Barndorff-Nielsen and Shephard (2006) Impact of Jumps on Returns and Realised Variances

Summary

Barndorff-Nielsen and Shephard (2006) derive the second-order properties of realized variance (RV) when log-prices follow a time-deformed Lévy process $Y_t = mt + Z_{\tau_t}$, where $Z$ is a Lévy process and $\tau_t$ is an independent stochastic time-change. The central finding is that RV is an inconsistent estimator of the time-change whenever $Z$ is non-Brownian: the irreducible variance of the RV error is $k_4\Delta\xi$ (driven by the 4th cumulant of $Z_1$) and does not vanish as the sampling frequency increases. Despite inconsistency, the bias of RV is $O(M^{-1})$ and small. The autocorrelation of RV underestimates that of the true time-change when jumps are present, meaning volatility is even more predictable than high-frequency studies suggest. The paper also constructs quasi-likelihood (QL) estimators for the parameters of time-deformation models, implemented via the Kalman filter for Markovian variance processes and the Durbin algorithm for the general case.

Key Claims

• RV is inconsistent under non-Brownian Lévy time-change. $[Y]_{d,i} \xrightarrow{p} [Y]_i \neq k_2\tau_i$ outside the Brownian case because quadratic variation (QV) includes jump contributions. Even observing the exact price path, one cannot recover the time-change when $Z$ has jumps.
• Bias-variance decomposition. $E([Y]_{d,i} - k_2\tau_i) = O(M^{-1})$ is negligible; $\text{Var}([Y]_{d,i} - k_2\tau_i)$ contains $k_4\Delta\xi$ (the 4th-cumulant contribution) which does not vanish — bias² is $O(M^{-2})$ and is dominated by variance in the mean-squared error (MSE).
• Autocorrelation function (ACF) inequality. $\text{Cor}([Y]_{d,i}, [Y]_{d,i+s}) \leq \text{Cor}(\tau_i, \tau_{i+s})$ when $k_4 > 0$: RV systematically underestimates the autocorrelation of the true variance process, so volatility is even more predictable than RV-based estimates suggest.
• 4th cumulant jump test. For a symmetric Lévy process $Z$, $k_4 = 0$ if and only if $Z$ is scaled Brownian motion plus drift (by Lévy-Khintchine). Testing $k_4 = 0$ is therefore equivalent to testing for absence of jumps.
• QL estimation. A Gaussian QL in the second-order properties of $([Y]_{d,1},\ldots,[Y]_{d,n})$ is identified by constraining $k_2 = 1$; standard errors from the sandwich covariance $I^{-1}JI^{-1}$. Computed via Durbin ($O(n^2)$) for general stationary $\tau$, or Kalman filter ($O(n)$) for Ornstein-Uhlenbeck (OU)-based state-space models.
• Empirical results (USD/DEM, $M=288$, 1986–1996). Single OU fits poorly regardless of Lévy/Brownian. $J=2$ superposition much better; Lévy versions outperform Brownian for small $J$ models (logL −1195 vs. −1222 at $J=2$). Log-normal OU (LNOU) with Lévy gives logL −1219, most stable across $M$. Long-memory gamma parameter $H \approx 0.17$–$0.24$ in line with fractionally integrated GARCH (FIGARCH) estimates.

Concepts Introduced or Extended

• Time-Changed Lévy Process
• Realized Volatility
• Stochastic Volatility

Entities Mentioned

• Ole E. Barndorff-Nielsen
• Neil Shephard
• Torben G. Andersen
• Tim Bollerslev

Quotes

"The movement from Brownian motion to the Lévy time-change model has really only impacted the variability of the RV error. This is the most important point we make in this paper."

"Even though the RV is an inconsistent estimator of the time-change, it is an almost unbiased one."

"Volatility is even more predictable than has been shown by the recent econometric work on realised variance."

My Take

The paper makes three important contributions in one. First, the inconsistency result is sobering: no matter how many intraday observations you add, jump-driven Lévy components prevent RV from recovering the true latent variance. Second, the ACF inequality is a useful corrective to the Andersen-Bollerslev (AB, 1998) narrative — not only was squared-return evaluation noisy, but even 5-min RV understates true volatility persistence when jumps are present. Third, the QL framework is computationally practical and general enough to accommodate OU, superposition, LNOU, and long-memory models under a single estimation paradigm. The empirical application is illustrative rather than definitive (only USD/DEM), and the restriction $z \perp \tau$ rules out leverage — a limitation acknowledged by the authors.