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On the Relation Between Linearity-Generating Processes and Linear-Rational Models

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  • Damir Filipovic
  • Martin Larsson
  • Anders B. Trolle

Abstract

We review the notion of a linearity-generating (LG) process introduced by Gabaix (2007) and relate LG processes to linear-rational (LR) models studied by Filipovic, Larsson, and Trolle (2017). We show that every LR model can be represented as an LG process and vice versa. We find that LR models have two basic properties which make them an important representation of LG processes. First, LR models can be easily specified and made consistent with nonnegative interest rates. Second, LR models go naturally with the long-term risk factorization due to Alvarez and Jermann (2005), Hansen and Scheinkman (2009), and Qin and Linetsky (2017). Every LG process under the long forward measure can be represented as a lower dimensional LR model.

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  • Damir Filipovic & Martin Larsson & Anders B. Trolle, 2018. "On the Relation Between Linearity-Generating Processes and Linear-Rational Models," Papers 1806.03153, arXiv.org.
    • Handle: RePEc:arx:papers:1806.03153
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    References listed on IDEAS

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    1. Lars Peter Hansen & José A. Scheinkman, 2009. "Long-Term Risk: An Operator Approach," Econometrica, Econometric Society, vol. 77(1), pages 177-234, January.
    2. Likuan Qin & Vadim Linetsky & Yutian Nie, 2018. "Long Forward Probabilities, Recovery, and the Term Structure of Bond Risk Premiums," The Review of Financial Studies, Society for Financial Studies, vol. 31(12), pages 4863-4883.
    3. Xavier Gabaix, 2012. "Variable Rare Disasters: An Exactly Solved Framework for Ten Puzzles in Macro-Finance," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 127(2), pages 645-700.
    4. Pierre Collin‐Dufresne & Robert S. Goldstein, 2002. "Do Bonds Span the Fixed Income Markets? Theory and Evidence for Unspanned Stochastic Volatility," Journal of Finance, American Finance Association, vol. 57(4), pages 1685-1730, August.
    5. Emmanuel Farhi & Xavier Gabaix, 2016. "Editor's Choice Rare Disasters and Exchange Rates," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 131(1), pages 1-52.
    6. Xavier Gabaix, 2007. "Linearity-Generating Processes: A Modelling Tool Yielding Closed Forms for Asset Prices," NBER Working Papers 13430, National Bureau of Economic Research, Inc.
    7. Dybvig, Philip H & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1996. "Long Forward and Zero-Coupon Rates Can Never Fall," The Journal of Business, University of Chicago Press, vol. 69(1), pages 1-25, January.
    8. Fernando Alvarez & Urban J. Jermann, 2005. "Using Asset Prices to Measure the Persistence of the Marginal Utility of Wealth," Econometrica, Econometric Society, vol. 73(6), pages 1977-2016, November.
    9. Jérome Detemple & Marcel Rindisbacher, 2010. "Dynamic Asset Allocation: Portfolio Decomposition Formula and Applications," The Review of Financial Studies, Society for Financial Studies, vol. 23(1), pages 25-100, January.
    10. Damir Filipović & Martin Larsson & Anders B. Trolle, 2017. "Linear-Rational Term Structure Models," Journal of Finance, American Finance Association, vol. 72(2), pages 655-704, April.
    11. Likuan Qin & Vadim Linetsky, 2017. "Long‐Term Risk: A Martingale Approach," Econometrica, Econometric Society, vol. 85, pages 299-312, January.
    12. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.
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