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.
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 . According to the Taylor Theorem, G (Y, X) is approximated as
under the 4th-order expansion. The partial derivatives are computed as
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 is similar.
When the range for y is from to , define the following integrals as functions of x.
where r = 0, 1, 2, ….
Firstly, assuming ,
Firstly, solve for
Secondly, solve for
Thirdly, solve for
Substitute F3 (x) for (33) to obtain
The expressions for , , and can be written in a linear system,
where Λ’s denote the corresponding coefficients.
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
the ME conditional mean function and its response function can be traced out as well.
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.
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