. Let V (xt−1 ) be the contingent claim value of the firm at the end of period t − 1 given the vector of state variables xt−1 . The Bellman equation of producer i is given by V (xt−1 ) =

max

{Iit−1 ,

it }

{Dit−1 + Et−1 [Mt V (xt )]}

subject to the constraints, Dit−1 = Pit−1 Yit−1 − Iit−1 Yit−1 =

it−1 F

1 ≥ Et−1 Kit

i

(Kit−1 ) ∙µ ¶α ¸ 1

α

it

(6)

θit = (1 − δ i )Kit−1 + Iit−1

for all dates t. Et−1 [.] is the expectation operator conditional on the firms’ information set at the end of period t − 1, δ i is the depreciation rate of capital and F i (.) is the (certain) production function, which is increasing and concave. Substitute the law of motion for capital in the value function and let μit−1 be the Lagrange multiplier associated with the technological constraint (6), the first order conditions are ∂ : Et−1 [Mt Vk (xt )] = 1 ∂Iit−1 ∂ : Mt V i (xt ) = μit−1 Et−1 ∂ it

30

∙µ

it

θit

¶α ¸ 1 −1 α

(7)

α−1 −α it θ it

h³ ´α i = 1. SubstiSince in equilibrium the restriction (6) is naturally binding, we have Et−1 θitit tuting this in the previous equation, we can write the first order condition for the optimal choice of the productivity level it as ∂ : Mt V i (xt ) = μit−1 ∂ it

α−1 −α it θ it

(8)

(Kit−1 ) + Et−1 [Mt Vki (xt )](1 − δ i )

(9)

The envelope conditions are Vki (xt−1 ) = Pit−1

i it−1 Fki

V i (xt−1 ) = Pit−1 F i (Kit−1 )

(10)

Using equation (7), the envelope condition (9) can be written as Vki (xt−1 ) = Pit−1

i it−1 Fk

(Kit−1 ) + (1 − δ i )

(11)

Substituting the envelope condition (10) at time t back into (8) yields Mt Pit F i (Kit ) = μit−1

α−1 −α it θ it

(12)

Taking expectations on both sides of the previous equation yields £

Et−1 [Mt Pit ] F i (Kit ) = μit−1 Et−1

α−1 −α it θ it

¤

(13)

This equation defines the Lagrange multiplier. Substitute μit−1 from (13) back in (12) yields Mt Pit F i (Kit ) = Et−1 [Mt Pit ] F i (Kit )/Et−1 Rearranging terms h Mt = Et−1 [Mt Pit /Pit−1 ] /Et−1 ( it /

α−1 −α θit it−1 )

£

α−1 −α it θ it

i µP

it−1

Pit

¤

α−1 −α it θ it

¶µ

it it−1

¶α−1

(14)

θ−α it

which we can write compactly as Mt = φit−1

µ

Pit−1 Pit

¶µ

it it−1

¶α−1

θ−α it

h i . This is equation (7) in the text. where φit−1 = Et−1 [Mt Pit /Pit−1 ] /Et−1 ( it / it−1 )α−1 θ−α it Solving for the (growth rate) in the productivity level yields it it−1

1 1−α

1 α−1

= φit−1 Mt

α α−1

θit

31

µ

Pit Pit−1

¶

1 α−1

,

which is equation (5) in the text. Finally, to obtain the expression for investment returns, substitute (11) at time t back in (7) to obtain Et−1 [Mt RtI ] = 1 where RtI = (1 − δ i ) + Pit

i it Fk

(Kit )

is the (random) investment return. These are equations (8) and (9) in the text. The second order conditions are satisfied by the assumptions on the production technology, i.e., α > 1 and F i (.) increasing and concave.

A-3

Proof of Proposition 1

The proof is mainly algebra. From the producers’ i first order condition (see equation (7) in the text) we have µ ¶µ ¶α−1 P Y it−1 it ¯ it−1 θ−α (15) Mt = φ it Pit Yit−1 Since market are complete, the SDF Mt is unique. This implies that at an interior solution, the marginal rate of transformation is equalized across time and states across all technologies i = 1, .., N . Taking the log of both sides of the previous equation we have pit + (α − 1) ∆yit − α¯ θit for i = 1, .., N mt = γ it−1 − ∆¯

(16)

¡ ¢ ¯ where lowercase variables are the log of the corresponding uppercase variable, γ it−1 = ln φ it−1 , ¯θi = ln(θit ), ∆ is the first difference operator. I use a bar over the log relative price (¯ pi ) to emphasize that this a relative price (with respect to the numeraire good), not the actual price. According to the identification Assumption 1 we have α¯θit =

J X

j

λij ¯ θt

(17)

j=1

where J is the number of common productivity factors in the economy. Substituting (17) in (16) yields J X j mt = γ it−1 − ∆¯ pit + (α − 1) ∆yit − λij ¯ (18) θt for i = 1, .., N j=1

As specified in Assumption 1, I normalize λ1j = 1 for j = 1, .., J. Now, consider the first order conditions for J + 1 technologies, with J + 1 ≤ N . Taking the difference between equation (18) for

32

technologies i = 2, ..., J + 1 relative to the same equation for technology 1 yields J X £ ¤ j 0 = γ it−1 − γ 1t−1 − [∆pit − ∆p1t ] + (α − 1) [∆yit − ∆y1t ] − (λij − 1) ¯ θt for i = 2, .., J + 1 j=1

(19) where I’ve used the notation pit (without the bar on the top) to denote the actual price (not relative). The fact that I now use the actual price follows from the fact that the price of the numeraire good cancels out from the difference in the relative prices between any two goods. From now on, it is convenient to write all the i = 2, .., J + 1 equations defined in (19) in matrix form. Rearranging terms we have (20) L¯ θt = Ωt−1 − I∆Pt + (α − 1)I∆Yt where I is a [J × J] identity matrix, Ωt−1 is a dimension J column vector in which each element i is γ (i+1)t−1 −γ 1t−1 , L is a [J × J] matrix in which each row-i, column-j element is given by λ(i+1)j −1, ∆Pt is a dimension J column vector in which each element i is ∆p(i+1)t − ∆p1t , ∆Yt is a dimension θt is a dimension J column vector J column vector in which each element i is ∆y(i+1)t − ∆y1t and ¯ i in which each element i is ¯θt . With J ≥ 1 common productivity factors, it is easy to show that we can identify these common factors from price and output data from i = J + 1 technologies. Assuming the matrix L has full rank, we can solve (20) for the size J vector ¯ θt to obtain ¯ θt = L−1 Ωt−1 − L−1 ∆Pt + (α − 1)L−1 ∆Yt

(21)

This equation shows that we can identify the underlying common productivity factors from price and output data only. As an example, for the case of one common productivity factor (J = 1) we have L = λ21 − 1, and thus the single common productivity factor ¯θt can be recovered from £ ¤ ¯θt = (λ21 − 1)−1 γ 2t−1 − γ 1t−1 − (λ21 − 1)−1 [∆p2t − ∆p1t + (α − 1) (∆y2t − ∆y1t )]

(22)

To express the actual marginal rate of transformation in terms of observed price and output data, substitute (21) in the marginal rate of transformation (18) for technology 1 to obtain © ª p1t + (α − 1) ∆y1t − ιJ L−1 Ωt−1 − L−1 ∆Pt + (α − 1)L−1 ∆Yt mt = γ 1t−1 − ∆¯

where ιJ is a size J row vector of ones. Finally, the previous equation can be written more compactly as mt = κt−1 −

J+1 X i=2

[bpi (∆pit − ∆p1t ) + byi (∆yit − ∆y1t )] − ∆¯ p1t + (α − 1)∆y1t

(23)

where bpi and byi are the the (i − 1)th elements in the [1 × J] row vectors −ιJ L−1 and (α − 1)ιJ L−1 respectively, and κt−1 = γ 1t−1 − ιJ L−1 Ωt−1 is a variable pre-determined at t. Equation (23) is the 33

exact log marginal rate of transformation. For empirical purposes, this marginal rate of transformation can be further simplified. As I show empirically, the factor risk prices bp ’s are typically a large number and the parameter α is small (close to one). Thus, to an excellent approximation, the previous marginal rate of transformation can be written as mt ≈ κt−1 −

J+1 X i=2

[bpi (∆pit − ∆p1t ) + byi (∆yit − ∆y1t )]

(24)

which shows that in order to empirically identify the marginal rate of transformation, only the relative movements (with respect to the reference technology 1) in output and price growth matter. Taking the exponential of (24) yields the marginal rate of tranformation in Proposition 1. For the case of one common productivity factor (J = 1), we have L−1 = (λ − 1)−1 and thus we can write the marginal rate of transformation as mt ≈ κt−1 − bp (∆p2t − ∆p1t ) − by (∆y2t − ∆y1t ) where, to simplify notation, I’ve defined λ21 = λ, and the factor risk prices are given by "

bp by

#

=

"

1/(1 − λ) (α − 1)/(λ − 1)

#

This completes the proof.

A-4

Principal Components in the Cross Section of Relative Price and Output Growth

To do a principal components analysis of the cross section of relative price and output growth, I first linearize the marginal rate of transformation (see equation 12 in the text) by a first order Taylor expansion around the unconditional mean of the factors denoted by E [xt ] where xt is the factor. Normalizing the mean of the marginal rate of transformation to one (since the mean is not identified from the estimation of the model on excess returns) yields Mt∗

≈1−

J+1 X i=2

[bpi (∆pit − ∆p1t − E [∆pit − ∆p1t ]) + byi (∆yit − ∆y1t − E [∆yit − ∆y1t ])]

(25)

I then do a separate principal components analysis of the cross section of relative price growth J+1 (∆pit − ∆p1t )J+1 i=2 and of the cross section of relative output growth (∆yit − ∆y1t )i=2 . To extract the principal components I proceed as follows. Define ∆Yt as the 1 × J vector containing the realizations of the relative output growth (∆yit − ∆y1t )J+1 i=2 (procedure is similarly for the relative prices growth) in each technology i = 2, .., J + 1 at time t. The variance-covariance matrix of ∆Yt

34

can be written as var(∆Yt ) = ΩΛΩ> where Λ is a diagonal matrix of eigenvalues of the matrix var(∆Yt ) and Ω is an orthogonal matrix (i.e. Ω> = Ω−1 ) whose columns are standardized eigenvectors. The vector 1 × N of principal components pct is then defined by P Ct = (∆Yt − ∆Y¯ )Ω

(26)

where ∆Y¯ ∈ Rn is a vector with the sample means of the relative output growth rates. The variance of the kth principal component is equal to Λk , the kth eigenvalue of var(∆Yt ). Moreover, the total variation in the cross section of output growth tr(var(∆Yt )) is equal to the total variation of principal components tr(Λ), where tr denotes trace. Thus, the percentage variation in output growth explained by the first k principal components is 100 ×

Pk

i=1 Λi

tr(Λ)

By construction, the first principal component is the orthogonal component that explains most of the variation in output growth in all sectors, the second component explains most of the part not explained by the first component and so forth. Once the principal components have been extracted, each pricing factor in (25) can specified as a linear combination of the principal components as ∆pit − ∆p1t = ∆yit − ∆y1t =

J X

γ pij P P Cj

(27)

γ yij OP Cj

(28)

j=1

J X j=1

where P P Cj is the jth principal component of the cross section of relative prices growth, OP Cj is the jth principal component of the cross section of relative output growth and γ Pij and γ yij are the loadings of each pricing factor on the corresponding principal component. Here, I consider all the four sectors reported in NIPA, namely durable goods, nondurable goods, services and structures and, as in the one common factor specificaiton, I specify the durable goods sector as the technology 1 (the reference technology). Table 6 presents the results of the principal components analysis of the cross—section of relative price and output growth. Each principal component is a linear combination of the corresponding pricing factors. The top part in Panel A report the loadings of each principal component of the cross section of relative output growth on the relative output growth in each sector. The first principal is almost a "level" factor. It moves all sectors in the same direction but puts considerably less weight in the Services sector. The top part in Panel B reports the cumulative percentage in the variation in the cross section of output growth that is explained by the first k = 1, .., 3 principal components. The first principal component alone

35

explains almost 77% of the total variance. The bottom part in Panels A and B repeat the same analysis for the cross-section of relative price growth. Clearly, the first principal component of the cross section of relative price growth is a "level" factor moving all sectors in the same direction. Thus the price first principal component is approximately the same as the average relative price growth. The bottom part in Panel B shows that almost 64% of the total variance in the cross section of relative price growth is explained by the price first principal component alone. The previous analysis suggest that the marginal rate of transformation is well approximated by the first principal components of the output growth and of the relative price growth in the asset pricing. Thus, under the multiple common productivity factors specification, the marginal rate of transformation is approximately given by Mt∗ ≈ 1 − bp PFPCt − by OFPCt

(29)

as reported in the text, where PFPCt (price first principal component) is the first principal component of the cross-section of the relative price growth and OFPCt (output first principal component) is the first principal component of the cross-section of relative output growth. This approximation is obtained by first noting that using only the first principal component, the relative output and relative price factors in (27) and (28) are approximated by ∆pit − ∆p1t ≈ γ pi1 P F P Ct and ∆yit − ∆y1t = γ yi1 OF P Ct . Substituting these expressions in equation (25) yields equation (29), where the factor risk prices are given by "

bp by

#

⎡

J+1 X

p p

b γ ⎢ ⎢ i=2 i i1 =⎢ ⎢ J+1 ⎣ X y y bi γ i1 i=2

A-5

Additional description of the data

A-5.1

Macro data

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

The consumption of nondurable and services data, the population and the consumption price index data used in the tests of the standard consumption-based asset pricing model is from the BEA. Nondurable plus Services consumption is obtained from Table 2.3.5, sum of lines 6 and 13. Population is from Table 2.1, line 38. Real consumption is obtained by deflating nominal consumption by the Consumer Price Index, obtained from Table 2.3.4, line 2. Per capita real consumption is obtained by dividing real consumption by the population. In matching returns with output and price data, I use the follow convention: match returns at time t with output (and price) growth at time t + 1. A convention is needed because the level of output is a flow during a year rather than a point-in-time observation as the returns; that is, output data are time averaged. This convention follows the Campbell’s (2003) (on consumption) beginning of the period timing convention and assumes that output data for year t measures the

36

output at the beginning of the year. In this case, output growth for a given year is next year output divided by this year output. This matching assumption slightly improves the performance of the model relative to the assumption that output data for year t measures the output at the end of the year.10

A-5.2

Asset data

The data for the three Fama-French factors (SMB, HML and Market excess returns) and the six Fama-French factors is from Prof. Kenneth French’s webpage. The three factors are: (i) the Market excess return on a value-weighted portfolio of NYSE, AMEX, and Nasdaq stocks minus the T-bill rate; (ii) SMB is the return on the Small-minus-Big portfolio; and (iii) HML, is the return on the High-minus-Low portfolio. The SMB and HML portfolios are based on the six FamaFrench benchmark portfolios sorted by size (breakpoint at the median) and book-to-market equity (breakpoints at the 30th and 70th percentiles). The SMB return is the difference in average returns between three small and three big stock portfolios. The HML return is the difference in average returns between two high and two low book-to-market portfolios. See Fama and French, 1993, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of Financial Economics, for a complete description of these factor returns. The data for the 25 portfolios sorted on size and book to market and for the 10 momentum portfolios is from Prof. Kenneth French’s webpage. The data used to compute the 9 risk double sorted portfolios is from CRSP, available at the Wharton Research Data Services (WRDS) website. The data for the 5 industry portfolios is from Gomes, Kogan and Yogo (2007). Excess returns are computed by subtracting the risk free rate, as measured by the US treasury bill return rate, from CRSP. The description of each set of portfolio sorts is the following: 25 portfolios sorted on size and book-to-market: according to the description provided at Prof. Kenneth French’s webpage, these portfolios, which are constructed at the end of each June, are the intersections of 5 portfolios formed on size (market equity, ME) and 5 portfolios formed on the ratio of book equity to market equity (BE/ME). The size breakpoints for year t are the NYSE market equity quintiles at the end of June of t. BE/ME for June of year t is the book equity for the last fiscal year end in t-1 divided by ME for December of t − 1. The BE/ME breakpoints are NYSE quintiles. 9 risk-sorted portfolios (double sorted on "pre-ranking" relative output growth and relative price growth betas): the relative output growth and the relative price growth factors that I use here have an annual frequency and thus it is infeasible to use these factors to create "pre-ranking" betas due to the small sample size. To address this issue, I create two mimicking portfolios of these factors which I label price mimicking portfolio (PMPt ) and output mimicking 10

Jagannathan and Wang (2007) show that even though the standard consumption-based model does not perform well with annual averages, it performs significantly better when annual consumption growth is measured based only on the fourth quarter of each year. Jagannathan and Wang’s paper emphasizes the effect of different matching assumption between returns and macroeconomic variables on asset pricing tests.

37

portfolio (OMPt ). The PMPt is obtained by first estimating the following regression, ∆pNDt − ∆pDt = a + b0 Rte + εt

(30)

where∆pNDt − ∆pDt is the relative price growth factor and Rte are the excess returns on the base assets. The coefficients b can be interpreted as the weights in a zero-cost portfolio. The return on the PMPt is then (31) PMPt = b0 Rte which is the minimum variance combination of assets that is maximally correlated with the relative price growth factor. Regression (30) (and similarly for the relative output growth factor) is estimated using annual data from 1932 to 2006. Then, assuming that the coefficients b are relatively stable over time and within the year, I use Equation (31) to extend the sample before 1932 and to generate observations of the mimicking portfolio at a monthly frequency. The base test assets I employ are the Fama-French 6 benchmark portfolios and the 10 momentum portfolios. An identical procedure is used to obtain the OMPt factor. Following Fama and French (1992) I create nine double sorted risk based portfolios of NYSE, AMEX and NASDAQ stocks as follows. For every calendar year, I first estimate the PMP and the OMP betas for each firm, using 24 to 60 months of past return data. As in Fama and French (1992), I denote this beta as the "pre-ranking" PMP and OMP beta estimate. I then do the following double sorting procedure: I sort stocks into three bins (cutoffs at the 33 and 66 percentile) based on their "pre-ranking" PMP beta and repeat the procedure based on each stock’s "pre-ranking" OMP beta. The intersection of these bins gives 9 portfolios. I then compute the return on each of these portfolios for the next 12 calendar months by a value weighted average of the returns of the stocks in the portfolio. This procedure is repeated at the end of June for each calendar year. 10 momentum portfolios: according to the description provided at Prof. Kenneth French’s webpage, these portfolios, which are constructed monthly using NYSE prior (2-12) return decile breakpoints. The portfolios are constructed each month and include NYSE, AMEX, and NASDAQ stocks with prior return data. To be included in a portfolio for month t (formed at the end of the month t-1), a stock must have a price for the end of month t-13 and a good return for t-2. In addition, any missing returns from t-12 to t-3 must be -99.0, CRSP’s code for a missing price. Each included stock also must have ME for the end of t-1. 5 industry portfolios: these portfolios were constructed by Gomes, Kogan and Yogo (2007). The universe of stocks to construct these portfolios is the ordinary common equity traded in NYSE, AMEX, or Nasdaq, which are recorded in the Center for Research in Securities Prices (CRSP) Monthly Stock Database. In June of each year t, the universe of stocks is sorted into five industry portfolios based on their SIC code: services, nondurable goods, durable goods, investment, and other industries. Other industries include the wholesale, retail, and financial sectors as well as industries whose primary output is to government expenditures or net exports. The stock must have a non-missing SIC code in order to be included in a portfolio. Once the portfolios are formed,

38

their value-weighted returns are tracked from July of year t through June of year t + 1.

A-6

Description of the benchmark models

Lucas-Breeden standard consumption-based model: The consumption-based model is based on a measure of consumer’s marginal rate of substitution whereas the production-based model is based on a measure of the firm’s marginal rate of transformation. Thus this model is a natural benchmark for the production-based model. In a SDF linear representation, this model is described by Mt = 1 − b1 ∆ct+1 where b1 =Coefficient of Relative Risk Aversion and ∆ct is the per capita consumption of nondurable and services goods growth rate. In matching returns and consumption growth, I follow Campbell (2003) timing convention as in the production-based model. Thus I match the returns at time t with the consumption growth at time t + 1. This timing convention improves the fit of the consumption-based model thus providing a better benchmark for the production-based model. Fama-French (1993) three factor model: This model uses the returns on three factor mimicking portfolios to explain expected returns. In a SDF representation, this model is described by Mt = 1 − b1 Markett − b2 SMBt − b3 HMLt where Market is the excess returns on the market portfolio, SMB is the returns on the Smallminus-Big portfolio, and HML is the returns on the High-minus-Low portfolio. The excess market return is the return on a value-weighted portfolio of NYSE, AMEX, and Nasdaq stocks minus the one-month T-bill rate. See Appendix A-5.2 for an additional description of these portfolios.

References [1] Apgar, David, 2006, Risk Intelligence: Learning to Manage What We Don’t Know (Harvard Business School Press) [2] Balvers, Ronald.and Dayong Huang, 2006, Productivity-Based Asset Pricing: Theory and Evidence,Journal of Financial Economics, forthcoming [3] Berk, Jonathan B, Richard C. Green and Vasant Naik, 1999, Optimal Investment, Growth Options and Security Returns, Journal of Finance 54, 1553 − 1607 [4] Boldrin, Michele, Lawrence J. Christiano and Jonas Fisher, 2001, Habit Persistence, Asset Returns and the Business Cycle, American Economic Review 91, 149 − 166 [5] Breeden, Douglas T., 1979, An Intertemporal Capital Pricing Model with Stochastic Investment Opportunities, Journal of Financial Economics 7, 265 − 296 39

[6] Brock, William, 1982, Asset Prices in a Production Economy, in John J. McCall, ed.: The Economics of Information and Uncertainty University of Chicago Press [7] Burnside, Craig, Marting Eichenbaum, and Sergio T. Rebelo, 1996, Sectoral Solow Residuals, European Economic Review 40, 861 − 869 [8] Campbell, John, 2003, Consumption-Based Asset Pricing, in George Constantinides, Milton Harris, and Rene Stulz, ed.: Handbook of the Economics of Finance, Vol. IB, Chapter 13 , North-Holland, Amsterdam, 803 − 887, 2003 [9] Chambers, Robert, and John Quiggin, 2000, Uncertainty, Production, Choice, and Agency-The State-Contingent Approach (Cambridge University press) [10] Chen, Nai-Fu, Richard Roll and Steven Stephen A. Ross, 1986, Economic Forces and the Stock Market, Journal of Business 59, 383 − 403 [11] Cochrane, John, 1991, Production-Based Asset Pricing and the Link Between Stock Returns and Economic Fluctuations, Journal of Finance 461, 209 − 237 , 1993, Rethinking Production Under Uncertainty, Working paper, University of

[12] Chicago [13]

, 1996, A Cross Sectional Test of an Investment Based Asset Pricing Model, Journal of Political Economy 1043, 572 − 621 , 2002, Asset Pricing (Princeton University Press, NJ)

[14] [15]

, 2005, Financial Markets and the Real Economy,in Richard Roll, ed.: The International Library of Critical Writings in Financial Economics, forthcoming

[16] Cooper, Russell, and John Haltiwanger, 2005, On the Nature of Capital Adjustment Costs, forthcoming in Review of Economic Studies [17] Daniel, Kent and Sheridan Titman, 2005, Testing Factor-Model Explanations of Market Anomalies,Working Paper, Northwestern University and University of Texas, Austin [18] D’Avolio, Gene, 2002, The market for borrowing stock, Journal of Financial Economics 25, 23 − 49 [19] Fama, Eugene F., and Kenneth R. French, 1992, The Cross-Section of Expected Stock Returns, Journal of Finance 47, 427 − 466 [20]

, 1993, Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics 33, 3 − 56

[21]

, 1996, Multi-Factor Explanations of Asset Pricing Anomalies, Journal of Finance 51, 55 − 84 40

[22] Feenstra, Robert C., and Hiau Looi Kee, 2004, Export Variety and Country Productivity,Working paper, NBER, No. 10830 [23] Gala, Vito, 2005, Investment and Returns: Theory and Evidence, Working paper, University of Chicago GSB [24] Gomes, João F., Leonid Kogan, and Lu Zhang, 2003, Equilibrium Cross Section of Returns, Journal of Political Economy 111, 693 − 732 [25] Gomes, João F., Leonid Kogan, and Motohiro Yogo, 2006, Durability of Output and Expected Stock Returns, Working paper, Wharton School of the University of Pennsylvania [26] Gomes, João F., Amir Yaron, and Lu Zhang, 2006, Asset Pricing Implications of Firms’ Financing Constraints, Review of Financial Studies, 19(4), 1321 − 1356 [27] Gourio, Francois, 2005, Operating Leverage, Stock Market Cyclicality and the Cross-Section of Returns, Working paper, University of Chicago [28] Hall, Robert E.,1988, Intertemporal Substitution in Consumption, Journal of Political Economy 96, 339 − 357 [29] Hansen, Lars Peter and Kenneth J. Singleton, 1982, Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models, Econometrica 55, 1163 − 1198 [30] Jagannathan, Ravi and Zhenyu Wang, 1996, The Conditional CAPM and the Cross-Section of Expected Returns, Journal of Finance 51, 3 − 53 [31] Jagannathan, Ravi and Yong Wang, 2007, Lazy Investors, Discretionary Consumption, and the Cross-Section of Stock Returns, Journal of Finance 62, 1623 − 1661 [32] Jermann, Urban, 1998, Asset Pricing in Production Economies, Journal of Monetary Economics 41, 257 − 275 [33]

, 2007, The Equity Premium Implied by Production, Working paper, Wharton School of the University of Pennsylvania

[34] Kandel, Shmuel, and Robert F. Stambaugh, 1995, Portfolio Inneficiency and the Cross-Section of Expected Returns, Journal of Finance 50, 157 − 184 [35] Lamont, Owen and Richard H. Thaler, 2003, Can the Market Add and Subtract? Mispricing in Tech Stock Carve-Out, Journal of Political Economy 111, 227 − 268 [36] Lewellen, Jonathan, Stefan Nagel and Jay Shanken, 2006, A Skeptical Appraisal of AssetPricing Tests, NBER working paper No. 12360 [37] Li, Erica X. N., Dmitry Livdan, and Lu Zhang, 2007, Anomalies, Working paper

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[38] Li, Qing, Maria Vassalou, and Yuhang Xing, 2003, Sector Investment Growth Rates and the Cross-Section of Equity Returns,Working paper, Columbia University [39] Liu, Laura X. L., Toni M. Whited, and Lu Zhang, 2007, Regularities, Working Paper [40] Lucas, Robert E. Jr, 1979,Asset Prices in an Exchange Economy, Econometrica 46, 1426−1446 [41] Panageas, Stavros, and Jianfeng Yu, 2006, Technological Growth, Asset Pricing, and Consumption Risk, Working paper, Wharton School of the University of Pennsylvania [42] Papanikolaou, Dimitris, 2007, Investment-Specific Technology Shocks and Asset Prices, Working paper, Massachusetts Institute of Technology [43] Roll, Richard and Stephen A. Ross, 1995, On the Cross-sectional Relation between Expected Returns and Betas, Journal of Finance 49, 101 − 121 [44] Rouwenhorst, Geert, 1995, Asset Pricing Implications of Equilibrium Business Cycle Models, in T. F.Cooley, ed.: Frontiers of Business Cycle Research [45] Sheffi, Yossi, 2005, The Resilient Enterprise: Overcoming Vulnerability for Competitive Advantage (MIT Press)

42

Table 1 Descriptive Statistics for Selected Macroeconomic Variables This table reports the descriptive statistics of the output growth rate of output (∆Yi ) and the growth rate of prices (∆Pi ) in sector i =D (durable goods) and sector i =ND (nondurable goods), the first principal components of the cross section of relative price growth (PFPC), the first principal component of relative output growth (OFPC), and the growth rate in the per capita consumption of non-durables and services (∆C) for comparison. The data are annual and the sample is 1932 − 2006.

Var

Mean (%)

∆YD ∆YND ∆PD ∆PND ∆C PFPC OFPC

7.16 3.13 1.74 3.39 2.50 0 0

Total S.D. Autocorrel. (%)

NBER expansions Mean S.D. (%) (%)

NBER recessions Mean S.D. (%) (%)

12.03 2.64 3.34 3.64 1.62 4.07 20.60

9.69 3.41 1.55 3.55 2.81 0.30 −2.08

−2.63 2.06 2.46 2.78 1.29 −1.37 9.08

0.18 −0.01 0.62 0.59 0.27 0.68 0.14

43

11.57 2.45 3.26 3.59 1.54 3.86 21.61

8.39 3.14 3.65 3.89 1.35 4.70 13.25

Figure 1 Production Possibilities Frontier Across States of Nature This figure plots the production possibilities frontier across states of nature (bold line) for a standard representation of technology (left panel) and for a smooth (differentiable) representation of technology (right panel) in a two states of nature economy. The firm is producing at point A. Mi is the slope of the production possibilities frontier at point A. Standard Technology

Smooth Technology 2 Output in state of nature 2

Output in state of nature 2

2

1.5

1

A M2

0.5

1.5

1

A

0.5

M1 0

0

0.5 1 1.5 Output in state of nature 1

M1 2

44

0

0

0.5 1 1.5 Output in state of nature 1

2

Table 2 GMM Estimation of the Production-Based Model on Several Portfolio Sorts This table reports the second stage GMM estimates and tests of the production-based model. The estimated model is E[Ret ]=cte +by Cov(Factor1,Ret )+bp Cov(Factor2,Ret ) in which Ret is a vector with the excess returns of the following portfolio sorts: (i) the Fama-French 25 portfolios sorted on size and book to market (25-SIZE-BM); (ii) 9 risk double sorted portfolios (9-RISK); (iii) 10 momentum portfolios (10-MOM); (iv) the Gomes, Kogan and Yogo 5 industry portfolios (5-IND); (v) all portfolios together (ALL); and (vi) all portfolios minus the 10 momentum portfolios together (ALL-MOM). Panel A reports results for the one common productivity factor specification in which Factor1 is the difference between the growth rate of output in the nondurable goods and the durable goods sectors and Factor2 is the difference between the growth rate of the price in the nondurable goods sector and the durable goods sector. Panel B reports results for the multiple common productivity factors specification in which Factor1 is the first principal component of the cross section of the relative growth rate of output and Factor2 is the first principal component of the cross section of the relative growth rate of prices. The table reports the estimates of the factor risk prices by , bp and the intercept (cte ) and the corresponding GMM standard errors. Values for the factor risk prices with (∗ ) are statistically different from zero at the 5% level. In addition, the table reports the following measures of the goodness of fit and tests of the model (Diagnostics): the GMM first stage cross-sectional R-squared (R2 ), the first stage mean absolute pricing error (MAE, in %) and the second stage J-test of overidentifying restrictions with the corresponding p-value (in %). In Panel A, the table also reports the estimates of the curvature parameter α and the sensitivity of the underlying productivity level in the nondurable goods sector to the common productivity factor (λ) implied by the second stage GMM estimates of the risk prices with the corresponding standard errors obtained by the delta-method. The data are annual and the sample is 1932 − 2006.

Panel A: One Common Productivity Factor Specification 25-SIZE-BM 9-RISK 10-MOM 5-IND ALL ALL-MOM Parameters α s.e. (α) λ se(λ) Risk Prices

1.03 (0.03) 0.98 (0.00)

1.27 (0.42) 0.94 (0.06)

0.85 (0.07) 0.98 (0.01)

1.18 (0.14) 0.97 (0.02)

0.98 (0.02) 0.98 (0.00)

1.07 (0.02) 0.98 (0.00)

cte s.e. (cte ) by s.e. (by ) bp s.e. (bp ) Diagnostics

0.02 (0.02) −1.45 (1.62) 47.48∗ (8.85)

0.06 (0.04) −4.36 (2.90) 16.34 (16.84)

0.01 (0.04) 9.34∗ (4.39) 62.20 (33.35)

0.05 (0.03) −6.61 (4.32) 36.78 (25.97)

0.00 (0.00) 1.46∗ (0.43) 60.86∗ (4.08)

−0.00 (0.01) −3.53∗ (1.25) 48.65∗ (4.80)

R2 MAE J-Test p-value (J)

75.30 1.39 13.14 92.91

68.37 0.76 7.14 30.78

71.83 1.59 4.91 67.05

94.69 0.19 2.58 27.48

58.28 1.81 14.45 99.98

74.87 1.42 13.85 99.97

45

Table 2 (cont.) Panel B: Multiple Common Productivity Factors Specification 25-SIZE-BM 9-RISK 10-MOM 5-IND ALL ALL-MOM Risk Prices cte s.e. (cte ) by s.e. (by ) bp s.e. (bp ) Diagnostics

0.01 (0.02) −3.37∗ (1.39) 28.30∗ (5.21)

0.07 (0.04) −2.12 (2.75) 24.26 (14.34)

0.06 (0.07) 7.23 (5.75) 75.11 (58.19)

0.05 (0.04) −7.20 (4.77) 27.25 (24.20)

0.00 (0.00) 0.27 (0.29) 41.83∗ (4.04)

0.00 (0.01) −3.54∗ (0.75) 34.32∗ (3.09)

R2 MAE J-Test p-value (J)

66.53 1.60 11.38 96.89

85.25 0.56 5.52 47.90

87.13 1.18 4.81 68.28

75.88 0.48 2.25 32.43

50.40 1.96 13.85 100.00

72.30 1.35 13.65 99.97

46

Table 3 Average Returns, Fitted Marginal Rate of Transformation Covariances and Relative Price and Relative Output Growth Covariances of the 25 Fama-French Portfolios Panel A reports the average annual excess returns on the 25 Fama-French portfolios sorted on size and book-to-market equity (BE/ME). Panel B reports the opposite of the covariance between the fitted marginal rate of transformation and the returns of each one of the 25 Fama-French Portfolios. The fitted marginal rate of transformation is given by Mt∗ = 1−by Factor1−bp Factor2, where Factor1 is the difference between the growth rate of output in the nondurable goods and the durable goods sectors and Factor2 is the difference between the growth rate of the price in the nondurable goods sector and the durable goods sector, and the factor risk prices b0 s are the GMM second stage estimates of the production-based model using the 25 Fama-French portfolios as test assets. Panel C and D reports the covariance of Factor1 (Output) and Factor2 (Price) and the returns of each one of the 25 Fama-French Portfolios respectively. Avg. is the corresponding row or column average. The data are annual and the sample is 1932 − 2006.

BE/ME Sort Growth 2 3 4 Value avg. Growth 2 3 4 Value avg.

Panel A: Average Excess Returns (%) Size Sort Small 2 3 4 Big avg. 6.6 9.7 10.5 9.5 8.3 8.9 12.1 14.0 13.3 10.7 8.4 11.7 16.0 16.0 13.8 13.5 10.5 14.0 19.0 17.3 15.1 14.4 11.0 15.3 21.6 18.2 16.6 15.9 12.8 17.0 15.1 15.0 13.9 12.8 10.2 Panel C: Output Covariance ×10−3 0.2 −5.0 −6.8 −4.4 −5.1 −4.2 −3.1 −7.7 −6.6 −5.3 −4.4 −5.4 −5.4 −6.7 −5.7 −5.9 −4.2 −5.6 −6.3 −7.2 −5.7 −4.2 −6.5 −6.0 −6.2 −5.3 −6.1 −5.5 −6.4 −5.9 −4.2 −6.4 −6.2 −5.0 −5.3

47

Panel B: MRT Covariance × − 10−2 Size Sort Small 2 3 4 Big avg. −2.4 5.1 3.2 1.7 3.1 2.1 3.7 3.0 4.5 3.2 0.8 3.1 5.6 6.1 2.5 3.5 3.0 4.1 6.2 9.0 6.7 6.0 5.5 3.9 10.5 7.2 7.2 8.3 6.3 7.9 5.3 5.6 4.7 4.5 3.4 Panel D: Price Covariance ×10−3 −0.5 0.8 0.4 0.2 0.4 0.3 0.6 0.3 0.7 0.5 −0.0 0.4 0.9 1.0 0.3 0.5 0.4 0.6 1.6 1.1 1.0 1.0 0.5 1.0 1.9 1.3 1.2 1.5 1.0 1.4 0.9 0.9 0.7 0.7 0.5

Table 4 Nine Risk Double Sorted Portfolios This table reports the mean excess returns and the post-ranking covariances with the relative output growth factor (Output) and the relative price growth factor (Price) of 9 risk double sorted portfolios formed by pre-ranking relative output growth and relative price growth betas. Portfolio "High" is a portfolio of stocks whose pre-ranking beta of the corresponding factor is in the top 33 percentile and portfolio "Low" is a portfolio with stocks whose pre-ranking beta of the corresponding factor is on the bottom 33 percentile. The portfolios are value weighted and are rebalanced annually. Avg. is the corresponding row or column average. The data are annual and the sample is 1932 − 2006.

Price Sort Low Medium High avg. Low Medium High avg.

Panel A: Average Excess Returns (%) Output Sort Low Medium High avg. 10.5 9.5 7.7 9.2 12.3 10.5 8.8 10.5 15.1 11.5 9.6 12.1 12.6 10.5 8.7 Panel C: Output Covariance ×10−3 −5.4 −5.8 −2.4 −4.5 −6.8 −5.1 −2.3 −4.7 −5.4 −5.1 −4.3 −5.0 −5.8 −5.3 −3.0

48

Panel B: Price Covariance Output Sort Low Medium High −0.3 −0.1 −0.2 0.4 0.0 0.0 1.1 0.6 0.8 0.4 0.2 0.2

×10−3 avg. −0.2 0.2 0.8

Table 5 Comparison of Three Asset Pricing Models on the 25 Fama-French Portfolios This table presents the second stage GMM estimates and tests of three asset pricing models using the 25 FamaFrench Portfolios sorted on size and book-to market equity as test assets. The three models are have the following specification. (i) Production—based model (PBM): E[Ret ]=cte +by Cov(OUTPUTt ,Ret )+bp Cov(PRICEt ,Ret ) where the OUTPUTt factor is the difference between the growth rate of output in the nondurable goods and the durable goods sectors and the PRICEt factor is the difference between the growth rate of the price in the nondurable goods sector and the durable goods sector; (ii) Consumption-CAPM (C-CAPM): E[Ret ]=cte +bc Cov(CONSUMPTIONt ,Ret ) where CONSUMPTIONt is the real per capita growth rate of consumption of nondurables+services; and (iii) Fama-French three factor model (FF3F): E[Ret ]=cte +b1 Cov(MARKETt ,Ret )+b2 Cov(SMBt ,Ret )+b3 Cov(HMLt ,Ret ) where Market, SMB and HML are the three Fama-French factors. Appendix A-6 provides a description of these models. Panel A reports the estimates of the factor risk prices b and the corresponding GMM standard errors in parenthesis. Values with (∗ ) are statistically different from zero at the 5% level. Panel B reports measures of the goodness of fit and tests of each model (Diagnostics). It reports the GMM first stage cross-sectional R-squared (R2 ), the first stage mean absolute pricing error (MAE, in %) and the second stage J-test of overidentifying restrictions with the corresponding p-value (in %). The data are annual and the sample is 1932 − 2006.

PBM

C—CAPM

FF3F

Risk Prices OUTPUT PRICE

1.45 (1.62) −47.48∗ (8.85) 41.28∗ (4.21)

CONSUMPTION

−2.37∗ (0.68) 3.74∗ (0.73) 4.04∗ (0.56)

MARKET SMB HML Diagnostics R2 MAE J-Test p-value (J)

75.30 1.39 13.14 92.91

49

31.94 2.25 14.61 91.78

87.25 0.97 15.06 81.97

Table 6 Principal Component Analysis of the Cross-Section of Relative Output and Relative Price Growth Panel A reports the loadings of each principal component of the cross section of relative output growth on the relative growth rate of output in each sector (PC-Output) as well as the loadings of each principal component of the cross section of relative price growth on the relative price growth in each sector (PC-Price). Relative output and price growth are measured relative to the reference technology which is the durable goods sector. Panel B reports the cumulative percentage variation in the cross section of relative output growth and relative price growth that is explained by the first k = 1, 2, 3 principal components. The data are annual and the sample is 1932 − 2006.

k Nondurable Goods Services Structures

k Nondurable Goods Services Structures

Panel A: Loadings PC-Output 1 2 3 0.47 −0.47 −0.75 0.17 −0.79 0.59 0.87 0.40 0.29

k Percentage

PC-Price 1 2 3 0.68 0.12 0.73 0.37 −0.91 −0.19 0.64 0.39 −0.66

Panel B: Explained Variance PC-Output 1 2 3 0.77 0.21 0.02

PC-Price k Percentage

50

1 0.64

2 0.23

3 0.13

Figure 2 Predicted vs Realized Excess Returns Across Several Portfolio Sorts The figure shows the plot of realized versus predicted excess returns (per year) for the estimation of the productionbased asset pricing model on the following portfolios sorts: (i) top-left: the Fama-French 25 portfolios sorted on size and book to market; (ii) top-right: 9 risk couble sorted portfolios; (iii) bottom-left: 10 momentum portfolios and (iv) bottom-right: the Gomes, Kogan and Yogo 5 industry portfolios. The figure also reports the cross sectional R-squared (R2 ) and the mean absolute pricing error (MAE) in %. The data are annual and the sample is 1932 − 2006.

2

2

25−SIZE and BM R = 0.75 MAE (%) = 1.4

9−RISK SORTED R = 0.68 MAE (%) = 0.76 16

15 20

Actual mean excess return

Actual mean excess return

25

14 25 24 35 13 23 34 45 44 3322 4332 55 12 54 42 31 53 21 41 52 51

15 10 11 5

13

14 12 10 32 31

8

23 1122 2133

12

6 4 2

0

0

5 10 15 20 Predicted mean excess return

0

25

0

10−MOMENTUM R2 = 0.72 MAE (%) = 1.6

15

5−INDUSTRY R2 = 0.95 MAE (%) = 0.19

20

14 Actual mean excess return

10 Actual mean excess return

5 10 Predicted mean excess return

15 9 8 10 2 3

5

7 6 4

5

12

4 15 3

10 8 2

6 4 2

1 0

0

5 10 15 Predicted mean excess return

20

51

0

0

2

4 6 8 10 12 Predicted mean excess return

14

Figure 3 Times Series of the Innovations in the Common Agregate Productivity Factor and in the Marginal Rate of Transformation Panel A plots the innovations in the log common productivity factor and Panel B plots the inovations in the log marginal rate of transformation implied by the second stage GMM estimation of the production-based model on the Fama-French 25 portfolios sorted on size and book-to-market. Shaded bars are NBER recession years. The data are annual and the sample is 1932 − 2006.

Panel A: Innovations in the Aggregate Productivity Factor 6

wθ

Innovation

4 2 0 −2 −4

1940

1950

1960

1970

1980

1990

2000

Year Panel B: Innovations in the Marginal Rate of Transformation 4

wMRT

Innovation

2 0 −2 −4 −6

1940

1950

1960

1970

52

1980

1990

2000

Figure 4 Smooth Production Possibilities Frontier Across States of Nature: an Aggregation Result This left panel in the figure plots the production possibilities frontier across states of nature for two standard rep√ resentations of the technology of the form yi (s) = i (s) ki where ki = 1, s = wet, dry, 1 (wet) = 2 (dry) = 0.5 and 1 (dry) = 2 (wet) = 1. The right panel plots the production possibilities frontier of the resulting aggregate production function Y = y1 + y2 , in which k1 + k2 ≤ 1. Individual Technologies yi=eik1/2 i

Aggregate Technology Y=y1+y2 1.5 Output in the state of nature dry

Output in the state of nature dry

1.5 PPF of Technology 2 1 PPF of Technology 1 0.5

0

0

0.5 1 Output in the state of nature wet

1.5

53

PPF of the Aggregate Technology

1

0.5

0

0

0.5 1 Output in the state of nature wet

1.5