### 统计代写|金融统计代写financial statistics代考|Aı¨t-Sahalia and Jacod test

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## 统计代写|金融统计代写financial statistics代考|Aı¨t-Sahalia and Jacod test

In Aitt-Sahalia et al. (2009), the authors develop a testing methodology for jumps in the (log) price process by comparing two higher order realized power variations with different sampling intervals, $k \Delta$ and $\Delta$, respectively. In this context $\Delta=\frac{1}{M}, M$ is the number of intra-daily observations and $k$ is a given integer. The $p$ th order realized power variation can be given as
$$\hat{B}(p, \Delta)=\sum_{j=1}^{M-1}\left|X_{t+(j+1) / M}-X_{t+j / M}\right|^{p}$$
The ratio of the two realized power variations with different sampling intervals takes the following form
$$\hat{S}(p, k, \Delta)=\frac{\hat{B}(p, k \Delta)}{\hat{B}(p, \Delta)}$$
The corresponding jump test statistic can then be defined as:
$$A S J=\frac{k^{(p / 2)-1}-\hat{S}(p, k, \Delta)}{\sqrt{V_{t, M}}} \stackrel{d}{\rightarrow} N(0,1)$$
where $V_{t, M}$ can be estimated using either a truncation technique as in
$$\widehat{V}{t, M}=\Delta \frac{\hat{A}(2 p, \Delta) M(p, k)}{\hat{A}(p, \Delta)^{2}}$$ where $$\hat{A}(2 p, \Delta)=\frac{\Delta^{1-p / 2}}{\mu{p}} \sum_{j=1}^{M-1}\left|X_{t+(j+1) / M}-X_{t+j / M}\right|^{p} 1_{\left{\left|X_{l+(j+1) / M}-X_{l+j / M}\right| \leq \alpha \Delta^{w}\right}}$$

or using multipower variation as in
$$\hat{V}{t, M}=\Delta \frac{M(p, k) \bar{A}(p /([p]+1), 2[p]+2, \Delta)}{\bar{A}(p /([p]+1),[p]+1, \Delta)^{2}}$$ where $$\begin{gathered} \bar{A}(r, q, \Delta)=\frac{\Delta^{1-q r / 2} M-q+1 q-1}{\mu{r}^{q}} \sum_{j=q} \prod_{i=0}\left|X_{t+(j+i) / M}-X_{t+(j+i-1) / M}\right|^{r}, \ M(p, k)=\frac{1}{\mu_{p}^{2}}\left(k^{p-2}(1+k) \mu_{2 p}+k^{p-2}(k-1) \mu_{p}^{2}-2 k^{p / 2-1}-\mu_{k, p}\right) \end{gathered}$$
and $\mu_{r}=E\left(|U|^{r}\right)$ and $\mu_{k, p}=E\left(|U|^{p}\left|U+\sqrt{\left.(k-1) V\right|^{p}}\right|\right)$ for $U, V \sim N(0,1)$. The null hypothesis of no jumps is rejected when the test statistic ASJ is significantly positive.

## 统计代写|金融统计代写financial statistics代考|Podolskij and Ziggel (PZ) test

In Podolskij and Ziggel $(2010)$ the concept of truncated power variation is used to construct test statistics which diverge to infinity if jumps are present and have a normal distribution otherwise. The jump testing procedure in this chapter is valid (under weak assumptions) for all semimartingales with absolute continuous characteristics and general models for the noise processes. The methodology followed by the authors is a modification of that proposed in Mancini (2009). In particular they consider
$$T(X, p)=M^{\frac{p-1}{2}} \sum_{j=1}^{M-1}\left|X_{t+(j+1) / M}-X_{t+j / M}\right|^{p}\left(1-\eta_{i} 1_{\left{\left|X_{t+(j+1) / M}-X_{t+j / M}\right| \leq \alpha \Delta^{w}\right}}\right)$$
where $\left{\eta_{i}\right}_{i \in[1,1 / \Delta]}$ is a sequence of positive i.i.d random variables. The test statistic has the following form
$$P Z=\frac{T(X, p)}{V a r^{*}(\eta) \hat{A}(2 p, \Delta)} \stackrel{d}{\rightarrow} N(0,1)$$
where $\hat{A}(2 p, \Delta)$ is the same as in (53).

## 统计代写|金融统计代写financial statistics代考|Corradi, Silvapulle, and Swanson test

Building on previous work by Aït-Sahalia (2002) and Corradi et al. (2018) design “long time span” jump tests based on realized third moments or “tricity” for the null hypothesis that the probability of a jump is zero. This jump testing methodology is used to detect jumps by examining the “jump intensity” parameter in the data generating process rather than realized jumps over a “fixed time span.” This test is of immense value when one is interested

in using jump diffusion processes for valuation problems like options pricing and default modeling. Let,
\begin{aligned} \hat{\mu}{3, T, \Delta}=& \frac{1}{T} \sum{j=1}^{n-1}\left(X_{t+(j+1) / M}-X_{t+j / M}-\frac{X_{t+n / M}-X_{t+1 / M}}{n}\right)^{3} \ &-\frac{1}{T^{+}} \sum_{j=1}^{n^{+}-1}\left(X_{t+(j+1) / M}-X_{t+j / M}-\frac{X_{t+n^{+} / M}-X_{t+1 / M}}{n^{+}}\right)^{3} \ & 1\left{\left|X_{t+(j+1) / M}-X_{t+j / M}\right| \leq \tau(\Delta)\right} \end{aligned}
where we have $n^{+}$observations over an increasing time span of $T^{+}$, a shrinking discrete sampling interval $\Delta=\frac{1}{M}$, so that $n^{+}=\frac{T^{+}}{\Delta}, T^{+} \rightarrow \infty$ and $\Delta \rightarrow 0 . \tau(\Delta)$ is the truncation parameter and one example for the choice of such truncation can be given as follows. If $\sigma_{s}$ as in (1) is a square root process, so that all moments exist, we can set $\tau(\Delta)=c \Delta^{\eta}$ with $\frac{2}{7}<\eta<\frac{1}{2}$. The authors define $n=\frac{T}{\Delta}=n^{+}-\frac{T^{+}-T}{\Delta}$, with $T^{+}>T$ and $\frac{T^{+}}{T} \rightarrow \infty$. Then, the test statistic for the null hypothesis of no jumps can be given as
$$C S S=\frac{T^{1 / 2}}{\Delta} \hat{\mu}{3, T, \Delta} \stackrel{d}{\rightarrow} N\left(0, \omega{0}\right)$$
where $\omega_{0}$ is defined in Corradi et al. (2018). Since, under the alternative hypothesis of positive jump intensity, the variance of the statistic is of larger order, it is difficult to construct a variance estimator which is consistent under all hypotheses. The authors use a threshold variance estimator, which removes the contribution of the jump component thus developing an estimator for the variance of Corradi, Silvapulle, and Swanson test $(C S S)$ which is consistent under the null hypothesis of no jumps. Thus we have
\begin{aligned} \hat{\sigma}{C S S}^{2}=& \frac{1}{\Delta^{2}} \sum{j=0}^{n-1}\left(X_{t+(j+1) / M}-X_{t+j / M}-\frac{X_{t+n / M}-X_{t+1 / M}}{n}\right)^{3} \ & 1\left{\left|X_{t+(j+1) / M}-X_{t+j / M}\right| \leq \tau(\Delta)\right} \end{aligned}
Thus the $t$-statistic version of the jump test is
$$t_{C S S}=\frac{C S S}{\hat{\sigma}_{C S S}}$$

## 统计代写|金融统计代写financial statistics代考|Podolskij and Ziggel (PZ) test

T(X, p)=M^{\frac{p-1}{2}} \sum_{j=1}^{M-1}\left|X_{t+(j+1) / M}-X_ {t+j / M}\right|^{p}\left(1-\eta_{i} 1_{\left{\left|X_{t+(j+1) / M}-X_{t+j / M}\right| \leq \alpha \Delta^{w}\right}}\right)T(X, p)=M^{\frac{p-1}{2}} \sum_{j=1}^{M-1}\left|X_{t+(j+1) / M}-X_ {t+j / M}\right|^{p}\left(1-\eta_{i} 1_{\left{\left|X_{t+(j+1) / M}-X_{t+j / M}\right| \leq \alpha \Delta^{w}\right}}\right)

## 统计代写|金融统计代写financial statistics代考|Corradi, Silvapulle, and Swanson test

\begin{aligned} \hat{\mu}{3, T, \Delta}=& \frac{1}{T} \sum{j=1}^{n-1}\left(X_{t+(j +1) / M}-X_{t+j / M}-\frac{X_{t+n / M}-X_{t+1 / M}}{n}\right)^{3} \ &- \frac{1}{T^{+}} \sum_{j=1}^{n^{+}-1}\left(X_{t+(j+1) / M}-X_{t+j / M}-\frac{X_{t+n^{+} / M}-X_{t+1 / M}}{n^{+}}\right)^{3} \ & 1\left{\left |X_{t+(j+1) / M}-X_{t+j / M}\right| \leq \tau(\Delta)\right} \end{对齐}\begin{aligned} \hat{\mu}{3, T, \Delta}=& \frac{1}{T} \sum{j=1}^{n-1}\left(X_{t+(j +1) / M}-X_{t+j / M}-\frac{X_{t+n / M}-X_{t+1 / M}}{n}\right)^{3} \ &- \frac{1}{T^{+}} \sum_{j=1}^{n^{+}-1}\left(X_{t+(j+1) / M}-X_{t+j / M}-\frac{X_{t+n^{+} / M}-X_{t+1 / M}}{n^{+}}\right)^{3} \ & 1\left{\left |X_{t+(j+1) / M}-X_{t+j / M}\right| \leq \tau(\Delta)\right} \end{对齐}

C小号小号=吨1/2Δμ^3,吨,Δ→dñ(0,ω0)

\begin{对齐} \hat{\sigma}{C S S}^{2}=& \frac{1}{\Delta^{2}} \sum{j=0}^{n-1}\left(X_ {t+(j+1) / M}-X_{t+j / M}-\frac{X_{t+n / M}-X_{t+1 / M}}{n}\right)^{3 } \ & 1\left{\left|X_{t+(j+1) / M}-X_{t+j / M}\right| \leq \tau(\Delta)\right} \end{对齐}\begin{对齐} \hat{\sigma}{C S S}^{2}=& \frac{1}{\Delta^{2}} \sum{j=0}^{n-1}\left(X_ {t+(j+1) / M}-X_{t+j / M}-\frac{X_{t+n / M}-X_{t+1 / M}}{n}\right)^{3 } \ & 1\left{\left|X_{t+(j+1) / M}-X_{t+j / M}\right| \leq \tau(\Delta)\right} \end{对齐}

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