#### A. Calibration

Calibration of an economic model involves restricting some parameter values exogenously and setting others to replicate a benchmark data set as a model solution (e.g., Dawkins, Srinivasan, and Whalley (2001)). Once calibrated, the model can be used to assess the effects of an unobservable change in exogenous parameter values. The model solution provides predictions of the way in which the economy is likely to respond to the change, while the pre-change solution serves as the reference point.

Table I summarizes the key parameter values in the model. All model parameters are calibrated at the monthly frequency to be consistent with the empirical literature. I break down all the parameters into three groups. The first group includes parameters that can be restricted by prior empirical or quantitative studies: capital share α; depreciation δ; persistence ρ_{x} and conditional volatility σ_{x} of aggregate productivity; and inverse price elasticity of demand η. Because of the general consensus concerning their numerical values, these parameters provide essentially no degrees of freedom for generating the quantitative results.

Table I. ** Benchmark Parameter Values** This table lists the benchmark parameter values used to solve and simulate the model. I break all the parameters into three groups. Group I includes parameters whose values are restricted by prior empirical or quantitative studies: capital share, α; depreciation, δ; persistence of aggregate productivity, ρ_{x}; conditional volatility of aggregate productivity, σ_{x}; and inverse price elasticity of demand, η. Group II includes parameters in the pricing kernel, β, γ_{0}, and γ_{1}, which are tied down by matching the average Sharpe ratio and the mean and volatility of real interest rate. The final group of parameters is calibrated with only limited guidance from prior empirical studies. I start with a reasonable set of parameter values and conduct extensive sensitivity analysis in Tables III and IV. Group I | Group II | Group III |
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α | δ | ρ_{x} | σ_{x} | η | β | γ_{0} | γ_{1} | θ^{−}/θ^{+} | θ^{+} | ρ_{z} | σ_{z} | *f* |
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0.30 | 0.01 | 0.95^{1/3} | 0.007/3 | 0.50 | 0.994 | 50 | − 1000 | 10 | 15 | 0.97 | 0.10 | 0.0365 |

The capital share α is set to be 30%, similar to that in Kydland and Prescott (1982) and in Gomes (2001). The monthly rate of depreciation, δ, is set to be 0.01, which implies an annual rate of 12%, the empirical estimate of Cooper and Haltiwanger (2000). The persistence of the aggregate productivity process, ρ_{x}, is set to be 0.95^{1/3}= 0.983, and its conditional volatility, σ_{x}, 0.007/3 = 0.0023. With the AR(1) specification for *x*_{t} in (1), these monthly values correspond to 0.95 and 0.007 at the quarterly frequency, respectively, consistent with Cooley and Prescott (1995). Finally, I follow Caballero and Pindyck (1996) and set the inverse price elasticity of demand η to be 0.50.

The second group of parameters includes those in the pricing kernel: β, γ_{0}, and γ_{1}. These parameters can be tied down by aggregate return moments implied by the pricing kernel. The log pricing kernel in (4) and (5) implies that the real interest rate *R*_{ft} and the maximum Sharpe ratio *S*_{t} can be written as, respectively,

- (20)

and

- (21)

where

- (22)

- (23)

I thus choose β, γ_{0}, and γ_{1} to match (i) the average Sharpe ratio; (ii) the average real interest rate; and (iii) the volatility of real interest rate.^{11}

This procedure yields β= 0.994, γ_{0}= 50, and γ_{1}=−1000, and they deliver an average Sharpe ratio of 0.41, an average real interest rate of 2.2% per annum, and an annual volatility of real interest rate of 2.9%. These moments are very close to those in the data reported by Campbell and Cochrane (1999) and by Campbell, Lo, and MacKinlay (1997). As these parameters are pinned down tightly by the aggregate return moments, they provide no degrees of freedom in matching cross-sectional moments of returns, which are my focus here.

Importantly, a γ_{0} of 50 does not necessarily imply extreme risk aversion, nor does a γ_{1} of −1,000. Because the pricing kernel is exogenously specified in the model, the criterion of judging whether its parameters are representative of reality should be whether the aggregate return moments implied by the pricing kernel mimic those in the data. After all, I do not claim any credits in explaining time series predictability; my contribution is to endogenize cross-sectional predictability of returns, given time series predictability.

The calibration for the third group of parameters has only limited guidance from prior studies and I have certain degrees of freedom in choosing their values. There are five parameters in this group: (i) the adjustment cost coefficient, θ^{+}; (ii) the degree of asymmetry, θ^{−}/θ^{+}; (iii) the conditional volatility of idiosyncratic productivity, σ_{z}; (iv) the persistence of idiosyncratic productivity, ρ_{z}; and (v) the fixed cost of production, *f*. I first choose their benchmark values by using available studies and by matching key moments in the data. I then conduct extensive sensitivity analysis.

First, θ^{+} can be interpreted as the adjustment time of the capital stock given one unit change in marginal *q* (e.g., Shapiro (1986) and Hall (2001)). The first-order condition with respect to investment for the value-maximization problem says that θ^{+}· (*i*/*k*) =*q*− 1, where *q* is the shadow price of additional unit of capital. If *q* rises by one unit, the investment-capital ratio (*i*/*k*) will rise by 1/θ^{+}. To cumulate to a unit increase, the flow must continue at this level for θ^{+} periods.

The empirical investment literature has reported a certain range for this adjustment time parameter. Whited (1992) reports this parameter to be between 0.5 and 2 in annual frequency, depending on different empirical specifications. This range corresponds to an adjustment period lasting from 6 to 24 months. Another example is Shapiro (1986), who finds the adjustment time to be about eight calendar quarters or 24 months. I thus set the benchmark value of θ^{+} to be 15, which corresponds to the average empirical estimates, and conduct sensitivity analysis by varying θ^{+} from 5 to 25.

The empirical evidence on the degree of asymmetry, θ^{−}/θ^{+}, seems scarce. Here I simply follow Hall (2001) and set its benchmark value to be 10 (Table III contains comparative static experiments on this parameter).

Table III. ** Properties of Portfolios Sorted on Book-to-Market** This table reports summary statistics for HML and 10 book-to-market portfolios, including mean, *m*, volatility, σ, and market beta, β. Both the mean and the volatility are annualized. The average HML return (the value premium) is in annualized percent. Panel A reports results from historical data and benchmark model with asymmetry and countercyclical price of risk (θ^{−}/θ^{+}= 10 and γ_{1}=− 1000). Panel B reports results from two comparative static experiments. Model 1 has symmetric adjustment cost and constant price of risk (θ^{−}/θ^{+}= 1 and γ_{1}= 0), and Model 2 has asymmetry and constant price of risk (θ^{−}/θ^{+}= 10 and γ_{1}= 0). All the model moments are averaged across 100 artificial samples. All returns are simple returns. HML | Panel A: Data and Benchmark | Panel B: Comparative Statics |
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Data | Benchmark | Model 1 | Model 2 |
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*m* | β | σ | *m* | β | σ | *m* | β | σ | *m* | β | σ |
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**4.68** | 0.14 | 0.12 | **4.87** | 0.43 | 0.12 | **2.19** | 0.09 | 0.04 | **2.54** | 0.11 | 0.04 |
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Low | 0.11 | 1.01 | 0.20 | 0.09 | 0.85 | 0.23 | 0.08 | 0.95 | 0.30 | 0.08 | 0.94 | 0.30 |

2 | 0.12 | 0.98 | 0.19 | 0.10 | 0.92 | 0.24 | 0.09 | 0.97 | 0.31 | 0.09 | 0.97 | 0.31 |

3 | 0.12 | 0.95 | 0.19 | 0.10 | 0.95 | 0.25 | 0.09 | 0.99 | 0.31 | 0.09 | 0.98 | 0.31 |

4 | 0.11 | 1.06 | 0.21 | 0.11 | 0.98 | 0.26 | 0.09 | 1.00 | 0.32 | 0.10 | 0.99 | 0.31 |

5 | 0.13 | 0.98 | 0.20 | 0.11 | 1.01 | 0.27 | 0.10 | 1.00 | 0.32 | 0.10 | 1.00 | 0.32 |

6 | 0.13 | 1.07 | 0.22 | 0.12 | 1.04 | 0.28 | 0.10 | 1.01 | 0.32 | 0.10 | 1.01 | 0.32 |

7 | 0.14 | 1.13 | 0.24 | 0.12 | 1.08 | 0.28 | 0.10 | 1.02 | 0.32 | 0.10 | 1.02 | 0.32 |

8 | 0.15 | 1.14 | 0.24 | 0.12 | 1.12 | 0.30 | 0.10 | 1.03 | 0.33 | 0.11 | 1.04 | 0.33 |

9 | 0.17 | 1.31 | 0.29 | 0.13 | 1.18 | 0.31 | 0.11 | 1.04 | 0.33 | 0.11 | 1.05 | 0.33 |

High | 0.17 | 1.42 | 0.33 | 0.15 | 1.36 | 0.36 | 0.11 | 1.07 | 0.34 | 0.12 | 1.08 | 0.34 |

To calibrate parameters ρ_{z} and σ_{z}, I follow Gomes (2001) and Gomes et al. (2003) and restrict these two parameters using their implications on the degree of dispersion in the cross-sectional distribution of firms. One direct measure of the dispersion is the cross-sectional volatility of individual stock returns. Moreover, since disinvestment in recessions is intimately linked to the value premium, as argued in Section II.C below, it is important for the model to match the average rate of disinvestment as well.

These goals are accomplished by setting ρ_{z}= 0.97 and σ_{z}= 0.10. These values imply an average annual cross-sectional volatility of individual stock returns of 28.6%, approximately the average of 25% reported by Campbell et al. (2001) and 32% reported by Vuolteenaho (2001). Furthermore, the average annual rate of disinvestment is 0.014, close to 0.02 in the data reported by Abel and Eberly (2001).

The value of σ_{z} is also in line with the limited empirical evidence. Pástor and Veronesi (2003) show that the average volatility of firm-level profitability has risen from 10% per year in the early 1960s to about 45% in the late 1990s.^{12} The calibrated conditional volatility of firm-level productivity is 10% per month, corresponding to 35% per year, which seems reasonable given the range estimated by Pástor and Veronesi.

The unconditional volatility of idiosyncratic productivity is about 32 times that of aggregate productivity. Such a high idiosyncratic shock is necessary to generate a reasonable amount of dispersion in firm characteristics within the model. However, even with such a high firm-level shock, firm value and investment rate are much more sensitive to changes in aggregate productivity *x*_{t} than to changes in idiosyncratic productivity *z*_{t}.^{13} The reason is that *x*_{t} affects the stochastic discount factor, while *z*_{t} does not; shocks at the firm-level are mainly cash flow shocks that can be integrated out, while shocks at the aggregate level consist primarily of discount rate shocks, consistent with Vuolteenaho (2001).

Finally, I am left with the fixed cost of production, *f*. Since *f* deducts the firm's profit given in (7), it has a direct impact on the market value of the firm. I thus calibrate *f* to be 0.0365 such that the average book-to-market ratio in the economy is 0.54, which matches approximately that in the data, 0.67, reported by Pontiff and Schall (1999).

Table II reports the set of key moments generated using the benchmark parameters. I simulate 100 artificial panels each with 5,000 firms and 900 months. I then compute the return and quantity moments for each sample and report the cross-sample averages in Table II. The corresponding moments in the data are also reported for comparison.

Table II. ** Key Moments under the Benchmark Parametrization** This table reports a set of key moments generated under the benchmark parameters reported in Table I. The data source for the average Sharpe ratio is the postwar sample of Campbell and Cochrane (1999). The moments for the real interest rate are from Campbell et al. (1997). The data moments for the industry returns are computed using the 5-, 10-, 30-, and 48-industry portfolios in Fama and French (1997), available from Kenneth French's web site. The numbers of the average volatility of individual stock return in the data are from Campbell et al. (2001) and Vuolteenaho (2001). The data source for the moments of book-to-market is Pontiff and Schall (1999), and the annual average rates of investment and disinvestment are from Abel and Eberly (2001). Moments | Model | Data |
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Average annual Sharpe ratio | 0.41 | 0.43 |

Average annual real interest rate | 0.022 | 0.018 |

Annual volatility of real interest rate | 0.029 | 0.030 |

Average annual value-weighted industry return | 0.13 | 0.12–0.14 |

Annual volatility of value-weighted industry return | 0.27 | 0.23–0.28 |

Average volatility of individual stock return | 0.286 | 0.25–0.32 |

Average industry book-to-market ratio | 0.54 | 0.67 |

Volatility of industry book-to-market ratio | 0.24 | 0.23 |

Annual average rate of investment | 0.135 | 0.15 |

Annual average rate of disinvestment | 0.014 | 0.02 |

Table II suggests that the model does a reasonable job of matching these return and quantity moments. Importantly, the fit seems reasonable not only for the moments that serve as immediate targets of calibration, but also for other moments. The mean and volatility of industry return are comparable to those computed using the industry portfolios of Fama and French (1997). The volatility of aggregate book-to-market ratio is 0.24, close to that of 0.23 reported by Pontiff and Schall (1999). The average rate of investment is 0.135 in the model, close to 0.15 in the data reported by Abel and Eberly (2001). In sum, the calibrated parameter values seem reasonably representative of the reality.

#### B. Empirical Predictions

I now investigate the empirical predictions of the model concerning the cross section of returns. I show that (i) the benchmark model with asymmetry and a countercyclical price of risk is capable of generating a value premium similar to that in the data. And (ii) without these two features, an alternative parameter set does not exist that can produce the correct magnitude of the value premium. Therefore, at least in the model, asymmetry and countercyclical price of risk are necessary driving forces of the value premium.

Table III reports summary statistics, including means, volatilities, and unconditional betas for portfolio HML and for 10 portfolios sorted on book-to-market, using both the historical data and 10 artificial data simulated in the model.^{14} The book value of a firm in the model is identified as its capital stock, and the market value is defined as the *ex* dividend stock price, as in footnote 10. I follow Fama and French (1992, 1993) in constructing HML and 10 book-to-market portfolios for each simulated panel. I repeat the entire simulation 100 times and report the cross-simulation averages of the summary statistics in Table III. From Panel A, the benchmark model is able to generate a positive relation between book-to-market and average returns. The benchmark model generates a reliable value premium, measured as the average HML return, which is quantitatively similar to that in the data.

To evaluate the role of asymmetry and the countercyclical price of risk, I conduct comparative static experiments in Panel B of Table III by varying two key parameters governing the degree of asymmetry, θ^{−}/θ^{+}, and the time-variation of the log pricing kernel, γ_{1}. Two cases are considered: Model 1 has symmetric adjustment cost and the constant price of risk (θ^{−}/θ^{+}= 1 and γ_{1}= 0) and Model 2 has asymmetry and constant price of risk (θ^{−}/θ^{+}= 10 and γ_{1}= 0). All other parameters remain the same as in the benchmark model.

Panel B of Table III shows that, without asymmetry or time-varying price of risk, Model 1 displays a small amount of the value premium. Introducing asymmetry in Model 2 increases the amount somewhat, but it is still lower than that in the benchmark model. In short, asymmetry and the time-varying price of risk seem indispensable for generating the value premium in the benchmark model.

However, the importance of these features established in Table III is conditional on the benchmark calibration of Model 1. It is possible that even without these two features, an alternative parameter set may exist in Model 1 that will produce the correct magnitude for the value premium. I thus conduct extensive sensitivity analysis on Model 1 by varying its parameter values from the benchmark calibration.

Panels A–H of Table IV report the results from the following eight comparative static experiments on Model 1: Low Volatility (σ_{z}= 0.08, Panel A); High Volatility (σ_{z}= 0.12, Panel B); Fast Adjustment (θ^{+}= 5, Panel C); Slow Adjustment (θ^{+}= 25, Panel D); Low Fixed Cost (*f*= 0.0345, Panel E); High Fixed Cost (*f*= 0.0385, Panel F); Low Persistence (ρ_{z}= 0.95, Panel G); and High Persistence (ρ_{z}= 0.98, Panel H). These experiments cover a wide range of empirically plausible parameter values. A conditional volatility of 12% per month for the idiosyncratic productivity corresponds to 42% per year, close to the upper bound of 45% estimated by Pástor and Veronesi (2003). As argued in Section II.A, the two alternative values of θ^{+} cover the range of its empirical estimates. The two values of fixed cost of production imply a wide range of industry book-to-market, from 0.29 to 9.58. Finally, a persistence level of 0.98 for the idiosyncratic productivity is close to that of the aggregate productivity, and is likely to be an upper bound.^{15}

Table IV. ** The Performance of Model 1 (θ**^{−}/θ^{+}= 1 and γ_{1}= 0) under Alternative Parameter Values This table reports summary statistics for HML and 10 book-to-market portfolios, including annualized mean, *m*, and volatility, σ, and market beta, β, generated from Model 1 without asymmetry and countercyclical price of risk. The average HML returns are in annualized percent. Nine alternative parameter values are considered: Low Volatility (σ_{z}= 0.08); High Volatility (σ_{z}= 0.12); Fast Adjustment (θ^{+}= 5); Slow Adjustment (θ^{+}= 25); Low Fixed Cost (*f*= 0.0345); High Fixed Cost ( *f*= 0.0385); Low Persistence (ρ_{z}= 0.95); High Persistence (ρ_{z}= 0.98); and High Volatility, Slow Adjustment, High Fixed Cost, and High Persistence (Panel I). All moments are averaged across 100 artificial samples. All returns are simple returns. HML | Panel A. Low Volatility | Panel B. High Volatility | Panel C. Fast Adjustment | Panel D. Slow Adjustment |
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*m* | β | σ | *m* | β | σ | *m* | β | σ | *m* | β | σ |
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**1.78** | 0.07 | 0.03 | **2.28** | 0.10 | 0.04 | **1.57** | 0.07 | 0.04 | **2.31** | 0.08 | 0.03 |
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Low | 0.08 | 0.95 | 0.30 | 0.08 | 0.94 | 0.29 | 0.09 | 0.96 | 0.30 | 0.07 | 0.95 | 0.29 |

2 | 0.09 | 0.98 | 0.31 | 0.09 | 0.97 | 0.30 | 0.10 | 0.98 | 0.31 | 0.08 | 0.98 | 0.30 |

3 | 0.09 | 0.99 | 0.31 | 0.10 | 0.99 | 0.31 | 0.10 | 0.99 | 0.31 | 0.09 | 0.99 | 0.31 |

4 | 0.09 | 1.00 | 0.31 | 0.10 | 1.00 | 0.31 | 0.10 | 0.99 | 0.32 | 0.09 | 1.00 | 0.31 |

5 | 0.10 | 1.00 | 0.31 | 0.10 | 1.01 | 0.31 | 0.10 | 1.00 | 0.32 | 0.09 | 1.00 | 0.31 |

6 | 0.10 | 1.01 | 0.32 | 0.10 | 1.02 | 0.32 | 0.10 | 1.01 | 0.32 | 0.10 | 1.01 | 0.32 |

7 | 0.10 | 1.02 | 0.32 | 0.11 | 1.02 | 0.32 | 0.11 | 1.02 | 0.32 | 0.10 | 1.02 | 0.32 |

8 | 0.10 | 1.02 | 0.32 | 0.11 | 1.04 | 0.32 | 0.11 | 1.02 | 0.33 | 0.10 | 1.03 | 0.32 |

9 | 0.10 | 1.03 | 0.32 | 0.11 | 1.05 | 0.33 | 0.11 | 1.04 | 0.33 | 0.11 | 1.04 | 0.32 |

High | 0.11 | 1.05 | 0.33 | 0.12 | 1.08 | 0.34 | 0.12 | 1.07 | 0.34 | 0.11 | 1.06 | 0.33 |

HML | Panel E. Low Fixed Cost | Panel F. High Fixed Cost | Panel G. Low Persistence | Panel H. High Persistence | Panel I. |
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*m* | β | σ | *m* | β | σ | *m* | β | σ | *m* | β | σ | *m* | β | σ |
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**1.89** | 0.07 | 0.03 | **2.34** | 0.12 | 0.05 | **1.88** | 0.07 | 0.03 | **2.63** | 0.12 | 0.05 | **3.13** | 0.12 | 0.05 |
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Low | 0.08 | 0.95 | 0.30 | 0.09 | 0.93 | 0.30 | 0.09 | 0.95 | 0.30 | 0.07 | 0.94 | 0.29 | 0.05 | 0.93 | 0.28 |

2 | 0.09 | 0.98 | 0.31 | 0.09 | 0.97 | 0.31 | 0.09 | 0.98 | 0.30 | 0.08 | 0.97 | 0.30 | 0.07 | 0.97 | 0.29 |

3 | 0.10 | 0.99 | 0.31 | 0.10 | 0.98 | 0.31 | 0.10 | 0.99 | 0.31 | 0.09 | 0.98 | 0.31 | 0.07 | 0.98 | 0.30 |

4 | 0.10 | 1.00 | 0.31 | 0.10 | 0.99 | 0.32 | 0.10 | 1.00 | 0.31 | 0.09 | 0.99 | 0.31 | 0.08 | 1.00 | 0.30 |

5 | 0.10 | 1.00 | 0.31 | 0.10 | 1.00 | 0.32 | 0.10 | 1.00 | 0.31 | 0.09 | 1.00 | 0.31 | 0.08 | 1.01 | 0.31 |

6 | 0.10 | 1.01 | 0.32 | 0.11 | 1.01 | 0.32 | 0.10 | 1.01 | 0.31 | 0.09 | 1.01 | 0.32 | 0.08 | 1.02 | 0.31 |

7 | 0.10 | 1.02 | 0.32 | 0.11 | 1.02 | 0.33 | 0.10 | 1.02 | 0.32 | 0.10 | 1.03 | 0.32 | 0.09 | 1.03 | 0.31 |

8 | 0.11 | 1.02 | 0.32 | 0.11 | 1.04 | 0.33 | 0.11 | 1.02 | 0.32 | 0.10 | 1.04 | 0.32 | 0.09 | 1.04 | 0.32 |

9 | 0.11 | 1.03 | 0.32 | 0.11 | 1.05 | 0.33 | 0.11 | 1.03 | 0.32 | 0.10 | 1.06 | 0.33 | 0.10 | 1.06 | 0.32 |

High | 0.12 | 1.05 | 0.33 | 0.12 | 1.09 | 0.35 | 0.11 | 1.05 | 0.33 | 0.11 | 1.11 | 0.35 | 0.11 | 1.10 | 0.33 |

Importantly, Table IV shows that the amount of value premium generated from the eight alternative parameter sets of Model 1 is uniformly much lower than that in the data and that in the benchmark model. The table also indicates that the magnitude of the value premium increases with the persistence and conditional volatility of idiosyncratic productivity, the adjustment time parameter, and the fixed cost of production.^{16} A natural question is then whether Model 1 can generate the correct magnitude of the value premium by combining all the extreme parameter values used in Panels B, D, F, and H. Panel I in Table IV reports that this is not true. The value premium generated from this parameter set is still lower than that in the data by 1.5% per annum.

In sum, the simulation results indicate that (i) an alternative parameter set does not exist that will produce the correct magnitude for the value premium in Model 1 without asymmetry and the countercyclical price of risk. And (ii) once these two ingredients are incorporated, the benchmark model is able to generate a value premium consistent with the data. I conclude that, at least in the model, asymmetry and the countercyclical price of risk are necessary driving forces of the value premium.