Pre-Averaging Returns to Reduce High-Frequency Microstructure Noise

Summary

The document introduces pre-averaging as a way to address microstructure noise in high-frequency observations of an underlying price process. It defines a statistic as a weighted sum of consecutive price increments, with weights obtained by evaluating a function over a short observation window. The proposed setup takes the observed price to be the latent price plus noise and applies the weighted increments to that observed series. The window length is described as growing with the cube root of the number of observations.

The weight function is required to be nonzero, continuous, piecewise continuously differentiable, zero outside the unit interval, and to have a piecewise Lipschitz derivative. The document does not choose a specific function or show how to implement the estimator; it asks how to do so for a simulated series. It also gives no simulation results or discussion of bias, variance, boundary handling, or later noise correction, so it serves as a statement of the setup rather than a complete procedure.

Key ideas

• Pre-averaging forms a statistic from weighted consecutive increments of observed prices.
• The observed series is modeled as a latent price plus microstructure noise.
• Weights come from a function evaluated across a finite window and must satisfy stated smoothness and support conditions.
• The proposed window length scales with the cube root of the observation count.
• The document leaves the choice and implementation of the weight function unresolved.

Tags

# 77502


# Non-zero real-valued function continuous and piecewise $C^1$ that vanishes outside (0,1) with piecewise Lipschitz derivative












In this paper the authors to overcome the presence of microstructure noise which "contaminates" the ito-semimartingale in high-frequency data uses the idea of pre-averaging.

For an arbitrary process $V$ the pre-average statistics is defined as: $$V_k^n=\sum_{j=1}^{m_n} g\left( \frac{j}{m_n}\right) \Delta_{k+j}^n V =\sum_{j=1}^{m_n} g\left( \frac{j}{m_n}\right) \left( V_{t(n,k+j)-V_{t(n,k+j-1)}} \right)$$ Here $g:\mathbb{R}\to \mathbb{R}$ is a non-zero real-valued function which is continuous and piecewise $C^1$ that vanishes outside of the open interval $(0, 1)$, and has a piecewise Lipschitz derivative $g'$, $m_n = n^{\frac{1}{3}}$ where n is the total number of observation

If I simulate a coloumn vector of log-price(let'say 1000 observations) in high-frequency such that $Y = X+U$, ( $X$ is the latent price and $U$ is the microstructure noise) from which I want to eliminate the noise thanks to the pre-averaging method above, How can I implement the above function $g$? The pre-average statistics in this case would be : $$Y_k^n=\sum_{j=1}^{m_n} g\left( \frac{j}{m_n}\right) \Delta_{k+j}^n Y$$ I do not know how to use this function $g$ to apply this method

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.