REGLAMENTO (CE) 2023/2006, DE 22 DE DICIEMBRE DE 2006, DE LA COMISIÓN, SOBRE BUENAS PRÁCTICAS DE

As explained in Jacod et al. (2009) the literature has provided different methods to filter for microstructure noise, including resampling and pre-averaging techniques. Microstructure noise is only present at very high frequency data and has been shown to dissipate at five-minute or lower sampling frequencies. Resampling techniques are based on these findings and choose a resampling frequency accordingly. Pre-averaging techniques, in contrast, average prices over a window whose length increases with the number of intraday observations. While pre-averaging was initially applied on non-overlapping windows (Podolskij and Vetter 2009), procedures have been recently developed to allow for a moving-average window (Jacod et al. 2009) that limits the number of observations that are lost. It is important to select a proper filtration, as not all filtration methods are compatible for all statistical analyses. We apply two main techniques in our article, the Lee and Mykland (2008) jump test and the Christensen et al. (2014) method to identify the bipower and jump variation components of realized variance. Below we describe the microstructure noise cleaning methods that we use, depending on the technique applied.

3.8.1.1. Lee and Mykland’s (2012) pre-averaging approach

Lee and Mykland’s (2008) test assumes returns to have stationary and independent increments. Non-overlapping pre-averaging techniques yield asymptotically independent returns (Gonçalves, Hounyo and Meddahi 2014), which is not the case with overlapping methods that create moving average structures. Hence, we employ the non-overlapping noise-filtering approach by Lee and Mykland (2012), a technique also applied in recent research for jumps detection (e.g. Brogaard et al. 2018). This method combines resampling and pre-averaging methods. First, autocorrelation in

Second, subsampled prices are averaged within non-overlapping windows of size 𝑀⁡ (see Lee and Mykland 2012). In our study, daily autocorrelation functions of the transactions price identify an average serial correlation of four on average. As a result, we subsample prices every five ticks and we then smooth the subsampled prices over the non-overlapping intervals of 𝑀⁡ = ⁡𝐶⁡⌊𝑁/𝑘⌋1/2 observations each, where N is the number of intraday observations and C is a parameter whose optimal value needs to be identified. We choose the C parameter so that the annualized realized volatility (ARV) of the filtered price is the closest to the ARV at 5-minute sampling frequency. The latter is supposed to reflect the efficient price realized volatility, as 5-minute resampled prices are commonly accepted noise-free prices (Hansen and Lunde, 2006; Wu et al., 2015). The distance between the two measures is minimized at M = 3.1 (see figure 3.6. below). Note that the resampling and non-overlapping pre-averaging results yield to noise-filtered prices observed, on average, every 30 to 60 seconds.

Figure 3.6. Comparison of annualized realized volatilities for prices subsampled every 5-minutes and for the noise-filtered tick transaction prices

Note: The 𝐴𝑅𝑉𝑐 is an average over the different days of the annualized daily realized volatility (squared root of ∑𝑁𝑖=1(𝑟𝑡_𝑖)2 ) of noise-filtered prices. The benchmark ARV 5-min is constructed using 5-minute sampled transaction

3.8.1.2. Christensen et al.’s (2014) pre-averaging approach

To identify the bipower and jump variation components of realized variance, Christensen et al. (2014) pre-average observed prices in an overlapping local neighborhood of K observations.

𝑟̂_{𝑡𝑖,𝐾}= ⁡1 𝐾( ∑ 𝑃̂𝑡𝑖+𝑗−⁡ 𝐾−1 𝑗=𝐾/2 ⁡ ∑ 𝑃̂_{𝑡𝑖+𝑗}⁡ 𝐾 2−1 𝑗=0 )

where 𝐾 = ⁡𝜃√𝑁 + 𝑜(𝑁−1⁄4). Their asymptotics are based on overlapping pre-averaging techniques and we thus follow their filtering approach. Note the similarities between 𝑀⁡ = ⁡𝐶⁡⌊𝑁/𝑘⌋1/2 _{and K; the main difference between the two parameters is k which corresponds to the} observations lost due to autocorrelation resampling. The 𝜃 parameter is chosen so that the annualized realized volatility (ARV) of the filtered price is the closest to the ARV at 5-minute sampling frequency. We find that ⁡ = ⁡0.4 is the optimal value to obtain a noised filtered price series (see figure 3.7.).

Figure 3.7. Robustness check on 𝜃 and M regarding the annualized realized volatility.

Notes: The 𝐴𝑅𝑉𝑐

is an average over the different days of the annualized squared root of

𝑁 𝑁−𝐾+2⁡ 1 𝐾𝜓_𝐾⁡∑ (𝑟̂𝑡𝑖,𝐾) 2 𝑁−𝐾+1 𝑖=0 − ⁡ 𝜔̂ 2

3.8.2. Robustness analysis of the jump test results with a longer window size: 𝑊 = 185.

Figure 3.8. Distribution of intraday jumps out across intraday time intervals in each period with 𝑊 = 185, January 15, 2008 – December 4, 2015.

With 𝑊 = 185 (which is approximately equivalent to using the data over the preceding 100 minutes), a total 259 days have at least one jump, and a total of 439 jumps are detected. On announcement days, the average number of jumps per day is 0.20 in period 1, 0.67 in period 2, and 3.12 in period 3; while on non-announcement days, the average number of jumps per day is 0.10, 0.12, and 0.24 in periods 1, 2, and 3, respectively.

0 1 0 2 0 3 0 4 0 5 0 Period 1 % 09:30:00−10:29:59 11:30:00−12:29:59 36.5 20.9 23.5 19.1 0 1 0 2 0 3 0 4 0 5 0 Period 2 intraday intervals % 07:30:00−09:29:59 11:30:00−12:29:59 17.4 30.4 21.7 17.4 13.1 0 1 0 2 0 3 0 4 0 5 0 Period 3 % 08:30:00−09:29:59 11:30:00−12:29:59 7.7 16.8 53.5 8.1 13.9

CHAPTER 4