Model Calibration

In order to derive the asset pricing implications from the model outlined in the previous section, we calibrate aggregate consumption and dividend growth dynamics to annual U.S. data from 1930-2013. Details on the data used for the calibration are provided in Appendix A.3. Our baseline calibration is reported in Table 1, along with alternative calibrations for a few special cases to be discussed in subsequent sections. The dividend leverage parameter is chosen to be 4.5, which is a bit higher than the value used by others, but the same value as used in Lettau et al. (2008).19 Furthermore, this choice of calibration results in a model implied value for the standard deviation of dividends that is within the 95% confidence interval from the data estimate over our sample as reported in Table 2. We also calibrate the conditional probabilitiesband din the transition matrix to 5% and 12.5% respectively.

19_{For example, Abel (1999) uses 2.74 while Bansal and Yaron (2004) use 3. However, unlike here, these}

Recall the regular (γ0, ψ0) and irregular (γelev, ψdepr) preference parameters. We define the

parameter kγ to be the elevated risk aversion factor where γelev =γ0kγ and kψ to be the

depressed EIS offset where ψdepr = ψ0 −kψ. The model calibration implies a constant

regular level of risk aversion of 7.25 for the vast majority of time periods. However, on rare occasion (b = 0.05) an exogenous shock elevates investor risk aversion by a factor of

kγ= 3.24 times the regular level. This calibration is consistent with the empirical evidence

presented in Guiso et al. (2013) who estimate that after the 2008 financial crisis, risk aver- sion increased 2.0-3.5 times the pre-crisis level.20 Mehra and Prescott (1985) argue that a value of 10 for risk aversion is the maximum feasible value, a rule of thumb often invoked in the asset pricing literature using the class of representative agent preferences. In our baseline calibration, in the majority of periods, we require a value for risk aversion of only 7.25. However, the elevated value for risk aversion isγelev=γ0kγ= 23.5, which is high but

still much lower than, for example, the risk aversion of about 80 implied by Campbell and Cochrane (1999) when consumption surplus is at its steady state (and in the hundreds for low-consumption surplus ratios, which correspond with“recessions”) and in the low-end of the range for this parameter reported in Table 4 of Melino and Yang (2003). Furthermore, risk aversion is rarely this high in our model, elevated by the plausible factor of 3.24 only once every 42 years on average and then reverting to its regular level in under four years on average. If preference parameters, such as risk aversion, actually vary over time then prior estimates of risk aversion from the data are estimates of the mean level of a random variable. In the bottom panel of Table 1 we report the model implied unconditional means of γt and ψt. Under this interpretation of previous acceptable benchmark values of risk

aversion being the mean of a random variable, our model requires E[γt] to be 8.60, which

is less than the plausible maximum benchmark of 10.

20

They estimate a change in risk aversion by a factor of 2 for the average investor in their data and 3.5 for the median investor.

We also calibrate the regular level of the EIS parameter to be 0.956. The “correct” value of the EIS is widely debated and there is an extensive empirical literature that attempts to estimate it with estimates ranging from very small (even negative) to larger than one.21 In our calibration, we did not set out with a particular value for this parameter in mind, however, our model matches more regularities of asset prices with the EIS parameter at just below one (E[ψt] = 0.9556), which implies a preference for early resolution of uncertainty

since 1/EISt,t+1 < γt in all states of our model. In the end, among other consumption

based asset pricing models in the literature, our calibration of ψt fits right between the

small values of around 0.1-0.3 (Campbell and Cochrane (1999) and Guvenen (2009)) and the larger values of 1.0-1.5 (Wachter (2013), Bansal and Yaron (2004) and Bansal et al. (2012)).

The calibrated level of the EIS parameter is 0.956 for the vast majority of years, how- ever, on rare occasions (d = 0.125) an exogenous shock depresses the EIS parameter by

kψ = 0.002. It may seem striking that our calibration only requires such a small movement

in the EIS parameter. Since we have little empirical guidance on plausible variation in the EIS, small movements are a conservative assumption. Moreover, given the difficulty in statistical estimation of the EIS in the literature under the assumption of constant EIS, the small movement in our calibration is very likely not to be rejected by statistical test. However, this magnitude is in the range of values that Kamstra et al. (2014) require in seasonal fluctuation of the EIS parameter. Also, the magnitude of kψ implies a 20 basis

point change in the sensitivity of consumption growth to a change in interest rates, which does not strike us as implausibly small. The probabilitybin the baseline calibration implies

21_{Havranek, Horvath, Irsova, and Rusnak (2013) provides a recent broad survey of estimates across mul-}

tiple studies and countries and show a wide variation in estimates. Campbell (1999) reports widely varying and often imprecise estimates. Hall (1988), Vissing-Jorgensen (2002) and Guren, Manoli, Weber, and Chetty (2011) estimate EIS to be small (around zero or less than one). While Hansen and Singleton (1982), Attana- sio and Weber (1989), Beaudry and Wincoop (1996), Gruber (2006) and Engegelhardt and Kumar (2009) estimate the EIS to be large (around one or greater than one). Guren et al. (2011) provides a good survey on the various estimates of the EIS and how these estimates vary depending on if they are micro or macro estimates.

that risk aversion transitions into an irregular state about once every 42 years, with an average duration of irregular risk aversion of about 3.8 years. Likewise, the probabilitydin the baseline calibration implies that the EIS parameter transitions into an irregular state about once every 18 years, with an average duration of about 3.8 years. If we assume the Great Depression and the Great Recession were periods of low consumption growth that corresponded with a state of fear shifting investor preferences, then our baseline calibration is roughly consistent with the frequency and duration of these states of the U.S. economy over the sample period 1930-2013.22

Before discussing the model’s ability to match the stylized facts of aggregate asset prices, it is important to note the number of degrees of freedom our model exhibits. Of all the parameters in Table 1, only the discount factor (δ), the regular values of risk aversion and the EIS parameter (γ0, ψ0), the depressed EIS parameter offset (kψ) and the conditional

probabilities (b, d) are not calibrated directly to data or from empirical evidence. However, the EIS parameter offset is very small, which would not obviously amplify the results of the model a priori. Furthermore, b and dare closely tied to the average levels of RA and EIS generated by the model, and those levels were targeted under the constraints that the relevant literature implies a plausible risk aversion be less that 10 and estimates of the EIS to be somewhere between 0 and 2. Hence, we only have six degrees of freedom in calibrating our model. Given this feature, coupled with the fact that we assume very simple consumption and dividend dynamics with a very coarse state space, it is actually quite surprising the model does well in producing as large a number of asset price features as it does, lending credibility to the model’s basic insights.

22_{This assumption is consistent with the empirical studies of Malmendier and Nagel (2011) and Guiso}