# 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

• The function $g$ does not imply one particular function, but a family of functions that satisfy the described mathematical conditions. One such example is $g(x) = \min\left(x, 1-x\right)$ which is also used in the original paper (Vetter, Podolskij et al. (2009)) describing the pre-averaging estimator (see the example around eq. 3.11). Unless otherwise specified, it is safe to assume that the authors of your linked paper, used the same function for their own study.
– Pleb
Nov 27, 2023 at 9:55
• yes I just find out 2 sec this paper . To deal with noise in Y I can use this estimators as they do in the paper $\bar{Y}_{i}^n = \left( \sum_{j=k_n/2}^{k_n-1} Y_{i+j}^n- \sum_{j=0}^{k_n/2-1} Y_{i+j}^n \right)$. Am I right?
– XY0
Nov 27, 2023 at 10:02
• Yes. Remember to specify $\frac{1}{k_n}$ infront of the bracket, as done in the paper :-)
– Pleb
Nov 27, 2023 at 10:13
• is $Y_{i}^n$ assumed to the first difference ? in equation 3.13 they said $\bar{Z}_{i}^n = \left( \sum_{j=k_n/2}^{k_n-1} Z_{i+j}^n- \sum_{j=0}^{k_n/2-1} Z_{i+j}^n \right)$ in equ 3.5 they write $Z_i^n = Z_{i \Delta_n}$ what is the meaning of the subscript $\Delta_n$?
– XY0
Nov 27, 2023 at 10:40
• $Y_i^n$ or $Z_i^n$ which are used within the brackets, are the noisy prices (usually log-prices) such that $\bar{Y}_i^n$ or $\bar{Z}_i^n$ denotes pre-averaged log-returns. The notation $\Delta_n$ denotes a generalized way of describing the intraday frequency of the log-prices, since we can define more than one sampling scheme. Usually we use Calendar Time Sampling (CTS) where we have equidistant intraday time intervals (eg. 1 second intervals) and thus $\Delta_n$ becomes $\Delta_n = \frac{t-(t-1)}{n} = \frac{1}{n}$. Ie. between day $t-1$ and day $t$, we have $n$ intraday samples.
– Pleb
Nov 27, 2023 at 12:15