### 统计代写|金融统计代写financial statistics代考| Jump testing

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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|Barndorff-Nielsen and Shephard test

To test for the existence of jumps in the sample path of asset prices, Barndorff-Nielsen and Shephard (2006) propose nonparametric Hausman (1978) type tests using the difference between Realized Quadratic Variation, an estimator of integrated volatility which is not robust to jumps, and $B P V$, which is a jump robust estimator of integrated volatility. Realized Quadratic Variation is considered to be the same as $R V$. The adjusted jump ratio test statistic can be given as:
$$B N S=\frac{M^{1 / 2}}{\sqrt{\vartheta \max \left(1, \frac{Q P V}{\left(\mu_{1}^{2} B P V\right)^{2}}\right)}}\left(1-\frac{B P V}{R V}\right) \stackrel{d}{\rightarrow} N(0,1)$$
where $B P V$ is the same as in $(13), R V$ is the same as in $(12), \theta=\left(\left(\pi^{2} / 4\right)+\right.$ $\pi-5) \approx 0.6090$. The realized quadpower variation $Q P V$ is used to estimate integrated quarticity $\left(\int_{0}^{t} \sigma_{s}^{4} d s\right)$ and can be given as:
$$Q P V=M \sum_{j=4}^{M}\left|\Delta_{j} X\left|\Delta_{j-1} X\right| \Delta_{j-2} X | \Delta_{j-3} X\right| \stackrel{d}{\rightarrow} \mu_{1}^{4} \int_{0}^{t} \sigma_{s}^{4} d s$$
The authors show that the null hypothesis of no jumps is rejected if the test statistic Barndorff-Nielsen and Shephard test (BNS) is significantly positive.

## 统计代写|金融统计代写financial statistics代考|Lee and Mykland test

Lee and Mykland (2007) use the ratio of realized return to estimated instantaneous volatility, and further construct a nonparametric jump test to detect the exact timing of jumps at the intra-day level. The test statistic which identifies whether there is a jump during $(t+j / M, t+(j+1) / M]$ can be given as:
16 PART | I Finance
$$L_{(t+(j+1) / M)}=\frac{X_{t+(j+1) / M}-X_{t+j / M}}{\sigma_{t+(j+1) / M}}$$
where
$$\sigma_{t+\widetilde{(j+1)} / M^{2}} \equiv \frac{1}{K-2} \sum_{i=j-K+1}^{j-2}\left|X_{t+(i+1) / M}-X_{t+i / M}\right|\left|X_{t+i / M}-X_{t+(i-1) / M}\right|$$
Here $K$ is the window size of a local movement of the process. It is chosen in a way such that the effect of jumps on volatility estimation is eliminated. The authors suggest a value of $K=10$ when the sampling frequency is $5 \mathrm{~min}$. Thus, it can be asymptotically shown that
$$\frac{\max {j \in \bar{A}{M}}\left|L_{(t+(j+1) / M)}\right|-C_{M}}{S_{M}} \rightarrow \varepsilon, \text { as } \Delta t \rightarrow 0$$
where $\varepsilon$ has a cumulative distribution function $P(\varepsilon \leq x)=\exp \left(-e^{-x}\right)$,
$$C_{M}=\frac{(2 \log M)^{1 / 2}}{c}-\frac{\log \pi+\log (\log M)}{2 c(2 \log M)^{1 / 2}} \text { and } s_{M}=\frac{1}{c(2 \log M)^{1 / 2}}$$
$M$ is the number of intra-daily observations, $c \approx 0.7979$ and $\bar{A}_{M}$ is the set of $j$ $\in{0,1, \ldots, M}$ so that there are no jumps in $(t+j / M, t+(j+1) / M]$.

## 统计代写|金融统计代写financial statistics代考|Jiang and Oomen test

Jiang and Oomen (2008) compare a jump sensitive variance measure to $R V$ in order to test for jumps. Their idea is based on the fact that in the absence of jumps the accumulated difference between the simple return and log return (called the swap variance) captures one-half of the integrated volatility in the continuous time limit. Consequently it can be stated, in the absence of jumps the difference between swap variance and $R V$ should be zero, while in the presence of jumps the same difference reflects the replication error of variance swap thus detecting jumps. The swap variance can be given as:
$$S V_{t, M}=2 \sum_{j=1}^{M-1}\left(\Delta_{j} P-\Delta_{j} X\right)$$
where $Y=\log (P)$ and $Y$ is the same as in (1). $\Delta_{j} P=\frac{P_{f+(i+1) / M}}{P_{t+j / M}}-1$ and $\Delta_{j} X$ is the same as in (10). The three different swap variance tests proposed by the authors can be given as:
(i) The difference test:
$$\frac{M}{\Omega_{S V}}\left(S V_{t, M}-R V_{t, M}\right) \stackrel{d}{\rightarrow} N(0,1)$$

(ii) The logarithmic test:
$$\frac{B P V_{t, M} M}{\Omega_{S V}}\left(\log \left(S V_{t, M}\right)-\log \left(R V_{t, M}\right)\right) \stackrel{d}{\rightarrow} N(0,1)$$
(iii) The ratio test:
$$\frac{B P V_{t, M} M}{\Omega_{S V}}\left(1-\frac{R V_{t, M}}{S V_{t, M}}\right) \stackrel{d}{\rightarrow} N(0,1)$$
where $\Omega_{S V}=\frac{\mu_{6} M^{3} \mu_{6 j p}^{-p}}{9 M-p+1} \sum_{j=1}^{M-p} \prod_{k=0}^{p}\left|\Delta_{j+k} X\right|^{6 / p}$ for $p \in{1,2, \ldots}, \mu_{z}=E\left(|x|^{z}\right)$ for $z \sim N(0,1) .$

## 统计代写|金融统计代写financial statistics代考|Lee and Mykland test

Lee 和 Mykland (2007) 使用已实现收益与估计瞬时波动率的比率，并进一步构建了非参数跳跃检验来检测当日水平跳跃的确切时间。确定是否有跳跃的检验统计量(吨+j/米,吨+(j+1)/米]可以给出：
16 PART | 我金融

σ吨+(j+1)~/米2≡1ķ−2∑一世=j−ķ+1j−2|X吨+(一世+1)/米−X吨+一世/米||X吨+一世/米−X吨+(一世−1)/米|

C米=(2日志⁡米)1/2C−日志⁡圆周率+日志⁡(日志⁡米)2C(2日志⁡米)1/2 和 s米=1C(2日志⁡米)1/2

## 统计代写|金融统计代写financial statistics代考|Jiang and Oomen test

Jiang 和 Oomen (2008) 将跳跃敏感方差测量与R在为了测试跳跃。他们的想法是基于这样一个事实，即在没有跳跃的情况下，简单收益和对数收益之间的累积差异（称为掉期方差）在连续时间限制内捕获了综合波动率的一半。因此可以说，在没有跳跃的情况下，互换方差和R在应该为零，而在存在跳跃的情况下，相同的差异反映了方差交换的复制误差，从而检测到了跳跃。互换方差可以表示为：

(i) 差异检验：

(ii) 对数检验：

(iii) 比率测试：

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## MATLAB代写

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