Published by De Gruyter August 31, 2020

# Maximum Entropy Analysis of Consumption-based Capital Asset Pricing Model and Volatility

• Tae-Hwy Lee , Millie Yi Mao and Aman Ullah

## Abstract

Based on the maximum entropy (ME) method, we introduce an information theoretic approach to estimating conditional moment functions with incorporating a theoretical constraint implied from the consumption-based capital asset pricing model (CCAPM). Using the ME conditional mean/variance functions obtained from the ME density, we analyze the relationship between asset returns and consumption growth under the theoretical constraint of the CCAPM. We evaluate the predictability of asset return using consumption growth through in-sample estimation and out-of-sample prediction in the ME mean regression function. We also examine the ME variance regression function for the asset return volatility as a function of the consumption growth. Our findings suggest that incorporating the CCAPM constraint can capture the nonlinear predictability of asset returns in mean especially in tails, and that the consumption growth has an effect on reducing stock return volatility, indicating the counter-cyclical variation of stock market volatility.

JEL Classification: C1; C5; G1

Corresponding author: Tae-Hwy Lee, Department of Economics, University of California, Riverside, CA, 92521, USA, E-mail:

## A Appendix

### A.1 4th-Order Taylor Expansion of Theoretical Constraint

In this subsection, we show the mathematical derivation of the 4th-order Taylor expansion of consumption Euler equation in (10) to obtain (11). After we drop the subscripts of X and Y, we define G(Y,X)βXαY1. According to the Taylor Theorem, G (Y, X) is approximated as

G(Y,X)G(Y0,X0)+GY(YY0)+GX(XX0)+2GY2(YY0)22+2GX2(XX0)22+2GYX(YY0)(XX0)+3GY3(YY0)33!+3GY2X(YY0)22(XX0)+3GYX2(YY0)(XX0)22+3GX3(XX0)33!+4GY4(YY0)44!+4GY3X(YY0)33!(XX0)+4GY2X2(YY0)22(XX0)22+4GYX3(YY0)(XX0)33!+4GX4(XX0)44!,

under the 4th-order expansion. The partial derivatives are computed as

GY=βXα,GX=βαXα1Y,2GY2=0,2GX2=βα(α+1)Xα2Y,2GYX=βαXα1,3GY3=3GY2X=0,3GYX2=βα(α+1)Xα2,3GX3=βα(α+1)(α+2)Xα3Y,4GY4=4GY3X=4GY2X2=0,4GYX3=βα(α+1)(α+2)Xα3,4GX4=βα(α+1)(α+2)(α+3)Xα4Y.

All the above partial derivatives are evaluated at (X0, Y0). After rearranging all the terms, (11) is obtained.

### A.2 Recursive Integration

In this subsection, we explain the mathematical details of the recursive integration method in subsection 2.3 with indefinite range. When the integral range of y is definite from a to b, the procedure to compute m (x) and β(x) is similar.

When the range for y is from to +, define the following integrals as functions of x.

Fr(x)+yrexp[λ20y2+λ10(x)y]dy,

where r = 0, 1, 2, ….

F0(x)+exp[λ20y2+λ10(x)y]dy,F1(x)+yexp[λ20y2+λ10(x)y]dy,F2(x)+y2exp[λ20y2+λ10(x)y]dy.

Firstly, assuming λ20>0,

0=+d exp[λ20y2+λ10(x)y]=+(2λ20yλ10(x))e[λ20y2+λ10(x)y]dy=2λ20F1(x)λ10(x)F0(x).

Secondly,

F0(x)=+exp[λ20y2+λ10(x)y]dy=ye[λ20y2+λ10(x)y]|++yde[λ20y2+λ10(x)y]=+(2λ20y2+λ10(x)y)e[λ20y2+λ10(x)y]dy=2λ20F2(x)+λ10(x)F1(x).

Thirdly,

F1(x)=+yexp[λ20y2+λ10(x)y]dy=+exp[λ20y2+λ10(x)y]d(12y2)=12y2e[λ20y2+λ10(x)y]|++12y2de[λ20y2+λ10(x)y]=+12y2(2λ20y+λ10(x))e[λ20y2+λ10(x)y]dy=λ20F3(x)+12λ10(x)F2(x).

Thus,

(33)F3(x)=1λ20F1(x)λ10(x)2λ20F2(x).

Define

F0(x)dF0(x)dx,F1(x)dF1(x)dx,F2(x)dF2(x)dx,andλ10(x)dλ10(x)dx.

Firstly, solve for F0(x)

F0(x)ddx+exp[λ20y2+λ10(x)y]dy=+ddxexp[λ20y2+λ10(x)y]dy=+λ10(x)ye[λ20y2+λ10(x)y]dy=λ10(x)F1(x).

Secondly, solve for F1(x)

F1(x)ddx+yexp[λ20y2+λ10(x)y]dy=+ddxyexp[λ20y2+λ10(x)y]dy=+λ10(x)y2e[λ20y2+λ10(x)y]dy=λ10(x)F2(x).

Thirdly, solve for F2(x)

F2(x)ddx+y2exp[λ20y2+λ10(x)y]dy=+ddxy2exp[λ20y2+λ10(x)y]dy=+λ10(x)y3e[λ20y2+λ10(x)y]dy=λ10(x)F3(x).

Substitute F3 (x) for (33) to obtain

F2(x)=λ10(x)λ10(x)2λ20F2(x)λ10(x)λ20F1(x).

The expressions for F0(x), F1(x), and F2(x) can be written in a linear system,

F0(x)=Λ01(x)F1(x),F1(x)=Λ12(x)F2(x),F2(x)=Λ21(x)F1(x)+Λ22(x)F2(x),

where Λ’s denote the corresponding coefficients.

Since

F0(x0+h)F0(x0)+F0(x)h,F1(x0+h)F1(x0)+F1(x)h,F2(x0+h)F2(x0)+F2(x)h,

for a given initial value x0 and a small increment h, the functions of x, F0 (x), F1 (x) and F2 (x) can be traced out. Given that

m(x)=F1(x)F0(x),
β(x)=F1(x)F0(x)F1(x)F0(x)F02(x),

the ME conditional mean function and its response function can be traced out as well.

## Acknowledgments

The authors are thankful to Raffaella Giacomini (the editor) and two anonymous referees for many valuable comments, to Amos Golan, Claudio Morana, and Liangjun Su for discussions on the subject matter of this paper, and to the seminar participants at the World Finance & Banking Symposium, New Delhi.

## References

Bera, A. K., and Y. Bilias. 2002. “The MM, ME, ML, EL, EF and GMM Approaches to Estimation: A Synthesis.” Journal of Econometrics 107: 51–86, https://doi.org/10.1016/s0304-4076(01)00113-0.Search in Google Scholar

Campbell, J. Y., and J. H. Cochrane. 1999. “By Force of Habit: A Consumption Based Explanation of Aggregate Stock Market Behavior.” Journal of Political Economy 107: 205–51, https://doi.org/10.1086/250059.Search in Google Scholar

Chib, S., and E. Greenberg. 1995. “Understanding the Metropolis-Hastings Algorithm.” The American Statistician 49: 327–35, https://doi.org/10.1080/00031305.1995.10476177.Search in Google Scholar

Cochrane, J. 1991. “Production-Based Asset Pricing and the Link Between Stock Returns and Economic Fluctuations.” The Journal of Finance 46: 209–37, https://doi.org/10.1111/j.1540-6261.1991.tb03750.x.Search in Google Scholar

Cochrane, J. 2000. Asset Pricing. Princeton: Princeton University Press.Search in Google Scholar

Giacomini, R., and G. Ragusa. 2014. “Theory-Coherent Forecasting.” Journal of Econometrics 182: 145–55, https://doi.org/10.1016/j.jeconom.2014.04.014.Search in Google Scholar

Giacomini, R., and H. White. 2006. “Tests of Conditional Predictive Ability.” Econometrica 74: 1545–78, https://doi.org/10.1111/j.1468-0262.2006.00718.x.Search in Google Scholar

Golan, A.. 2017. Foundations of Info-Metrics. New York: Oxford University Press.10.1093/oso/9780199349524.001.0001Search in Google Scholar

Golan, A., G. Judge, and D. Miller. 1996. Maximum Entropy Econometrics: Robust Estimation with Limited Data. New York: John Wiley & Sons.Search in Google Scholar

Hahn, J., G. Kuersteiner, and M. Mazzocco. 2019. “Estimation with Aggregate Shocks.” The Review of Economic Studies 87 (3): 1365–98, https://doi.org/10.1093/restud/rdz016.Search in Google Scholar

Hansen, L. P., and K. J. Singleton. 1983. “Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns.” Journal of Political Economy 91: 249–65, https://doi.org/10.1086/261141.Search in Google Scholar

Imbens, G. W., R. H. Spady, and P. Johnson. 1998. “Information Theoretic Approaches to Inference in Moment Condition Models.” Econometrica 66: 333–57, https://doi.org/10.2307/2998561.Search in Google Scholar

Kitamura, Y., and M. Stutzer. 1997. “An Information-Theoretic Alternative to Generalized Method of Moments Estimation.” Econometrica 65: 861–74, https://doi.org/10.2307/2171942.Search in Google Scholar

Kitamura, Y., G. Tripathi, and H. Ahn. 2004. “Empirical Likelihood-Based Inference in Conditional Moment Restriction Models.” Econometrica 72: 1667–714, https://doi.org/10.1111/j.1468-0262.2004.00550.x.Search in Google Scholar

Komunjer, I., and G. Ragusa. 2016. “Existence and Characterization of Conditional Density Projections.” Econometric Theory 32: 947–87, https://doi.org/10.1017/s0266466615000158.Search in Google Scholar

Kullback, S., and R.A. Leibler. 1951. “On Information and Sufficiency.” The Annals of Mathematical Statistics 22: 79–86, https://doi.org/10.1214/aoms/1177729694.Search in Google Scholar

Mao, Y., and A. Ullah. 2020. “Information Theoretic Estimation of Econometric Functions.” In Forthcoming in Advances in Info-Metrics: Information and Information Processing across Disciplines, edited by M. Chen, J. M. Dunn, A. Golan, and A. Ullah. Oxford University Press.10.1093/oso/9780190636685.003.0019Search in Google Scholar

Pagan, A., and A. Ullah. 1999. Nonparametric Econometrics Cambridge: Cambridge University Press.10.1017/CBO9780511612503Search in Google Scholar

Robertson, J. C., E. W. Tallman, and C. H. Whiteman. 2005. “Forecasting Using Relative Entropy.” Journal of Money, Credit, and Banking 37: 383–401, https://doi.org/10.1353/mcb.2005.0034.Search in Google Scholar

Ullah, A. 1996. “Entropy, Divergence and Distance Measures with Econometric Applications.” Journal of Statistical Planning and Inference 49: 137–62, https://doi.org/10.1016/0378-3758(95)00034-8.Search in Google Scholar

Wu, X. 2003. “Calculation of Maximum Entropy Densities with Application to Income Distribution.” Journal of Econometrics 115: 347–54, https://doi.org/10.1016/s0304-4076(03)00114-3.Search in Google Scholar

Wu, X. 2010. “Exponential Series Estimator of Multivariate Densities.” Journal of Econometrics 156: 354–66, https://doi.org/10.1016/j.jeconom.2009.11.005.Search in Google Scholar