A Labor-Augmented Investment-Based Asset Pricing Model

Transcription

1 A Labor-Augmented Investment-Based Asset Pricing Model Frederico Belo Carlson School of Management University of Minnesota Lu Zhang Stephen M. Ross School of Business University of Michigan and NBER September 009 Chen Xue Stephen M. Ross School of Business University of Michigan Abstract We introduce labor adjustment costs in the q-theory model of expected returns and test the labor-augmented model using moments of the cross-section of expected stock returns as well as stock valuation ratios. Adding labor substantially reduces the pricing errors of the baseline q-theory model across portfolios sorted on investment-to-assets, book-to-market, asset growth, and labor hiring. The labor-augmented model also substantially outperforms the baseline model in explaining the cross-section of stock valuation ratios, especially across investment-to-assets portfolios. However, neither model can fully capture the large spread in the valuation ratio observed in the data, especially across the book-to-market portfolios. Finance Department, Carlson School of Management, University of Minnesota, CarlSMgmt, 31 19th Avenue South, Minneapolis MN Tel: (61) , and Finance Department, Stephen M. Ross School of Business, University of Michigan, 701 Tappan Street, Ann Arbor MI Finance Department, Stephen M. Ross School of Business, University of Michigan, 701 Tappan Street, R 4336, Ann Arbor MI ; and NBER. Tel: (734) , fax: (734) , and zhanglu@bus.umich.edu. 1

2 1 Introduction We introduce labor adjustment costs into the q-theory model of asset pricing (e.g., Cochrane (1991) and Liu, Whited, and Zhang (009)) and test the labor-augmented model using moments for the cross-section of expected stock returns and stock valuation ratios. Under constant returns to scale, stock returns equal the weighted average of levered investment and labor hiring returns. In addition, stock valuation ratios (market value of equity-to-capital) are linked to the marginal benefits of investment and hiring. We use Generalized Methods of Moments (GMM) to match the observed average stock returns and average stock valuation ratios of portfolios formed on investment-to-assets (I/A), book-to-market (B/M), asset growth (AG), and labor hiring (LH) rate with the moments implied by the model. To gauge the model s performance we examine the magnitude of the model s errors and the plausibility of the estimated parameters. To assess the importance of labor for asset pricing, we compare the performance of the labor-augmented model with that from the baseline model, as well as with the performance of standard asset pricing models such as the Capital Asset Pricing Model (CAPM) and the Fama-French (1993) three-factor model. Adding labor into the baseline model substantially improves the explanatory power of the model for the cross-section of expected stock returns, even when both models are forced to simultaneously match the cross section of average stock returns and stock valuation ratios. Across the portfolios sorted on investment-to-assets, the annual average pricing error of the labor-augmented model is only 1.5%, which is considerably lower than the.9% of the baseline model. Across other testing assets, the labor-augmented model reduces the annual pricing errors of the baseline model from.4% to 1.94% across the book-to-market portfolios, from.99% to 1.90% across the asset growth portfolios, from 1.7% to 1.53% across the labor hiring portfolios, and from 3.03% to.14% when all portfolio sorts are considered simultaneously. The reduction in the pricing errors of the labor augmented model is impressive given the already good fit of the baseline model in explaining the cross section of average stock return, as documented

3 in Liu, Whited and Zhang (009), and especially when the results for both models are compared with those from standard asset pricing models. The annual pricing errors of the CAPM across these portfolios range from 4.46% to 5.98%, which are two to three times larger than the pricing errors of the labor model. The annual pricing errors of the the Fama-French model are also higher than in the labor model across all test assest: 3.9% versus 1.49% across the I/A portfolios, 1.46% versus 1.17% across the B/M portfolios, 4.04% versus % across the AG portfolios and 1.93% versus 1.47% across the LH portfolios. Adding labor into the baseline model also improves the explanatory power of the model for the cross-section of stock valuation errors, but the quality of the fit is more modest than for the cross-section of stock returns. Across the investment-to-assets portfolios, the stock valuation ratios of the labor model are less than half the size of the valuation errors of the model. Both models however, cannot fully capture the large spread in the valuation ratios across the B/M portfolios and, to a lesser degree, across the AG and LH portfolios. We also find that including the moment conditions for the stock valuation ratios in the estimation of the both the labor and the baseline model helps identify the structural parameters. The parameters estimates obtained when each model is required to simultaneously match the cross section of stock returns and valuation ratios are more stable across test assets and more precisely estimated than when only the moment conditions for expected stock returns are used. Barring a few exceptions, the effect of labor on stock returns and firm value has largely been overlooked in asset pricing. 1 Merz and Yashiv (007) show that adding labor into Cochrane s (1991) model substantially improve the model s performance in matching the level and time series of stock market valuation ratios. Bazdresch, Belo, and Lin (008) add labor adjustment costs into Zhang s (005) model and show that labor hiring negatively predicts future returns in the cross section both 1 A partial list of studies linking labor market variables to asset prices include Mayers (197), Fama and Schwert (1977) Campbell (1996), Jagannathan and Wang (1996), Jagannathan, Kubota and Takeharaet (1998), Santos and Veronesi (006) Boyd, Hu and Jagannathan (005), and Lustig and Van Nieuwerburgh (008). The interpretation of the empirical facts in these studies is silent about the production-side of the economy (technology).

4 in model simulations and in the data. Danthine and Donaldson (00) shows that frictions in the determination of the wage rate magnifies the risk premium of equity returns at the aggregate level. Our work differs because we simultaneously focus on the cross-section of stock returns and stock valuation ratios, and we provide a structural estimation of the adjustment cost parameters. Our work is also related to the literature on labor demand and investment which investigates the importance of capital and labor adjustment costs to explain investment and hiring dynamics. Our work differs because we estimate the adjustment cost parameters at the portfolio level and using information from stock market data. Section introduces labor adjustment costs in the standard model and derives implications for the cross-section of stock returns and stock valuation ratios. Section 3 presents the estimation methodology, and Section 4 presents the empirical results, and Section 5 concludes. The Labor-Augmented Model We describe the firm s maximization problem and derive testable implications for the cross-section of average stock returns and stock valuation ratios..1 Technology Time is discrete and the horizon infinite. Firm j uses capital inputs, k j t, labor inputs, nj t, and a vector of costlessly adjustable inputs z j t to produce a homogeneous output. The production function is given by y j t y(kj t,nj t,zj t,xj t ), in which xj t is a vector of exogenous aggregate and firm specific shocks, and displays constant returns to scale, that is, y j t = kj t yj t / kj t + nj t yj t / kj t + zj t yj t / zj t. See, for example, Cooper and Haltiwanger (1997), Caballero, Engel, and Haltiwanger (1995), Cooper, Haltiwanger, and Power (1999) and Cooper and Haltiwanger (003) on capital, and Hamermesh (1989), Bertola and Bentolila (1990), Davis and Haltiwanger (199), Caballero and Engel (1993), Caballero, Engel, and Haltiwanger (1997), Cooper, Haltiwanger, and Willis (004) on labor, and Shapiro (1986), Galeotti and Schiantarelli (1991), Hall (004), Merz and Yashiv (007), and Bloom (009) on joint estimation of capital and labor adjustment costs. 3

5 The marginal products of capital and labor are parameterized as in Love (003): y(k j t,nj t,zj t,xj t ) k j t y(k j t,nj t,zj t,xj t ) n j t y j t = γ k k j, (1) t y j t = γ n n j. () t in which y j t is sales, γ k > 0 is capital s share, and γ n > 0 is labor s share. In every period t, the capital stock evolves as k j t+1 = ij t + (1 kδ j t )kj t, (3) and the labor stock evolves as n j t+1 = hj t + (1 nδ j t )nj t, (4) in which i j t is capital investment, kδ j t is the depreciation rate of capital, hj t is labor hiring, and nδ j t is the employee separation rate. Firms incur adjustment costs when investing and hiring. The adjustment cost function, denoted φ(i j t,kj t,hj t,nj t ), is increasing and convex in ij t and h j t, decreasing in kj t and n j t, and linearly homogeneous in i j t,kj t,hj t, and nj t. We consider a quadratic adjustment cost function: φ(i j t,kj t,hj t,nj t ) = η k ( i j t in which η k,η n > 0 are adjustment cost parameters. k j t ) k j t + η n ( h j t n j t ) n j t, (5). Firm s Maximization Problem Following Hennessy and Whited (007), at the beginning of time t firm j can issue one-period debt, denoted b j t+1, which must be repaid at the beginning of t + 1. The gross corporate bond return on b j t, denoted rj Bt, can vary across firms and through time. Taxable corporate profits equal operating profits less capital depreciation, adjustment costs, and interest expenses: y(k j t,nj t,zj t,xj t ) wj t nj t e tz j t δj t kj t φ(ij t,kj t,hj t,nj t ) (rj Bt 1)bj t, 4

6 in which w j t is the wage rate and e t is the price vector of the z j t inputs, both exogenous to the firm. Adjustment costs are expensed, consistent with treating them as foregone operating profits. Let τ t denote the corporate tax rate at time t. The payout of firm j is: d j t (1 τ t)[y(k j t,nj t,zj t,xj t ) wj t nj t e tz j t φ(ij t,kj t,hj t,nj t )] ij t +bj t+1 rj Bt bj t + kδ j t kj t τ t+τ t (r j Bt 1)bj t in which k δ j t kj t τ t is the depreciation tax shield and τ t (r j Bt 1)bj t is the interest tax shield. Let m t+1 be the stochastic discount factor from t to t + 1, which is correlated with the aggregate component of x j t+1. Firm s choose optimal investment, hiring and debt decisions in order to maximize vj t, the cum-dividend market value of equity. The maximization problem can be formulated as follows: v j t max {i j t+s,kj t+s+1,hj t+s,nj t+s+1,bj t+s+1 } s=0 [ ] E t m t+s d j t+s, (6) subject to the capital and labor accumulation equations (3) and (4) and a transversality condition ] that prevents firms from borrowing an infinite amount of debt: lim T E t [m t+t b j t+t+1 = 0. Let s=0 kq j t and nq j t be the present value multipliers associated with equations (3) and (4), which are the marginal benefit of an additional unit of capital and labor, respectively..3 Stock Returns and Valuation Ratios Proposition 1 states the firm s equilibrium value and stock return. Proposition 1 Define p j t vj t dj t as the ex-dividend equity value. Firms value-maximization implies that ( in which k q j t 1 + (1 τ i j ) t t)η k k j t p j t + bj t+1 = kq j t kj t+1 + nq j t nj t+1, (7) ( and n q j t (1 τt)η h j t n n j t ). In addition, firms value-maximization implies that E t [m t+1 r j It+1 ] = 1, in which rj It+1 is the 5

7 investment return, defined as: r j It+1 (1 τ t+1) [ y γ j t+1 k + η k j k t+1 ( i j t+1 k j t+1 Similarly, E t [m t+1 r j Ht+1 ] = 1, in which rj Ht+1 ) ] + k δ jt+1 τ t+1 + (1 k δ jt+1 ) [1 + (1 τ t+1 )η k ( 1 + (1 τ t )η k ( i j t k j t i j t+1 k j t+1 )] ). is the hiring return, defined as [ ( ) ] y (1 τ t+1) γ j t+1 n + η h j ( ) r j n j n t+1 w j Ht+1 t+1 n j t+1 + (1 n δ jt+1 )(1 τ h t+1)η j t+1 n t+1 n j t+1 ( h j ). (9) t (1 τ t )η n n j t (8) Denote the after-tax corporate bond return as r j Bat+1 = r j Bt+1 (rj Bt+1 1)τ t+1, then ] E t [m t+1 r j Bat+1 = 1. Define r j St+1 (pj t+1 + dj t+1 )/pj t as the stock return, νj t bj t+1 /(pj t + bj t+1 ) as market leverage, and µ j t kq j t kj t+1 /( kq j t kj t+1 + nq j t nj t+1 ) as the value weight of capital in the firm value. The weighted-average of investment and hiring returns is equal to the weighted average of stock and bond returns: r j It+1 µj t + rj Ht+1 (1 µj t ) = rj Bat+1 νj t + rj St+1 (1 νj t ). (10) Proof. See Appendix A. Equation (10) establishes a link between the firm s stock return, and the firm s investment, hiring and after-tax corporate bond returns. Rearranging terms, equation (10) implies that the firm s stock return r j St+1 is given by: r j St+1 rj IHw,t+1 = rj It+1 µj t + rj Ht+1 (1 µj t ) rj Bat+1 νj t (1 ν j t ). (11) Thus, in equilibrium, stock returns are equal to a weighted average of levered investment and hiring returns, which we define as r j IHw,t+1. 6

8 Equation (10) has implications for equilibrium valuation ratios. Rearranging terms, we can express the market to capital ratio (which is closely related to the market to book ratio) as: p j t k j t+1 = ( 1 + (1 τ t )η k i j t k j t ) h j t n j t+1 + (1 τ t )η n n j t k j bj t+1 t+1 k j t+1 (1).4 Drivers of Stock Returns and Valuation Ratios Equations (11) and (1), together with the investment return equation (8) and the hiring return equation (9) suggest several economic forces driving the cross section of average stock returns and stock valuation ratios. For stock returns, the first two drivers are the marginal product of capital, measured by the sales to capital ratio (y t+1 /k j t+1 ), and the marginal product of labor, measured by the sales to labor ratio (y j t+1 /nj t+1 ). Higher marginal products of capital and labor are associated with higher returns. The third and fourth drivers, which are the second element in the numerator of the investment and hiring return divided by the corresponding denominator, are roughly proportional to the growth rate of investment and hiring. These drivers correspond to the capital gain component of the investment and hiring returns. Here, higher growth rates on investment and hiring are associated with higher returns. The fifth and sixth drivers for returns works through the investment rate and the hiring rate in the denominator of the investment and the hiring return. Here, higher investment rates and hiring rates are associated with lower stock returns. The seventh and eight driver for returns are the depreciation rate of capital ( k δ j t+1 ) and the employee separation rate ( nδ j t+1 ). Collecting terms involving k δ j t+1 in the numerator of the investment return equation (8) yields ( ) k δ j t+1 (1 τ t+1) 1 + η k i j t+1 /kj t+1 and collecting terms involving n δ n t+ n in the hiring return equation ( ) (9) yields δ n t+1 (1 τ t+1) 1 + η k i j t+1 /kj t+1. Thus higher rates of capital depreciation or employee separation imply lower investment and hiring returns and hence lower average stock returns. The ninth driver for returns is market leverage (v j t ). Taking the first-order derivative of (11) with respect to market leverage shows that returns should increase with market leverage today. 7

9 The tenth driver for returns in the wage rate (w j t+1 ) in the hiring return equation (9). Here, an higher wage rate decreases the marginal benefit of hiring, which in turn decreases the hiring return and hence average stock returns. Finally, the eleventh driver for returns is the labor to capital ratio (n j t+1 /kj t+1 ) which controls the relative weight (µ j t ) of the investment and hiring return in the stock return decomposition in equation (11). The effect of this driver for expected returns is ambiguous because it depends on the relative magnitude of the investment and hiring return. Because the weight of the investment return µ j t decreases with the labor to capital ratio, stock returns will also be decreasing in the labor to capital ratio if the investment return is higher than the hiring return. Turning to the analysis of the drivers of the stock valuation ratio in equation (1), the first and second drivers are the investment rate and the hiring rate. All else equal, firms with higher investment and hiring rates have higher valuation ratios (i.e. are growth firms). The third driver of the valuation ratio is the labor to capital ratio. All else equal, because the hiring rate is on average positive, firms that are more labor intensive have higher valuation ratios than capital intensive firms. Finally, the fourth driver is the debt to capital ratio (b j t+1 /kj t+1 ), which is positively related to the stock valuation ratio. 3 Empirical Methodology 3.1 Moment Conditions Equations (11) and (1) hold ex-post state by state, and so they also hold ex-ante in expectation. For estimation and testing, we follow Liu, Whited and Zhang (009) and study the ex-ante restrictions implied by these equations. Formally, we test if expected stock returns equal expected levered weighted average of investment and hiring returns: E [ ] r j St+1 rj IHw,t+1 = 0. (13) 8

10 In addition, we test if the average predicted stock valuation ratios in the model equal the average stock valuation ratios observed in the data: E [ p j t k j t+1 ( 1 + (1 τ t )η k i j t k j t ) h j t n j t+1 (1 τ t )η n n j t k j + bj t+1 t+1 k j t+1 ] = 0. (14) To construct a formal test of the model, we define the model errors from the empirical moments: e S i E T [r j St+1 rj IHw,t+1 [ ( ) e V p j t i j t i E T 1 + (1 τ t )η k k j t k j t+1 ] h j t n j t+1 (1 τ t )η n n j t k j + bj t+1 t+1 k j t+1 ] (15) (16) in which E T [.] is the sample mean of the series in brackets. Following Liu, Whited and Zhang (009), the key identification assumption for estimation and testing is that both model errors have a mean of zero, a standard assumption that underlies most Euler equation tests (see discussion in Cochrane, 1991). Because the capital and labor share parameters γ k, and γ n cannot be separately identified using the equations (13) and (14), we estimate the sum of these parameters (γ k + γ n ) which measures the total capital and labor share in the production function Estimation Method We estimate the technological parameters (γ k + γ n ), η k and η n using the Generalized Methods of Moments (GMM) by minimizing a weighted average of (15), or a weighted average of both (15) and (16). Following the recommendation in Cochrane (001), we use an identity weighting matrix in the GMM estimation to preserve the economic structure of the testing portfolios. When both moment conditions (15) and (16) are used, we adjust the weight of the stock valuation ratios moments so that both set of moment conditions are measured in comparable units and thus have efectively the same weight in the estimation. To conduct inference, we compute the optimal weighting matrix, using a standard Bartlett kernel with a window length of four. To test wether all (or a subset of) 3 The parameters γ k, and γ n are not separately identified because they enter additively, in the stock return equation. Specifically, we can rearrange terms to express the stock return equation as: r j St+1 = (γ k +γ n )yt+other, in other which other are terms do not depend on γ k, and γ n. In addition, the parameters γ k, and γ n do not enter the stock valuation ratio equation (14). 9

11 model errors are jointly zero, we use the a χ test from Hansen (198, Lemma 4.1). All GMM estimation tests are conducted at the portfolio level. This approach has advantages because portfolio level hiring and investment data is smoother than firm level data, consistent with the smooth adjustment cost function considered here. In addition, the portfolio level estimation allows us to connect our findings to the large empirical asset pricing literature documenting large differences in average returns and valuation ratios at the portfolio level. Liu, Whited and Zhang (009) provide additional motivation for this approach. 3.3 Data We construct annual levered weighted investment and hiring returns to match annual stock returns. Firm-level data is from the Center for Research in Security Prices (CRSP) monthly stock file and the annual Standard and Poor s Compustat industrial files. We select our sample by first deleting any firm-year observations with missing data or for which total assets, total employees, gross capital stock, debt, or sales are either zero or negative. We include only firms with year-end in December to align the data across firms. Following the standard convention, we exclude firms with primary SIC classifications between 4900 and 4999 or between 6000 and 6999 since the q-theory is unlikely to be applicable to regulated or financial firms Variable Definitions For data on capital, investment, output, debt, leverage, capital depreciation and corporate bond returns, we follow Liu, Whited, and Zhang (009). Appendix B describes briefly each data item. The labor market data on wage rate and labor separation rate is not available at the firm level (the firm level wage bill data in COMPUSTAT is missing for more than 80% of the firms on our sample). We measure these variables at the industry level as follows. Wage rate per worker: We measure w j t using annual data from the Bureau of Economic Analysis (BEA), GDP-by-Industry accounts. We compute the industry level (annual) wage rate per worker 10

12 as the ratio of the total compensation of employees (which includes wage and salary accruals and supplements to wages and salaries) to the total number of full time employee in the industry. This data is available at the digits Standard Industry Classification (SIC) level and is available from 1949 to From 1987 to 006, this data is available at the digits North American Industry Classification (NAICS) level. We assign the industry level annual wage rate to the firm level using the firm level SIC code (before 1987) and the firm level NAICS code (after 1987) reported in COMPUSTAT (whenever available) or in CRSP. Employee separation rate: We measure annual employee separation rate n δ j t using data for 16 major industry groups based on NAICS codes from the Job Openings and Labor Turnover Survey (JOLTS) available from the Bureau of Labor Statistics (BLS). Because this data is only available since 001, we proceed as follows. First, for each major industry group we compute the average annual labor separation rate over the period from 001 to the present. Then, we assign this average value to each firm in our sample for the entire period, using the firm level NAICS code. This procedure thus allows us to capture some heterogeneity in the employee separation rate across industries, but it does not capture variation over time. Since there is no NAICS data in Compustat before 1985, we convert the NAICS code into a SIC code in order to assign the labor separation data to the firm level for this earlier period Testing Portfolios and Portfolio Characteristics The testing portfolios that we use in the estimation are ten investment-to-assets portfolios (I/A), ten book-to-market portfolios (B/M), ten asset-growth portfolios (AG) and ten labor hiring portfolios (LH). Appendix B provides a description of how these portfolios are constructed. Table 1, Panel A, reports the average stock returns, stock valuation ratios and corresponding volatilities, for each portfolio and for the H-L portfolio, which is a portfolio that goes long in the high portfolio and short in the low portfolio. As previously documented in the empirical asset pricing literature, average stock returns are decreasing in the I/A ratio, AG and the LH ratio, and 11

13 increasing in the B/M ratio, and the opposite pattern holds for the stock valuation ratios. As discussed in Section.4, the stock return equation (11) and the stock valuation ratio equation (1) suggests several economic forces driving the cross section of average stock returns and stock valuation ratios. Table 1, Panel B reports the average of each driver of stock return and stock valuation ratio for each portfolio sort. The pattern of the characteristics across portfolios is, in general, consistent with the analysis in section.4. First, we observe a negative relationship between the average returns of these portfolios and the investment and hiring rates and a positive relationship between these variables and average stock valuation ratios. This observation is consistent with the analysis in Bazdresch, Belo and Lin (009), who find that the firm level investment and hiring rate is negative related to expected stock returns in the cross-section. Second, average investment growth, hiring growth and market leverage, are in general positively related with average stock returns and negatively related to stock valuation ratios. Third, the capital depreciation rate and the employee separation rate are negatively related with average stock returns and positively related to stock valuation ratios. The remaining characteristics do not exhibit a clear pattern across the portfolios. In the empirical section below, we quantify the importance of each characteristic for explaining the cross-section of average stock returns and stock valuation ratios. 4 Empirical Results We test the importance of labor for asset pricing by evaluating if the labor model can simultaneously explain the cross-section of valuation ratios and average stock returns of the test portfolios. In order to assess the marginal economic importance of labor, we compare the performance of the labor model with that from the baseline model studied in Liu, Whited and Zhang (009). Our framework conveniently nests the baseline model since this model corresponds to the parameterization of the labor model in which the labor adjustment cost parameter (η n ) is set to zero. Thus examining the statistical and economic significance of the labor adjustment cost parameter provides a convenient way of examining the marginal explanatory power of labor for asset prices. We also compare the 1

14 ability of the labor model in explaining stock returns with that from other traditional asset pricing models such as the CAPM and the Fama-French (1993) three factor model. In order to assess the importance of labor for asset pricing, we explicitly focus on specifications of the labor model and the baseline model with parsimonious representations of the technology. The simple specification of the marginal product of labor and capital and the quadratic adjustment cost function in both models allows us to keep a low number of parameters in each model. The baseline model has only two technological parameters (the share of capital parameter γ k and the capital adjustment cost parameter η k ). The quadratic adjustment cost specification of the labor model maintains this parsimonious framework by only adding one additional parameter, the labor adjustment cost η n. By minimizing the degrees of freedom available to match the data, this specification allows us to provide an economically meaningful comparison between the baseline model and the labor model, and hence for the importance of labor, not of different functional forms, for asset pricing. We provide estimation results across two different sets of moment conditions. In Section 4.1, we only use the moment conditions for average stock returns. Because these moment conditions have been studied before by Liu, Whited and Zhang (009) for the baseline model, they constitute a natural benchmark against which to compare the importance of labor in explaining the crosssection of stock returns. In Section 4., we augment the set of moment conditions studied in Liu Whited and Zhang by examining if the labor model and the baseline model can simultaneously explain the cross section of expected stock returns and stock valuation ratios. 4.1 Does Labor Help Explain the Cross-Section of Expected Stock Returns? Point Estimates Table, Panel A, reports the point estimates and overall model performance of both the baseline model and the labor model when we use the moment condition (13) to match the average stock returns of the four sets of portfolios separately or all the portfolios simultaneously. 13

15 For the labor model, the capital and labor share parameter estimate (γ k +γ n ) ranges from 0.63 to 0.89, excluding the results for the B/M portfolios, and is precisely estimated with a reasonable economic magnitude. Across B/M portfolios, the estimate is unreasonably large, 1.11, suggesting increasing returns to scale. Here however, this parameter is not precisly estimated, so we cannot reject the hypothesis that it is smaller than one. The estimate of the labor adjustment cost parameter η n ranges from 19.0 to 56.6 and is statisically significant when all assets are considered. The estimate of the capital adjustment cost parameter η k ranges from.63 to 5.8 if we exclude the results across the B/M portfolios. Across the B/M portfolios, the capital adjustment cost parameter is estimated to be considerably larger, 44.1, but is not statistically significant. Finally, the evidence implies that the firm s optimization problem has an interior solution because the positive estimates of both the capital and labor adjustment cost parameters mean that the adjustment cost function is increasing and convex in both investment and hiring. For the baseline model, the capital share parameter ranges from 0. to 0.35 when we exclude the results across the B/M portfolios. These values are consistent with the capital share values used in Rotemberg and Woodford (199) and estimeted in Liu, Whited and Zhang (009). Across the B/M portfolios, the capital share is estimated to be considerably larger, 0.76, but it is not statistically significant. The estimate of the capital adjustment cost parameter η n ranges from 1.3 to 11.8, if we exclude the B/M portfolios. As in the labor model, across the B/M portfolios, the capital adjustment cost parameter is estimated to be considerably larger, 53.9, but is not statistically significant. In order to interpret the magnitude of the estimated adjustment cost parameters, Table, Panel A reports the implied proportion of output that is lost due to capital and labor adjustment costs. ) These values are computed as η ) k (i j t /kj t /y j t for capital and η n (h j t /nj t /y j t for labor, and are evaluated at the sample mean of the investment rate, the hiring rate, and output. The estimated 14

16 magnitude of the capital and labor adjustment costs is reasonable across all test assets, except across the B/M portfolios. In the labor model, the fraction of output that is lost due to capital adjustment costs ranges from 1% to % if we exclude the B/M portfolios, and is 17% across B/M portfolios. The fraction of output that is lost due to labor adjustment costs ranges from 3% to 10%. For the baseline model, the fraction of output that is lost due to capital adjustment costs ranges from 1% to 5% if we exclude the B/M portfolios, and is 1% across B/M portfolios. Except for the results across B/M portfolios, these values are within the empirical estimates surveyed in Hamermesh and Pfann (1996), and discussed in Merz and Yashiv (007) Overall Model Performance Table, Panel A, reports two measures of overall performance: the average absolute pricing errors in percent per annum (a.a.p.e), and the χ test. Both the labor model and the baseline model perform reasonably well in accounting for the average returns of all portfolios. In the labor model, the a.a.p.e. ranges from 1.17% across the B/M portfolios to.08% when all portfolios are considered. For the baseline CAPM, the a.a.p.e are higher. Here, the a.a.p.e ranges from 1.47% across the B/M portfolios to.9% when all portfolios are considered. None of the two models is rejected by the χ test across any of the test assets considered here. The fit of the labor model compares favorably with the baseline model, especially when all portfolios are considered simulateneously. Here, adding labor to the baseline model decreases the a.a.p.e. from.9% to.08%, a significant difference of almost 1%. Across each individual portfolio sort, the difference in a.a.p.e. of the labor model and the baseline model is 0.6% across the I/A portfolios, 0.3% across the B/M portfolios, 0.8% across AG portfolios and 0.13% across LH portfolios. The difference of almost 1% in the annual a.a.p.e. of the labor model relative to the baseline model is impressive, given the already good fit of the baseline model in explaining average stock returns, as documented in Liu, Whited and Zhang (009). To put the results into perspective, Table 15

17 1, Panel A, reports the asset pricing test results of the standard CAPM and the Fama-French (1993) three factor model on these portfolios. To test the CAPM, we regress annual portfolio returns in excess of the risk-free rate on market excess returns. The risk-free rate, denoted r ft+1, is the annualized return on the one-month Treasury bill from Ibbotson Associates. The regression intercept (α CAPM ) measures the model error from the CAPM. To test the Fama-French model, we regress annual portfolio excess returns on annual returns of the market factor, a size factor, and a book-to-market factor (the factor returns data are from Kenneth French s Web site). The regression intercept (α FF ) measures the error of the Fama-French model. The annual a.a.p.e of the CAPM range from 4.46% across the LH portfolios, to 5.98% across the AG portfolios. The magnitude of these a.a.p.e are two to three times larger than the a.a.p.e reported for the labor model. The a.a.p.e. of the Fama-French model are smaller but they are still considerably higher than the a.a.p.e. of the labor model across all test assest: 3.9% versus 1.49% across the I/A portfolios, 1.46% versus 1.17% across the B/M portfolios, 4.04% versus % across the AG portfolios and 1.93% versus 1.47% across the LH portfolios Euler Equation Pricing Errors The a.a.p.e. and χ tests only indicate overall model performance. To provide a more complete picture, in Table 3, Panel A we report each individual portfolio pricing error, e S i, defined in equation (15), in which the levered weighted average of investment and hiring returns are constructed using the estimates from Table, Panel A. We also report the t-statistic (in square brackets) for each individual pricing errors, following Liu, Whited and Zhang (009). Across the I/A portfolios, the individual pricing errors for the labor model range from 3.39% to.79% and only one of the mean portfolio pricing errors is significant. In the baseline model, the mean pricing errors are larger and statistically significant for three portfolios. In particular, the high-minus-low I/A portfolio has a mean pricing error of 1.81% (t-stat = 0.86) in the labor model, considerably smaller than the pricing error of 3.78% (t-stat = 1.67) in the baseline model. As 16

18 reported in Table 1, the mean pricing errors of both models across these portfolios are considerably smaller than the mean pricing errors for the CAPM and Fama-French model. For example, the high-minus-low I/A portfolio has a mean pricing error of 14.7% (t-stat = 5.63) in the CAPM and a mean pricing error of 11.67% (t-stat = 3.63) in the Fama-French model. Across the B/M portfolios, none of mean portfolio pricing errors is significant, for both the labor model and the baseline model, and the performance of these two models compares favorably with that of the standard CAPM and the Fama-French model. According to Table 1, the high-minus-low B/M portfolio has a mean pricing error of 17.59% (t-stat = 4.08) in the CAPM and 5.85% (t-stat = 3.7) in the Fama-French model, which is considerably higher than the mean pricing error of 1.59% (t-stat = 0.7) in the labor model. Across the AG and the LH portfolios, none of the mean portfolio pricing errors are significant for the labor model. In contrast, in the baseline model, half of the mean portfolio pricing errors across AG portfolios are statistically significant, and three of the mean portfolio pricing errors are significant across the LH portfolios. Once again, the mean portfolio pricing errors in both models are considerably smaller than the mean portfolio pricing errors in both the CAPM and the Fama- French model. The high-minus-low AG portfolio has a large mean pricing error of 1.36% (t-stat = 6.36) in the CAPM and 3.14% (t-stat = 4.4) in the Fama-French model. Similarly, the high-minus-low LH portfolio has a mean pricing error of 11.45% (t-stat 4.43) in the CAPM and 10.70% (t-stat of -.6) in the Fama-French (1993) model. The mean pricing errors of these high-minus-low portfolios are considerably larger than those reported for the labor model,.69% (t-stat = 1) and 4.11% (t-stat = 1.53) for the high-minus-low AG and LH portfolio, respectively. Taken together, the results in this section show that, adding labor to the baseline model significantly decreases the magnitude of the mean portfolio pricing errors, especially across the I/A and the AG portfolios. In addition, the magnitude of the mean portfolio pricing errors is considerably smaller than that of standard asset pricing models such as the CAPM and the Fama- 17

19 French model. 4. Does Labor Help Jointly Explain the Cross-Section of Expected Stock Returns and Stock Valuation Ratios? 4..1 Point Estimates Table, Panel B, reports the point estimates and overall model performance of both the baseline model and the labor model, when we use the moment conditions (13) and (14) to simultaneously match expected stock returns and stock valuation ratios. For the labor model, the capital and labor share parameter estimate (γ k + γ n ) across all test assets ranges from 0.65 to 0.83, and is now always significant and with reasonable economic magnitude. The estimate of the labor adjustment cost parameter η n ranges from 1.08 to 50.46, and is significant when all portfolios are considered simulateneously. The estimate of the capital adjustment cost parameter η k ranges from.44 to 8.10 and is also statistically significant when all portfolios are included. Taken together, we cannot reject the hypothesis that labor and capital help for jointly explaining the cross-section of stock returns and stock valuation ratios. For the baseline model, the capital share parameter estimate ranges from 0.5 to 0.33, and is significant across all test assets. These values are again reasonably close to the capital share values used in Rotemberg and Woodford (199) and estimated in Liu, Whited and Zhang (009). The estimate of the capital adjustment cost parameter η k ranges from 4.58 to 11.54, and thus it is estimated to be considerably larger than in the labor model. In order to interpret the magnitude of the estimated adjustment cost parameters, Table, Panel B reports the reports the implied proportion of output that is lost due to capital and labor adjustment costs, computed as in Section The estimated magnitude of the capital and labor adjustment costs are now reasonable across all test assets, including across the B/M portfolios. In the labor model, the fraction of output that is lost due to capital adjustment costs ranges from 1% to 3%. The fraction of output that is lost due to labor adjustment costs ranges from % to 9%. For 18

20 the baseline model, the fraction of output that is lost due to capital adjustment costs ranges from % to 4%. As discussed in Section 4.1.1, these values are within the empirical estimates surveyed in Hamermesh and Pfann (1996), and Merz and Yashiv (007). Overall, the parameters estimates obtained here are more stable across test assets than when only the moment conditions for expected stock returns are used. In particular, the parameters estimates across the B/M portfolios are now in line with the parameters estimates obtained across the other test assets. Thus adding the stock valuation ratio moments helps for the identification of the structural parameters. 4.. Overall Model Performance Table, Panel B, reports three tests of overall model performance. χ (v) is the χ test that all the mean stock valuation ratios (v) errors are jointly zero, χ (r) is the χ test that all the mean stock returns (r) errors are jointly zero, and χ is the test that all the mean stock valuation ratios and stock returns errors are jointly zero. In addition, it reports the mean absolute valuation ratio moment errors in decimals (a.a.v.e.) as well as the average absolute pricing errors in annual percent (a.a.p.e.). Overall, adding labor to the baseline model decreases the annual a.a.p.e. by a value that range from 0.19% to 1.4%, depending on the test assets. As discussed in Section 4.1., this improvement is impressive given the already good performance of the baseline model in explaining the average stock returns of these portfolios, especially when compared with the performance of the standard CAPM and Fama-French model. When all test assets are considered, the labor model has an a.a.p.e of.14%, which is considerably lower than the 3.03% a.a.p.e of the baseline model. The improvement of the labor model over the baseline model is more pronounced across the I/A and the AG portfolios. Across the I/A portfolios, adding labor in the baseline model, reduces the a.a.p.e. from.9% to 1.5%. Similarly, across the AG portfolios, adding labor reduces the a.a.p.e. from.99% to 1.90%. Across 19

21 the B/M portfolios and the LH portfolios, the improvement of the labor model over the baseline model is more modest. Here, adding labor into the baseline model decreases the a.a.p.e from.4% to 1.94% across the B/M portfolios, and from 1.7% to 1.53% across the LH portfolios. Turning to the analysis of the stock valuation ratio errors, across all test assets, the mean valuation ratio errors in the labor model are all smaller than in the baseline model. In particular, across the I/A portfolios, the valuation ratio errors of the labor model are less than half the valuation errors of the baseline model (a.a.v.e. of 0.1 versus 0.09). When all test assets are considered jointly, the hypothesis that the mean stock valuation ratio errors are jointly zero is not rejected for both the labor and the baseline model. Both investment models however, cannot explain the large spread in the average valuation ratios of the B/M portfolios. Here, even though the hypothesis that the average valuation errors are jointly zero is not rejected, the size of the a.a.v.e. is large, approximately 0.6 in both models. As reported in Table 1, this value is twice the size of the 0.7 spread in the mean valuation ratios of the high and the low B/M portfolios. Finally, across the AG and the LH portfolios, the performance of both labor and the model is similar in terms of a.a.v.e Euler Equation Pricing and Valuation Errors Table 3, Panel B, reports the individual portfolio stock return error, e S i, defined in equation (15), in which levered weighted average investment and hiring returns are constructed using the estimates from Table, Panel B. Table 3, Panel C, reports the individual portfolio stock valuation ratio error, e V i, defined in equation (16), in which the portfolio stock valuation ratios are also constructed using the estimates from Table, Panel B. The t-statistics for each individual pricing and stock valuation errors are reported in square brackets. The annual individual return errors for the labor model range from 3.34% to.90% across the I/A portfolios, and none of mean portfolio pricing errors is significant. In the baseline model, the mean pricing errors are larger and statistically significant for three portfolios. The high-minus-low 0

22 I/A portfolio has a mean error of 1.4% (t-stat = 0.66) which is smaller than the pricing error of.46% (t-stat = 0.59) in the baseline model. As in the results reported in Section (estimation of the model on the expected stock return moments only), the mean pricing errors in the two models across these portfolios are still considerably smaller than the mean pricing errors in the standard CAPM and Fama-French model (see Table 1), despite the additional requirement to match the moments of the cross-section of the stock valuation ratios. Figure 1, Panels A and B, provides a visual description of the explanatory power for stock returns of both the labor model and the baseline model across the I/A portfolios. This figure plots the predicted versus realized returns implied by the parameters estimates reported in Table, Panel B. The straight line is the 45 line along which all assets should lie. The deviations from this line are the pricing errors, which provides the economic counterpart to the statistical analysis. In the figure, each digit number represents one portfolio (in addition L stands the lowest portfolios and H stands for the highest portfolio). In both models, the I/A portfolios lie relatively close to the 45 line. As in the data, the low I/A portfolios have higher returns than the high I/A portfolios, hence both models generate relatively small pricing errors. In the labor model, most portfolios lie closer to the 45 line than in the baseline model, and thus the labor model generates smaller average pricing errors. Across the B/M portfolios, the pricing errors of the labor model and the baseline model are not significant and are comparable in magnitude. The comparable fit of both models is clear in Figure, Panels A and B, which plots the predicted versus realized returns implied by the estimation of each model on these portfolios. In contrast with the estimation results using the moment conditions for returns only, the magnitude of the pricing error for the high-minus-low B/M portfolio is relatively large in both models, 5.44% in the labor model and 6.45% in the baseline model. Importantly however, for the labor model, these pricing errors are still smaller than in both the standard CAPM (17.59%) and the Fama-French model (5.85%), as reported in Table 1. 1

23 Across the AG portfolios, the pricing errors are in general smaller in the labor model than in the baseline investment model. Figure 3, Panels A and B provides a visual description of the fit of both models on these portfolios. Excluding the high portfolio, the fit of the labor model on these portfolios is impressive. According to Table 3, Panel B, none of the portfolio pricing errors is significant in the labor model, but three of the portfolio pricing errros are significant in the baseline model. For both models, the individual pricing errors are considerably smaller than those reported for the CAPM and the Fama-French model, as in the results reported in Section (estimation of the model on the expected return moments only). Finally, across the LH portfolios, the performance of both models is comparable, as it is clear from Figure 4, Panels A and B. In addition, according to Table 3, Panel B, only one of the portfolio pricing errors is significant in both models. The pricing error of the high-minus-low portfolios however, is smaller in the labor model than in the baseline model (.94% versus 4.7%), and considerably smaller than in the standard CAPM and the Fama-French model ( 11.45% and 10.7%, respectively). Turning to the analysis of the portfolio stock valuation ratio errors in the labor model, these errors range from 0.3 to 0.1 across I/A portfolios, and none of the valuation ratio errors is significant. In contrast, in the baseline model the valuation errors are larger (range from 0.4 to 0.09) and in general are significant. The better fit on the labor model in explaining the valuation ratios of these portfolios is clear in Figure 1, Panels C and D, which plots the predicted versus realized valuation ratios implied by the estimation of each model on these portfolios. The baseline model is unable to match not only the average level but also the spread in the valuation ratios of the I/A portfolios. In contrast, the labor model can match the average level of the valuation ratios of these portfolios, and is partially successful at capturing the spread in the valuation ratios. This result is consistent with the findings in Merz and Yashiv (007), who finds that adding labor adjustment costs into a standard production-based model with adjustment costs in capital inputs, substantially improves the fit of the model in matching the time series properties of aggregate stock

24 market prices. The valuation ratio errors across the B/M portfolios are large and significant for both the labor and the baseline model. As it is clear from Figure, Panels C and D, none of the two investment models is able to capture the large spread in the variation of the valuation ratios across the B/M portfolios. Finally, the valuation ratio errors across the AG and the LH portfolios are also comparable in magnitude for both the labor and the baseline model, as reported in Figures 3 and 4, Panels C and D. According to Table 3, Panel B, approximately half of the valuation errors are significant in both models. Taken together, the results in this section suggest that, when forced to simultaneously match both the cross section of average stock returns and stock valuation ratios, both the labor and the baseline model maintain the good explanatory power for stock returns. The pricing errors of the labor model are smaller that those reported for the baseline model, and considerably smaller than those reported for the standard CAPM and the Fama-French model. The fit of both models on the cross-section of valuation ratios is more modest. The labor model significantly outperforms the baseline model across the IA portfolios. Both models however, cannot fully capture the large spread in the valuation ratios across the B/M portfolios and, to a lesser degree, across the AG and LH portfolios Expected Returns Accounting As discussed in Section.4, the equality of stock returns and levered weighted average of investment and hiring returns in equation (1) suggest several economic forces driving the cross section of average stock returns. Here, we quantify the importance of each driver. We focus our analysis on the effect of the labor characteristics, namely the cross-sectional variation in the hiring rate (h t /n t ), the labor to capital ratio (n t+1 /k t+1 ), the growth rate in the hiring marginal n q ( n q t+1 ), the employee separation rate ( n δ t+1 ) and the wage rate (w t+1 ). Liu, Whited and Zhang (009) provide a comprehensive analysis of the effects of the remaining characteristics on average returns 3