Designing Realised Kernels to Measure the Ex-Post Variation of Equity Prices in the Presence of Noise
Social Science Research Network
This paper shows how to use realised kernels to carry out efficient feasible inference on the expost variation of underlying equity prices in the presence of simple models of market frictions. The issue is subtle with only estimators which have symmetric weights delivering consistent estimators with mixed Gaussian limit theorems. The weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which is close to that of the maximum likelihood
... likelihood estimator in the parametric version of this problem. Realised kernels can also be selected to (i) be analysed using endogenously spaced data such as that in databases on transactions, (ii) allow for market frictions which are endogenous, (iii) allow for temporally dependent noise. The finite sample performance of our estimators is studied using simulation, while empirical work illustrates their use in practice. for valuable comments. Neil Shephard's research is supported by the UK's ESRC. The Ox language of Doornik (2001) was used to perform the calculations reported here. with h = −H, ..., −1, 0, 1, ..., H and n = ⌊t/δ⌋. We will think of δ as being small and so X δj − X δ(j−1) represents the j-th high frequency return, while γ 0 (X δ ) is the realised variance of X. Here K(X δ ) − γ 0 (X δ ) is the realised kernel correction to realised variance for market frictions. We show that if k(0) = 1, k(1) = 0 and H = cn 2/3 then the resulting estimator is asymptotically mixed Gaussian, converging at rate n 1/6 . Here c is a estimable constant which can be optimally chosen as a function of k, the variance of the noise and a function of the volatility path, to minimise the asymptotic variance of the estimator. The special case of a so-called flat-top Bartlett kernel, where k(x) = 1 − x, is particularly interesting as its asymptotic distribution is the same as that of the two scale estimator. When we additionally require that k ′ (0) = 0 and k ′ (1) = 0 then by taking H = cn 1/2 the resulting estimator is asymptotically mixed Gaussian, converging at rate n 1/4 , which we know is the fastest possible rate. When k(x) = 1 − 3x 2 + 2x 2 this estimator has the same asymptotic distribution as the multiscale estimator. We use our novel realised kernel framework to make three innovations to the literature: (i) we design a kernel to have an asymptotic variance which is smaller than the multiscale estimator, (ii) we design K(X δ ) for data with endogenously spaced data, such as that in databases on transactions (see Renault and Werker (2005) for the importance of this), (iii) we cover the case where the market frictions are endogenous. All of these results are new and the last two of them are essential from a practical perspective. Clearly these realised kernels are related to so-called HAC estimators discussed by, for example, Newey and West (1987) and Andrews (1991) . The flat-top of the kernel, where a unit weight is imposed on the first autocovariance, is related to the flat-top literature initiated by Politis and Romano (1995) and Politis (2005) . However, the realised kernels are not scaled by the sample size, which has a great number of technical implications and makes their analysis subtle. The econometric literature on realised kernels was started by Zhou (1996) who proposed K(X δ ) with H = 1. This suffices for unbiasedness under a simple model for frictions where the population values of higher-order autocovariances of the market frictions are zero. However, the estimator is inconsistent. Hansen and Lunde (2006) use realised kernel type estimators, with k(x) = 1 for general H to characterize the second order properties of market microstructure noise. Again these are inconsistent estimators. Some analysis of the finite sample performance of a type of inconsistent realised kernel is provided by Bandi and Russell (2005b), who focus on the selection of H in the case where k(x) = 1 − x, the Bartlett kernel.