Statement
The Campbell-Shiller (1988) log-linearization used to solve asset pricing models with long-run risk and recursive preferences introduces economically significant approximation errors that can exceed 70% for key model moments when state processes are highly persistent. The errors are driven by the interaction between persistent long-run risk (x_t) and stochastic volatility (sigma_t), which creates nonlinearities that log-linearization cannot capture.
Evidence summary
Pohl, Schmedders, and Wilms (2018) compare log-linearized solutions to high-precision projection method solutions across six recent long-run risk models. Key findings:
- BKY (2012) calibration: equity premium overestimated by ~100 bps (13.5% relative error); P/D ratio volatility overestimated by 22%; return predictability overstated by approximately 2x.
- SSY (2018) 95% persistence estimates: errors up to 70%.
- Bond pricing models: log-linearization can invert the yield curve slope.
- Removing stochastic volatility from x_t reduces errors to near zero, confirming the interaction as the key error source.
Conditions and scope
- Applies to models with Epstein-Zin/recursive preferences and multiple persistent state processes.
- Requires gamma > 1/psi (preference for early resolution of uncertainty); without this, the model cannot generate a high equity premium regardless.
- Errors are non-monotone and can change dramatically with small parameter changes (e.g., rho from 0.975 to 0.98 nearly doubles the error).
- The CRE project’s Bansal approximation validation (67-72% Jensen gap at 50yr horizon) is consistent with these findings.
Counter-evidence
- For the original Bansal-Yaron (2004) calibration (lower persistence rho=0.979, nu=0.987), errors are more modest.
- When psi=1 (exact wealth-consumption ratio), errors in equity pricing alone are smaller (~8.6% in equity premium) but still significant.
Linked ideas
Open questions
- How do these errors propagate into structural estimation via Bayesian methods?
- Can Markov-switching representations avoid these errors while retaining tractability?
- Is there a diagnostic that can determine when log-linearization is adequate without computing the full solution?