WORKING PAPER SERIES. No 36 / Investment-Specific Shocks, Business Cycles, and Asset Prices

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1 BANK OF LITHUANIA. WORKING PAPER SERIES No 1 / 28 SHORT-TERM FORECASTING OF GDP USING LARGE MONTHLY DATASETS: A PSEUDO REAL-TIME FORECAST EVALUATION EXERCISE 1 WORKING PAPER SERIES Investment-Specific Shocks, Business Cycles, and Asset Prices By Giuliano Curatola, Michael Donadelli, Patrick Grüning and Christoph Meinerding No 36 / 216

2 ISSN (ONLINE) WORKING PAPER SERIES No 36 / 216 INVESTMENT-SPECIFIC SHOCKS, BUSINESS CYCLES, AND ASSET PRICES Giuliano Curatola *, Michael Donadelli *, Patrick Grüning ** and Christoph Meinerding * * Faculty of Economics and Business Administration and Research Center SAFE, Goethe University Frankfurt. s: curatola@safe.uni-frankfurt.de, donadelli@safe.uni-frankfurt.de, meinerding@finance.uni-frankfurt.de. ** Center for Excellence in Finance and Economic Research (CEFER), Bank of Lithuania, and Faculty of Economics, Vilnius University. PGruening@lb.lt. The authors acknowledge helpful discussions with Mykola Babiak, Roberto Casarin, Alain Coën, Fulvio Corsi, Ana-Maria Fuertes, Roberto Pancrazi, Antonio Paradiso, Aurelija Proškutė, Christian Schlag, Ctirad Slavík, and Peter Zorn. Moreover, we thank the seminar participants at the 11th Dynare Conference, the 19th Conference of the Swiss Society for Financial Market Research, the 5th Annual Lithuanian Conference on Economic Research, the 216 World Finance Conference, the 23rd Annual Meeting of the German Finance Association (DGF), and the University of Venice for their helpful comments. We gratefully acknowledge research and financial support from the Research Center SAFE, funded by the State of Hessen initiative for research LOEWE. The views expressed herein are solely those of the authors and do not necessarily reflect the views of the Bank of Lithuania or the Eurosystem. This paper is also available on SSRN as SAFE Working Paper No. 129:

3 Lietuvos bankas, 216 Reproduction for educational and non-commercial purposes is permitted provided that the source is acknowledged. Address Totorių g. 4 LT-1121 Vilnius Lithuania Telephone (8 5) Internet Working Papers describe research in progress by the author(s) and are published to stimulate discussion and critical comments. The Series is managed by Applied Macroeconomic Research Division of Economics Department. The views expressed are those of the author(s) and do not necessarily represent those of the Bank of Lithuania. ISSN (ONLINE)

4 Investment-Specific Shocks, Business Cycles, and Asset Prices November 24, 216 Abstract This paper proposes and tests a new source of time variation in real investment opportunities, namely long-run shocks to the productivity of the investment sector, to explain the joint behavior of macroeconomic quantities and asset prices. A twosector general equilibrium model with long-run investment shocks and wage rigidities produces both positive co-movement among key macroeconomic variables and a sizable return volatility differential between the investment and consumption sector. Moreover, positive long-run investment shocks are associated with low marginal utility and thus command a positive risk premium. We test our model using data on sectoral TFP and find evidence in support of our theoretical predictions. Keywords: General Equilibrium Asset Pricing, Production Economy, Long-Run Risk, Investment-Specific Shocks, Wage Rigidities. JEL: E32, G12. 4

5 1 Introduction Recent research has shown that investment-specific shocks play a key role in explaining the dynamics of business cycles and asset prices. 1 Typically, in the associated equilibrium models, investment-specific shocks affect the formation of new capital goods by changing the marginal efficiency of new investments (MEI) and, thus, they are often referred to as MEI shocks. Although this approach contributes a great deal to our understanding of the role of investment-specific shocks, it leaves some questions open. First, models that attribute a central role to MEI shocks tend to produce a counterfactual negative co-movement among macroeconomic variables (the so-called co-movement puzzle ). Second, with few exceptions, this literature focuses on the ability of investment-specific shocks to explain either macroeconomic dynamics or time-series and cross-sectional properties of stock returns. The joint implications of investment-specific shocks for asset prices and macroeconomic quantities are, however, not yet fully understood and appreciated. We argue that a consistent explanation of macroeconomic co-movements, relying on investment shocks, should also be able to provide a reasonable fit for key asset pricing moments and the dynamics of stock returns, and vice versa. In this paper we propose a different source of time variation in investment opportunities, namely long-run risk (LRR) in the total factor productivity (TFP) of the investment sector, and analyze its performance with respect to the two issues outlined above. We introduce LRR in a general equilibrium model with two sectors (consumption and investment), assuming that the TFP processes of both sectors are driven by short-run (SR) and long-run (LR) components. We show, both theoretically and empirically, that our decomposition of investment-specific shocks into SR and LR components makes a substantial 1 The different strands of literature on investment-specific shocks are summarized in Section 2. 5

6 contribution towards understanding and solving the issues described above. The co-movement of macroeconomic variables can be quantified in several ways. We start by looking at impulse response functions from our model and find that only SR investment shocks are responsible for the co-movement puzzle while LR investment shocks do not induce any negative co-movement between macroeconomic quantities. Moreover, in the long run (from about 5 quarters after the shock onwards), the responses of macro variables to LR investment shocks are all positive. The intuition behind this result is as follows. SR investment shocks increase the current profitability of the investment sector relative to the consumption sector. As a result, they naturally generate a substitution effect that induces households to reallocate resources towards the investment sector, thus inducing a negative co-movement among macroeconomic variables. Differently, LR investment shocks affect the future profitability of investment. In addition to the substitution effect, positive news about future investment profitability generate an additional wealth effect that creates positive conditional co-movement among macroeconomic variables in the long run. The conditional analysis using impulse responses illustrates only one aspect of the co-movement puzzle. The unconditional correlation of macro quantities, which comprises their response to all shocks in the economy, represents the other equally important aspect. Since SR shocks are quantitatively more important for unconditional correlations, our baseline model with SR and LR shocks still suffers from an unconditional co-movement puzzle. The typical way to fix this is to assume rigidities of some kind, for instance wage rigidities, and we follow the same route in this paper. However, since we do not include MEI shocks, which exacerbate the co-movement puzzle, the degree of wage rigidities necessary to match empirical moments in our benchmark model is much smaller and 6

7 more realistic than in a model with MEI shocks. The degree of wage rigidities necessary to make unconditional correlations positive is important because wage rigidities induce a trade-off between matching macroeconomic quantities and matching the equity return volatilites of the two sectors. More specifically, high wage rigidities imply a low volatility spread between the investment and consumption sector. First and foremost, they impair the ability of the productive sector to react to macroeconomic shocks and thus make firms cash flows riskier. On top, however, this additional cash flow risk spills over from the investment sector to the consumption sector through the formation of capital goods, and so the increase of the consumption sector volatility is higher than that of the investment sector volatility, which makes the volatility spread unrealistically low. LR investment risk offers a way out of this tension. It improves the model s ability to generate realistic properties of sectoral stock returns while at the same time not inducing negative co-movement among macroeconomic variables. MEI shocks also have the potential to induce a high volatility spread between the investment and consumption sector as Papanikolaou (211) shows. However, we document that MEI shocks induce negative co-movement between consumption and investment growth. From this point of view, LR investment shocks are, therefore, better able to jointly improve the dynamics of macroeconomic quantities and asset prices. The ability not to generate negative co-movement implies that LR investment risk needs to be associated with only a moderate degree of wage rigidities in order to obtain positive unconditional correlations among macroeconomic variables, thus preserving the asset pricing properties of the model. We also verify the main economic mechanism of our model empirically. We estimate the TFP processes of the consumption and the investment sector and extract their LR 7

8 component by applying a standard state-space approach to sectoral TFP data. We then estimate the dynamic impulse responses of consumption, output, investment, and labor to SR and LR sectoral shocks in a VAR model and find that LR investment shocks have a positive long-run impact on macro variables as predicted by our theoretical model. Moreover, we document that the SR and LR components of sectoral TFP are also quantitatively important. For instance, over a horizon of one quarter, SR shocks (together) account for 58 percent while LR shocks account for 12 percent of output fluctuations. The predictive power of SR shocks tends to decrease with the forecasting horizon while the predictive power of LR shocks tends to increase: over a horizon of 2 quarters SR shocks account for 48 percent while LR shocks account for 16 percent of output fluctuations. Our evidence also suggests that LR investment shocks are quantitatively more important than LR consumption shocks in explaining macroeconomic fluctuations. In fact, we find that LR investment shocks explain a fraction of output variance that can be up to 13 times bigger than the fraction of output variance explained by LR consumption shocks. The empirical identification strategy is also useful to test the cross-sectional asset pricing implications of LR investment shocks. In our model, these shocks are associated with economic states with low marginal utility and thus command a positive risk premium. We confirm this prediction empirically. Using the standard two-stage Fama-MacBeth procedure we document that LR investment shocks are priced in the cross-section of stock returns and carry a positive and significant risk premium. These results taken together suggest that LR investment risk is a plausible candidate to explain not only fluctuations of macroeconomic quantities, but also key asset pricing moments. The remainder of the paper is organized as follows. Section 2 discusses the relevant literature on this topic. Our two-sector production economy is presented in Section 3 8

9 and its quantitative implications are analyzed in Section 4. Section 5 provides empirical evidence on the main mechanism of the paper. Section 6 documents the cross-sectional asset pricing performance of LR investment shocks. Section 7 concludes. 2 Related Research This paper links two strands of literature. The first strand of literature argues that investment shocks are important drivers of business cycle fluctuations: Greenwood, Hercowitz, and Krussel (1997, 2), Christiano and Fisher (23), Fisher (26), Jaimovich and Rebelo (29), Justiniano, Primiceri, and Tambalotti (21, 211). Recently, a branch of this literature focuses on the co-movement puzzle generated by MEI shocks that represent the most popular way of modelling investment shocks: Khan and Tsoukalas (211), Furlanetto and Seneca (214a,b). A second strand of literature analyzes the asset pricing implications of investment shocks: Li (214), Kogan and Papanikolaou (214), Kogan, Papanikolaou, and Stoffman (215), Garlappi and Song (216b). Similarly to our paper, Papanikolaou (211) and Segal (215) analyze the joint implications of investment-specific shocks for macro quantities and asset prices. Our two-sector model is similar to that of Jaimovich and Rebelo (29), Papanikolaou (211), and Segal (215). We differ from Jaimovich and Rebelo (29) with respect to two important aspects: the utility function and the structure of sectoral shocks. Jaimovich and Rebelo (29) assume that the representative agent features habit formation in consumption and that the level of sectoral TFP is subject to permanent shocks which implies that the growth rate of sectoral TFP is a random walk. Differently, in our model the growth rate of sectoral TFP processes follows a random walk with time-varying drift. 9

10 Time variation in the drift induces long-run risk in consumption and investment. Therefore, we assume that the representative agent has Epstein-Zin preferences to ensure that sectoral long-run risk is priced in equilibrium. We also differ from Papanikolaou (211) with respect to investment-specific shocks: we consider LR shocks to the TFP process of the investment sector while Papanikolaou (211) considers MEI shocks. Segal (215) focuses on shocks to the conditional volatility of the investment sector TFP, while we analyze shocks to its conditional mean growth rate. Solutions for the co-movement problem have been proposed in the literature recently. Khan and Tsoukalas (211) show that the co-movement problem can be solved when the depreciation rate of capital is a convex function of its utilization or when the households preferences are such that there is no wealth effect of labor. 2 Furlanetto and Seneca (214a,b) solve the co-movement problem using a Calvo-style mechanism of price and wage rigidities, respectively. Similarly to Furlanetto and Seneca (214a,b), we also use wage rigidities to induce the right co-movement between macroeconomic variables, but differently from them we use wage rigidities as proposed by Uhlig (27). More technically, the presence of the LR components in sectoral TFP processes creates long-run consumption and investment risk that are both priced in our economy because the representative agent has Epstein-Zin preferences. In our framework, the SR and LR components of sectoral TFP processes are exogenous quantities that drive the dynamics of all endogenous variables (consumption, output, investment, and labor). Kaltenbrunner and Lochstoer (21) take a different perspective and analyze the conditions under which 2 More precisely, Khan and Tsoukalas (211) assume that the representative agent features habit formation in consumption as in Jaimovich and Rebelo (29). They eliminate the wealth effect of labor by assuming that consumption habits depend on past consumption only, but not on current consumption. 1

11 long-run risk arises endogenously in a one-sector production economy with Epstein-Zin preferences. They show that the consumption-savings decisions of households endogenously produces long-run risk in consumption even when the (log of) TFP follows a random walk. Besides the main research question we also differ from Kaltenbrunner and Lochstoer (21) with respect to the model characteristics. They consider a one-sector model where production depends on capital only. Differently, we build a two-sector model where production depends on both capital and labor. Thus, our equilibrium dynamics are also affected by the households labor choice and by the interactions between the two sectors. At the moment, there is no clear consensus in the literature concerning the implications of investment-specific shocks for the cross section of stock returns. Papanikolaou (211) and Kogan and Papanikolaou (213) find that investment-specific shocks are negatively priced in the cross section of stock returns. Differently, Li (214) and Segal (215) argue in favor of a positive market price of risk. Garlappi and Song (216a) suggest potential measurement problems in the existing proxies of investment specific shocks that might contribute to these mixed results. With respect to this literature we show both theoretically and empirically that LR investment shocks command a positive risk premium and we use the different TFP components themselves in the empirical tests, instead of other proxy variables. In summary, we contribute to these strands of literature by showing that a new, previously disregarded source of investment-specific technical change, namely LR investment risk, has important implications for asset prices and macroeconomic fluctuations. 11

12 3 The Model In the following subsections, we develop a dynamic stochastic general equilibrium (DSGE) model with two sectors that allows us to study the asset pricing implications of various shocks to investment good productivity and efficiency. The first sector is the consumption good sector. It admits a standard competitive representative firm that uses capital and labor to produce consumption goods which are supplied to the representative household for consumption. The second sector is the investment good sector that uses labor to produce investment goods which are then sold to the consumption good sector and used in the production of the consumption good. The representative household owns both sectors, has recursive preferences over consumption and leisure, and it freely allocates labor to the two sectors. The wage is subject to frictions in the form of sticky wages. The production technologies in both sectors are subject to both SR and LR productivity shocks. A summary of the equilibrium conditions and details about the solution of the model are provided in Appendix A. 3.1 Representative household Because we want to focus on the trade-off between short-run and long-run shocks, we assume that the representative agent has recursive preferences of the Epstein and Zin (1989) type: U t = [ (1 β)v 1 1 ψ t + β ( E t [ U 1 γ t+1 ] ]) 1 1/ψ 1 1 1/ψ 1 γ, (1) where γ denotes the relative risk aversion (RRA), ψ measures the elasticity of intertemporal substitution (EIS), and β (, 1) is the household s time discount factor. Note that this preference specification allows to separate the relative risk aversion from the elasticity 12

13 of intertemporal substitution. The utility flow v t is a Cobb-Douglas index of aggregate consumption C t and leisure L L t : v t := v(c t, L t ) = C ν t ( AC,t ( L L t ) ) 1 ν, where ν (, 1) reflects preferences for consumption versus leisure. A C,t (to be defined later) is the productivity of the consumption good sector and can be interpreted as the household s standard of living in the spirit of Croce (214). 3 In each period, the representative household chooses consumption C t and labor L t to maximize (1) subject to the following budget constraint: C t + B t+1 + ϑ C,t+1 (V C,t D C,t ) + ϑ I,t+1 (V I,t D I,t ) = W u t L t + B t R f t + ϑ C,t V C,t + ϑ I,t V I,t, (2) where ϑ C,t (ϑ I,t ) denotes equity shares held from time t 1 to time t in the representative firm producing the consumption (investment) good. V C,t (V I,t ) and D C,t (D I,t ) is the cum-dividend market value and the dividend of the firm producing the consumption (investment) good. B t denotes bond holdings from time t 1 to time t, R f t is the gross risk-free rate, and W u t represents the frictionless wage. Hence, the household chooses the amount of hours allocated to labor as if the wage was not sticky (wage rigidities are introduced later in Section 3.4). The first-order conditions of the maximization problem lead to the following expression for the stochastic discount factor (SDF): M t,t+1 = β ( Ct+1 C t ) 1 ( vt+1 v t ) ( 1 1 ψ U t+1 (E t [U 1 γ t+1 ]) 1 1 γ ) 1 ψ γ. (3) 3 Multiplying leisure by productivity in the utility flow also guarantees balanced growth. Defining the utility flow alternatively as either C t ( L L t ) ν or C ν t (A C,t 1 ( L L t )) 1 ν does not change the predictions of the model qualitatively, and the quantitative changes to the model-implied moments are relatively small. These results are available upon request. 13

14 The usual Euler equations of cum-dividend asset prices can be written as: V C,t = D C,t + E t [M t,t+1 V C,t+1 ], V I,t = D I,t + E t [M t,t+1 V I,t+1 ], 1 R f t = E t [M t,t+1 ]. Finally, the household s optimal labor allocation leads to the following condition: W u t = 1 ν ν C t L L t. 3.2 Consumption goods sector The consumption goods sector admits a representative perfectly competitive firm utilizing capital and labor to produce the consumption goods. The production technology is given by: Y C,t = K α C C,t (A C,tL C,t ) 1 α C, where α C is the capital share, labor L C,t is supplied by the household, and A C,t is the exogenous labor-augmenting productivity. We assume that A C,t is subject to both shortrun and long-run shocks: A C,t = e a C,t, a C,t = µ C + x C,t 1 + a C,t 1 + σ C ε C,t, x C,t = ρ C x C,t 1 + σ x,c ε x,c,t. The unconditional expected growth rate of productivity is µ C. Short-run productivity shocks are induced by ε C,t, whereas ε x,c,t indicates long-run shocks which affect the stochastic component in expected productivity growth x C,t. The persistence of long-run productivity shocks is measured by ρ C. Moreover, capital K C,t accumulates according to: 14

15 K C,t+1 = (1 δ K )K C,t + G (i C,t ) K C,t. (4) Here, i C,t = I C,t K C,t, and δ K is the depreciation rate of capital. The function G captures adjustment costs of investments as in Jermann (1998): G t := G (i C,t ) = α τ (i C,t ) 1 1 τ + α2, where the constants α 1 and α 2 are chosen such that there are no adjustment costs in the deterministic steady state. The net profit of the consumption goods sector D C,t is given by output minus the expenditure on investment goods and wages: D C,t = Y C,t P I,t I C,t W t L C,t. (5) The representative firm chooses labor, capital, and investment to maximize its value, i.e., the firm solves: V C, = max {K C,t+1,I C,t,L C,t } t= t= t= E [M,t D C,t ], (6) subject to the capital accumulation constraint (4). The first-order condition with respect to K C,t+1, [ ( )] 1 1 = E αc Y C,t+1 P I,t+1 I C,t+1 t Mt,t+1 + λ t+1 (G t δ K ), (7) λ t K C,t+1 determines the price of the investment good P I,t. The Lagrange multiplier attached to the capital accumulation constraint λ t is pinned down by the first-order condition with 15

16 respect to investment: λ t = P I,t, (8) G t where G t is the derivative of G t with respect to the investment-to-capital ratio i C,t. Finally, the first-order condition with respect to labor L C,t implies the following condition that determines the wage W t : W t L C,t = (1 α C )Y C,t. (9) 3.3 Investment goods sector The investment goods sector supplies investment goods to the consumption goods sector. It is populated by a representative firm that sells the demanded goods at the price P I,t. Investment goods are produced according to the technology: Y I,t = A I,t L 1 α I I,t, where L I,t is labor supplied by the household, 1 α I is the labor share, and A I,t is the stochastic total factor productivity of the investment goods sector, whose dynamics are given by the following process: A I,t = e a I,t, a I,t = µ I + x I,t 1 + a I,t 1 + σ I ε I,t, x I,t = ρ I x I,t 1 + σ x,i ε x,i,t. Thus, as in the consumption goods sector, the productivity of the investment goods sector is subject to both short-run (ε I,t ) and long-run (ε x,i,t ) shocks. The unconditional expected growth rate of investment goods sector s productivity is denoted by µ I, and ρ I denotes the persistence of long-run investment shocks. In Section 5.1 we provide empirical evi- 16

17 dence for the presence of LR components in the TFP processes of the consumption and investment sector. The LR component in the productivity of investments also captures the idea of investment hysteresis originally proposed by Dixit (1992). Classic theory postulates that firms should invest (or enter the market) when the price exceeds the average variable costs and disinvest (or exit the market) when the price falls below the average variable costs. However, empirical evidence indicates that, once firms have invested in a project, they tend to stay in business and continue their investment even when the underlying causes of investment are fully reversed. 4 This suggests that investment drivers may have long-lasting effects that, in our model, are captured by time variation in x I,t. The investment goods sector pays wages to the household and its total output Y I,t is sold to the consumption goods sector at the price P I,t. The net profit of the investment goods sector is therefore given by: D I,t = P I,t Y I,t W t L I,t. (1) The investment goods sector firm chooses labor L I,t to maximize the firm value: V I, = max {L I,t } t= t= { t= E [M,t D I,t ] The first-order condition with respect to labor L I,t is given by: }. (11) W t L I,t = (1 α I )P I,t Y I,t. (12) 4 The empirical evidence on investment hysteresis is still fragmented. A review of sectoral and experimental evidence on this topic can be found in Kogut and Chang (1996) and Bragger, Bragger, Hantula, and Kirnan (23). A deep economic motivation for a delayed exit strategy is provided by Bernanke (1983). Further theoretical studies that point to the importance of long-term trends in investment dynamics are provided by Dixit (1989), Bar-Ilan and Strange (1996), Bar-Ilan and Strange (1999), and Kogan (21). 17

18 3.4 Labor market frictions We assume that the labor supply is subject to frictions. In the spirit of Blanchard and Galí (27) and Uhlig (27), we impose that a fraction of the labor supply does not reach the market. As shown by Uhlig (27), this results in sticky wages, i.e., households wages are given by: W t = ( ) ξ A C,t 1 W t 1 (Wt u ) 1 ξ, A C,t 2 where W u t represents the frictionless wage. The intuition is that the household chooses labor hours as if there were no labor market frictions. Hence, W u t appears in the household budget constraint (2). However, the actual salary paid to households is W t, which therefore appears in the definition of firm s dividends (5), (1), and (14). The term A C,t 1 /A C,t 2 captures the fact that the wage is indexed to aggregate productivity if it cannot be chosen optimally. The parameter ξ controls the degree of wage rigidities. In particular, ξ = implies the absence of labor market rigidities. 3.5 Market clearing conditions and aggregate dividends The household supplies labor to the consumption and the investment goods sector. Thus, market clearing in the labor market dictates: L t = L C,t + L I,t. Equating the supply and demand for investment goods implies: I C,t = Y I,t. (13) 18

19 The output of the consumption goods sector is fully consumed by the household and therefore the consumption good market clears when: C t = Y C,t = W t L t + D M,t = W t L t + D C,t + D I,t. (14) The second equality is obtained by assuming that i) bonds are in zero net supply and the stocks of the consumption and the investment goods sector firms are in unit net supply (i.e., B t and ϑ C,t ϑ I,t 1), and ii) households receive the actual wage W t in exchange for their labor supply and not the frictionless wage W u t. The aggregate ( market ) dividends, D M,t = D C,t +D I,t, are given by the sum of the dividends distributed by the consumption and the investment goods sector. The market value at time, which defines the aggregate equity premium in our economy, is consequently given by: V M, = t= E [M,t D M,t ]. 3.6 Stochastic marginal efficiency of investment Justiniano, Primiceri, and Tambalotti (21) and Papanikolaou (211) introduce a stochastic marginal efficiency of investment (MEI) in order to explain macroeconomic fluctuations and the main features of asset prices. To highlight the different implications of MEI shocks and LR investment shocks, we therefore also analyze an augmented economy with stochastic MEI. This comparison helps us to assess how our model with LR investment specific shocks performs relative to Justiniano, Primiceri, and Tambalotti (21) and Papanikolaou (211). In the augmented economy, the marginal efficiency of investment goods is governed by 19

20 the stochastic process Z M,t. In order to increase the future capital stock by an absolute amount G(i c,t )K C,t, the representative firm needs to buy Z 1 M,t I C,t units of the investment good (and not I C,t as in our benchmark economy) at the relative price P I,t. Therefore, the total investment cost is given by Z 1 M,t I C,tP I,t. The log marginal efficiency of investment goods is stochastic and follows a strictly stationary AR(1) process: log(z M,t ) = ρ M log(z M,t 1 ) + σ M ε M,t. When the marginal efficiency of investment is stochastic some equations of our benchmark economy need to be changed. First, the net profit of the consumption goods sector D C,t is now given by: D C,t = Y C,t Z 1 M,t P I,tI C,t W t L C,t, (15) rather than by Equation (5). Correspondingly, the consumption goods firm s first-order condition with respect to capital (7) is replaced by: [ ( 1 1 = E t M αc Y C,t+1 Z 1 M,t+1 P )] I,t+1I C,t+1 t,t+1 + λ t+1 (G t δ K ), λ t K C,t+1 and the consumption goods firm s first-order condition with respect to investment, i.e., Equation (8), needs to be replaced by: λ t = Z 1 M,t P I,t G t. (16) Finally, the market clearing condition of the investment market (13) becomes: Z 1 M,t I C,t = Y I,t. 2

21 4 Quantitative Analysis We calibrate the model at quarterly frequency, consistent with the frequency of available data on sectoral output that we use to validate the empirical predictions of our model in Section 5. Our benchmark economy features SR and LR TFP shocks only, but no MEI shocks. We analyze the role of MEI shocks in the augmented economy in Section Calibration Our calibration is summarized in Table 1. The preference parameters are set in accordance with the long-run risk literature. The subjective discount factor β is set to.993 to help the model match the relatively low level of the risk-free rate observed in the data. We set the coefficients of relative risk aversion and elasticity of intertemporal substitution to values of 1 and 2, respectively. Similar values can be found in Bansal and Yaron (24), Croce (214), and Kung and Schmid (215). Note that γ > 1/ψ, i.e. our representative agent has a preference for early resolution of uncertainty, which is also in line with the recent experimental evidence of Brown and Kim (214). Following standard practice, the consumption share in the utility bundle ν is chosen such that the steady state supply of labor is one third of the total time endowment of the household. Given the other parameters, this is achieved by setting ν = The parameters of the sectoral TFP processes are selected to match as closely as possible the moments of asset prices and macroeconomic quantities as well as the empirical estimates of Section 5.1. µ C and µ I are such that the average annual growth rate of output is around.2, consistent with US data. We fix the persistence of sectoral LR components (i.e., ρ C,t and ρ I,t ) at.95 which corresponds to an annualized persistence of.8. This 21

22 Table 1: Quarterly benchmark calibration Parameter Description Source Value Preference parameters β Subjective discount factor γ Risk aversion 2 1 ψ Elasticity of intertemporal substitution 2 2 ν Consumption share in utility bundle Consumption goods sector Technology parameters α C Capital share in consumption good production 1.3 δ K Depreciation rate of physical capital 1.2 τ Elasticity of adjustment costs in investment 4.98 TFP parameters µ C Long-run mean of consumption goods sector TFP 2.18/4 σ C Volatility of short-run shocks to consumption goods sector TFP ε C 4.32/ 4 ρ C Autocorrelation of long-run shocks to consumption goods sector TFP x C 4.95 σ x,c Volatility of long-run shocks to consumption goods sector TFP ε x,c σ C Investment goods sector Technology parameters α I One minus the labor share in investment good production 1.1 TFP parameters µ I Long-run mean of investment goods sector TFP 2.18/4 σ I Volatility of short-run shocks to investment goods sector TFP ε I 4.32/ 4 ρ I Autocorrelation of long-run shocks to investment goods sector TFP x I 4.95 σ x,i Volatility of long-run shocks to investment goods sector TFP ε x,i σ I Labor market ξ Wage rigidities parameter 3.35 Notes: Parameters sources: 1=Papanikolaou (211), 2=Croce (214), 3=Uhlig (27), 4=own calibration. is also in line with Croce (214). Our empirical estimates and the previous literature (Croce, 214) suggest that the volatility of the LR shocks is a small percentage of the volatility of the SR shocks. Thus, we impose σ x,c =.1125 σ C and σ x,i =.1125 σ I. Finally, we set σ C = σ I =.32/ 4 to match the annualized volatility of output growth in the US for the period (i.e., 3.24 percentage points). The depreciation rate of physical capital in the consumption goods sector is standard and set to.2, similarly to Papanikolaou (211). On the production side, also in line with Papanikolaou (211), we set the capital share in consumption goods production α C equal to.3. The labor share in investment goods production, i.e., 1 α I, is equal to.9. 22

23 The elasticity of the supply curve of capital τ is equal to.98, a value in line with existing empirical evidence. 5 Finally, following Uhlig (27), we assume a moderate degree of wage stickiness by imposing ξ =.35 in the benchmark case. We also present robustness checks where we vary the parameter ξ. The model is solved in dynare using a second-order approximation. 4.2 Macro quantities and asset prices Table 2 reports key unconditional moments of macroeconomic and asset pricing quantities, obtained from model simulations. We consider several sub-cases that allow us to highlight the role played by sectoral SR and LR shocks. In column [1] we report the main features of an economy without long-run risk. Then, we introduce long-run risk in the consumption sector only (column [2]), and finally we allow for long-run risk in both the consumption and the investment sector (column [3]). In all of these cases, we assume the absence of wage rigidities. In the absence of long-run risk the model has difficulties in matching the basic properties of stock returns, most importantly the equity premium. The model reproduces the positive co-movement between consumption and output, but fails on all the other macroeconomic co-movements. Introducing long-run risk in the consumption sector (column [2]) makes this sector relatively riskier (as compared to the case of no long-run risk), which leads to a substantial increase in the risk premium required to hold the consumption sector equity. As a result, the market equity premium increases from 1.2 to 3.69 percentage points. However, longrun risk in the consumption sector produces even more counterfactual results for the 5 Eberly (1997), for instance, reports estimates that range between 1.8 and In an earlier empirical work, Abel (198) reports values for τ between.5 and

24 co-movement of consumption and investment and the co-movement of consumption and labor, as those numbers turn negative in column [2]. Introducing long-run risk in the investment sector (column [3]) further improves asset pricing quantities, especially the stock return volatility of the investment sector and the return differential between the two sectors. More precisely, the volatility spread between the investment and the consumption sector increases from 1.21 to 4.25 percentage points and gets closer to the value observed in the data (1.96 percentage points). Nevertheless, the risk premium of the consumption sector is higher than that of the investment sector. This can be explained by differences in the cyclical variation between the two sectors: due to adjustment costs, consumption is much more pro-cyclical than investments, i.e. corr( c, y) > corr( i, y). Therefore, the consumption sector equity exhibits more systematic risk and earns a higher risk premium in equilibrium. The correlations among macroeconomic variables remain basically unaffected, in particular the correlation between consumption and investment and the correlation between consumption and labor remain slightly negative. This means that LR investment risk improves the model s ability to fit sectoral stock returns but to explain the unconditional correlations among macroeconomic variable we need an additional ingredient, namely wage rigidities. Columns [4], [5], and [6] report results for the economy with long-run risk in both sectors and progressively higher degrees of wage rigidities, where column [5] refers to our benchmark economy with ξ =.35. The introduction of wage rigidities solves the comovement problem and the unconditional correlations between consumption and investment and between consumption and labor switch sign and turn positive. Wage rigidities also have implications for asset prices. They prevent wages from perfectly adjusting to TFP changes and thus impair the ability of firms to react to macroeconomic shocks. As 24

25 a result, firms equity becomes riskier. The equity volatility increases and households require a higher risk premium to hold stocks in equilibrium. Table 2: Model versus Data: Macroeconomic Quantities and Asset Prices The Role of LRR The Role of Wage Rigidities Model Data [1] [2] [3] [4] [5] [6] ASSET PRICES E[R M R f ] σ[r M R f ] E[R f ] σ[r f ] E[R I R C ] σ[r I ] σ[r C ] MACRO QUANTITIES σ( y) σ( y)/σ( c) σ( i)/σ( y) σ( l)/σ( y) σ( l) ρ( c, y) ρ( c, i) ρ( c, l) ρ( i, l) ρ( i, y) Notes: This table reports the main annual moments for the benchmark calibration ([5]) and five other model specifications. Means and volatilities are reported in percentage points. Model 1: No LRR in either sector (i.e., σ x,c = σ x,i = ) and no wage rigidities (i.e., ξ = ). Model 2: LRR in the consumption sector only (i.e., σ x,c >, σ x,i = ) and no wage rigidities (i.e., ξ = ). Model 3: LRR in both sectors (i.e., σ x,c >, σ x,i > ) and no wage rigidities (i.e., ξ = ). Models 4, 5, and 6: LRR in both sectors (i.e., σ x,c >, σ x,i > ) and different degrees of wage rigidities (ξ =.1, ξ =.35, and ξ =.5, respectively). The entries for the models are obtained by repetitions of small-sample simulations. All empirical moments are from Papanikolaou (211) for the time period Overall, our model is able to produce realistic moments for both macroeconomic and asset pricing quantities. Our benchmark calibration produces an equity premium of 3.97 percentage points and a low and stable risk-free rate (2.2 percentage points with a volatility of 1.5 percentage points). The model also succeeds in generating the average return spread between the investment and consumption sector (-1.44 percentage points vs percentage points in the data). Admittedly, like most DSGE models, our models encounters some difficulties in generating high stock return volatilities. Nevertheless, 25

26 our model generates the right sign for the volatility spread between the investment and consumption sector equity, and this spread is much larger than in any of the cases without LR investment shocks (see also Table 3 below). 6 The standard deviation of output growth is exactly as in US macroeconomic data. In addition, the relative standard deviations of consumption, investment, and labor to output are not far from their empirical counterparts. Finally, the model-implied correlations among macroeconomic variables are positive, ranging from.3 to.99, and are all close to the values in the data. 4.3 Impulse response analysis Figure 1 depicts the theoretical impulse response functions. 7 Our model features four types of shocks, and the four shocks contribute to the macroeconomic co-movement in different ways. It is instructive to start looking at the two investment-specific shocks. After a positive SR investment shock, the investment sector becomes relatively more profitable immedi- 6 The very recent asset pricing literature shows promising new economic channels which enable DSGE models to produce reasonable levels of return volatility. For instance, Nezafat and Slavík (215) develop a rich model that generates a sizeable stock market volatility through financial shocks affecting the tightness of firms financing constraints. Croce (214) obtains a volatility of about 11 percentage points by assuming a high persistence of long-run productivity shocks (.95 annually). Favilukis and Lin (216) obtain values of similar magnitude with stochastic firm leverage and labor rigidities in the form of infrequent renegotiation of wages. The model of Papanikolaou (211) generates high values of the stock market volatility when the volatility of shocks to the marginal efficiency of investments is set to 13.5 percentage points or the volatility of investment productivity shocks is set to 2 percentage points, but the volatility spread between the investment and consumption sectors is very similar to what we report in Table 2. 7 The impulse response functions are computed in dynare by perturbing the equilibrium system around the steady state. An alternative way to study the theoretical responses of equilibrium variables is to compute population impulse response functions (Carlstrom, Fuerst, and Paustian (29) and Hussain and Liu (216)) obtained by applying a VAR on model generated data. As the results from both approaches are similar, we discuss the impulse-response functions from perturbation here only. The results for the simulated VAR obtained from a long sample of model generated data are reported in Appendix B. Since we have long-run consumption and investment shocks in our model, using a number of short samples of model generated data (equal to the length of the empirical data) to estimate the VAR can be problematic (see, for instance, Colacito and Croce, 211). Therefore, we do not follow this approach. 26

27 ately. Households, thus, find it optimal to increase investment immediately. Consumption also increases slightly in the period of the shock, and it continues to fluctuate until period 5, whereas investment reverts to the steady state already in period 2. This diverging behavior of consumption and investment after period 1 in response to SR investment shocks causes negative conditional co-movement. To ensure that the consumption stream remains smooth over time, households decrease labor today and then gradually readjust their labor supply toward the steady state. SR investment shocks also have a negative initial impact on total output Y t = C t + P I,t I C,t. Given that consumption and investment both increase initially after the shock, the drop in output is fully explained by the decline in the price of investment goods. After a positive LR investment shock, the expected future profitability of the investment sector increases and households decide to invest more today and in the following periods. At the same time, households perceive themselves to be richer and want to increase current consumption. However, contrary to the reaction to short-run shocks, since the positive effect of LR investment shocks will materialize only over time, households have to finance their desired level of consumption by working more today. This increase in labor supply stimulates the total output of the economy. In summary, SR and LR investment shocks have different implications for the comovement of macroeconomic quantities. SR investment shocks affect the current profitability of the investment sector and generate a substitution effect that is the source of the negative co-movement among macroeconomic variables. Differently, LR investment shocks affect the future profitability of the investment sector and generate a positive wealth effect that induces positive co-movement among macroeconomic variables. The other two sources of variation have been studied extensively in the literature, and 27

28 our results basically replicate the ones by Croce (214). After a positive shock to the SR component of consumption TFP, households find it optimal to increase consumption, investment, and labor simultaneously. As a result, total output also increases. Shortrun consumption TFP shocks thus induce positive macroeconomic co-movement. The responses to LR shocks are slightly different and depend on the relation between the substitution effect and the wealth effect, which itself depends mainly on the IES. After a positive shock to the long-run consumption TFP, the continuation value of utility increases. In our parametrization, the IES is larger than one. Therefore, the substitution effect dominates the wealth effect, current consumption and continuation utility are substitutes, and households find it optimal to decrease consumption today and invest instead. However, consumption growth decreases only by a small amount and increases again after one quarter. The unconditional correlations among macroeconomic quantities result from the interaction of all four types of shocks described above. Since the positive co-movement induced by SR consumption shocks and LR investment shocks is quantitatively smaller than the negative co-movement induced by the other shocks (SR investment shocks and LR consumption shocks), the unconditional correlations between consumption and investment and between consumption and labor remain slightly negative unless we introduce wage rigidities. So, how do wage rigidities help reproducing the positive correlation between consumption and investment and between consumption and labor? As one can see from Figure 1, where the solid lines refer to the case with wage rigidities and the dotted lines to the case without them, the main effect comes from SR consumption TFP shocks. After such a shock, firms find it optimal to increase their labor demand. When the labor market is fully flexible this leads to an increase in wages that almost completely offsets 28

29 the incentive of firms to hire more workers. With wage rigidities, firms face an increase in the labor demand and a relatively smaller increase in salary, which increases the desire of firms to hire more workers. As a result, in the economy with wage rigidities the positive responses of labor, investment, and consumption with respect to SR shocks to the TFP process of the consumption sector are amplified as compared to an economy with fully flexible wages. In other words, with wage rigidities the negative co-movement between macroeconomic variables induced by SR shocks to the TFP of the investment sector is overcompensated by the positive co-movement resulting from SR shocks to the TFP of the consumption sector, and the overall correlation among key macroeconomic variables turns positive. 29

30 Figure 1: Impulse response functions Consumption Sector (SRR) Consumption Sector (LRR) Investment Sector (SRR) x Investment Sector (LRR) c x y i x l m Notes: This figure depicts the impulse-response functions for a length of 2 quarters of log consumption growth c, log output growth y, log investment growth i, log labor growth l, and log stochastic discount factor m. Impulse-response functions with respect to i) a positive one-standard deviation SR shock to the consumption sector TFP (ɛc), ii) a positive one-standard-deviation LR shock to consumption sector TFP (ɛx,c), iii) a positive one-standard deviation SR shock to investment sector TFP (ɛi) and iv) a positive one-standard-deviation LR shock to investment sector TFP (i.e., ɛx,i) are depicted in columns 1, 2, 3, and 4, respectively. The solid black line refers to the case with wage rigidities (ξ =.35) while the dotted blue line refers to the case with the absence of wage rigidities (ξ = ). The values reported are deviations from the steady state in percentage points. 3

31 4.4 Stochastic marginal efficiency of investment We argue that shocks to the TFP process of the investment sector are an alternative way to capture time variation in real investment opportunities. Thus, at this point it is instructive to compare the implications of SR and LR investment shocks with those of more established MEI shocks. In this section, we analyze an augmented economy that features MEI shocks in addition to SR and LR shocks to the TFP processes of consumption and investment sectors. The parameters related to the marginal efficiency of investment are calibrated following the recent literature. We fix σ M to an annualized value of.12 which is within the confidence intervals estimated by Justiniano, Primiceri, and Tambalotti (21). Moreover, we assume that the persistence of MEI shocks equals the persistence of the LR components of the sectoral TFP processes and therefore set ρ M =.95, similarly to Furlanetto and Seneca (214b). In Table 3, we compare our benchmark economy that features only SR and LR shocks in the sectoral TFP processes (column [1]) with an economy that features also MEI shocks in addition to sectoral SR and LR shocks (column [2]). MEI shocks decrease the correlation among key macroeconomic variables: the correlation between consumption and investment decreases from.3 to.1 (vs..39 in the data), the correlation between investment and labor decreases from.53 to.16 (vs..83 in the data) and the correlation between investment and output decreases from.36 to.1 (vs..67 in the data). The reasons for these results are illustrated in Figure 2 which depicts the impulse responses to MEI shocks. One can see that MEI shocks induce a negative co-movement between consumption, output, and labor. Actually, the impulse responses are similar to the impulse responses to short-run investment shocks depicted in Figure 1. This result suggests that MEI shocks and SR investment shocks have similar implications for the co-movement 31