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Higher Order Effects in Asset Pricing Models with Long-Run Risks

4 min read

Problem

The Campbell-Shiller (1988) log-linearization has become the standard method for solving asset pricing models with long-run risk and recursive preferences. But as models have evolved toward higher persistence and greater complexity, it is unclear whether the approximation remains adequate. This paper asks: do higher-order dynamics matter quantitatively, and if so, when and why?

Key idea

Log-linearization systematically underestimates the log price-dividend ratio and overestimates the equity premium, with errors that become economically significant (exceeding 70% for some moments) when the state processes are highly persistent. The critical mechanism is the interaction between persistent long-run risk and stochastic volatility: when both are persistent, agents can be trapped in undesirable states (low growth, high volatility) for extended periods, creating significant nonlinearities that log-linearization misses entirely.

Method

  1. Use projection methods (Chebyshev polynomials) as the high-precision benchmark, which are known to converge to the true solution.
  2. Compare log-linearized solutions against projection solutions across the state space for the Bansal-Kiku-Yaron (2012) calibration and five other recent long-run risk models.
  3. Analyze error sources by decomposing: (a) turning off stochastic volatility in x_t shows the interaction term is the dominant error source; (b) setting psi=1 (exact wealth-consumption ratio) isolates equity pricing errors.
  4. Examine sensitivity to persistence parameters rho (long-run risk) and nu (stochastic volatility), risk aversion gamma, and EIS psi.

Results

  • Bansal-Kiku-Yaron (2012): equity premium overestimated by ~100 bps; volatility of P/D ratio error exceeds 20%; true return predictability is approximately half of what log-linearization suggests.
  • Schorfheide-Song-Yaron (2018) 95% quantile estimates: errors as large as 70% for key moments.
  • Bansal-Shaliastovich (2013) and Koijen et al. (2010) bond models: log-linearization can invert the slope of the yield curve for high persistence.
  • The interaction between long-run risk and stochastic volatility is the key error source: with constant volatility in x_t, errors drop to near zero but the equity premium also collapses.
  • Even with psi=1 (exact SDF), linearizing equity returns alone produces ~8.6% error in the equity premium and ~17.5% error in P/D volatility.
  • Errors increase dramatically with small changes in persistence (rho from 0.975 to 0.98 nearly doubles the equity premium error).
  • The problem is only severe when gamma > 1/psi (preference for early resolution of risk), but without this preference the model cannot generate a high equity premium.

Limitations

  • Projection methods are computationally expensive, limiting their applicability to high-dimensional state spaces.
  • The analysis focuses on unconditional moments; conditional moment errors may behave differently.
  • The paper does not propose a simple closed-form alternative to log-linearization that would be as easy to use while avoiding the errors.
  • Results are model-specific; some calibrations (e.g., original Bansal-Yaron 2004 with lower persistence) show small errors.

Open questions

  • Can Markov-switching representations (as in generalized-disappointment-aversion-long-run-volatility) avoid these errors while maintaining analytical tractability?
  • How do these approximation errors propagate into structural estimation (e.g., Bayesian posteriors)?
  • Is there a systematic way to determine when log-linearization is adequate without computing the full nonlinear solution?
  • Do the errors affect the sign or magnitude of welfare calculations in these models?

My take

This is a critically important paper for anyone working with log-linear approximations in asset pricing. The 67-72% Jensen gap found in the CRE project’s Bansal approximation validation (see CLAUDE.md “Bansal Approx: FAIL”) is entirely consistent with Pohl et al.’s findings. The paper validates that the project’s decision to use exact Riccati recursions rather than log-linear approximations was correct. The key insight about the interaction between persistent risk channels creating extreme-state nonlinearities is directly relevant to the CRE model’s regime-switching structure, where different monetary policy regimes can trap the economy in states far from steady state. The connection to no-bubble-condition is important: the log-linear approximation may mask whether the no-bubble condition is actually satisfied, since the condition depends on the exact solution’s asymptotic behavior (related to exponential-quadratic-asset-pricing-factors).

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