Semiparametric Estimation in Continuous-Time: Asymptotics for Integrated Volatility Functionals with Small and Large Bandwidths

Abstract

This article studies the estimation of integrated volatility functionals, which is a semiparametric two-step estimation problem in the nonstationary continuous-time setting. We generalize the asymptotic normality results of Jacod and Rosenbaum to a wider range of bandwidths. Moreover, we employ matrix calculus to obtain a new analytical bias correction and variance estimation method. The proposed method gives more succinct expressions than the element-by-element analytical method of the above cited article. In addition, it has a computational advantage over the jackknife/simulation-based method proposed by Li, Liu, and Xiu. Comprehensive simulation studies demonstrate that our method has good finite sample performance for a variety of volatility functionals, including quadraticity, determinant, continuous beta, and eigenvalues.

KEYWORDS:

• Bias correction
• Integrated volatility functionals
• Matrix calculus
• Semiparametric two-step estimation

Supplementary Materials

The supplement provides further discussions of the main results and proofs of the theorems and propositions in the article.

Notes

1 Notes

Using the formulas given by Jacod and Rosenbaum (2013), Li, Todorov, and Tauchen (2016) studied the inference theory on volatility functional. With d = 10, the authors need to evaluate 990 distinct second-order partial derivatives for each functional form they consider. If d = 100, which is quite normal for principal component analysis, then this number will be close to 100 million. In the high frequency context, one also needs to re-evaluate these derivatives every time one computes a new spot volatility estimator.

2 In general, the volatility process can have a Poisson random measure different from the price process. However, one can always defined a Poisson random measure to capture the jumps of both X and c, and then adjust δ and $\tilde{\delta }$ accordingly. We can also have a decomposition for c.

3 It is easier to see from Propositions 1 and 3 that the estimator of the oversmoothing bias is a function of ${[{\widehat{\mathit{c}}}_i^n]}^{-1}\widehat{\Delta {\mathit{c}}_i^n}$. When all eigenvalues of this matrix are small, it is reasonable to expect the oversmoothing bias will be small as well.

4 We noted that Li, Liu, and Xiu (2019) compared their jackknife method with the one proposed by Jacod and Rosenbaum (2013, 2015) for quadraticity only. A potential reason is that it is rather tedious, if not practically impossible, to get the analytical forms for biases in the cases of continuous beta, eigenvalue, and determinant using the formulas provided by the latter cited articles.