Log-Linearization Approximation Error

Definition

The approximation error introduced by applying the Campbell-Shiller (1988) log-linear present-value relation to solve asset pricing models with recursive preferences. The method approximates the log price-dividend ratio as a linear function of state variables, which neglects higher-order nonlinear dynamics that become economically significant when state processes are highly persistent.

Intuition

Log-linearization works by expanding around a reference point (typically the unconditional mean of the log price-dividend ratio). When state processes are persistent, the economy can spend extended periods far from this reference point — for example, in regions of simultaneously low expected growth and high volatility. In these extreme regions, the true solution curves away from the linear approximation, and the errors accumulate. The key insight from higher-order-effects-asset-pricing-models is that the interaction between persistent long-run risk and stochastic volatility creates the largest errors, because high volatility increases the range of long-run risk realizations, pushing the economy into the nonlinear tails.

Formal notation

Let $p_t-d_t$ denote the log price-dividend ratio and $x_t,sigm{a}_t^2$ the state variables. The log-linearization approximates:

$p_t-d_t=A_0+A_1{x}_t+A_{2}sigm{a}_t^2($linear in states)

The true solution is a nonlinear function:

$p_t-d_t=f(x_t,sigm{a}_t^2)($convex in states for typical calibrations)

The approximation error |f(x,s) - $(A_0+A_{1}x+A_{2}s)|$ grows with distance from the linearization point, and scales with the persistence parameters rho, nu and the preference parameter $\theta$ = (1-gamma)/(1-1/psi).

Variants

1. Single linearization (Tsai-Wachter 2018): linearize only the wealth-consumption ratio; price equities exactly given the approximate SDF. Exact for $\psi$=1 and time-additive utility.
2. Double linearization (standard): linearize both the wealth-consumption ratio and the equity price-dividend ratio. Compounds errors from both steps.
3. No linearization (Bonomo et al. 2011): Markov-switching structure allows exact solutions without any linearization, but requires discretizing the state space.

Comparison

Method	Accuracy	Computational cost	Applicability
Double log-linear	Low for persistent models	Negligible	Any model
Single log-linear (Tsai-Wachter)	Moderate; exact for $\psi$=1	Low (Riccati ODEs)	Affine models
Projection methods	High	Moderate to high	Smooth state spaces
Markov switching (exact)	Exact (discretized)	Low to moderate	Finite state models
Value function iteration	High	High	General

When to use

Log-linearization remains useful for:

• Qualitative analysis of shock propagation mechanisms
• First-pass model exploration with modest persistence
• Models where persistence parameters are low $(\rho$ < 0.$97,\nu <$ 0.99)

Avoid log-linearization for:

• Quantitative model calibration with highly persistent processes
• Structural estimation where parameter inference depends on exact moments
• Term structure models where yield curve slope sign can be affected
• Any model where the equity premium magnitude is a key target

Known limitations

• Errors are non-monotone and can change dramatically with small parameter changes
• The errors are particularly severe for the volatility of the price-dividend ratio and return predictability
• Log-linearization can invert the yield curve slope in bond pricing models
• Cannot capture the no-bubble condition violation (the approximation may mask non-convergence)

Open problems

• No simple diagnostic exists to determine when log-linearization is adequate without computing the nonlinear solution
• Propagation of approximation errors into Bayesian structural estimation is not well understood
• The interaction between log-linearization errors and Monte Carlo estimation noise in applied work is unexplored

Key papers

• higher-order-effects-asset-pricing-models — systematic error analysis across 6 calibrations
• pricing-long-lived-securities-dynamic-endowment — single-linearization alternative
• generalized-disappointment-aversion-long-run-volatility — Markov-switching exact solutions

My understanding

This concept is directly load-bearing for the CRE project. The Bansal approximation validation (FAIL, 67-72% Jensen gap at 50yr) is an instance of this phenomenon. The project’s use of exact Riccati recursions for asset pricing and the RBPF for filtering avoids log-linearization entirely, which is the correct approach given the model’s regime-switching structure with persistent monetary policy regimes. The connection to exponential-quadratic-asset-pricing-factors is that the exact exponential-quadratic form captures the nonlinearities that log-linearization misses.