preaverage

hf.preaverage(data, K=None, g=None, return_K=False)

The preaveraging scheme of Podolskij and Vetter (2009). It uses the fact that if the noise is i.i.d with zero mean, then averaging a rolling window of (weighted) returns diminishes the effect of microstructure noise on the variance estimate.

Parameters

datapd.Series or pd.DataFrame

A time series of log-returns. If multivariate, the time series has to be synchronized (e.g. with refresh_time()).

Kint, default = None

The preaveraging window length. None implies \(K=0.4 n^{1/2}\) is chosen as recommended in Hautsch & Podolskij (2013).

gfunction, default = None

A weighting function. None implies \(g(x) = min(x, 1-x)\) is chosen.

Returns

data_papd.Series

The preaveraged log-returns.

Notes

The preaveraged log-returns using the window-length \(K\) are given by

\[\begin{equation} \begin{aligned} \bar{\mathbf{Y}}_{i}=\sum_{j=1}^{K-1} g\left(\frac{j}{K}\right) \Delta_{i-j+1}\mathbf{Y}, \quad \text { for } i=K, \ldots, n, \end{aligned} \end{equation}\]

where \(\mathbf{Y}_i\) have been synchronized beforehand, for example with refresh_time(). Note that the direction of the moving window has been reversed compared to the definition in Podolskij and Vetter (2009) to stay consistent within the package. \(g\) is a weighting function. A popular choice is

\[\begin{equation} g(x)=\min (x, 1-x). \end{equation}\]

References

Podolskij, M., Vetter, M., 2009. Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli 15 (3), 634–658.