Human Capital, Business Cycles and Asset Pricing∗ Min Wei† Columbia University This Draft: May 9, 2004
Abstract I investigate the asset pricing and business cycle implications of a dynamic stochastic general equilibrium model with human capital and education. Key features of the model are (1) a higher consumption risk resulting from the representative agent’s desire to smooth leisure and from the short-run inelasticities in physical and human capital investment, and (2) a countercyclical risk aversion induced by shocks to human capital. The model provides a good fit to a number of assetpricing facts including a low riskfree rate, an upward-sloping yield curve, a higher equity premium, countercyclical dividend yields, and long-horizon predictability of excess stock returns. On the macroeconomic side, the model matches conventional business-cycle statistics as well as a standard model does, while generating a high volatility in hours worked and a negative correlation between the output and the time spent in education, both of which are consistent with the data. Keywords: Human Capital; Education; Asset Pricing; Equity Premium Puzzle; Time-Varying Risk Aversion JEL Classifications: E32, E44, G12
∗
I am grateful to my thesis committee members Geert Bekaert, Andrew Ang and John Donaldson for their invaluable guidance and support throughout the course of this research. I thank Fernando Avalos, Monika Piazzesi, Tano Santos, Suresh Sundaresan, Lu Zhang, and seminar participants at Columbia University, the Federal Reserve Board, Lehman Brothers, Northwestern University, Stanford University, the University of North Carolina at Chapel Hill and University of Rochester for helpful comments and suggestions. Remaining errors are my own. † Graduate School of Business, Columbia University, 311 Uris Hall, 3022 Broadway, New York, NY 10027. Email:
[email protected].
1
1
Introduction
Human capital is by far the largest component of wealth for most people. For example, Kendrick (1976) finds that human capital makes up more than half of the total capital stock in the United States, while Jorgenson and Fraumeni (1989) document that human capital makes up a remarkable 93% of aggregate wealth in the United States. Becker (1993) estimates the value of human capital to be at least four times as large as the value of stocks, bonds, housing and all other assets combined. As part of the return to the human capital, labor income constitutes more than two-thirds of the total income in the U.S. and other developed countries. Apart from its sheer magnitude, the fact that the share of labor income in total income varies countercyclically over time suggests that human capital can also have important implications for asset pricing. Despite this importance, human capital has rarely been formally modelled in an assetpricing framework. The majority of the asset pricing literature follows the Robert E. Lucas (1978) tradition and proceeds in an exchange economy setting, where the representative agent derives utility from her consumption only and her total wealth equals her financial wealth. In this paper, I propose a dynamic equilibrium model with human capital and education. This model is different from the existing literature in five ways. First, the representative agent’s utility function is non-separable over consumption and leisure, where leisure is augmented by the human capital. The leisure-augmenting effect of human capital can be justified by considering the non-market benefits of human capital.1 This specification implies that an average investor may exhibit time-varying risk aversion towards uncertainty in consumption growth even if her utility function is completely standard in the consumption-leisure composite. In particular, when there is a negative human capital shock, output drops and consumption goes down, but human capital decreases even more, meaning that consumption will make up a larger share in the consumer’s utility bundle. In such a scenario the consumer might be more sensitive to consumption risks, hence exhibiting a countercyclical risk aversion to consumption. 1
See Michael (1982), McMahon (1997) and Wolfe and Haveman (2001).
2
Second, the agent is concerned about three activities: work, leisure and education. When consumption drops during an economic downturn, the agent would like to work more to raise her current consumption. This is not very rewarding, however, because productivity and wages are low during such a time. In my model, the agent has another alternative. She can invest in education, training or other activities that enhance her human capital, so that she can expect higher wage incomes and larger non-market benefits from human capital in the future. There is some empirical evidence that people spend more time in education during economic downturns by either going back to school or staying longer in school.2 These observations suggest that a richer model of consumption/labor/education/investment choices might have different asset-pricing implications from the standard models. Third, I allow both productivity shocks and shocks to human capital. The productivity shock is a common way of modelling sources of business cycle fluctuations in the dynamic general equilibrium literature,3 while the human capital shock is rarely modelled in a production economy setting.4 Examples of a negative human capital shock include the loss of firm- or sector-specific human capital due to job loss, while an improving labor market is an example of a positive human capital shock. Fourth, I introduce time-to-plan restrictions on human capital investment, in addition to similar restrictions on physical capital investment. Production-economy models typically generate a much lower equity premium than endowment-economy models, as the firm’s demand for capital – the asset supply – reacts to technology shocks and varies procyclically over time, which acts to decrease asset price fluctuations and the riskiness of equity claims. Adding the labor-leisure-education choice further aggravates the problem, because it gives the agent one extra channel to smooth her consumption. Imposing time-to-plan restrictions prevents a full adjustment in both types of capital in the short run, leading to larger swings in asset price and a higher expected equity return. 2
See Dellas and Sakellaris (1996) and Sakellaris and Spilimbergo (2000). See Kydland and Prescott (1982). 4 Exceptions include Pries (2001) and Krebs (2003). 3
3
Finally, this model separates consumption and dividends. In the Lucas-type economy, consumption is the same as dividends. In the data, dividend and consumption are only weakly correlated, and dividends are much more volatile than consumption. Most endowment economy studies either equate aggregate dividends with aggregate consumption, or assume an exogenous process for the dividend-consumption ratio. In this model both series are endogenously determined and different from each other. After a careful calibration of the model parameters, I simulate the model and compute the small-sample macroeconomic and asset return statistics. My main findings are: The model matches most macroeconomic statistics, while improving on several dimensions over the standard dynamic equilibrium model, including generating a higher volatility in hours worked and a countercyclical time spent in education. On the asset pricing side, at a moderate relative risk aversion coefficient of five, the model generates a low annual mean riskfree rate of 0.68%, an annual mean equity premium of 1.67%, an upward-sloping yield curve, and a Sharpe Ratio of 18.7%. It also generates plausible cyclical variations in asset returns, producing a countercyclical dividend yield and replicates long-horizon predictability of excess return by dividend yields. My model can be linked to the external habit literature, which models the representative consumer as deriving utility from her consumption in excess of a slow-moving external habit.5 The main intuition from these models is that the agent will exhibit different attitudes towards risk as her “habit” level – a reference point against which she measures her consumption – evolves over time and responds to past consumption fluctuations. The fluctuating risk aversion makes equity a less attractive investment compared to bonds, because the periods of high expected equity returns generally coincide with periods where the representative investor has higher risk aversion. It also generates cyclical variations in asset returns. Unlike the traditional habit literature,6 however, the “habit” in these models is no longer linked to past consumptions in a linear fashion, but follows a conveniently 5
See Campbell and Cochrane (1999), Bekaert, Engstrom, and Grenadier (2004), Wachter (2003) and
Menzly, Santos, and Veronesi (2002, 2003). 6 See Abel (1990), Constantinides (1990) and Sundaresan (1989).
4
specified exogenous process. Therefore, these models are silent on the more fundamental question of what economic factors are driving the “habit” process and the time-varying risk attitude in general. In my model, the human capital-scaled leisure-consumption ratio acts as a second risk factor like the habit factor in habit-persistence models, generating cyclical variations in the agent’s risk attitude and in asset prices. In contrast to an externally specified habit, the leisure-consumption ratio is endogenously determined. The rest of the paper is structured as follows. Section 2 reviews the related literature. The model is outlined in Section 3, and calibrated and solved in Section 4. Its business cycle and asset pricing implications are investigated in Section 5. Finally, Section 6 provides some concluding remarks. Data descriptions and all proofs are provided in the Appendix.
2
Related Literature
This paper builds on five strands of literature. Dynamic Stochastic General Equilibrium (DSGE ) Models simultaneously model the consumption and the production side of the economy, and have become the standard model to use in studying growth and business cycle issues in macroeconomics.7 The DSGE framework is a natural context in which asset pricing issues can be studied, where macroeconomic factors are allowed to affect both output and asset prices.8 Most of these models assume a standard CRRA utility for the representative agent and are susceptible to the same difficulties outlined by Mehra and Prescott (1985). Furthermore, Rouwenhorst (1995) shows that even an unusually high risk aversion does not necessarily leads to a high equity premium, because the firm’s ability to adjust its production plans 7
See Brock and Mirman (1972), Hansen (1985), King, Plosser and Rebelo (1988a, 1988b), Cooley and
Prescott (1995) and King and Rebelo (1999). They are also known as Real Business Cycle Models. 8 Such an investigation has been undertaken by Brock (1979), Donaldson and Mehra (1984), Naik (1994), Rouwenhorst (1995), Boldrin, Christiano, and Fisher (1995, 2001), Jermann (1998), Avalos (1999), Tallarini (2000), Danthine and Donaldson (2002), Lettau (2003), and Gomes, Yaron, and Zhang (2003), among others.
5
reduces stock price fluctuations and lowers the equity return. Few studies in this literature incorporate labor-leisure-education choices or feature an explicit consideration of human capital, both of which are among the main features of my model. This paper is also related to the literature on Human Capital Formation. BenPorath (1967) develops a model in which the agent invests in educational activities and earns an exogenous market rent on her human capital. This model is the main vehicle in studying the life cycle of earnings and the earnings distribution.9 Heckman (1976) further captures the notion that human capital may have non-market benefits by allowing it to interact with the agent’s leisure time. A number of papers incorporate human capital into the DSGE framework.10 In these models, the representative agent either enjoys her consumption only, or has an utility that is separable in consumption and leisure. Human capital is usually modelled like physical capital, acting as another input in the production process and accumulating through investment in the form of consumption goods. Finally, non-market benefits of human capital are rarely taken into account. In contrast, this paper assumes non-separable utility, models human capital as accumulating through time spent in education, and allow human capital to enhance the agent’s leisure time. A number of researchers study (C)CAPM Models with Exogenous Labor Income without modelling the production sector, and find that incorporating human capital does not improve the empirical support for Mayers’ (1972) CAPM,11 while others find that including human capital helps both in predicting future returns and in explaining returns in the cross section.12 While I follow a similar approach and estimate a latent-factor model of asset pricing, the full model endogenizes the labor income and labor supply decisions. The fourth strand of literature is the Production-Based Asset Pricing Models,13 9 10
Surveys in this area include Mincer (1997), Weiss (1986) and Neal and Rosen (2000). See Uzawa (1965), Lucas (1988), King and Rebelo (1990), Romer (1990), Mankiw, Romer, and Weil
(1992), Caball´e and Santos (1993), Einarssona and Marquis (1998), Perli and Sakellaris (1998), LadronDe-Guevara, Ortigueira, and Santos (1999), Ortigueira (2000), Pries (2001) and Krebs (2003), among others. 11 See Fama and Schwert (1977) and Campbell (1996). 12 See Jagannathan and Wang (1996), Santos and Veronesi (2003) and Palacios-Huerta (2003c). 13 See Cochrane (1991, 1996).
6
which look at the production side of the economy that is overlooked by the (C)CAPM models. However, they are still partial-equilibrium in nature. Because the consumer’s problem is not considered in the model, the pricing kernel cannot really be pinned down. For example, Cochrane (1996) only tests the hypothesis that the pricing kernel is a linear function of the returns on real investment. The current paper simultaneously models the consumption and the production side of the economy, and can fully characterize the pricing kernel and links between asset prices and macroeconomic variables. Finally, this paper is related to the vast exchange-economy Equity Premium Puzzle Literature,14 and in particular the related research on habit persistence and timevarying risk aversion. Despite the recent success of habit-persistence models, a number of authors identify problems with the habit formation specification.15 Empirical studies using aggregate-level data generally support the existence of habit,16 while studies using household-level data are less supportive.17 A number of authors model time variations in the representative agent’s risk attitude while being agnostic about the source of variations,18 while several recent papers begin to link changes in the risk aversion to underlying macro variables in an endowment economy setting.19 Using a production-economy model, the current paper argues that time variations in the relative shares of consumption and leisure can shift the risk attitude of a typical investor, and may constitute a systematic risk that needs to be priced. Two papers that are most closely related to mine are Avalos (1999) and Guvenen (2003), both of which conduct asset-pricing analysis in a DSGE framework. Avalos (1999) 14 15
Excellent reviews of this literature include Kocherlakota (1996) and Mehra and Prescott (2003). See Lettau and Uhlig (2000), Otrok, Ravikumar, and Whiteman (2002), Bansal and Yaron (2004),
Bansal, Gallant, and Tauchen (2003), and Chapman (2002) 16 See Osborn (1988), Ferson and Constantinides (1991), Ferson and Harvey (1992), Heaton (1995) and Daniel and Marshall (1999). 17 See Heien and Durham (1991), Naik and Moore (1996), Dynan (2000). 18 See Normandin and St-Amour (1998), Gordon and St-Amour (2000, 2003), Bekaert, Engstrom, and Grenadier (2004), Melino and Yang (2003) and Danthine, Donaldson, Giannikos, and Guirguis (2002). 19 See Bakshi and Chen (1996), Piazzesi, Schneider, and Tuzel (2003), Lustig and Van Nieuwerburgh (2003), Guvenen (2003) and Brandt and Wang (2003).
7
introduces time-to-plan restrictions on the physical investment, and finds his model able to generate a higher equity premium compared to the standard model. He does not consider non-separable utility over consumption and leisure, human capital, or educational activities. My model can generate a countercyclical risk aversion, a mechanism that is absent in his model. As a result, the equity premium in this model is two times as high as in his model for comparable parameterizations. Guvenen (2003) looks at a two-agent DSGE model, where the agents differ in whether or not they have access to the stock market and in their elasticities of intertemporal substitution (EIS ). In equilibrium, the high-EIS stockholder borrows from the low-EIS non-stockholder, essentially providing an insurance to the latter group against aggregate shocks. This has two effects on the asset prices. On the one hand, the stockholder’s consumption becomes leveraged and hence more volatile, leading to a higher market price of risk. On the other hand, the stockholder charges the non-shareholder an insurance premium by demanding a low riskfree rate. In contrast, this paper preserves the oneagent complete-market framework and shows that introducing human capital and a more general utility specification can also improve the model’s fit to various asset pricing facts.
3
Model
I assume that the economy is populated by a large number of homogeneous agents and that markets are complete. Directly marketable claims on human capital do not exist per se. However, as argued by Jagannathan and Wang (1996), existing financial claims like mortgage loans and life insurance, which are traded against or used to hedge future labor income shocks, already make human capital partly tradable. Furthermore, the ongoing progress in creating markets for claims on aggregate income and service flows as proposed by Shiller (1993) would make labor income and human capital more and more similar to other tradable financial assets.
8
3.1
Consumers
The representative consumer in this economy maximizes her lifetime utility over consumption and leisure:20 E0
∞ X t=0
(1−γ)/ρ
βi
[αCtρ + (1 − α) (Ht Lνt )ρ ] 1−γ
, 0 < α, β < 1, ρ ≤ 1, γ > 0
(1)
where Ct , Lt and Ht are consumption flow, leisure services and human capital level respectively, and the time period is one quarter. The single-period CES utility function (1−γ)/ρ
V
(Ct , Ht Lνt )
[αCtρ + (1 − α) (Ht Lνt )ρ ] = 1−γ
(1−γ)
reduces to a Cobb-Douglas utility function V (Ct , Ht Lνt ) =
[Ctα (Ht Lνt )1−α ] 1−γ
when ρ = 0.
Consumption and leisure are complements if ρ is negative and substitutes if ρ is positive. They are perfect substitutes if ρ equals 1. Two features of the utility function V set it apart from the standard specification. First, leisure enters V in a non-separable fashion, meaning that the consumer’s risk attitude towards consumption will vary over time and depend on the amount of leisure she enjoys. Second, following Heckman (1976) and Ortigueira (2000), people with a higher level of human capital Ht are modelled as deriving larger utility from leisure. Such a modelling choice can be motivated by the non-market benefits of human capital.21 There is extensive evidence that one’s own human capital positively affects the health status and the life expectancy of oneself and the spouse, the quality of one’s children,22 the marital stability,23 and the efficiency in consumption24 and in home production of non-market goods.25 Wolfe 20
Other studies that also employ a two-good non-separable CES utility include Piazzesi, Schneider,
and Tuzel (2003) and Lustig and Van Nieuwerburgh (2003). 21 Recent surveys of this literature include Michael (1982), McMahon (1997) and Wolfe and Haveman (2001). 22 See 23 See 24 See 25 See
Haveman and Wolfe (1994). Becker, Landes, and Michael (1977). Michael (1972) and Schultz (1975). Eisner (1988) and Benhabib, Rogerson, and Wright (1991).
9
and Haveman (2001) conclude that the magnitude of such effects can be as large as the marketed effects of human capital. These non-market benefits represent additional returns to human capital other than what is realized through the labor market. Here I capture these benefits by allowing human capital to directly enhance the leisure services. If there is a market for scaled leisure, its time-t price in terms of consumption goods, pLt , must be equal to the ratio of the two marginal utilities: pLt
∂Vt /∂ (Ht Lt ) (1 − α) (Ht Lt )ρ−1 = = , ∂Vt /∂ (Ct ) αCtρ−1
and the share of leisure-related expenditure is φLt =
(1 − α) (Ht Lt )ρ Ht Lt ∗ pLt = . αCtρ + (1 − α) (Ht Lt )ρ Ct + Ht Lt ∗ pLt
It can be shown that the time-t coefficient of relative risk aversion (RRA) in this economy is a function of φL : RRAt = −
¡ ¢ ∂ log (∂Vt /∂Ct ) = γ 1 − φLt + (1 − ρ) φLt , ∂ log Ct
On the one hand, when consumption drops by 1%, the marginal utility of consumption will increase because the agent is risk averse. In the standard model with power utility over consumption, the marginal utility of consumption will increase by γ%. In my ¡ ¢ model, it increases by a smaller percentage of γ 1 − φLt , because the consumption-leisure 1/ρ
composite, [αCtρ + (1 − α) (Ht Lνt )ρ ]
, only drops by a percentage of 1 − φLt . This risk
aversion effect is summarized in the first term. On the other hand, a lower consumption also lowers the consumption-leisure ratio, which leads to further increase in the marginal utility of consumption, because the agent wishes to maintain a balance between consumption and scaled leisure. This effect is more pronounced if consumption and leisure are less substitutable (ρ is smaller), or if consumption already makes up a smaller share in the total expenditure (φLt is larger). This substitution effect is summarized in the second term. In summary, the coefficient of relative risk aversion, RRAt , varies over time as the expenditure share of leisure, φLt , varies over time. The sign of 1 − γ − ρ determines how 10
RRA changes with φLt . In particular, RRAt will increase (decrease) with φLt if 1−γ −ρ > 0 (< 0). It is easier to track the quantity share rather than the expenditure share of each good. I define a new variable Qt as the scaled leisure-consumption ratio: Qt = Ht Lt /Ct . Ceteris paribus, a larger leisure-consumption ratio, Qt , raises the expenditure share of leisure, φLt . It also lowers the relative price of scaled leisure, pLt , which reduces the expenditure share of leisure, φLt . The relative importance of these two effects is controlled by 1 − ρ. If ρ > 0, the quantity effect outweighs the price effect, and the quantity ratio, Qt , will be positively related to the expenditure share of leisure, φLt . The opposite is true if ρ < 0. I can re-write RRA as a function of Qt : RRAt = γ +
1−γ−ρ . −ρ α Q + 1 t 1−α
(2)
The sign of ρ (1 − γ − ρ) then determines how RRA co-varies with Qt . If ρ (1 − γ − ρ) < 0, RRA decreases with Qt . This could happen in two cases. The first case is when 1 − γ − ρ > 0 and ρ < 0, a larger Qt induces a smaller φLt and a smaller RRA. The second case is when 1 − γ − ρ < 0 and ρ > 0, a larger Qt induces a larger φLt and a smaller RRA. The sign of ρ (1 − γ − ρ) also determines what is a bad state in this economy. The agent fears states with a low consumption growth. If ρ (1 − γ − ρ) < 0, she also fears states with a smaller Qt because her marginal utility of consumption will rise more rapidly in the event of a negative consumption shock. The agent dislikes such states and prefers assets that pay out more during such times. The opposite is true if ρ (1 − γ − ρ) > 0. In the data, both consumption and labor are highly procyclical (King and Rebelo (1999)), suggesting a countercyclical leisure if hours spent in education is relatively smooth. However, Qt can be either procyclical or countercyclical depending on the business cycle property of human capital level. The business cycle property of Qt further
11
decides that of RRA, with RRA being countercyclical if ρ (1 − γ − ρ) < 0 (> 0) and Qt is procyclical (countercyclical).
3.2
Technology
I specify two shocks in this economy: a shock to the human capital level AH t and a productivity shock AK t . The human capital shock evolves according to H H H ln AH t = (1 − ψH ) ln A + ψH ln At−1 + σH ²t
(3)
while the productivity shock evolves according to K K K ln AK t = (1 − ψK ) ln A + ψK ln At−1 + σK ²t
(4)
© ª K 0 where AH and AK are the steady-state values of these two shock and ²t = ²H is t , ²t standard normal with a correlation coefficient ϕKH . It is well known that we need restrictions on the elasticity of capital supply to generate any equity premium. Without such restrictions, firms will continue investing until the market value and the replacement cost of one additional unit of capital become equal, and the return to physical capital and the return to equity become equal. In such cases the equity return volatility and the equity premium will be minimal since the physical capital return is small and nearly constant in the data.26 In this paper, I adopt a time-to-plan specification for both physical and human capital following Kydland and Prescott (1982), Christiano and Todd (1996) and Avalos (1999). The time-to-plan restriction causes investment to be reflected in production not immediately but after multiple planning and building periods. In particular, the law of motion for the physical capital stock follows Kt = (1 − δK ) Kt−1 + S1,t Sj,t = Sj+1,t−1 , 26
(5)
j = 1, . . . , J − 1
See Jermann (1998), Boldrin, Christiano, and Fisher (1995, 2001), Lettau and Uhlig (2000) and
Danthine and Donaldson (2002).
12
where Kt is the physical capital stock at the beginning of time t + 1 but in the time-t information set, Sj,t is the value of the investment project j periods away from completion, and δK is the depreciation rate of the physical capital. Let φK j for j = 1, . . . , J be the fraction of resources allocated to the investment project Sj,t . In each period t, there exist up to J investment projects simultaneously: J − 1 projects started 1 to J − 1 periods earlier, plus one new project SJ,t started this period. Each project requires an investment worth φK j fraction of the total value Sj,t . Therefore, the total investment during this period is It =
J X
φK j Sj,t .
(6)
j=1
The human capital accumulation process is specified in a similar fashion: ¡ ¢ κ Ht = AH 1 − δH + τ U1,t−1 Ht−1 t Uj,t = Uj+1,t−1 ,
(7)
j = 1, . . . , J − 1
where Ht is the human capital stock at time t, Uj,t is the amount of time the consumer spends in all human-capital-enhancing projects (henceforth I call these activities “education”) j periods from completion, δH is the depreciation rate of the human capital, τ is a productivity or ability parameter, and κ is a return-to-scale parameter. The total amount of time spent on education in period t is Ut =
J X
φH j Uj,t ,
j=1
where φH j , j = 1, . . . , J is the fraction of time allocated to project Uj,t . I use J = 4 for both physical and human capital accumulation processes, representing H a building time of one year. There are two ways of specifying φK j (φj ) in the literature.
The original Kydland and Prescott (1982) formulation sets φK j = 0.25 for all j, meaning resources flow uniformly across all building periods. Christiano and Todd (1996), on the K other hand, set φK 4 = 0.1 and φj = 0.33, j = 1, 2, 3, representing one initial planning
period when few resources are used. Christiano and Todd (1996) found that using this
13
specification improves the model’s ability to fit some key business-cycle properties. I H adopt this latter specification for both φK j and φj .
I specify the technology as a standard Cobb-Douglas function. Given the beginningof-period physical capital stock Kt−1 and human capital stock Ht−1 , the current-period K human capital shock AH t and productivity shock At , and the quantity of labor Nt , current-
period output for time t is determined according to 1−θ K θ Yt = A K , 0 < θ < 1. t F (Kt−1 , Ht Nt ) = At Kt−1 (Ht Nt )
(8)
where the current-period human capital level, Ht , is determined according to equation (7).
3.3
Equilibrium
I assume that the agent has one unit of time, which is divided among leisure Lt , education Ut , and labor supply Nt : Lt + Ut + Nt = 1
(9)
Finally, the resource constraint in this economy is Ct + It ≤ Yt
(10)
Since markets are complete, we can exploit the second welfare theorem and find the equilibrium allocations by solving the corresponding central planner’s problem: maximize (1) subject to the constraints (3) through (10) given initial conditions K0 and H0 . The first-order conditions to this problem are given in Appendix C.
14
3.4
Asset Pricing
The optimal solution to the representative consumer’s problem requires that a single pricing kernel, Mt+1 , prices all real assets in this economy: Λt+1 Λt ¸ρ−1 · ¸ 1−γ−ρ · ρ ρ αCt+1 + (1 − α) (Ht+1 Lt+1 )ρ Ct+1 =β ρ ρ Ct αCt + (1 − α) (Ht Lt ) ¸−γ · · ¸ 1−γ−ρ ρ α + (1 − α) Qρt+1 Ct+1 =β ρ Ct α + (1 − α) Qt
Mt+1 =β
(11)
(12)
where Λt = ∂V (Ct , Lνt ) /∂Ct is the marginal utility of consumption at time t. The pricing kernel consists of two parts. The first part is the pricing kernel in all standard consumption-based asset pricing models with power utility, while the second part is unique to this model. I use small letters with tildes to denote the logarithm of corresponding capital letters and capital letters without time index to denote the steady-state values, and linearize the pricing kernel around the steady states:27 m e t+1 , ln (Mt+1 ) ≈ ln β − γ∆e ct+1 − ζ∆e qt+1
(13)
ρ
(1−α)Q where ζ = − (1 − γ − ρ) α+(1−α)Q ρ > 0 if γ > 1. It has a similar form as the pricing kernel
in Campbell and Cochrane (1999). The risk-free rate in this economy is 1 2 rtf = −Et m e t+1 − σm , 2 e
(14)
while the excess return of a risky asset depends on its covariance with the pricing kernel: 1 2 = −covt (rt+1 , m e t+1 ) = ζcovt (rt+1 , ∆qt+1 ) + γcovt (rt+1 , ∆ct+1 ) (15) Et rt+1 − rtf + σr,t 2 27
Details of the linearization are provided in Appendix B.
15
4
Solving the Model
4.1
Model Calibration
I calibrate the model to match some key long-run economic statistics at the quarterly frequency. Table 1 summarizes the calibration process. 4.1.1
Long-Run Behavior
Following the literature (see Campbell (1994) and Jermann (1998), for example), I set the long term output growth G to 1.004 and the quarterly riskfree rate under certainty R to 1.006. The capital share in the production function θ is fixed at 0.36, which implies a steadystate output-physical capital ratio Y = K0
P4
j ΦK j R = 0.0755 θ
j=0
and a investment-physical ratio of 4
X I j = ΦK j G = 0.0251, K0 j=0 both of which are close to the values commonly used in the literature. 4.1.2
Human Capital and Productivity Shocks
I normalize steady-state values of the productivity shock, AK , and the human capital shock, AH , to one. Estimations of the Solow residuals typically yield a highly persistent process that is close to a random walk.28 Here I set the persistence coefficient ψK to 0.98. I look at three types of economies: two economies with productivity shocks only (called “PShock” hereafter) or human capital shocks only (called “HShock” hereafter), and one economy with both shocks (called “Benchmark” hereafter). In the “PShock” model, I set the quarterly volatility of productivity shocks, σK , to 0.712%, as is standard in the 28
See Prescott (1986) for example.
16
literature. The volatility of human capital shocks, σH , is not a standard parameter. For the purpose of fair comparison, I fix shock volatilities in “HShock” and “Benchmark” models so as to generate the same output volatility as in the “PShock” model, and assign equal weights to productivity and human capital shocks in the “Benchmark” model. This leads to a quarterly volatility of 1.502% for the human capital shock in the “HShock” model and 0.645% for both shocks in the “Benchmark” model. 4.1.3
Preferences
At the steady state, returns on all assets equal the return under certainty R, and the pricing equations follow Gγ = βR, which determines the rate of time preference β = less than
log R log G
Gγ . R
For β to be less than one, γ must be
= 1.5. A smaller-than-unity discount rate is reasonable by introspection,
but not necessary for a finite expected utility (see Benninga and Protopapadakis (1990)). 4.1.4
Production and Capital Accumulation
Heckman (1976) estimates an annual depreciation rate of human capital of 0.037, which corresponds to a quarterly rate of δH = 1 − (1 − 0.037)1/4 = 0.009. The depreciation rate of the physical capital δ is fixed at 0.021, which is the standard value used in the literature. A final set of the parameters and steady-state statistics are determined by evaluating
17
the first-order conditions at the steady state: Steady-state output-physical capital ratio
Y K0
=
Steady-state physical investment-capital ratio
I K0
=
Steady-state education:
U=
Productivity parameter in Ht accumulation:
τ=
Steady-state human capital-physical capital ratio:
H K0
Steady-state Ht -scaled leisure-consumption ratio:
Q=
P4
j ΦK j R θ
j=0
P4
j=0
j ΦK j G
ν(1−βG1−γ ) G−1+δH Uκ 1 N
=
³
Y K0
1+(ν−1)N P4 G R j j=1 ϕj ( G ) +1 κ(G−1+δH )
1 ´ 1−θ
H/K0 ν L C/K0
"
³ ´− θ 1−θ (1−θ) KY
Consumption share in the utility bundle zt
α= 1+
Steady-state leisure:
L=1−U −N
#−1
0
νQρ−1 Lν−1
They depend on a set of free parameters Θ = {ν, ρ, κ, ψH , ϕKH , γ}, where ν is the extra risk aversion coefficient towards leisure, ρ controls the intratemporal substitution between consumption and leisure, κ is the return-to-scale parameter in ht−1 dynamics, ψH is the persistence of human capital shocks, ϕKH is the correlation coefficient between productivity and human capital shocks, and γ is the curvature coefficient in the utility function. In the main analysis, ν = 0.5, ρ = 0.5, κ = 0.05, ψH = 0, ϕKH = 0, and γ is fixed to set RRA to 5. These values are chosen informally rather than through a formal search over the parameter space. Therefore a more careful parameterization method might further improve the performance of the model. In Section 5.4, I also experiment with alternative values for these parameters and report the results.
4.2
Model Solution and Asset Prices
This section outlines the method used in solving the model and pricing the assets in a log-linear environment.29 The model is solved using the method of log-linear approximation and undetermined coefficients following King, Plosser and Rebelo (1988a), Campbell (1994), Uhlig (1997) and Jermann (1998). I first log-linearize the first-order conditions derived in Appendix C 29
More details are provided in Appendices D and E. A complete derivation is available upon request.
18
and collect them in a stylized system of equations, and then solve the system using the method of undetermined coefficients to obtain the model dynamics: It is straightforward to apply the Jermann (1998) log-normal asset pricing methodology in this framework. The basic pricing equation requires that the current price Pt of a stream of future dividends {Dt+j }∞ j=1 equals the present value of all dividends discounted using the pricing kernel: ³ Pt
{Dt+j }∞ j=1
´
· ¸ exp (λt+j ) = Et β Dt+j , exp (λt ) j=1 ∞ X
where the logarithm of the marginal utility of consumption, λt , and the logarithm of the dividend, dt , are both solved as part of model dynamics. I examine three types of artificial assets in this economy: an n-period real bond that yields one unit of consumption good n periods later, a claim to the physical capital – the equity, and a claim to the human capital. I assume that the firm does not issue new stocks and finances its capital solely through retained earnings. Under this assumption each period the equity holder earns a dividend DtK = Yt −
∂Yt Nt − It . ∂Nt
The “dividend” to the holder of an artificial claim to the human capital stock consists of three parts: (1) the marginal utility from human capital, measured in terms of consumption goods, multiplied by the human capital stock, (2) the marginal product of human capital multiplied by the human capital stock, and (3) minus the value of time spent in education in terms of consumption goods. DtH =
∂Yt ∂Yt ∂Vt /∂Ht Ht + Ht − Ut ∂Vt /∂Ct ∂Ht ∂Nt
A n-period zero coupon bond can be priced by setting Dt+n = 1 and Dt+j = 0, j 6= n.
5
Empirical Results
In this section I solve the model specified and calibrated in previous sections, and examine its business-cycle and asset pricing implications. I simulate 500 samples using the solved 19
model dynamics, each of 203 periods (the length of the data used to compute the data moments), compute various business-cycle and financial statistics, and compare them to their data counterparts. I first look at three basic cases where ν = 0.5, ρ = 0.5, κ = 0.05, ψH = 0, ϕKH = 0 γ : fixed to set RRA to 5, {0, 0.712%} (“PShock”), {σH , σK } = {1.502%, 0} (“HShock”), {0.645%, 0.645%} (“Benchmark”), and then conduct a sensitivity analysis with regard to various parameters in the final subsection. I estimate two other models for comparison purposes. The first one is a DSGE model with a flexible labor supply, where the representative agent’s utility function is multiplicatively separable over consumption and leisure (Ct Lνt )1−γ − 1 , U (Ct , Lt ) = 1−γ
(16)
and there is no restriction on the capital accumulation process. The extra risk aversion parameter ν is fixed to set the steady-state labor supply to 1/3. With other parameters set to the calibrated values, ν equals 1.92. The results from this model are reported in columns marked “Standard”. The second model follows the same setup as the “Standard” model but imposes timeto-plan restrictions on the physical capital accumulation, formulated as in Avalos (1999). The agent has the same utility as in equation (16), and the physical capital evolves according to (5) and (6). The results from this model are reported in columns marked “Standard TTP”.
5.1
Habit and the leisure-consumption Ratio
Before going to the details of model analysis, I first examine the properties of the factor ∆qt , which plays a central role in this model. It summarizes all the new information 20
of human capital for asset-pricing purposes. For example, how variable this factor is determines how different my model is from standard consumption-based asset pricing models. How it co-varies with the macroeconomic variables determines the business cycle property of the market price of risk, while how it co-varies with equity returns determines whether or not equity entails additional risk to hold. The lack of data on human capital stock and time spent in leisure prevents us from directly testing the validity of including ∆qt in the pricing kernel in the form of equation (13).30 However, I can follow a reverse-engineering approach to see what properties ∆qt needs to satisfy in order to fit the asset pricing data, and compare them to those implied by the model. We already know from Campbell and Cochrane (1999) that adding an external habit factor can greatly enhance the model’s fit to a number of asset pricing facts. If indeed this factor can be interpreted as the human-capital-augmented leisureconsumption ratio, as suggested by this model, they should exhibit similar characteristics. On the other hand, pinpointing the differences between them will help us understand the deficiencies of my model. To conduct such a comparison, I first assume that the state of the economy can be summarized by a vector of state variables including the consumption growth ∆e ct , a latent factor qet , the dividend growth ∆det and the inflation π et , which follows a VARMA(1,3) process. I estimate this model by Maximum Likelihood and invert out the latent factor qet from the price-dividend ratio. Results from such an exercise confirm Campbell and Cochrane (1999)’s finding that augmenting the standard CCAPM model with a second risk factor can improve the model’s fit. The key element is a procyclical latent factor that proxies for a countercyclical risk aversion of the representative agent. Details of this model and the estimation are given in Appendix F. Table 2 compares the characteristics of both ∆e qt inverted from the latent-factor model 30
In the labor economics literature the human capital stock is often proxied by the educational attain-
ment index (see, among others, Psacharopoulos and Arriagada (1986b), Psacharopoulos and Arriagada (1986a) and Barro and Lee (1993)). Such a proxy is appropriate for cross-section comparisons or earnings analysis, but not my analysis since education is only an input into the human capital formation process.
21
above and ∆qt simulated using the solved dynamics from the three production-economy models. Column 1 reports the first-order autocorrelation of ∆qt . The inverted ∆qt from the latent-factor model is mean reverting with a quarterly autocorrelation coefficient of 0.50, while the “HShock” model, the model with human capital shocks only, generates a less significant autocorrelation of 0.31. Columns 2 to 4 report results from decomposing the variance of the pricing kernel in equation (13) into two parts that are due to ∆ct and ∆qt respectively, and a covariance term. In the latent factor model, the variance of the pricing kernel is mainly due to ∆qt , and the contribution of the covariance term is negative but insignificant. In the “HShock” model, the consumption growth plays a much more important role, and the covariance term contributes a hefty 33% to the variance of the pricing kernel. Columns 5 to 8 report the response rate of various asset return variables to ∆qt , cov (rt , ∆qt ) , var (∆qt ) computed as the slope coefficients in a regression of rt onto the current-period ∆qt rt = α + β∆qt , where the regressand rt is specified on the top of each column. In the latent-factor model, the response of the short rate to a shock to ∆qt is negatively and marginally significant, the responses of the stock return and the excess stock return are positive and significant, while the response of the expected excess stock return is positive but insignificant. Similar patterns can be found in the model with human capital shocks only, but the responses of the equity return and the equity premium are less significant. Columns 9 to 12 report correlations of ∆qt with various macroeconomic and assetpricing variables. In both the latent-factor model and the “HShock” model, ∆qt is positively and significantly correlated with the consumption growth, the output growth, the dividend growth and the equity return. When we introduce productivity shocks, the similarities quickly disappear. The “PShock” and the “Benchmark” models generate a negative autocorrelation of around −0.3 in Col22
umn 1. In Columns 2 to 4, both consumption growth and ∆qt become much more volatile than the pricing kernel. The consumption growth remains a more important component, while the contribution of the covariance term become significantly negative. Models with productivity shocks also imply a positive and significant response in the riskfree rate, and negative and significant responses in the stock return and the excess stock return in Columns 5 to 8, contrary to what we see in the “Latent” and the “HShock” models. Finally, in columns 9 to 12, when we allow productivity shocks, all correlations become significantly negative. In summary, we observe similar patterns of ∆qt in the latent-factor model and the production-economy model with human capital shocks only, while models with productivity shocks generate ∆qt that exhibits very different characteristics. These results show that there could be a link between ∆qt inverted from the latent-factor estimation and that from a production-economy model where human capital shocks play a dominant role. In the following sections, I investigate the business cycle and asset pricing implications of the models in more detail.
5.2
Business Cycle Implications
Table 3 reports business cycle statistics from the data and five different models. All variables are quarterly, in logarithms and percentage terms, and detrended using the Hodrick and Prescott (1997) filter, to be consistent with the statistics reported by King and Rebelo (1999). The first two models are the “Standard” and “Standard TTP” models described above, while the rest are my proposed model with one or two shocks: productivity shocks in the “PShock” model, human capital shocks in the “HShock” model, and both shocks in the “Benchmark” model. All models are broadly consistent with the data, in the sense that the volatilities and the comovements of major macro series are quite similar to the stylized facts documented in the business cycle literature. In particular, consumption is less volatile than output, while investment is roughly three times as volatile as output. 23
In the “Standard” model with no restrictions on the capital accumulation process, the representative agent adjusts her consumption and investment plans fully and instantaneously as shocks occur, resulting in almost perfect correlations between output and other macroeconomic series. The time-to-plan restriction requires that the adjustment be made gradually over four periods of time, which reduces the correlations between those series to a level consistent with the data. Compared to the “Standard TTP” model, the “PShock” model generates higher economic fluctuations, both in the level of output volatility and in relative volatilities of consumption, hours worked and investment. Comparing the two one-shock economies, “PShock” and “HShock”, we see that human capital shocks generate a relatively more volatile consumption and a relatively less volatile investment. One statistic of particular interest is the volatility of hours worked. Standard DSGE models are unable to generate a sufficiently volatile labor as observed in the data.31 My results show that this is still the case with the “standard” model, which generates a relative volatility of 0.274 compared to 0.99 in the data. The relative volatility rises to 0.395 in the “Standard TTP” model and further to 0.418 in the model “PShock” with productivity shocks. In the comparative studies below I will show that this model is capable of generating a relative labor volatility as high as 0.96. A new variable introduced in this model is hours spent in education. There is some empirical evidence that people spend more time in education during economic downturns either by going back to school or by staying longer in school. For example, graduate and professional schools typically receive larger numbers of applications as the economy slows down, as people go back to school to brush up their skills or explore new career paths. Dellas and Sakellaris (1996) document that the propensity of 18- to 22-year old high school graduates to enroll in college is significantly countercyclical after controlling 31
See Prescott (1986) and Einarssona and Marquis (1998). Hansen (1985) studies a model of Rogerson
(1988)-type indivisible labor and finds it able to generates a high relative volatility of labor (0.77). However, Mulligan (2001) shows that when aggregated over a continuum of heterogeneous consumers, the indivisible labor model becomes equivalent to a divisible labor model.
24
for observable characteristics. Sakellaris and Spilimbergo (2000) provide international evidence that U.S. university enrolment of students from OECD countries is positively correlated with business cycles of the sending countries. Consistent with this evidence, all three variations of my model produce negative correlations between the education and output, with a very negative value when human capital shocks are the only type of uncertainty in the economy (−0.41 in “PShock”, −0.40 in “Benchmark”, and −0.76 in “HShock”).
5.3
Asset Pricing Implications
Table 4 reports financial implications of the model and their data counterparts. All statistics are in annualized percentage terms. 5.3.1
The Short Rate
Panel A of Table 4 looks at the short rate. The column labelled “Standard” reports the model with multiplicatively separable utility and perfect short-run elasticity of investment. The “Standard TTP” model introduces time-to-plan restrictions on physical capital accumulation into the “Standard” model. The last three sets of results come from the full model with one or two types of shocks. In the two standard models and the “PShock” model, the small-sample mean of the short rate is comparable to that in the data. In the “HShock” model, however, the short rate turns significantly negative with a volatility that is less than one-fourth of that in the “PShock” model. From equation (14) we know that the first two moments of the short rate are given by h i 1 E rtf = −E [m e t+1 ] − var [∆λt+1 ] , 2 h i var rtf = var [Et [∆λt+1 ]] . The much lower short rate comes from a much more volatile ∆λt+1 in the “HShock” model, which is 10 times as volatile as that in the “PShock” model, as we shall see in 25
Table 7. The lower volatility of the short rate, on the other hand, comes from the fact that Et [∆λt+1 ] is eight times more sensitive to productivity shocks than to human capital shocks. One problem shared by all models with short-run inelasticities is that the volatility of the short rate is too high. In the data, the equity return is five times more volatile than the short rate, whereas all models except the “Standard” model predict a volatility of short rate more than half as high as the equity return. In the “HShock” model, the absolute levels of these two volatilities are much lower, but the relative volatility of the short rate stays counterfactually high. 5.3.2
The Term Structure
Compared to its widely-documented difficulty in generating a high equity premium, it is less well known that the “Standard” model also produces a counterfactual downwardsloping term structure, as shown in Panel B of Table 4. The unconditional mean of a n-period zero-coupon bond yield depends on the conditional variance of the n-period pricing kernel, divided by the number of periods: " Ã n !# X £ b ¤ 1 mt+i . E rt,n = constant − E vart 2n i=1 Without restrictions on the short-term elasticity of physical capital, a positive productivity shock raises both consumption and investment today. Higher consumption lowers the pricing kernel today, while higher investment leads to a higher physical capital stock, higher consumption, and a lower pricing kernel tomorrow. This positive correlation between the current-period and the expected next-period pricing kernels makes the longer bond less risky, and induces a negative term spread. After imposing the time-to-plan restrictions on physical capital investment, the “Standard TTP” model generates an upward-sloping yield curve. In this economy, the agent raises her current consumption and investment in response to a positive productivity shock. However, under the time-to-plan restrictions, the new investment will not turn
26
into productive capital until four periods later, while at the same time extracting resources away from consumption and into investment during these four periods, resulting in a lower consumption growth and a higher pricing kernel. This negative correlation between the current-period and the expected next-period pricing kernels translates into a positive term spread. The yield curve remains upward-sloping in all three productioneconomy models with time-to-plan restrictions on physical and human capital investment. 5.3.3
Excess Equity Return
Panel C of Table 4 looks at the equity pricing implications. Consistent with previous studies, the “Standard” model generates almost no equity premium and low volatilities of returns. The agent is extremely successful in smoothing her consumption over time, resulting in an economy with very low aggregate risk. The “Standard TTP” model with time-to-plan restrictions on physical capital accumulation generates a higher equity premium, although it is still well below what is observed in the data. As explained by Avalos (1999), the firm is restricted in its ability to adjust the level of capital supply in the short run, resulting in more volatile stock prices and a higher equity premium that is required to compensate for this elevated risk. The equity premium is much higher than in previous models when we move to my models, reaching 1.7% in the “Benchmark” model. Similar to what we see in the short rate, the volatility of equity returns in the “HShock” model is only one-sixth of that in the “PShock” model. Combining both shocks, the “Benchmark” model generates a low riskfree rate of 0.2%, an equity premium of 1.7%, and a high Sharpe Ratio of 18.7%. We can see that the “Benchmark” results are much closer to the ”PShock” results rather than to the “HShock” results. This is because human capital shocks have a smaller effect on the economy compared to productivity shocks of the same magnitude, as can be seen from the fact that to generate the same output volatility, we need a standard deviation of 1.502% for human capital shocks in the “HShock” model, but a mere 0.712% for productivity shocks in the “PShock” model.
27
5.3.4
Why is the Equity Premium Higher?
To understand why the production-economy models generate a higher equity premium, we need to look at the impulse responses to different shocks. Figures 1 and 2 graph the impulse response functions of various variables to a 1% positive shock to productivity and human capital respectively. The first two rows graph the responses of various macro variables, while the last rows graph the responses of various asset-pricing variables. In Figure 1, we see a number of patterns frequently documented in the literature: when there is a positive productivity shock, output increases contemporaneously and for several periods afterwards. The consumer consumes more and invests more to smooth her consumption over time. The shock affects the consumer’s decision to work in two ways. She wants to spend more time in leisure because the output is higher, which is a “wealth” effect; she also wants to work more because working is more rewarding when productivity is high, which is a “substitution” effect. The substitution effect dominates and the consumer spends more time working and in education, while spending less time in leisure activities. The higher education converts to a higher human capital level only after four periods and even then has only a small effect on the human capital level under the current specification. Consequently, the logarithm of the augmented leisure-consumption ratio qt drops and the relative risk aversion RRA increases after the shock. Hence the et , ln (Λt ) productivity shock has two counteracting effects on the Arrow-Debreu price λ et down, (see Equations (11) and (13)). The higher consumption growth, ∆ct , pushes λ while the lower qt pushes it up. The first effect dominates in this economy, and the et drops significantly upon the shock before gradually coming back Arrow-Debreu price λ to its original level. Figure 2 graphs the impulse response functions of the same set of variables to a 1% positive shock to the human capital level. The responses of consumption, investment and output are similar to those in Figure 1. Two main differences are the responses of the time spent in education and the asset return variables. Education time drops in response to the shock and stays low throughout the period, consistent with what we observe in the 28
data. The augmented leisure-consumption ratio qt increases after the shock as a result of the increased human capital level, reducing the relative risk aversion. In this case, a higher consumption growth and a higher qt acts in the same direction to push the marginal et significantly lower compared to in Figure 1. utility of consumption λ The different asset pricing implications of productivity shocks and human capital shocks are further borne out in Figures 3 and 4, which graph the impulse responses of the riskfree rate and the equity return to each shock. Note what are graphed are percentage deviations from the steady-state values, not percentages in absolute terms. We already see in Figure 1 that, in response to a positive productivity shock, the price et drops 7% immediately, then comes back to around 1% below the steady-state or risk λ level in the next period. It then fluctuates between −1% and −2% before eventually returning to the original level. In Figure 3, this translates to a sharp increase followed by a sharp decrease in both the bond and the equity prices, followed by increasingly smaller fluctuations. In return terms, the riskfree rate first drops 20%, comes back to approximately the before-shock level, and then oscillates between 1% and 5% before stabilizing at the steady-state level. In contrast, the equity return first jumps up to over 25% and then sharply decreases to −20%, before coming back to slightly above the steady-state level. From the second quarter onward, the two returns closely track each other and fluctuate within the same range. The higher equity premium comes primarily from the dramatic upward/downward movement in the equity return and the riskfree rate right after the shock, which in turn comes from the larger consumption growth response than those in the “Standard” or the “Standard TTP” models (1.6% versus 0.4% in the “Standard” model and 0.8% in the “Standard TTP” model). Similar to Figure 3, the riskfree rate moves downward while the equity return moves upward immediately after a positive human capital shock in Figure 4. They both return to approximately their respective original levels in the next period, and follow similar fluctuating patterns thereafter. However, while in Figure 3 the two returns closely track each other after a productivity shock, in Figure 4, the stock return stays almost 2% above 29
the riskfree rate throughout the period, resulting in a much higher mean equity premium than what is obtained after a positive productivity shock. This difference comes from the distinct responses of the coefficient of relative risk aversion RRA to these two types of shocks. The countercyclical risk aversion induced by human capital shocks makes stocks especially bad hedges against such shocks because when the stock return is high, not only is consumption relatively plentiful, but the agent is also less risk averse. 5.3.5
Excess Return on Human Capital
This framework also allows us to examine an artificial claim to human capital. Studies on the effect of education on earnings typically yield an estimate of the monetary return to human capital of 7% to 12%, is approximately the same magnitude as the returns to financial assets.32 Palacios-Huerta (2003a) finds that for specific demographic groups the return can be 5% to 20% higher than financial asset returns. Furthermore, Elias (2003) considers the non-monetary benefits of education and reaches an estimate of 16% for the total return to human capital assets. All these suggest a real return on human capital in the range of 3% to 12%. Panel D of Table 4 reports the first two moments of excess human capital returns. In all three versions of my model excess human capital returns exhibits very similar properties as excess equity returns. This results from my assumption of market completeness, while in reality human capital is illiquid and subject to various market frictions. For example, Palacios-Huerta (2003b) shows that accounting for market frictions such as borrowing constraints and short-sale constraints can account for up to two thirds of the human capital return premium. Introducing such constraints into the production economy model remains a challenge. 32
See Mincer (1958) and Becker (1993) for example.
30
5.4
Comparative Statics
I do not conduct a formal search routine in selecting the non-standard parameters in Θ in Section 4.1. In this section I investigate how alternative parameter values might change the results of the model. Table 5 reports the key statistics when Θ in the “Benchmark” model are changed one parameter at a time. For an easy comparison, the corresponding statistics from the data and the “Benchmark” model are listed in the first panel. Extra Risk Aversion on Leisure: ν The parameter ν controls the risk aversion towards leisure in excess of that towards consumption. For a given RRA towards consumption, a higher ν means the agent is more tolerant of fluctuations in leisure, and leads to a higher volatility in hours worked, output and consumption as we can see from Table 5. The relative volatility of hours worked also increases with ν. When ν equals 3, it is more than tripled compared to the “standard” model, and becomes close to what we observe in the data (0.96% versus 0.99% in the data). Higher economic fluctuations also translates to higher bond yields and equity returns. The equity premium, however, is not monotonic in ν. Intratemporal Substitution: ρ A higher ρ raises the intratemporal elasticity of substitution between consumption and leisure,
1 , 1−ρ
and makes consumption and leisure more substitutable within the period.
It also induces a higher volatility for all variables in Table 5. This is intuitive because when consumption and leisure are better substitutes, the agent can more easily maintain a relatively stable consumption-leisure bundle even if the two series move around frequently. The relative volatilities of consumption and labor (investment) are increasing (decreasing) in ρ, although the largest value of investment volatility relative to the output volatility (1.93 when ρ = 0) still falls short of the actual value in the data (2.93). ρ also controls how the RRA varies with the augmented leisure-consumption ratio Qt . When ρ equals zero, consumption and leisure are multiplicatively separable, and the amount of leisure affects the marginal utility of, but not the risk aversion towards, con31
sumption. In this case the RRA is constant over time and equals γ +(1 − γ) (1 − α). With a nonzero ρ, the deviation of the logarithm of RRA from its steady state can be written as rrat , ln (RRAt ) − ln (RRA) =
RRA − γ ρ (qt − zt ) . RRA
Disregarding the small difference in γ, a higher ρ makes RRA more variable. in Table 5 we see that as ρ increases, the equity return increases while bond yields decreases, resulting in a higher equity premium. Return to Education: κ The parameter κ controls the scale of the return to education, with education having a larger effect on human capital level when κ increases. Table 5 lists the results for two more values of κ. As κ increases, the volatility of output drops, while the relative volatilities of consumption and investment increase. This is intuitive because a model with a very low κ is similar to a model without an education choice, where the agent is restricted to use savings or labor supply to smooth her consumption. All asset returns are increasing in κ, but bond yields increase faster and the equity premium drops as κ increases. Persistence of human capital shocks: ψH The parameter ψH controls the persistence of human capital shocks. Table 5 reports two more sets of results when ψH equals 0.1 and 0.2 respectively. As ψH increases, the yield curve becomes downward-sloping. This is because the second factor in the pricing kernel of equation (13) is largely driven by human capital and inherits the time series property of human capital shocks. A positive autocorrelation between human capital shocks generates a positive autocorrelation in the second risk factor, As human capital shocks become more persistent, the pricing kernel is increasingly dominated by the second factor. Correlation between the Shocks: ϕKH The parameter ϕKH controls the correlation between human capital shocks and productivity shocks. The fact that unemployment and GDP growth generally move in op32
posite directions suggests that there might exist a positive correlation between these two types of shocks. Table 5 reports two more sets of results allowing ϕKH to assume nonzeros values 0.1 and 0.2 respectively. Allowing correlated shocks has little effect on the macroeconomic statistics while at the same time generating lower asset returns and higher equity premium and Sharpe Ratio. Role of Time-to-plan The third panel in Table 5 investigates how imposing the time-to-plan restriction on human capital and/or physical capital accumulations changes the model’s implications. The first row reports the results from the model with no time-to-plan restriction. Output is relatively more variable compared to the “Benchmark” model, while the consumption is much less variable. On the asset pricing side, the average yield curve is inverted and the equity premium (0.66%) is almost the same as in the “Standard TTP” model (0.73%). The results change little when we impose time-to-plan on human capital only. When we impose the same restriction on physical capital instead, however, we get less variable output, more variable consumption, hours worked and investment, while the average yield curve becomes upward-sloping and the equity premium increases to 0.82%. Comparing this case to the “Benchmark” results, we see that imposing the time-to-plan restriction on both physical and human capital processes has an important effect on the asset returns, doubling the equity premium to 1.71% annually. The volatilities of the asset returns increase ten times when the restriction is imposed on physical capital, and quadruple again when it is also imposed on human capital.
5.5
Cyclical Variation in Asset Returns
In this section I investigate the cyclical behavior of model-implied asset returns using model-simulated data.
33
5.5.1
Cyclicality of Returns and Expected Returns
Table 6 reports the correlations of various financial variables with the output, with standard errors reported in parentheses. The first two rows of the table report the correlations computed using the data, with standard errors computed using the Generalized Method of Moments (GMM). The rest of the rows report the results from the same five models as in Table 3 to Table 4 and the small-sample standard errors. In the data, the riskfree rate is positively correlated with output, while the equity return and the excess equity return are negatively correlated with output, although not significantly. In comparison, the correlations of realized returns with output are insignificant in all five models, as are the correlations of the expected (excess) stock return with the output. The dividend yield dyt =
Dt Pte
is strongly negatively correlated with output in the
data and in all models, with correlations from the last three models within two standard errors of the data result. The correlation between the intertemporal marginal rate of substitution (IMRS ), ∆λt , and output assumes positive values in the “Standard TTP” and “PShock” models, and negative values in the “Standard”, “HShock” and “Benchmark” models. None of them is significantly different from zero. From equation (15) we know that the mean excess return on any financial asset, adjusted for the Jensen’s inequality term, can be decomposed into two parts: the covariance of the excess return with the IMRS, ∆λt , normalized by the variance of ∆λt , and the variance of ∆λt : h E rt+1 −
rtf
i
³ ´ 1 2 f e t+1 + σr = −cov rt+1 − rt , m 2 ³ ´ cov rt+1 − rtf , ∆λt+1 × var (∆λt+1 ) , =− var (∆λt+1 )
(17)
where the second equality comes from equation (11). The first term reflects how much the excess returns change in response to a 1% change in the pricing kernel, while the second term reflects how volatile the pricing kernel is. For a given volatility of the pricing kernel, the more negative the response rate of the excess return to the pricing kernel is, 34
the higher premium such an asset will earn over the riskfree rate. In Table 7 I report the response rate of various asset return variables together with the unconditional variance of ∆λt+1 , where the response rate is computed as the slope coefficient β in a regression of rt onto the current-period change in the Arrow-Debreu price ∆λt rt = α + β∆λt , where the regressand rt represents one of the various asset return and expected asset return variables specified on the top of each column. The main difference between the “Standard” model and the other models is the way the short rate responds. In the event of a 1% increase in ∆λt+1 , the short rate drops 1.8% in the “Standard” model, but increases by 4.5% to 47.2% in all other models. Although in all except the “Standard” model, the short rate increases while the equity return and the equity premium drop upon the shock, they differ in how strong these responses are. The responses are the strongest in the “PShock” and the “Standard TTP” model, and much weaker in the “HShock” model. In none of these models does the expected stock return respond significantly to a shock to ∆λt . The excess stock return, however, responds positively and significantly in the “Standard TTP”, the “PShock” and the “Benchmark” models. The variance of the pricing kernel exhibits a monotonic pattern. Starting from the “Standard” model, it increases tenfold when I introduce short-run physical capital inelasticities in the “Standard TTP” model, and another tenfold when human capital shocks become the only source of economic uncertainties in the “HShock” model. 5.5.2
Predictive Regressions
A number of papers show that the excess stock return can be predicted using various financial and macro variable, most significantly by the nominal short rate and the log
35
dividend yield.33 In Panel A of Table 8 I regress k-period cumulative excess stock returns ´ Pk ³ e f 1 r − r onto the log dividend yield, dyt , and report the slope coefficients t+i t+i−1 i=1 k and the R2 statistics. For data regressions, the standard errors of the slope coefficients are computed using the Hodrick (1992) method, while standard errors of the R2 statistics are computed using the General Method of Moments (GMM ). For regressions using modelsimulated data, I report small-sample standard errors for both slope coefficients and R2 statistics, and compute the dividend yield assuming that dividends are re-invested in the stock market. Panel A of Table 8 shows that the data and the model regressions show roughly the same pattern, with positive slope coefficients on dyt , meaning a low dyt predicts a high k-period excess stock return in the future, and an R2 increasing with horizon. In the data, the R2 statistics climbs from 2.3% at the 1-quarter horizon to 73.3% at the 15year horizon. The increasing pattern is less pronounced in model-simulated regressions, especially in the “HShock” model with human capital shocks, where the maximum R2 is a mere 6.9% at the 15-year horizon. The nominal short rate and the term spread are also frequently used in the literature to predict real GDP growth.34 Panel B of Table 8 regress the k-quarter cumulative real P GDPt+i b onto the 5-year term spread, rt,20 GDP growth k1 ki=1 ln GDP − rtf . In the data, a t+i−1 higher term spread predicts a higher GDP growth, and the R2 statistics exhibit a hump shape, peaking at the 3-year horizon. The model-implied coefficients are mostly positive. The R2 statistics are strongly increasing in horizon in the “Standard” model, reaching 33.3% at the 15-year horizon, while in the rest of the models, R2 is generally largest at the 1-quarter horizon. 33
See, among others, Campbell (1987), Fama and French (1988), Breen, Glosten, and Jagannathan
(1989), Bekaert and Hodrick (1992), Campbell and Hamao (1992), Hodrick (1992), Lee (1992), Shiller and Beltratti (1992), Goetzmann and Jorion (1993, 1995), Ang and Bekaert (2001), Goyal and Welch (2003) and Valkanov (2003). 34 See Ang, Piazzesi, and Wei (2003) and the many references therein.
36
6
Conclusion
In this paper I explore the implications of a novel DSGE model with human capital and education. Key features of the model include non-separable utility over consumption and leisure, non-market benefits of human capital, education choices, human capital shocks, and time-to-plan restrictions on both physical and human capital investment. A latent-factor analysis of the consumers’ problem shows that adding an unobservable stochastic risk factor can potentially resolve the equity premium puzzle and the riskfree rate puzzle. The analysis of the full model confirms that a model with human capital and short-run inelasticities in capital accumulation is capable of producing a much higher equity premium compared to the standard DSGE model and generating plausible business-cycle fluctuations in asset returns. There are three channels that lead to a higher equity premium. First, the leisuresmoothing motive leads to larger consumption fluctuations as the representative consumer is concerned about smoothing the consumption-leisure bundle rather than consumption itself. Second, short-run restrictions on physical and human capital accumulations prevent a full adjustment in the consumer’s human capital level and the firm’s physical capital level immediately after the shock, leading to higher fluctuations in equity demand and supply. Finally, stocks are bad hedges especially against human capital shocks because such shocks induce a countercyclical risk aversion and a highly countercyclical IMRS. On the macroeconomic side, the model is able to match a wide array of conventional macroeconomic statistics, while improving on a few dimensions over the standard DSGE model, including generating a higher labor volatility and producing a negative correlation between output and time spent in education, both of which are consistent with the data. This paper is a first attempt at incorporating human capital and education choices into the standard DSGE model. Given the dominating status of labor income and human capital in a common investor’s income and wealth portfolios, I believe that human capital is an important component that economic models must incorporate in order to give a realistic description of the world. In this paper, I show that incorporating human capital 37
also has important asset pricing implications, while at the same time provides a framework for studies on labor-leisure choices and human capital investment decisions that are otherwise impossible to explore.
38
Appendix A
Data
I use chained consumer expenditures for nondurable and services for the period of 1950:Q1 through 2002:Q4, and CPI-U All Items for the period of June 1951 to March 2003, both obtained from DRI. The data on real GDP (chained 1996 dollar) for the period of 1948:Q1 to 2002:Q4 is obtained from Bureau of Economic Analysis. The NYSE value-weighted returns with and without dividend (RD and RX respectively) are from CRSP for the period of 1950:Q2 to 2002:Q4. The dividend growth and the price-dividend series are constructed using the methodology in Hodrick (1992). In particular, I normalize the price level at 1950:Q2 to 1 and compute the price series recursively by e Pte = RX ∗ Pt− 1 4
and the quarterly dividends by e Dtq = (RDt − RXt ) ∗ Pt− 1. 4
I use annualized dividend to avoid the seasonality problem in the dividend data. There are three ways to do this in the literature: Campbell and Shiller (1988) simply sum up dividends over the past four quarters, Hodrick (1992) assumes that dividends are reinvested at the quarterly riskfree rate, while Cochrane (1992) assumes that dividends are re-invested in the stock market. I form the annual dividends assuming they are reinvested in the stock market: Dt =
3 X i=0
q Dt− i
4
i−1 ³ Y
´ 1 + RDt− j .
j=0
4
This also allows computing the annualized stock return as the sum of past quarterly stock returns. Finally I compute the annualized log real dividend growth ∆dt = ln (Dt ) − ln (Dt−1 ) 39
and the annualized log dividend yield dyt = ln (Dt ) − ln (Pte ) . I use zero-coupon yield data for maturities 1, 4, 8, 12, 16 and 20 quarters from CRSP spanning 1952:Q2 to 2002:Q4. The 1-quarter rate is from the CRSP Fama risk-free rate file. All other bond yields are from the CRSP Fama-Bliss discount bond file.
B
Linearization of the Pricing Kernel
I define a new variable Zt by Ztρ = α + (1 − α) Qρt
(18)
and re-write the pricing kernel in equation (12) as · ¸−γ · ¸1−γ−ρ Ct+1 Zt+1 Mt+1 = β . Ct Zt Taking logarithm of both sides of the equation yields m e t+1 , ln (Mt+1 ) = ln β − γ∆e ct+1 + (1 − γ − ρ) ∆e zt+1 ,
(19)
where small letters with tildes represent the logarithm of corresponding capital letters. Using the formula ex ≈ ex0 [1 + (x − x0 )] we can log-linearize both sides of equation (18) around their respective steady-state values Z ρ [1 + ρ (e zt − ln Z)] ≈ α + (1 − α) Qρ [1 + ρ (e qt − ln Q)] ,
(20)
while evaluating equation (18) at the steady-state gives Z ρ = α + (1 − α) Qρ . Substituting the last equation into equation (20) and taking the lag operate ∆ on both sides yield ∆e zt =
(1 − α) Qρ (1 − α) Qρ ∆e q = ∆e qt . t Zρ α + (1 − α) Qρ
Combining equation (19) and (21) we get the log-linearized pricing kernel: m e t+1 ≈ ln β − γ∆e ct+1 + (1 − γ − ρ) 40
(1 − α) Qρ ∆e qt+1 . α + (1 − α) Qρ
(21)
C
First-Order Conditions
Let Kj,t be the capital stock that will be in place in period t+j, after projects Sj,t mature. Then from (5) we have K0,t = Kt−1 Sj,t = S1,t+j−1 = Kj,t − (1 − δ) Kj−1,t Substituting into (6), we can re-write the current-period investment in terms of current and future capital stocks: It =
4 X
φj [Kj,t − (1 − δ) Kj−1,t ]
j=1
=
4 X
ΦK j Kj,t
j=0
where ΦK 0 = − (1 − δ) φ1 ΦK i = φi − (1 − δ) φi+1 , i = 1, . . . , 3 ΦK J = φJ The central planner’s problem in Section 3.3 can be represented using the Lagrange Multiplier: max
{Ct ,Lt ,U4,t ,K4,t ,Nt }
E0
∞ X
( β t V (Ct , Ht Lνt )
t=0 4 h X ¢i ¡ K − Λt Ct + ΦK K − A F K , H N j,t 0,t t t j t
à − Ξt
j=0
Lt +
4 X
! ϕj Uj,t + Nt − 1
j=1
−
3 X
Λjt (Kj,t − Kj+1,t−1 ) −
j=0
3 X j=1
with the following first-order conditions: 41
¤ £ ¡ ¢ κ 1 − δH + τ U1,t−1 Ht−1 − Πt Ht − AH t )
Ξjt (Uj,t − Kj+1,t−1 )
Ct :
Λt = V1t
Lt : K0,t
Ξt = νV2t Ht Lν−1 t ¡ K ¢ = Λ0t : Λt At F1t − ΦK 0
K1,t :
0 1 Λt ΦK 1 = βEt Λt+1 − Λt
K2,t :
1 2 Λt ΦK 2 = βEt Λt+1 − Λt
K3,t :
2 3 Λt ΦK 3 = βEt Λt+1 − Λt
K4,t :
3 Λt ΦK 4 = βEt Λt+1
U1,t :
¡ ¢ £ ¤ κ H Πt = V2t Lνt + Λt AK t F2t Nt + β 1 − δH + τ U1,t Et Πt+1 At+1 £ ¤ κ−1 1 ϕ1 Ξt = βτ κU1,t Ht Et Πt+1 AH t+1 − Ξt
U2,t :
ϕ2 Ξt = βEt Ξ1t+1 − Ξ2t
U3,t :
ϕ3 Ξt = βEt Ξ2t+1 − Ξ3t
U4,t :
ϕ4 Ξt = βEt Ξ3t+1
Nt :
Πt :
Λt AK t F2t Ht = Ξt P K Ct + 4j=0 ΦK j Kj,t = At F (K0,t , Ht Nt ) P Lt + 4j=1 ϕj Uj,t + Nt = 1 ¡ ¢ κ Ht = AH 1 − δ + τ U H t 1,t−1 Ht−1
Λjt :
Kj,t = Kj+1,t−1 , j = 0 . . . 3
Ξjt :
Uj,t = Uj+1,t−1 , j = 1 . . . 3
Ht :
Λt : Ξt :
where Fit and Vit , i = 1, 2 stand for partial derivatives of the production function F and the utility function V with respect to the i-th argument respectively.
D
Model Solution
To solve the model, I first log-linearize the first-order conditions derived in Appendix C and collect them in a stylized system of equations: 0 = Ae xt + Be xt−1 + C yet + De zt 0 = Et [F x et+1 + Ge xt + H x et−1 + J yet+1 + K yet + Le zt+1 + M zet ]
42
where x et is the vector of endogenous state variables such as the physical and human capital stock, yet is the vector of control variables such as consumption and investment, and zet = {a1t , a2t } is the vector of exogenous state variables. This system is then solved using the method of undetermined coefficients to obtain the following model dynamics: x et = P x et−1 + Qe zt
(22)
yet = Re xt−1 + Se zt
(23)
zet+1 = N zet + ²t+1 , ²t+1 |It ∼ N (0, Σ)
(24)
I combine endogenous and exogenous state variables into a single state vector st = {e xt , zet } and rewrite equations (22) to (24) in a state-space representation: st = M st−1 + W ²t yet = T st−1 + S²t where
M =
P QN 0
E
,
W =
N
Q
,
h T =
I
i R SN
.
Asset Pricing
In this economy, the marginal utility of consumption is a function of the consumption, Ct , and the augmented leisure-consumption ratio, Qt : Λt+1 = [α + (1 − α) Qρt ]
1−γ−ρ ρ
−γ Ct+1 ,
while the pricing kernel is given by Mt+1 = β
Λt+1 = βG−γ exp (λt+1 − λt ) . Λt
The basic pricing formula requires that the return to any asset in this economy Rt+1 must satisfy 1 = Et [Mt+1 (1 + Rt+1 )] 43
(25)
The logarithm of the marginal utility on consumption λt and the logarithm of the dividend dK et in 23: t are solved as part of the endogenous control variables y λt = (1 − γ − ρ) (1 − α)
Qρ (ht−1 + lt − ct ) − γct α + (1 − α) Qρ
= Tλ st−1 + Sλ ²t K K dK t = Td st−1 + Sd ²t H H dH t = Td st−1 + Sd ²t
for properly defined Tλ , Sλ , TdK , SdK , TdH and SdH .
E.1
Long-Term Bond
b Denote the quarterly continuously-compounded yield of a n-period real bond by rt,n .
b rt,n
n Y 1 = − ln Et Mt+i n i=1 £¡ −γ ¢n ¤ 1 = − ln Et βG exp (λt+n − λt ) n · ¸ 1 1 = − ln β + γ log G − Et (λt+n − λt ) + vart (λt+n − λt ) n 2
(26)
Using λt+n − λt = Tλ st+n−1 + Sλ ²t+n − (Tλ st−1 + Sλ ²t ) n
= Tλ M st−1 +
n X
Tλ M i−1 W ²t+n−i + Sλ ²t+n − (Tλ st−1 + Sλ ²t )
i=1 n
¡
= Tλ (M − I) st−1 + Tλ M
n−1
¢
W − Sλ ²t +
n−1 X i=1
44
Tλ M i−1 W ²t+n−i + Sλ ²t+n
we have that the unconditional moments are 1 E [vart (λt+n − λt )] 2n 1 = − ln β + γ log G − var (λt+n − Et λt+n ) 2n " Ã ! # n−1 X ¡ ¢ 1 0 = − ln β + γ log G − Tλ M i−1 W ΣW 0 M i−1 Tλ0 + Sλ ΣSλ0 2n i=1
b Ert,n = − ln β + γ log G −
¡ b ¢ 1 = 2 var [Et (λt+n − λt )] var rt,n n ¡ ¢ ¡ ¢0 i 1 h = 2 Tλ (M n − I) var (st ) (M n − I)0 Tλ0 + Tλ M n−1 W − Sλ Σ Tλ M n−1 W − Sλ n The simple bond yield is ¡ b ¢ b Rt,n = exp rt,n −1 with unconditional moments µ ¶ ¡ b¢ 1 b = exp Ert + var rt −1 2 ¡ ¢ ¡ ¡ ¢¢ ¡ ¡ ¢¢ var Rtb = exp 2Ertb + 2var rtb − exp 2Ertb + var rtb . ERtb
E.2
Riskfree Rate
Time-t log riskfree rate is the 1-period bond yield rtf = − ln Et [Mt+1 ] 1 = − ln β + γ log G − Et (λt+1 − λt ) − vart (λt+1 − λt ) 2
1 = − ln β + γ log G − Tλ (M − I) st−1 − (Tλ W − Sλ ) ²t − Sλ ΣSλ0 2
(27)
Its unconditional moments are 1 Ertf = − ln β + γ log G − Sλ ΣSλ0 , 2 ³ ´ var rtf = Tλ (I − M ) var (st ) (I − M )0 Tλ0 + (Tλ W − Sλ ) Σ (Tλ W − Sλ )0 , ³ ´ f cov rtf , rt−1 = Tλ (I − M ) M var (st ) (I − M )0 Tλ0 + Tλ (M − I) W Σ (Tλ W − Sλ )0 The simple riskfree rate is Rtf
³ ´ = exp rtf − 1 45
with unconditional moments µ
³ ´¶ 1 = exp + var rtf −1 2 ³ ´ ³ ³ ´´ ³ ³ ´´ f f f f var Rt = exp 2Ert + 2var rt − exp 2Ert + var rtf ERtf
E.3
Ertf
Equity Returns
I simulate the model and compute the asset returns for each period. The bond yields are computed according to (27) and (26). Equity returns are computed as follows: The timet stock price is the present value of all future dividends discounted by the appropriate pricing kernel Pte = Et
∞ X
K Mt,t+i Dt+i
i=1
In this model we have K Et Mt,t+i Dt+i
= Et
i Y
K Mt+j Dt+i
j=1
¡ ¢i ¡ ¢ K K = Et βG−γ exp (λt+i − λt ) Gi exp dK Dt t+i − dt ¡ ¢i £ ¡ ¢¤ K = βG1−γ DtK Et exp λt+i − λt + dK t+i − dt Thus the quarterly simple stock return is e K Pt+1 + Dt+1 Pte P K K Et+1 ∞ i=1 Mt+1,t+1+i Dt+1+i + Dt+1 P∞ = K Et i=1 Mt,t+i Dt+i ¡ ¢ P ∞ 1−γ i K K ) Et+1 exp λt+1+i − λt+1 + dK Dt+1 t+1+i − dt+1 + 1 i=1 (βG = K ¡ ¢ P∞ K K 1−γ )i E exp λ Dt t t+i − λt + dt+i − dt i=1 (βG ¢ ¡ K ¢ ¡ P∞ 1−γ i ) Et+1 exp λt+1+i − λt+1 + dK t+1+i + exp dt+1 i=1 (βG =G ¡ ¢ P∞ K 1−γ )i E exp λ (βG − λ + d t t+i t t+i i=1
e Rt+1 =
Using the formula we have ¡
Et exp λt+i − λt +
dK t+i
¢
·
¡ ¢ ¡ ¢ 1 K = exp Et λt+i − λt + dK t+i + vart λt+i − λt + dt+i 2
46
¸
Since ¡ ¡ ¢ ¢ K λt+i − λt + dK st+i−1 + Sλ + SdK ²t+i − (Tλ st−1 + Sλ ²t ) t+i = Tλ + Td ¡ ¢ ¡ ¢ = Tλ + TdK M i st−1 + Sλ + SdK ²t+i + ¡
Tλ +
TdK
i ¢X
M j−1 W ²t+i−j − Tλ st−1 − Sλ ²t
j=1
=
£¡ ¡
¢ ¤ £¡ ¢ ¤ Tλ + TdK M i − Tλ st−1 + Tλ + TdK M i−1 W − Sλ ²t +
Tλ +
TdK
i−1 ¢X
¡ ¢ M j−1 W ²t+i−j + Sλ + SdK ²t+i
j=1
The two elements in this expression can be computed as: ¡ ¢ £¡ ¢ ¤ £¡ ¢ ¤ Et λt+i − λt + dK Tλ + TdK M i − Tλ st−1 + Tλ + TdK M i−1 W − Sλ ²t t+i = ¡
vart λt+i − λt +
dK t+i
¢
¡
= Tλ +
TdK
i−1 ¢X
¡ ¢0 ¡ ¢0 M j−1 W ΣW 0 M j−1 Tλ + TdK +
j=1
¡
E.4
¢ ¡ ¢0 Sλ + SdK Σ Sλ + SdK
Human Capital Returns
H The return to human capital can be computed similarly as the equity return. Let Dt+1 be
the “dividend” type of return to the human capital, which is the sum of human capital share of the output and the marginal utility from the human capital, minus the value of time spent in education, all expressed in terms of the consumption goods: ∂V (Ct , Ht Lνt ) /∂Ht ∂Yt ∂Yt Ht + H t − Ut ν ∂Ht ∂V (Ct , Ht Lt ) /∂Ct ∂Nt µ ¶ Yt 1 Nt + Lt − Ut Ht = (1 − θ) K0,t ν
DtH =
Following similar steps as in the previous section, I obtain the following expression for the 1-period human capital return: H H + Dt+1 Pt+1 PtH ¢ ¡ H ¢ ¡ P∞ 1−γ i ) Et+1 exp λt+1+i − λt+1 + dH t+1+i + exp dt+1 i=1 (βG =G ¢ ¡ P∞ H 1−γ )i E exp λ t t+i − λt + dt+i i=1 (βG
H = Rt+1
47
where ¡ ¢ £¡ ¢ i ¤ £¡ ¢ i−1 ¤ H H Et λt+i − λt + dH = T + T M − T s + T + T M W − S ²t λ λ t−1 λ λ t+i d d ¡
vart λt+i − λt +
dH t+i
¢
¡
= Tλ +
TdH
i−1 ¢X
¡ ¢0 ¡ ¢0 M j−1 W ΣW 0 M j−1 Tλ + TdH +
j=1
¡
¢ ¡ ¢0 Sλ + SdH Σ Sλ + SdH
and TdH and SdH are defined so that H H dH t = Td st−1 + Sd ²t .
I simulate the model to find the first two moments of RtH .
F
A Latent-Variable Estimation
The state of the economy is summarized by a vector of state variables ³
Xt = ∆e ct , qet , ∆det , π et
´0
,
t where det = ln Dt is the logarithm of the dividend and π et = log PPt−1 is the logarithm of the
inflation. I specify the model at an annual horizon but use data in quarterly frequencies to avoid the seasonality problem in dividend and to use the data more efficiently. I assume that Xt follows a V ARM A (1, 3) process where the M A terms are specified to accommodate the overlapping observations: Xt = µ + ΦXt−1 + Σ²t + ΨΣ²t− 1 + Ψ2 Σ²t− 1 + Ψ3 Σ²t− 3 , 4
2
4
(28)
and the time subscript t − 1 to t refers to an annual horizon. The log of the real pricing kernel can be rewritten as m e t+1 ≈ ln β − ζ∆e qt+1 − γ∆e ct+1 = ln β − (ζe02 + γe01 ) Xt+1 + ζe02 Xt ,
48
(29)
while the log of the nominal pricing kernel is formulated in the standard way as m b t+1 = m e t+1 − π et+1 = ln β − (ζe02 + γe01 + e04 ) Xt+1 + ζe02 Xt ,
(30)
where ei is an indicator row vector that selects the i-th element. It can be shown35 that both the n-period real bond yield ytn and the n-period nominal bond yield ybtn are affine functions of the state variables Xt : 1 1 ytn , − log (Ptn ) = − (An + Bn Xt ) , n n ³ ´ ´ 1 1 ³b bn Xt , ybtn , − log Pbtn = − An + B n n
(31) (32)
while the price-dividend ratio can be written as a sum of exponential affine functions: ∞ X P Dt = exp (Cn + Dn Xt ) (33) n=1
The affine forms of (31) to (32) and the exponential affine form of (33) imply that this model falls into the class of affine PV models.36 I can also show that the log price-dividend ratio and the equity return rte are both approximately affine in Xt : pdt , log (P Dt ) ≈ C + DXt rte ≈ F + GXt − HXt−1
(34) (35)
The model is estimated by Maximum Likelihood using data described in Appendix A. I assume that the price-dividend ratio series is observed without error and use it to invert the unobserved factor qt using the linear approximation (34). I also observe a set of yields {ytn }N n=1 with error, which provide over-identifying restrictions of the model. I further impose restrictions E [qt ] = 0 and var [qt ] = 1 to insure identification, and a positivity constraint on ξ based on its definition. The parameters to be estimated are the VAR parameters {µ, Ψ, Σ}, the preference parameters {β, γ, ξ}, and a set of measurement error volatilities. 35 36
Details are available upon request. See, among others, Ang and Liu (2001), Bakshi and Chen (2001), Bekaert, Engstrom, and Grenadier
(2004) and Mamaysky (2002).
49
References Abel, A. B. (1990): “Asset Prices under Habit Formation and Catching up with the Joneses,” American Economic Review Papers and Proceedings, 80(2), 38–42. Ang, A., and G. Bekaert (2001): “Stock Return Predictability: Is It There?,” NBER Working Paper No. 8207. Ang, A., and J. Liu (2001): “A General Affine Earnings Valuation Model,” Review of Accounting Studies, 6(4), 397–425. Ang, A., M. Piazzesi, and M. Wei (2003): “What Does the Yield Curve Tell Us About GDP Growth?,” Working Paper. Avalos, F. (1999): “Time to Build, Aggregate Fluctuations and Asset Pricing,” Working Paper, University of Chicago. Backus, D. K., and S. E. Zin (1994): “Reverse Engineering the Yield Curve,” NBER Working Paper 4676. Bakshi, G., and Z. Chen (2001): “Stock Valuation in Dynamic Economics,” Working Paper. Bakshi, G. S., and Z. Chen (1996): “The Spirit of Capitalism and Stock-Market Prices,” American Economic Review, 86(1), 133–157. Bansal, R., A. R. Gallant, and G. Tauchen (2003): “Rational Pessimism, Rational Exuberance, and Markets for Macro Risks,” Working Paper. Bansal, R., and A. Yaron (2004): “Risks for the long run: a potential resolution of asset pricing puzzles,” Journal of Finance, Forthcoming. Barro, R. J., and J.-W. Lee (1993): “International Comparisons of Educational Attainment,” Journal of Monetary Economics, 32(3), 363–394. Becker, G. S. (1993): Human Capital : A Theoretical and Empirical Analysis, with Special Reference to Education. The University of Chicago Press, Chicago, 3rd edn. Becker, G. S., E. M. Landes, and R. T. Michael (1977): “An Economic Analysis of Marital Instability,” Journal of Political Economy, 85(6), 1141–1188. Bekaert, G., E. Engstrom, and S. R. Grenadier (2004): “Stock and Bond Returns with Moody Investors,” Working Paper, Columbia University. Bekaert, G., and R. J. Hodrick (1992): “Characterizing Predictable Components in Excess Returns on Equity and Foreign Exchange Markets,” Journal of Finance, 47(2), 467–509. Ben-Porath, Y. (1967): “The Production of Human Capital and the Life Cycle of Earnings,” Journal of Political Economy, 74(4, Part 1), 352–365. 50
Benhabib, J., R. Rogerson, and R. Wright (1991): “Homework in Macroeconomics: Household Production and Aggregate Fluctuations,” Journal of Political Economy, 99(6), 1166–1187. Benninga, S., and A. Protopapadakis (1990): “Leverage, Time Preference and the ‘equity Premium Puzzle’,” Journal of Monetary Economics, 25(1), 49–58. Boldrin, M., L. J. Christiano, and J. D. M. Fisher (1995): “Asset Pricing Lessons for Modeling Business Cycles,” NBER Working Paper No. 5262. (2001): “Habit Persistence, Asset Returns, and the Business Cycle,” American Economic Review, 91(1), 149–166. Boudoukh, J. (1993): “An Equilibrium Model of Nominal Bond Prices with InflationOutput Correlation and Stochastic Volatility,” Journal of Money, Credit and Banking, 25(3), 636–665. Brandt, M. W., and K. Q. Wang (2003): “Time-Varying Risk Aversion and Unexpected Inflation,” Journal of Monetary Economics, 50(7), 1457–1498. Breen, W., L. R. Glosten, and R. Jagannathan (1989): “Economic Significance of Predictable Variations in Stock Index Returns,” Journal of Finance, 44(5), 1177–1189. Brock, W. A. (1979): “An Integration of Stochastic Growth Theory and the Theory of Finance, Part I: The Growth Model,” in General Equilibrium, Growth & Trade, ed. by J. Green, and J. Scheinkman, pp. 165–192. Academic Press, Chicago. Brock, W. A., and L. J. Mirman (1972): “Optimal Economic Growth and Uncertainty: The Discounted Case,” Journal of Economic Theory, 4(3), 479–513. ´, J., and M. S. Santos (1993): “On Endogenous Growth with Physical and Caballe Human Capital,” Journal of Political Economy, 101(6), 1042–1067. Campbell, J. Y. (1987): “Stock Returns and the Term Structure,” Journal of Financial Economics, 18(2), 373–399. (1994): “Inspecting the Mechanism: An Analytical Approach to the Stochastic Growth Model,” Journal of Monetary Economics, 33(3), 463–506. (1996): “Understanding Risk and Return,” Journal of Political Economy, 104(2), 298–345. (2000): “Asset Pricing at the Millennium,” Journal of Finance, 55(4), 1515– 1567. (2003): “Consumption-Based Asset Pricing,” in Handbook of Economics of Finance, ed. by G. Constantinides, M. Harris, and R. M. Stulz. North-Holland, Amsterdam.
51
Campbell, J. Y., and J. H. Cochrane (1999): “By Force of Habit: A ConsumptionBased Explanation of Aggregate Stock Market Behavior,” Journal of Political Economy, 107(2), 205–251. Campbell, J. Y., and Y. Hamao (1992): “Predictable Stock Returns in the United States and Japan: A Study of Long-Term Capital Market Integration,” Journal of Finance, 47(1), 43–69. Campbell, J. Y., and R. J. Shiller (1988): “The Dividend-Price Ratio and Expectations of Future Dividends and Discount Factors,” Review of Financial Studies, 1(3), 195–228. Chapman, D. A. (2002): “Does Intrinsic Habit Formation Actually Resolve the Equity Premium Puzzle?,” Review of Economic Dynamics, 5(3), 618–645. Christiano, L. J., and R. M. Todd (1996): “Time to Plan and Aggregate Fluctuations,” Quarterly Review (Federal Reserve Bank of Minneapolis), 20(1), 14–27. Cochrane, J. H. (1991): “Production-Based Asset Pricing and the Link between Stock Returns and Economic Fluctuations,” Journal of Finance, 46(1), 209–237. (1992): “Explaining the Variance of Price-Dividend Ratios,” Review of Financial Studies, 5(2), 243–280. (1996): “A Cross-Sectional Test of an Investment-Based Asset Pricing Model,” Journal of Political Economy, 104(3), 572–621. Constantinides, G. M. (1990): “Habit Formation: A Resolution of the Equity Premium Puzzle,” Journal of Political Economy, 98(3), 519–543. Cooley, T. F., and E. C. Prescott (1995): “Economic Growth and Business Cycles,” in Frontiers of Business Cycle Research, ed. by T. H. Cooley, pp. 1–38. Princeton University Press, Princeton, New Jersey. Daniel, K., and D. Marshall (1999): “Consumption-Based Modeling of Long-Horizon Returns,” Working Paper. Danthine, J.-P., and J. B. Donaldson (2002): “Labor Relations and Asset Returns,” Review of Economic Studies, 69(1), 41–64. Danthine, J.-P., J. B. Donaldson, C. Giannikos, and H. Guirguis (2002): “On the Consequences of State Dependent Preferences for the Pricing of Financial Assets,” Working Paper. Dellas, H., and P. Sakellaris (1996): “On the Cyclicality of Schooling: Theory and Evidence,” Universit´e catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES) Discussion Paper # 1997002. Den Haan, W. J. (1995): “The Term Structure of Interest Rates in Real and Monetary Economies,” Journal of Economic Dynamics and Control, 19(5-7), 909–940. 52
Donaldson, J. B., and R. Mehra (1984): “Comparative Dynamics of an Equilibrium Intertemporal Asset Pricing Model,” Review of Economic Studies, 51(3), 491–508. Dynan, K. E. (2000): “Habit Formation in Consumer Preferences: Evidence from Panel Data,” American Economic Review, 90(3), 391–406. Einarssona, T., and M. H. Marquis (1998): “An RBC Model with Growth: The Role of Human Capital,” Journal of Economics and Business, 50(5), 431–444. Eisner, R. (1988): “Extended Accounts for National Income and Product,” Journal of Economic Literature, 26(4), 1611–1684. Elias, J. J. (2003): “The Education Premium Puzzle,” Working Paper. Fama, E. F., and K. R. French (1988): “Dividend Yields and Expected Stock Returns,” Journal of Financial Economics, 22, 3–25. Fama, E. F., and G. W. Schwert (1977): “Human capital and capital market equilibrium,” Journal of Financial Economics, 4(1), 115–146. Ferson, W. E., and G. M. Constantinides (1991): “Habit Persistence and Durability in Aggregate Consumption: Empirical Tests,” Journal of Financial Economics, 29(2), 199–240. Ferson, W. E., and C. R. Harvey (1992): “Seasonality and Consumption-Based Asset Pricing,” Journal of Finance, 47(2), 511–552. Goetzmann, W. N., and P. Jorion (1993): “Testing the Predictive Power of Dividend Yields,” Journal of Finance, 48(2), 663–679. (1995): “A Longer Look at Dividend Yields,” Journal of Business, 68(4), 483– 508. Gomes, J. F., A. Yaron, and L. Zhang (2003): “Asset Prices and Business Cycles with Costly External Finance,” Review of Economic Dynamics, 6(4), 767–788. Goyal, A., and I. Welch (2003): “Predicting the Equity Premium With Dividend Ratios,” Management Science, 49(5), 639–654. Guvenen, M. F. (2003): “A Parsimonious Macroeconomic Model for Asset Pricing: Habit Formation or Cross-Sectional Heterogeneity?,” Working Paper. Hansen, G. D. (1985): “Indivisible Labor and the Business Cycle,” Journal of Monetary Economics, 16(3), 309–327. Haveman, R. H., and B. L. Wolfe (1994): Succeeding Generations : On the Effects of Investments in Children. Russell Sage Foundation, New York. Heaton, J. (1995): “An Empirical Investigation of Asset Pricing with Temporally Dependent Preference Specifications,” Econometrica, 63(3), 681–717. 53
Heckman, J. J. (1976): “A Life-Cycle Model of Earnings, Learning, and Consumption,” Journal of Political Economy, 84(4, Part 2), S11–S44. Heien, D., and C. Durham (1991): “A Test of the Habit Formation Hypothesis Using Household Data,” Review of Economics and Statistics, 73(2), 189–199. Hodrick, R. J. (1992): “Dividend Yields and Expected Stock Returns: Alternative Procedures for Inference and Measurement,” Review of Financial Studies, 5(3), 357– 386. Hodrick, R. J., and E. C. Prescott (1997): “Postwar U.S. Business Cycles: An Empirical Investigation,” Journal of Money, Credit and Banking, 29(1), 1–16. Jagannathan, R., and Z. Wang (1996): “The Conditional CAPM and the CrossSection of Expected Returns,” Journal of Finance, 51, 3–53. Jermann, U. J. (1998): “Asset Pricing in Production Economies,” Journal of Monetary Economics, 41(2), 257–275. Jorgenson, D. W., and B. M. Fraumeni (1989): “The Accumulation of Human and Nonhuman Capital, 1948-1984,” in The Measurement of Saving, Investment and Wealth, Studies in Income and Wealth, ed. by R. E. Lipsey, and H. S. Tice, vol. 52, pp. 227–282. University of Chicago Press, Chicago. Kendrick, J. W. (1976): The Formation and Stocks of Total Capital. Columbia University Press for NBER, New York. King, R. G., C. I. Plosser, and S. T. Rebelo (1988a): “Production, Growth and Business Cycles: I. The Basic Neoclassical Model,” Journal of Monetary Economics, 21(2/3), 195–232. (1988b): “Production, Growth and Business Cycles: II. New Directions,” Journal of Monetary Economics, 21(2/3), 309–341. King, R. G., and S. T. Rebelo (1990): “Public Policy and Economic Growth: Developing Neoclassical Implications,” Journal of Political Economy, 98(5, Part 2), S126– S150. (1999): “Resuscitating Real Business Cycles,” in Handbook of Macroeconomics, ed. by J. B. Taylor, and M. Woodford, vol. 1B, chap. 14, pp. 927–1007. Elsevier Science Publications, Amsterdam; New York and Oxford. Kocherlakota, N. R. (1996): “The Equity Premium: It’s Still a Puzzle,” Journal of Economic Literature, 34(1), 42–71. Krebs, T. (2003): “Human Capital Risk and Economic Growth,” Quarterly Journal of Economics, 118(2), 709–744. Kydland, F. E., and E. C. Prescott (1982): “Time to Build and Aggregate Fluctuations,” Econometrica, 50(6), 1345–1370. 54
Ladron-De-Guevara, A., S. Ortigueira, and M. S. Santos (1999): “A TwoSector Model of Endogenous Growth with Leisure,” Review of Economics and Statistics, 66(3), 609–631. Lee, B.-S. (1992): “Causal Relations Among Stock Returns, Interest Rates, Real Activity, and Inflation,” Journal of Finance, 47(4), 1591–1603. Lettau, M. (2003): “Inspecting the Mechanism: Closed-Form Solutions for Asset Prices in Real Business Cycle Models,” The Economic Journal, Forthcoming. Lettau, M., and H. Uhlig (2000): “Can Habit Formation Be Reconciled with Business Cycle Facts?,” Review of Economic Dynamics, 3(1), 79–99. Lucas, Robert E., J. (1988): “On the Mechanics of Economic Development,” Journal of Monetary Economics, 22(1), 3–42. Lustig, H., and S. Van Nieuwerburgh (2003): “Housing Collateral, Consumption Insurance and Risk Premia,” Working Paper. Mamaysky, H. (2002): “A Model for Pricing Stocks and Bonds,” Working Paper. Mankiw, N. G., D. Romer, and D. N. Weil (1992): “A Contribution to the Empirics of Economic Growth,” Quarterly Journal of Economics, 107(2), 407–437. Mayers, D. (1972): “Nonmarketable Assets and Capital Market Equilibrium under Uncertainty,” in Studies in the Theory of Capital Markets, ed. by M. C. Jensen, pp. 223–248. Praeger Publishers, New York. McMahon, W. W. (1997): “Conceptual Framework for Measuring the Total Social and Private Benefits of Education,” International Journal of Educational Research, 27(6), 453–481. Mehra, R., and E. C. Prescott (1985): “The Equity Premium: A Puzzle,” Journal of Monetary Economics, 15(2), 145–61. (2003): “The Equity Premium in Retrospect,” in The Handbook of Economics and Finance, ed. by G. Constantinides, M. Harris, and R. M. Stulz. North-Holland, Amsterdam. Melino, A., and A. X. Yang (2003): “State Dependent Preferences Can Explain the Equity Premium Puzzle,” Review of Economic Dynamics, Forthcoming. Menzly, L., T. Santos, and P. Veronesi (2002): “The Time Series of the Cross Section,” Working Paper. (2003): “Understanding Predictability,” Journal of Political Economy, Forthcoming. Michael, R. T. (1972): The Effect of Education on Efficiency in Consumption. National Bureau of Economic Researh, New York, NY. 55
(1982): “Measuring Non-Monetary Benefits of Education: A Survey,” in Financing Education: Overcoming Inefficiency and Inequality, ed. by W. W. McMahon, and T. G. Geske, pp. 119–149. University of Illinois Press, Urbana. Mincer, J. (1958): “Investment in Human Capital and Personal Income Distribution,” Journal of Political Economy, 66(4), 281–302. (1997): “The Production of Human Capital and the Life Cycle of Earnings: Variations on a Theme,” Journal of Labor Economics, 15(1, Part 2), S26–S47. Mulligan, C. B. (2001): “Aggregate Implications of Indivisible Labor,” Advances in Macroeconomics, 1(1), Article 4. Naik, N. Y., and M. J. Moore (1996): “Habit Formation and Intertemporal Substitution in Individual Food Consumption,” Review of Economics and Statistics, 78(2), 321–328. Naik, V. (1994): “Asset Prices in Dynamic Production Economies with Time-Varying Risk,” Review of Financial Studies, 7(4), 781–801. Neal, D., and S. Rosen (2000): “Theories of the Distribution of Earnings,” in Handbook of Income Distribution, ed. by A. Atkinson, and F. Bourguignon, chap. 7, pp. 379–427. North-Holland, Amsterdam. Normandin, M., and P. St-Amour (1998): “Substitution, Risk Aversion, Taste Shocks and Equity Premia,” Journal of Applied Econometrics, 13(3), 265–281. Ortigueira, S. (2000): “A Dynamic Analysis of an Endogenous Growth Model with Leisure,” Economic Theory, 16(1), 43–62. Osborn, D. R. (1988): “Seasonality and Habit Persistence in a Life Cycle Model of Consumption,” Journal of Applied Econometrics, 3(4), 255–266. Otrok, C., B. Ravikumar, and C. H. Whiteman (2002): “Habit Formation: A Resolution of the Equity Premium Puzzle?,” Journal of Monetary Economics, 49(6), 1261–1288. Palacios-Huerta, I. (2003a): “An Empirical Analysis of the Risk Properties of Human Capital Returns,” American Economic Review, 93(3), 948–964. (2003b): “Risk and Market Frictions as Determinants of the Human Capital Premium,” Working Paper. (2003c): “The Robustness of the Conditional CAPM with Human Capital,” Journal of Financial Econometrics, Forthcoming. Perli, R., and P. Sakellaris (1998): “Human Capital Formation and Business Cycle Persistence,” Journal of Monetary Economics, 42(1), 67–92.
56
Piazzesi, M., M. Schneider, and S. Tuzel (2003): “Housing, Consumption, and Asset Pricing,” Working Paper. Prescott, E. C. (1986): “Theory Ahead of Business Cycle Measurement,” CarnegieRochester Conference Series on Public Policy, 25, 11–66. Pries, M. J. (2001): “Uninsured Idiosyncratic Risk and Human Capital Accumulation,” Working Paper. Psacharopoulos, G., and A. M. Arriagada (1986a): “The Educational Attainment of the Labor Force: An International Comparison,” Report no. EDT 38, Education and Training Department, The World Bank. (1986b): “The Educational Composition of the Labor Force: An International Comparison,” International Labor Review, 125(5), 561–574. Robert E. Lucas, J. (1978): “Asset Prices in an Exchange Economy,” Econometrica, 46(6), 1429–1445. Rogerson, R. (1988): “Indivisible Labor, Lotteries and Equilibrium,” Journal of Monetary Economics, 21(1), 3–16. Romer, P. M. (1990): “Endogenous Technological Change,” Journal of Political Economy, 98(5, Part 2), S71–S102. Rouwenhorst, K. G. (1995): “Asset Pricing Implications of Equilibrium Business Cycle Models,” in Frontiers of Business Cycle Research, ed. by T. H. Cooley, pp. 294– 330. Princeton University Press, Princeton, New Jersey. Sakellaris, P., and A. Spilimbergo (2000): “Business Cycles and Investment in Human Capital: International Evidence on Higher Education,” Carnegie-Rochester Conference Series on Public Policy, 52(1), 221–256. Santos, T., and P. Veronesi (2003): “Labor Income and Predictable Stock Returns,” Working Paper. Schultz, T. W. (1975): “The Value of the Ability to Deal with Disequilibria,” Journal of Economic Literature, 13(3), 827–846. Shiller, R. (1993): Macro Markets: Creating Institutions for Managing Society’s Largest Economic Risks. Oxford University Press. Shiller, R. J., and A. E. Beltratti (1992): “Stock Prices and Bond Yields: Can their Comovements Be Explained in Terms of Present Value Models?,” Journal of Monetary Economics, 30(1), 25–46. Stock, J. H., and M. W. Watson (1999): “Business Cycle Fluctuations in U.S. Macroeconomic Time Series,” in Handbook of Macroeconomics, ed. by J. B. Taylor, and M. Woodford, vol. 1A, pp. 3–64. Elsevier Science Publications, Amsterdam; New York and Oxford. 57
Sundaresan, S. M. (1989): “Intertemporally Dependent Preferences and the Volatility of Consumption and Wealth,” Review of Financial Studies, 2(1), 73–89. Tallarini, T. (2000): “Risk-Sensitive Real Business Cycles,” Journal of Monetary Economics, 45(3), 507–532. Uhlig, H. (1997): “A Toolkit for Analyzing Nonlinear Dynamic Stochastic Models Easily,” Working Paper. Uzawa, H. (1965): “Optimum Technical Change in An Aggregative Model of Economic Growth,” International Economic Review, 6(1), 18–31. Valkanov, R. (2003): “Long-Horizon Regressions: Theoretical Results and Applications,” Journal of Financial Economics, Forthcoming. Wachter, J. A. (2003): “A Consumption-Based Model of the Term Structure of Interest Rates,” Working Paper, New York University. Weiss, Y. (1986): “The Determination of Life Cycle Earnings: A Survey,” in Handbook of Labor Economics, Vol., ed. by P. R. G. Layard, and D. E. Card, chap. 11, pp. 603–640. North-Holland, Amsterdam. Wolfe, B., and R. Haveman (2001): “Accounting for the Social and Non-Market Benefits of Education,” in The Contribution of Human and Social Capital to Sustained Economic Growth and Well-Being, ed. by J. Helliwell, pp. 221–250. University of British Columbia Press, Vancouver.
58
Table 1: Full Model Calibration ν ρ κ ψH ϕKH γ G R N θ δK δH φK i φH i ψK σK σH
Free Parameters (values used in main analysis) extra risk aversion to leisure 0.5 intratemporal substitution between consumption 0.5 and leisure return-to-scale factor in ht−1 dynamics 0.05 persistence of human capital shocks 0 correlation between productivity and 0 human capital shocks curvature coefficient in the utility function fixed to set RRA to 5 Calibrated Parameters mean economic growth rate (quarterly) 1.004 return under certainty (quarterly) 1.006 Steady-state labor 1/3 capital share in the production function 0.36 physical capital depreciation rate 0.9% human capital depreciation rate 2.1% K time-to-plan parameters for physical capital φK 4 = 0.01, φi = 0.33, i = 1, 2, 3 H = 0.01, φ time-to-plan parameters for human capital φH 4 i = 0.33, i = 1, 2, 3 persistence coefficient of productivity shocks 0.98 standard deviation of productivity shocks 0.712% (“P”), 0% (“H”), 0.645% (“B”) standard deviation of human capital shocks 0% (“P”), 1.502% (“H”), 0.645% (“B”) Derived Parameters and Steady-State Statistics
β
annual discount rate
β=
Y K0 I K0
steady-state output-physical capital ratio steady-state physical investment-capital ratio steady-state education:
Y K0 I K0
U
Gγ RP4
Φj R j
= j=0θ P4 = j=0 Φj Gj 1+(ν−1)N U= P4 G 1−γ ν(1−βG
)
κ(G−1+δH )
L τ
steady-state leisure productivity parameter in Ht accumulation
H K0
steady-state human capital-physical capital ratio
L=1−U −N H τ = G−1+δ Uκ 1 ³ ´ 1−θ H 1 Y K0 = N K0
Q
steady-state Ht -scaled leisure-consumption ratio
Q=
α
consumption share in the utility bundle zt
59
H/K0 ν C/K0 L
"
α= 1+
´− θ 1−θ Y K0 νQρ−1 Lν−1
(1−θ)
³
j=1
#−1
j
R ϕj ( G ) +1
60
0.306 (0.072)
−0.329 (0.061)
−0.302 (0.062)
HShock
PShock
Benchmark
(3)
(4)
2.847 (0.233)
7.123 (0.075)
0.605 (0.010)
0.033 (0.159)
∆ct
1.034 (0.135)
2.875 (0.058)
0.063 (0.004)
0.982 (0.125)
∆qt
−2.881 (0.354)
−8.995 (0.133)
0.333 (0.006)
−0.014 (0.034)
cov.
Variance Decomposition
(2)
0.312 (0.015)
0.333 (0.014)
−0.166 (0.008)
−0.012 (0.006)
rtf
(5)
(7)
−0.520 (0.016)
−0.503 (0.014)
0.073 (0.017)
−
f rt−1
−0.369 (0.014)
−0.348 (0.011)
0.136 (0.004)
0.449 (0.073)
rte
Response Rate
0.442 (0.075)
rte
(6)
0.146 (0.028)
0.134 (0.025)
0.040 (0.010)
0.078 (0.126)
h i e Et rt+1 − rtf
(8)
−0.839 (0.023)
−0.994 (0.001)
0.856 (0.006)
0.216 (0.071)
∆ct
(9)
(11)
−0.668 (0.029)
−0.837 (0.006)
0.970 (0.007)
0.300 (0.069)
∆yt
−0.695 (0.038)
−0.765 (0.029)
0.484 (0.039)
0.232 (0.065)
∆dt
Correlations
(10)
−0.918 (0.011)
−0.928 (0.009)
0.291 (0.042)
0.605 (0.050)
rte
(12)
Note: This table reports small-sample properties of changes in the scaled leisure-consumption ratio, ∆qt , computed using inverted factors for the latent-factor model and simulated data for production-economy models. Column 1 reports the first-order autocorrelations of ∆qt . Column 2 to 4 report the variance decomposition of the pricing kernel in equation 13 into two parts that are due to ∆ct and ∆qt respectively, and a covariance term. Column 5 to 8 report the response rate of asset return variables to ∆qt . Column 9 to 12 report the correlations between ∆qt and various macro variables. Standard errors are reported in parentheses. For model specifications of production-economy models please refer to Table 3.
0.503 (0.057)
Latent
correlation
Auto-
(1)
Table 2: Habit and the leisure-consumption Ratio
Table 3: Full Model - Business Cycle Results
Data
Standard
Standard TTP
PShock
HShock
Benchmark
Output Volatility (in %) Persistence
1.810 0.840
1.096 0.704
0.988 0.808
1.159 0.772
1.159 0.798
1.159 0.776
Consumption Volatility (in %) Corr. w/ output Relative Vol.
1.350 0.880 0.750
0.647 1.000 0.590
0.839 0.806 0.849
1.066 0.773 0.920
1.229 0.928 1.060
1.099 0.802 0.949
Hours Worked Volatility (in %) Corr. w/ output Relative Vol.
1.790 0.880 0.990
0.300 0.999 0.274
0.391 0.529 0.395
0.485 0.911 0.418
0.405 -0.019 0.350
0.473 0.759 0.408
Education Volatility (in %) Corr. w/ output Relative Vol.
n/a -ve n/a
n/a n/a n/a
n/a n/a n/a
0.074 -0.411 0.064
0.588 -0.758 0.507
0.262 -0.398 0.226
Investment Volatility (in %) Corr. w/ output Relative Vol.
5.300 0.800 2.930
1.996 1.000 1.821
1.897 0.848 1.919
2.282 0.799 1.969
1.507 0.790 1.300
2.163 0.789 1.866
Note: This table reports business cycle results computed from the data and various models. “Relative Vol.” measures the volatility of individual variable relative to the volatility of output. The “Data” column reports business cycle results computed by King and Rebelo (1999) using U.S. data from Stock and Watson (1999) for the period of 1947/Q1 to 1996/Q4. The columns marked “Standard” and “Standard TTP” report results from two models without human capital as specified at the beginning of Section 5. The columns marked “PShock”, “HShock” and “Benchmark” report results from my model with productivity shocks only, human capital shocks only and both shocks, respectively.
61
Table 4: Full Model – Asset Pricing Results Data
Mean
1-qtr
SD
1-qtr
Mean
4-qtr 12-qtr 20-qtr
SD
4-qtr 12-qtr 20-qtr
Mean Std. Dev.
PShock
HShock
Benchmark
1.553 (0.189) 2.569 (0.002)
Panel A: Short Rate 2.335 2.141 2.156 (0.307) (1.107) (1.269) 0.182 7.125 8.989 (0.056) (0.372) (0.441)
-5.712 (0.144) 1.979 (0.099)
0.682 (1.151) 8.163 (0.400)
0.418 (0.103) 0.708 (0.149) 0.856 (0.159)
Panel B: Term Spreads −0.003 0.108 0.093 (0.012) (0.725) (0.928) −0.009 0.165 0.171 (0.041) (0.852) (1.097) −0.014 0.176 0.186 (0.064) (0.866) (1.121)
0.534 (0.214) 0.577 (0.237) 0.536 (0.228)
0.182 (0.841) 0.254 (0.954) 0.260 (1.016)
6.843 (0.337) 8.297 (0.407) 8.579 (0.420)
1.522 (0.766) 1.845 (0.093) 1.906 (0.096)
6.215 (0.306) 7.536 (0.370) 7.792 (0.382)
Panel C: Excess Equity Return 0.007 0.732 1.024 (0.111) (1.032) (1.251) 0.783 7.128 8.945 (0.288) (0.684) (0.445)
7.342 (0.220) 1.488 (0.075)
1.671 (1.258) 9.116 (0.450)
3.359
0.187
7.340 (0.205) 1.384 (0.069)
1.670 (1.257) 9.111 (0.450)
1.392 (0.001) 2.026 (0.001) 2.159 (0.001)
6.335 (2.279) 15.458 (0.125)
Sharpe Ratio
0.530
Mean
3–12 (n/a) n/a (n/a)
Std. Dev.
Standard
0.011 (0.003) 0.039 (0.011) 0.062 (0.017)
0.009
Std. TTP
5.428 (0.584) 6.581 (0.344) 6.806 (0.355)
0.103
0.116
Panel D: Excess Human Capital Return n/a n/a 1.080 (n/a) (n/a) (1.308) n/a n/a 9.428 (n/a) (n/a) (0.466)
Note: This table reports the first two moments of 1-quarter real short rate, 1-, 3- and 5-year real term spreads, real excess equity returns, and real excess human capital returns, computed using the data and from various models. The “Data” column reports results computed using the ex-post real yields and real stock returns in the U.S. for the period of 1952/Q3 to 1998/Q2. For model specifications please refer to Table 3. Small-sample standard errors are reported in parentheses.
62
63
0 0.7 0.9
0.1 0.3
0.1 0.2
0.1 0.2
no h k
ρ
κ
ψH
ϕKH
TTP
1.308 1.243 1.141
1.154 1.140
1.178 1.199
1.134 1.063
1.051 1.243 1.384
1.202 1.337 1.719
1.810 1.159
0.603 0.636 0.790
0.949 0.950
0.947 0.952
0.958 1.042
0.915 0.975 1.016
0.934 0.900 0.848
0.750 0.949
0.425 0.415 0.527
0.408 0.408
0.404 0.402
0.358 0.366
0.345 0.496 0.635
0.487 0.669 0.962
0.990 0.408
1.819 1.826 1.929
1.866 1.863
1.853 1.842
1.932 1.960
1.930 1.827 1.784
1.807 1.717 1.634
2.930 1.866
Relative Volatility Consp. Hr. Work. Invest.
0.819 0.581 0.267
0.201 0.220
0.181 0.172
0.122 0.111
0.167 0.197 0.207
0.124 0.114 0.111
0.530 0.187
Sharpe Ratio
0.987 1.012 0.853
0.518 0.360
0.697 0.785
1.391 1.977
0.889 0.602 0.528
1.346 1.729 1.929
1.536 0.682
rtf
0.946 0.969 0.957
0.903 0.866
0.647 0.281
1.622 2.185
1.147 0.854 0.761
1.573 1.913 2.029
2.499 0.942
b rt,20
0.625 0.606 0.824
1.827 1.962
1.660 1.670
1.085 0.892
1.394 1.803 1.888
1.119 1.018 1.052
5.221 1.671
Mean rte − rtf −
rtf
0.625 0.606 0.824
1.827 1.962
1.661 1.672
1.097 1.007
1.394 1.806 1.889
1.129 1.042 1.035
3–12 1.670
rtH
0.171 0.177 1.963
8.140 8.014
8.211 8.772
8.133 7.893
7.612 8.423 8.681
8.302 8.617 9.638
2.361 8.184
rtf
0.117 0.126 0.140
0.404 0.398
0.412 0.446
0.405 0.399
0.383 0.416 0.427
0.411 0.426 0.484
2.691 0.407
0.822 0.150 3.085
9.070 8.931
9.164 9.721
8.909 8.037
8.798 9.173 9.141
9.042 8.901 9.494
15.731 9.112
Std. Dev. rte − rtf
b rt,20
0.822 1.155 3.086
9.065 8.927
9.169 9.739
8.950 9.024
8.797 9.185 9.133
9.086 8.962 9.050
n/a 9.111
rtH − rtf
Note: This table reports a key set of results from data and models with different parameterizations. The first panel reports results from the data and the “Benchmark” model. The second panel reports results from models that differ from the “Benchmark” parameterization in one parameter only, where the parameter name and the new value are given at the beginning of each row. The third panel reports results from models with time-to-plan restrictions imposed on neither type of capital, human capital, or physical capital respectively. In the “Benchmark” model, ν = 0.5, ρ = 0.5, κ = 0.05, ψH = 0 and ϕKH = 0.
1 2 3
ν
Data Benchmark
Output Volatility (%)
Table 5: Comparative Statics
Table 6: Correlations with Output £ e ¤ Et rt+1
h i e Et rt+1 − rtf
-0.048 (0.068)
n/a
n/a
0.127 (0.155)
0.053 (0.128)
0.158 (0.125)
-0.067 (0.033)
-0.035 (0.020)
0.076 (0.037)
PShock
-0.113 (0.035)
-0.018 (0.015)
HShock
0.035 (0.047)
Benchmark
-0.089 (0.047)
rtf
rte
f rte − rt−1
Data
0.194 (0.068)
-0.032 (0.070)
Standard
0.318 (0.349)
Std. TTP
dyt
∆λt
-0.587 (0.102)
n/a
0.089 (0.094)
-0.994 (0.004)
-0.152 (0.054)
0.035 (0.038)
0.140 (0.043)
-0.953 (0.028)
0.012 (0.018)
0.121 (0.037)
0.038 (0.018)
0.162 (0.042)
-0.839 (0.074)
0.003 (0.014)
0.027 (0.028)
0.022 (0.051)
0.148 (0.094)
0.043 (0.054)
-0.339 (0.236)
-0.023 (0.050)
-0.005 (0.014)
0.098 (0.050)
0.073 (0.031)
0.131 (0.055)
-0.690 (0.207)
-0.009 (0.049)
Note: This table reports correlations of various financial variables with output from data and different models. Data results are computed using (ex-post) real returns in the U.S. for the period of 1952/Q3 to 1998/Q2. For model specifications please refer to Table 3.
64
Table 7: Decomposition of Excess Asset Returns £ e ¤ Et rt+1
h i e Et rt+1 − rtf
VAR(∆λt ) × 102
-0.056 (0.011)
-0.004 (0.017)
-0.000 (0.017)
0.027 (0.003)
-0.803 (0.011)
-0.471 (0.024)
-0.027 (0.016)
0.305 (0.034)
0.385 (0.048)
0.472 (0.032)
-0.913 (0.006)
-0.492 (0.030)
-0.037 (0.017)
0.384 (0.036)
0.480 (0.056)
HShock
0.045 (0.001)
-0.057 (0.003)
-0.054 (0.000)
0.000 (0.002)
0.003 (0.004)
4.087 (0.407)
Benchmark
0.191 (0.023)
-0.372 (0.031)
-0.226 (0.026)
0.003 (0.011)
0.150 (0.028)
1.151 (0.120)
rtf
rte
Standard
-0.018 (0.004)
-0.060 (0.012)
Std. TTP
0.448 (0.026)
PShock
f rte − rt−1
Note: This table reports decomposition results of equation (17). The second to the sixth columns report the response rate of asset returns to changes in the price of risk, ∆λt , computed as slope coefficients in the contemporary regression rt = α + β∆λt , where the regressand rt is one of various asset return and expected asset return variables specified in the first row. The last column reports the unconditional variance of ∆λt . Small-sample standard errors are reported in parentheses. For model specifications please refer to Table 3.
65
66
0.669 (0.281) 0.466 (0.193) 0.156 (0.123) 0.070 (0.052)
R2
0.032 (0.029) 0.130 (0.189) 0.030 (0.058) 0.027 (0.038)
Data
0.023 (0.022) 0.110 (0.184) 0.127 (0.209) 0.733 (0.512)
R2
2.332 (2.551) 2.091 (1.888) 1.977 (1.618) 1.539 (0.967)
0.008 (0.010) 0.086 (0.090) 0.136 (0.128) 0.333 (0.220)
0.065 (0.013) 0.004 (0.005) 0.002 (0.003) −0.001 (0.002)
0.120 (0.040) 0.006 (0.008) 0.004 (0.006) 0.004 (0.007)
0.028 (0.021) 0.004 (0.006) 0.004 (0.006) 0.008 (0.011)
0.036 0.008 (0.045) (0.010) 0.030 0.063 (0.032) (0.065) 0.028 0.096 (0.027) (0.094) 0.021 0.236 (0.017) (0.182) Term Spread PShock β R2
0.029 (0.013) 0.001 (0.004) −0.000 (0.003) −0.001 (0.001)
0.037 0.009 (0.036) (0.010) 0.034 0.091 (0.028) (0.087) 0.032 0.146 (0.025) (0.127) 0.024 0.342 (0.017) (0.235) Panel B: GDP growth on Standard TTP β R2
Panel A: Excess Stock Return on Dividend Yield Standard TTP PShock β R2 β R2
0.006 (0.001) 0.065 (0.086) 0.107 (0.129) 0.288 (0.231)
Standard β R2
0.001 (0.004) 0.001 (0.004) 0.000 (0.004) 0.000 (0.003)
Standard β R2
R2 0.029 (0.023) 0.009 (0.011) 0.007 (0.010) 0.004 (0.006)
HShock
0.124 (0.062) 0.023 (0.022) 0.015 (0.019) 0.002 (0.008)
β
R2 0.004 (0.006) 0.021 (0.029) 0.031 (0.041) 0.069 (0.081)
HShock
−0.000 (0.015) 0.003 (0.009) 0.003 (0.008) 0.003 (0.006)
β
0.007 (0.010) 0.063 (0.066) 0.095 (0.096) 0.232 (0.183)
0.030 (0.014) 0.001 (0.005) −0.000 (0.003) −0.001 (0.002)
0.025 (0.020) 0.004 (0.006) 0.004 (0.006) 0.008 (0.010)
Benchmark β R2
0.037 (0.047) 0.032 (0.033) 0.029 (0.028) 0.022 (0.018)
Benchmark β R2
f GDPt b where yt equals rte − rt−1 in Panel A and ln GDP in Panel B, and xt equals dyt in Panel A and rt,20 − rtf in Panel B. For data regressions, t−1 standard errors are computed using the Hodrick (1990) method for slope coefficients and the Generalized Method of Moments for the R2 statistics, and are reported in parentheses. For model regressions, small-sample standard errors for slope coefficients and R2 statistics are reported in parentheses. For model specifications please refer to Table 3.
k 1X yt+i = α + βxt + ²t,t+k , k i=1
Note: This table reports slope coefficients and the R2 statistics from univariate regressions of the k-period cumulative excess stock b return on the log dividend yield dyt , or univariate regressions of the k-period cumulative GDP growth on the 5-year term spread rt,20 − rtf :
60
20
12
1
k
60
20
β
0.036 (0.017) 0.023 (0.016) 0.021 (0.019) 0.032 (0.019)
1
12
β
k
Data
Table 8: Predictive Regressions
Response to Technology
Response to Technology 1.2
1
Consumption
0.5
2
% deviation from SS
% deviation from SS
% deviation from SS
Response to Technology
2.5
1.5
1.5 1
Investment
0.5 0 2
4
6
8
0.6
Output
0.4 0.2
0 0
1 0.8
0 0
2
4
6
8
0
2
4
6
Years
Years
Years
Response to Technology
Response to Technology
Response to Technology
8
0.1 0.3
0.4 0.3 0.2
Labor
0.1
0
% deviation from SS
% deviation from SS
% deviation from SS
0.5
0.25 0.2 0.15
Education
0.1 0.05
−0.1
Leisure
−0.2 −0.3 −0.4 −0.5
0
0 0
2
4 Years
6
8
0
Response to Technology
2
4 Years
6
8
0
Response to Technology
4 Years
6
8
Response to Technology 0
0.5
0
2
qt
−1
0.4
% deviation from SS
% deviation from SS
% deviation from SS
−1 −0.5
0.3 0.2
RRA
0.1
−2
Lambda
−3 −4 −5 −6
−1.5 −7
0 0
2
4 Years
6
8
0
2
4 Years
6
8
0
2
4 Years
Figure 1: Impulse Responses to Productivity Shocks
67
6
8
Response to HShock
Response to HShock
Response to HShock 0.8
0.5 0.4 0.3
Consumption
0.2
% deviation from SS
0.8
0.6
% deviation from SS
% deviation from SS
0.7
0.6
0.4
Investment 0.2
0.6
0.4
Output 0.2
0.1 0
0 0
2
4
6
8
0 0
2
4
Years
Response to HShock
8
0
2
4
6
8
Years
Response to HShock
Response to HShock
0.1
0
Labor −0.1
0
−0.1
% deviation from SS
% deviation from SS
0.1
% deviation from SS
6
Years
Education
−0.2
0.2 0.15 0.1
Leisure
0.05
−0.3 −0.2
0 0
2
4 Years
6
8
0
Response to HShock
2
4 Years
6
8
0
Response to HShock
2
4 Years
6
8
Response to HShock
0.1
0
0.4 0.3
qt
0.2 0.1
−2
0.05
% deviation from SS
% deviation from SS
% deviation from SS
0.5
0
RRA
−0.05 −0.1
−4
Lambda
−6 −8 −10 −12
−0.15
0 0
2
4 Years
6
8
0
2
4 Years
6
8
0
2
4 Years
Figure 2: Impulse Responses to Human Capital Shocks
68
6
8
Impulse responses to a shock in Technology Equity Return Riskfree Rate
25
Percent deviation from steady state
20
15
10
5
0
−5
−10
−15
−20 −1
0
1
2
3 4 Years after shock
5
6
7
Figure 3: Impulse Responses of Asset Prices to Productivity Shocks
69
8
Impulse responses to a shock in HShock Equity Return Riskfree Rate
Percent deviation from steady state
3
2
1
0
−1
−2 −1
0
1
2
3 4 Years after shock
5
6
7
8
Figure 4: Impulse Responses of Asset Prices to Human Capital Shocks
70
