Dennis Kristensen

Nonparametric Filtering of the Realised Spot Volatility: A Kernel-Based Approach

A kernel weighted version of the standard realised integrated volatility es- timator is proposed. By different choices of the kernel and bandwidth, the measure allows us to focus on specific characteristics of the volatility process. In particular, as the bandwidth vanishes, an estimator of the realised spot volatility is obtained. We denote this the filtered spot volatility. We show con- sistency and asymptotic normality of the kernel smoothed realised volatility and the filtered spot volatility. The choice of bandwidth is discussed and data- driven selection methods proposed. A simulation study examines the finite sample properties of the estimators.

Asymptotics of the QMLE for a Class of ARCH(q) Models

Strong consistency and asymptotic normality are established for the quasi-maximum likelihood estimator for a class of ARCH(q) models. The conditions are that the ARCH process is geometrically ergodic with a moment of arbitrarily small order. Furthermore for consistency, we assume that the second-order moment exists for the nondegenerate rescaled errors and, similarly, that the fourth-order moment exists for asymptotic normality to hold. Contrary to existing literature on (G)ARCH models the parameter space is not assumed to be compact; we only impose a lower bound for the constant term in our parameterization of the conditional variance. It is demonstrated that the general conditions are satisfied for a range of specific models.We are grateful to the editor and the referees for their very helpful and detailed suggestions, which, we believe, improved the paper substantially. We thank SA¸ren T. Jensen for stimulating discussions and Jonathan Dennis for helpful research assistance. Rahbek acknowledges continuing financial support from the Danish Social Sciences Research Council. Kristensen received funding from the Danish Research Agency and the Financial Markets Group, LSE, during this research.

Estimation of Dynamic Models with Nonparametric Simulated Maximum Likelihood

We propose an easy-to-implement simulated maximum likelihood estimator for dynamic models where no closed-form representation of the likelihood function is available. Our method can handle any simulable model without latent dynamics. Using simulated observations, we nonparametrically estimate the unknown density by kernel methods, and then construct a likelihood function that can be maximized. We prove that this nonparametric simulated maximum likelihood (NPSML) estimator is consistent and asymptotically efficient. The higher-order impact of simulations and kernel smoothing on the resulting estimator is also analyzed; in particular, it is shown that the NPSML does not suffer from the usual curse of dimensionality associated with kernel estimators. A simulation study shows good performance of the method when employed in the estimation of jump–diffusion models.

A CLOSED-FORM ESTIMATOR FOR THE GARCH(1,1) MODEL

We propose a closed-form estimator for the linear GARCH(1,1) model. The estimator has the advantage over the often used quasi-maximum-likelihood estimator (QMLE) that it can be easily implemented, and does not require the use of any numerical optimisation procedures or the choice of initial values of the conditional variance process. We derive the asymptotic properties of the estimator, showing T^{(K-1)/K}-consistency for some K(1,2) when the 4th moment exists and the square root of T-asymptotic normality when the 8th moment exists. We demonstrate that a finite number of Newton-Raphson iterations using our estimator as starting point will yield asymptotically the same distribution as the QMLE when the 4th moment exists. A simulation study confirms our theoretical results.

UNIFORM CONVERGENCE RATES OF KERNEL ESTIMATORS WITH HETEROGENEOUS DEPENDENT DATA

The main uniform convergence results of Hansen (2008, Econometric Theory 24, 726–748) are generalized in two directions: Data are allowed to (a) be heterogeneously dependent and (b) depend on a (possibly unbounded) parameter. These results are useful in semiparametric estimation problems involving time-inhomogeneous models and/or sampling of continuous-time processes. The usefulness of these results is demonstrated by two applications: kernel regression estimation of a time-varying AR(1) model and the kernel density estimation of a Markov chain that has not been initialized at its stationary distribution.

Asymptotic Theory for the QMLE in GARCH-X Models With Stationary and Nonstationary Covariates

This article investigates the asymptotic properties of the Gaussian quasi-maximum-likelihood estimators (QMLE’s) of the GARCH model augmented by including an additional explanatory variable—the so-called GARCH-X model. The additional covariate is allowed to exhibit any degree of persistence as captured by its long-memory parameter dx; in particular, we allow for both stationary and nonstationary covariates. We show that the QMLE’s of the parameters entering the volatility equation are consistent and mixed-normally distributed in large samples. The convergence rates and limiting distributions of the QMLE’s depend on whether the regressor is stationary or not. However, standard inferential tools for the parameters are robust to the level of persistence of the regressor with t-statistics following standard Normal distributions in large sample irrespective of whether the regressor is stationary or not. Supplementary materials for this article are available online.

Uniform Convergence Rates of Kernel Estimators with Heterogenous, Dependent Data

The main uniform convergence results of Hansen (2008, Econometric Theory 24, 726–748) are generalized in two directions: Data are allowed to (a) be heterogeneously dependent and (b) depend on a (possibly unbounded) parameter. These results are useful in semiparametric estimation problems involving time-inhomogeneous models and/or sampling of continuous-time processes. The usefulness of these results is demonstrated by two applications: kernel regression estimation of a time-varying AR(1) model and the kernel density estimation of a Markov chain that has not been initialized at its stationary distribution.

Semi-Nonparametric Estimation and Misspecification Testing of Diffusion Models

We propose novel misspecification tests of semiparametric and fully parametric univariate diffusion models based on the estimators developed in Kristensen (Journal of Econometrics, 2010). We first demonstrate that given a preliminary estimator of either the drift or the diffusion term in a diffusion model, nonparametric kernel estimators of the remaining term can be obtained. We then propose misspecification tests of semparametric and fully parametric diffusion models that compare estimators of the transition density under the relevant null and alternative. The asymptotic distribution of the estimators and tests under the null are derived, and the power properties are analyzed by considering contiguous alternatives. Test directly comparing the drift and diffusion estimators under the relevant null and alternative are also analyzed. Markov Bootstrap versions of the test statistics are proposed to improve on the finite-sample approximations. The finite sample properties of the estimators are examined in a simulation study.

Geometric Ergodicity of a Class of Markov Chains with Applications to Time Series Models

We set up general conditions for a general non-linear Markov model to be geometrically ergodic (which implies beta-mixing of the stationary solution) and existence of certain moments. The conditions are fairly general and can be applied to most known time series models. We demonstrate the usefulness of our general result by applying it to various popular time series models. For each model, we give conditions for beta-mixing and existence of certain moments. In many cases, our conditions are weaker than those found elsewhere in the literature. In particular, we derive sufficient conditions for a class of univariate GARCH models to be geometrically ergodic without having a 2nd moment. In certain cases, the conditions are also sufficient. We also consider multivariate GARCH models and give conditions for stationarity with finite 2nd moment.

Adding and subtracting Black-Scholes: A new approach to approximating derivative prices in continuous-time models

We develop a new approach to approximating asset prices in the context of continuous-time models. For any pricing model that lacks a closed-form solution, we provide a closed-form approximate solution, which relies on the expansion of the intractable model around an “auxiliary” one. We derive an expression for the difference between the true (but unknown) price and the auxiliary one, which we approximate in closed-form, and use to create increasingly improved refinements to the initial mispricing induced by the auxiliary model. The approach is intuitive, simple to implement, and leads to fast and extremely accurate approximations. We illustrate this method in a variety of contexts including option pricing with stochastic volatility, computation of Greeks, and the term structure of interest rates.