In this paper, we introduce a general method for estimating the quadratic covariation of one or more spot parameter processes associated with continuous time semimartingales, and present a central limit theorem that has this class of estimators as one of its applications. The class of estimators we introduce, that we call Two-Scales Quadratic Covariation ( TSQC ) estimators, is based on sums of increments of second differences of the observed processes, 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 based on only one partition of the observational window. Moreover, a two-scales approach is employed to deal with asymptotic bias terms in a systematic manner, thus automatically giving consistent estimators without having to work out the form of the bias term on a case-to-case basis. We highlight the versatility of our central limit theorem by applying it to a novel leverage effect estimator that does not belong to the class of TSQC estimators. The principal empirical motivation for the present study 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 spot-volatility process. As an application of the TSQC estimators, 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 and hence observed.

Hjort, Nils Lid & Stoltenberg, Emil Aas (2021)

The partly parametric and partly nonparametric additive risk model

Aalen’s linear hazard rate regression model is a useful and increasingly popular alternative to Cox’ multiplicative hazard rate model. It postulates that an individual has hazard rate function h(s)=z1α1(s)+⋯+zrαr(s) in terms of his covariate values z1,…,zr. These are typically levels of various hazard factors, and may also be time-dependent. The hazard factor functions αj(s) are the parameters of the model and are estimated from data. This is traditionally accomplished in a fully nonparametric way. This paper develops methodology for estimating the hazard factor functions when some of them are modelled parametrically while the others are left unspecified. Large-sample results are reached inside this partly parametric, partly nonparametric framework, which also enables us to assess the goodness of fit of the model’s parametric components. In addition, these results are used to pinpoint how much precision is gained, using the parametric-nonparametric model, over the standard nonparametric method. A real-data application is included, along with a brief simulation study.

Stoltenberg, Emil Aas & Hjort, Nils Lid (2020)

Models and inference for on–off data via clipped Ornstein–Uhlenbeck processes