Mahieu-Schotman (1998) An Empirical Application of Stochastic Volatility Models

Summary

Mahieu and Schotman (1998) compare four estimation strategies for discrete-time stochastic volatility (SV) models applied to weekly exchange rate data for six USD/GBP/DEM/JPY currency pairs (1975–1991). The central problem is that the log-squared-return transformation produces a linear state-space model with non-Gaussian log chi-square(1) measurement error, making exact Kalman filtering and quasi-maximum likelihood (QML) biased. The paper extends the Kim-Shephard-Chib (1998) fixed-mixture-of-normals approximation to a flexible mixture whose parameters are estimated from the data, and documents that QML severely underestimates volatility persistence and volatility-of-volatility. An option pricing application shows that the resulting implied volatility uncertainty is far larger than QML suggests.

Key Claims

The log-squared return $y_t = \ln s_t^2 = h_t + \ln \varepsilon_t^2$ is a linear state-space model, but the measurement error $\ln \varepsilon_t^2 \sim \log\chi^2(1)$ has variance $\pi^2/2 \approx 4.93$ and severe left skewness. Treating it as Gaussian (QML) produces significant downward bias in $\hat\sigma_\eta$ and $\hat\phi$ .
A 7-component fixed mixture of normals (Kim, Shephard, and Chib 1998) approximates the log chi-square density; conditional on discrete mixture indicators $Z_t \in \{1,\ldots,7\}$ , the model becomes Gaussian and exact Kalman filtering/smoothing applies.
Mahieu-Schotman extend to a flexible mixture: the component weights, means, and variances are free parameters estimated jointly with the SV parameters via Simulation expectation-maximization (SIEM). The flexible mixture fits exchange rate fat tails better than the fixed mixture.
Variance decomposition: $\text{Var}(y_t) = \sigma_h^2/(1-\phi^2) + \pi^2/2$ . The log chi-square noise accounts for 60–80% of total variation in $y_t$ , leaving only 20–40% attributable to true log-volatility variation. Latent volatility is therefore very imprecisely estimated.
QML produces standard errors of volatility estimates that are too small; Bayesian/SIEM standard errors are substantially larger.
Option pricing: the expected option payoff integrates over simulated future volatility paths. The SV-implied volatility has very large standard errors — consistent with the high measurement noise — while QML understates these errors.
SV log-volatility and GARCH conditional variance are highly correlated across methods, but SV paths are smoother (Kalman smoother averages over shocks); deviations are concentrated at return outliers.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"The log chi-square distribution has a variance of $\pi^2/2 \approx 4.93$ and is strongly skewed. Ignoring this non-normality in QML leads to a significant underestimation of both the variance and persistence of the volatility process."

My Take

The key practical lesson is stark: QML is badly biased for SV models and should not be used in empirical work. The bias is especially severe for the vol-of-vol parameter $\sigma_\eta$ , which determines how much volatility varies and hence how much option prices vary. The flexible-mixture contribution is incremental relative to Kim-Shephard-Chib (1998) but shows that the fixed 7-component mixture does not fully capture exchange-rate tails. The option pricing section is somewhat informal — no confidence intervals on implied vol curves are plotted — but the qualitative message (QML understates uncertainty) is well-supported by the variance decomposition.