### 统计代写|金融统计代写financial statistics代考|Empirical experiments

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|金融统计代写financial statistics代考|Data description

The empirical experiments are conducted with six stocks and two ETFs. The six individual stocks, which include the Boeing Company (BA), Exxon Mobile Corporation (XOM), Johnson \& Johnson (JNJ), JPMorgan Chase \& Co. (JPM), Microsoft Corporation (MSFT), and Walmart Inc.(WMT), have the highest weight in their corresponding SPDR market sector ETFs such as XLI (industrial sector), XLE (energy sector), XLV (healthcare sector), XLF (finance sector), XLK (technology sector), and XLP (consumer staples sector). The two SPDR sector ETFs chosen are the energy and technology sector ETFs and XLE $\&$ XLK. The dataset is obtained from the Trade and Quote Database (TAQ) of Wharton Research Data Service (WRDS) and it covers the period from January 1, 2006 to December 31, 2013 for a total of 2013 days. We select trade data ranging from 9:30 am to $4 \mathrm{pm}$ on regular trading days. Overnight transactions are excluded from our dataset. We mainly use a 5 -min sampling frequency to eradicate the effect of market microstructure noise in the data which yields 78 total observations per day. We also use a 1-min sampling frequency in specific cases which yields 390 observations per day. It should be noted that all empirical experiments are carried out on the logarithmic values of the stock and ETF prices.

## 统计代写|金融统计代写financial statistics代考|Methodology

Our empirical experiment consists of three sections: (i) integrated volatility measures, (ii) jump tests, and (iii) co-jump tests. For each of the different parts, we conduct analysis involving the most widely used measures and tests, respectively. A detailed description of the different measures and tests used and the empirical methodologies thereof is given as follows.

First, we use six different measures to estimate Integrated Volatility for all the stocks and ETFs: (1) RV (Section 3.1), (2) BPV (Section 3.2), (3) TPV (Section 3.3), (4) TRV (Section 3.7), (5) MedRV, and (6) MinRV (Section 3.11). Second, to test for price jumps in the data three different jump tests are used: (1) ASJ jump test (Section 4.4), (2) BNS jump test (Section 4.1), and (3) LM jump test (Section 4.2). Lastly, co-jump tests are carried out using (1) JT co-jump test (Section 5.2), (2) BLT co-jump test (Section 5.1), and (3) GST co-exceedance rule (Section 5.4).

Estimation of integrated volatility, BNS and LM jump tests as well as all the co-jump tests are carried out using 5 -min data where $\Delta$ is set to $\frac{1}{78}$. However, for the ASJ jump test, both $1-\left(\Delta=\frac{1}{390}\right)$ and 5 -min frequencies are used as a basis for comparative study.

When conducting analysis using jump tests, we calculate the percentage of days identified as having jumps. For both the BNS and ASJ tests, it can be given as:
Percentage of jump days $=\frac{100 \sum_{i=1}^{T} I\left(Z_{i}>c_{\alpha}\right)}{T} \%$
where $I(\cdot)$ is the jump indicator function, $c_{\alpha}$ is the critical value at $\alpha$ significance level and $Z_{i}$ is the BNS or ASJ jump test statistics. For the LM jump test on the other hand it can be derived as:
$$\text { Percentage of jump days }=\frac{100 \sum_{i=0}^{T} I\left(\exists t \in i,\left|L_{t}\right|>c_{\alpha}\right)}{T} \%$$
where $L_{t}$ is the LM jump test statistic at the intra-day level within a particular day, $t$ refers to the 78 intra-day intervals and $c_{\alpha}$ is the critical value at $\alpha$ significance level.

Once jumps are detected, we follow Andersen et al. (2007) and Duong and Swanson (2011) to construct risk measures by separating out the variation due to daily jump component and the continuous components. This is done by using volatility measures $R V$ and $T P V$. It can be given as:

Variation due to jump component $=J V_{t}=\max \left[R V_{t}-T P V_{t}, 0\right] * I_{j u m p, t}$
Consequently the ratio of jump to total variation for all three jump tests can be calculated as:
Ratio of jump variation to total variation $=\frac{J V_{t}}{R V_{t}}$
For BLT co-jump test, the percentage of days identified as having co-jumps is calculated using:

Percentage of co – jump days $=\frac{100 \sum_{i=0}^{T} I\left(\exists j, z_{m c p, i, j}c_{m c p, \alpha, r}\right)}{T} \%$
where $c_{m c p, \alpha, l}$ and $c_{m c p, \alpha, r}$ are left and right tail critical values derived from bootstrapping the null distribution. $\alpha$ is the significance level. For the JT co-jump test, the percentage of days identified as having co-jumps is calculated as:
$$\text { Percentage of co }-\text { jump days }=\frac{100 \sum_{i-0}^{T} I\left(\Phi_{n}^{(d)} \geq c_{n}^{(d)}\right)}{T} \%$$
In the co-exceedance rule proposed by Gilder et al. (2014), we use the BNS jump test and the LM jump test to identify co-jumps. The percentage of days identified as having co-jumps can be given as:

Percentage of co -jump days $=\frac{100 \sum_{i=0}^{T} I\left(\left|Z_{i}\right| \geq \Phi_{\alpha}\right) * I\left(\exists t \in i,\left|L_{t}\right|>c_{\alpha}\right)}{T} \%$
where $Z_{i}$ is the BNS jump test statistic and $L_{t}$ is the LM jump test statistic.
In addition to reporting the findings of our empirical experiment on the entire sample, we also conduct analysis after splitting the data set into two periods. The first sample consists of the period from January 2006 to June 2009 and the second sample consists of the period from July 2009 to December 2012. This is done to inspect whether the jump activity in the stocks and the ETFs changes considerably over time. The break date of our sample (June 2009) roughly corresponds to the end of the business cycle contraction after the financial crisis as given by NBER.

## 统计代写|金融统计代写financial statistics代考|Findings

Table 1 gives the summary statistics for integrated volatility which is estimated using six volatility measures $R V, B P V, T P V$, MedRV, MinRV, and $T R V$. The sample period considered for the six stocks and the two ETFs is January 2006-December 2013 . The mean, standard deviation, minimum, and maximum values are all in terms of $10^{-4}$. Among all the stocks and ETFs, JPMorgan seems to have undergone maximum price fluctuations across the sample period as it displays the highest mean and max values across all the volatility measures. On the other hand Johnson \& Johnson and XLK appear to be tied in terms of having undergone least amount of price fluctuations as they display the lowest mean and max volatility estimates. Among all the volatility measures, $B P V$ reports the lowest mean volatility estimate while $R V$ reports the highest mean volatility estimate for any given stock or ETF. This can be explained by the fact that in the presence of frequent jumps, $R V$ overestimates integrated volatility. To get a clearer idea of how volatility differs across the stocks and ETFs, we turn to Figs. 1 and 2 which display the estimated volatility for the stocks Boeing and Exxon with respect to the six aforementioned volatility measures. Similar figures for four other stocks and two ETFs have not been given for the purpose of brevity and can be provided upon request. In general stocks and ETFs achieve their highest volatility in the

fourth quarter of 2008 during the financial crisis with a few exceptions. For XLE, in case of all four volatility measures apart from $T P V$ and $T R V$, volatility reaches its peak in the second quarter of 2009. For XLK on the other hand, only in case $R V$ the volatility peak is reached in the first quarter of 2008 while for the other measures it is the fourth quarter of 2008 .

We now look at Tables $2-5$ which display the descriptive statistics of the three jump tests. For the ASJ jump test we consider both 5- and 1-min frequencies while for the BNS and the LM jump tests we only consider 5 -min frequency. Panel A in the tables refers to the prefinancial crisis sample period, January 2006-June 2009 and panel B refers to the postcrisis period July 2009December 2012. In case of the ASJ jump tests, we find noticeable differences between 5- (Table 2) and 1-min (Table 3) frequencies. Overall the mean value of the statistics is higher for the 1-min data compared to the $5-\mathrm{min}$ frequency suggesting that more jumps would be identified in the 1-min case. The skewness values are all negative irrespective of the sample period, type of stock and frequency of sampling suggesting that the ASJ test statistics are leftskewed. Panel A for both frequencies appear to have overall higher mean and max values again suggesting more jump activity in the financial crises period.

## 统计代写|金融统计代写financial statistics代考|Methodology

跳跃天数百分比 =100∑一世=0吨一世(∃吨∈一世,|大号吨|>C一种)吨%

百分比 − 跳天 =100∑一世−0吨一世(披n(d)≥Cn(d))吨%

## 统计代写|金融统计代写financial statistics代考|Findings

2008 年第四季度在金融危机期间，除了少数例外。对于 XLE，在所有四种波动性措施的情况下，除了吨磷在和吨R在，波动性在 2009 年第二季度达到顶峰。另一方面，对于 XLK，只有以防万一R在波动性峰值在 2008 年第一季度达到，而其他指标则是在 2008 年第四季度。

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