Solving Euler Equations via Two-Stage Nonparametric Penalized Splines

1 Dec 2020


Liyuan Cui, Yongmiao Hong, Yingxing Li

Published in Journal of Econometrics, 2020

Euler equations have been widely adopted as a main vehicle in finance and macroeconomics to investigate the connection between agent preferences, asset prices, and economic fundamentals. However, solving models of this class is a widely acknowledged challenge since most nonlinear rational expectation models do not allow for analytic solutions.  Significant efforts, mainly through numerical approximation techniques, have been made to solve general equilibrium asset pricing models. Traditionally, this is often done by working on the conditional population moments implied by Euler equations, and hence approximating policy functions by matching means and variances for state variables and policy functions’ coefficients under some pre-specified distributional and functional form assumptions. However, there is no assurance that these distributional and functional form assumptions made for computational convenience will capture the true dynamics of the underlying processes. 

This study by Dr Liyuan Cui, Assistant Professor, Department of Economics and Finance and co-authors proposes a novel estimation-based approach to solving asset pricing models for both stationary and time-varying observations, in which structural changes of various types may occur during the sampling period. The proposed method is robust to misspecification errors while inheriting a closed-form solution. By representing the Euler equation, implied by asset pricing models, into a well-posed integral equation of the second kind, this paper proposes a penalized two-stage nonparametric estimation method and establish its optimal convergence under mild conditions. This newly designed penalized splines regression also distinguishes itself in the nonparametric literature by weakening the impact of the spline setting and instead letting the penalty play the key role in smoothing. Through empirical analysis on return predictability, this paper docuemnts that higher implied dividend yields predict lower future cash flows, but predict higher future interest rates at short horizons. Moreover, the implied and the observed dividend yields have opposite impacts on cash flow predictions, which may indicate that they represent different sets of information. 

This work opens several avenues for future research. Although the current method is designed to solve asset pricing models with time-separable preferences, it may be extended to recursive preference models. With the use of iteration, the proposed methodology may enable estimation of multiple unknown policy functions without solution misspecification and accumulative approximation errors. Cui and co-authors expect that their approach will provide more reliable and accurate impulse functions for economic shocks, thus facilitating the formation of effective policies in the real economy.