# Smoothing Data for Further Analysis and Preliminary Observations

In this work several microstructure variables were researched, including

• Stock return and price

• Price change and its absolute value

• Spread and relative spread (ratio of spread to price)

Due to systematic noise in microstructure data, it is necessary to smooth the data for further analysis. In this work, one of the modern wavelet methods was used. The basic principle of wavelet smoothing is performing wavelet decomposition and applying a “smoothing” transformation for wavelet coefficients for a certain threshold level. By looking at the smoothed trajectory of a variable, we can already discern whether its behavior is regular or not (Antoniou and Vorlow 2005). One of the results is the visible regular dynamics of stock return, price changes and relative spread, while other variables show randomness. To illustrate the effect, Fig. 1 demonstrates the dependence of Lukoil stock characteristics (intraday data aggregated by 10 s, 13th January 2006) from their delayed values with a rather

Fig. 1 Phase trajectory for (a) price; (b) return; (c) price change; (d) absolute value of price change; (e) spread; (f) relative spread |

large lag of x = 200 s. Similar results hold for other lag lengths. Obtained results are consistent with the work of Antoniou and Vorlow (2005) for FTSE100 stock returns (daily data). As estimator of price we take arithmetic mean of best bid and ask quotes; return means price change during 10 s divided by price value at the beginning of the period; spread means simple bid-ask spread and relative spread is ratio of spread to price value.

Phase trajectories (b)-(d), (f) produce somewhat regular patterns [unlike (a) and (e)], which can be considered as indirect evidence of nonlinear dynamics.[2] To verify this we calculate BDS statistic for each series which shows at 0.5 % significance level rejection of hypothesis that increments are i. i.d. which means that the data can be generated by a low-dimensional chaotic or nonlinear model. Unfortunately BDS statistic still cannot be a reliable criterion for short samples of data, even for whole trading day (see below about BDS). It is important to note that any practical use of the model can be achieved only in the case of low dimension. High-dimensional systems usually have too many unknown parameters and are quite unstable, thus unpredictable even for the short horizon. In this case, stochastic modeling will be more appropriate. A fine illustration is given by Poincare (1912), describing atmospheric effects. It is theoretically possible to calculate the distribution of rain drops on the pavement, but due to the complex nature of the generating process, their distribution seems uniform; thus it is much easier to prove this hypothesis by assuming that the generating system is purely stochastic (Poincare 1912).

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