A CLT for second difference estimators with an application to volatility and intensity

In this paper we introduce a general method for estimating the quadratic covariation of one or more spot parameters processes associated with continuous time semimartingales. This estimator is applicable to a wide range of spot parameter processes, and may also be used to estimate the leverage effect of stochastic volatility models. The estimator we introduce is based on sums of squared increments of second differences of the observed process, and the intervals over which the differences are computed are rolling and overlapping. This latter feature lets us take full advantage of the data, and, by sufficiency considerations, ought to outperform estimators that are only based on one partition of the observational window. The main result of the paper is a central limit theorem for such triangular array rolling quadratic variations. We highlight the wide applicability of this theorem by showcasing how it might be applied to a novel leverage effect estimator. The principal motivation for the present study, however, is that the discrete times at which a continuous time semimartingale is observed might depend on features of the observable process other than its level, such as its (non-observable) spot-volatility process. As the main application of our estimator, we therefore show how it may be used to estimate the quadratic covariation between the spot-volatility process and the intensity process of the observation times, when both of these are taken to be semimartingales. The finite sample properties of this estimator are studied by way of a simulation experiment, and we also apply this estimator in an empirical analysis of the Apple stock. Our analysis of the Apple stock indicates a rather strong correlation between the spot volatility process of the log-prices process and the times at which this stock is traded (hence observed).