Nonparametric Stochastic Discount Factor Decomposition

Abstract

We assume that the state process X = {X t : t ∈ T } is either beta-mixing or rho-mixing. The beta-mixing coefficient between two σ-algebras A and B iswith the supremum taken over all A-measurable finite partitions {A i } i∈I and Bmeasurable finite partitions {B j } j∈J . The beta-mixing coefficients of X are defined asWe say that X is exponentially beta-mixing if β q ≤ Ce −cq for some C c > 0. The rho-mixing coefficients of X are defined asWe say that X is exponentially rho-mixing if ρ q ≤ e −cq for some c > 0. We use the sequence ξ k = sup x G −1/2 b k (x) to bound convergence rates. When X has bounded rectangular support and Q has a density that is bounded away from 0 and ∞, ξ k is known to be O( √ k) for (tensor-product) spline, cosine, and certain wavelet bases and O(k) for (tensor-product) polynomial series (see, e.g., Newey (1997), Chen andChristensen (2015)). It is also possible to derive alternative sufficient conditions in terms of higher moments of) by extending arguments in Hansen (2015) to accommodate weakly dependent data and asymmetric matrices. Timothy M. Christensen:

(Almost) Model‐Free Recovery

“…Bansal and Lehmann (1997), Alvarez and Jermann (2005), and Backus, Chernov, and Zin (2014) obtain a direct bound on the expected log return of any traded asset using the entropy of the pricing 4 Chabi-Yo, Bakshi, and Gao (2015) test the path-dependence assumption in the bond market, while Qin and Linetsky (2017) and Qin and Linetsky (2016) give extended recovery theorems to general recurrent Markovian processes and general semimartingale state dynamics, respectively. Christensen (2017) introduces a general nonparametric method for estimating the long-term factorization of the pricing kernel in dynamic Markov environments. kernel.…”

Necessary and Sufficient Conditions for Existence and Uniqueness of Recursive Utilities

“…Action of K on the corresponding eigenfunction of the valuation operator is then representative of the long run dynamics induced by K on positive functions. These objects-the principal eigenpairs of valuation operators associated with future cash and utility payoffs-have increasingly been used to understand long run risks and long run values in macroeconomic and financial applications by inducing a decomposition of the stochastic discount factor (see, e.g., Alvarez and Jermann (2005); Hansen and Scheinkman (2009);Qin and Linetsky (2017);Christensen (2017b)). In connecting the role of principal eigenpairs of the valuation operator to the theory of monotone concave operators, we link two active strands of research on present values associated with cash and utility flows.…”

The Long Bond, Long Forward Measure and Long-Term Factorization In Heath-Jarrow Morton Models

“…In addition to the references above, the growing literature on the long-term factorization and its applications includes Hansen and Scheinkman (2012), Hansen and Scheinkman (2017), , Borovička et al (2011), Borovička and, Bakshi and Chabi-Yo (2012), Bakshi et al (2015), Christensen (2017), Christensen (2016), , , Backus et al (2015), Filipović et al (2017), Filipović et al (2016), Lustig et al (2016). Empirical investigations in this literature show that the martingale M ∞ t is highly volatile and economically significant.…”

“…Bakshi and Chabi-Yo (2012) provide theoretical and empirical bounds on the volatility of the martingale component. Christensen (2017) estimates the long-term factorization in a structural asset pricing model connecting to the macro-economic fundamentals. estimate the long-term factorization in a dynamic term structure model (DTSM) and show how the martingale component in the long-term factorization controls the term structure of the risk-return trade-off in the bond market.…”