### 统计代写|金融统计代写financial statistics代考|Bubbles and crises

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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|The Augmented Dickey–Fuller test

It is well known in the unit root literature that the limit distribution of the ADF statistic depends on both the null hypothesis and the precise regression model specification. ${ }^{c}$ Appropriate choices of both therefore have a material impact in practical implementation.

The null hypothesis $\left(H_{0}\right)$ of the PSY test captures normal market behaviors and states that asset prices follow a martingale process with a mild drift function such that (Phillips et al., 2014)
$$y_{t}=g_{T}+y_{t-1}+u_{t},$$
where $g_{T}=k T^{-\gamma}$ (with constant $k, \gamma>1 / 2$, and sample size $T$ ) captures any mild drift that may be present in prices but which is of smaller order than the martingale component and is therefore asymptotically negligible.
The regression model chosen for the PSY procedure is
$$\Delta y_{t}=\mu+\rho y_{t-1}+\sum_{j=1}^{p} \phi_{j} \Delta y_{t-j}+v_{t},$$

where for implementation purposes the regression error $v_{t}$ is assumed to satisfy $v_{t}^{i . i . d} \sim\left(0, \sigma^{2}\right)$. The $p$ lag terms of $\Delta y_{t}$ are included to take care of potential serial correlation. The lag order $p$ is often selected by information criteria. The regression model includes an intercept but no time trend and nests the null hypothesis as a special case with $\mu=g_{T}$ and $\rho=0$. The ADF statistic is simply the $t$-ratio of the least squares estimate of the coefficient $\rho$.

The i.i.d error condition may be replaced with a martingale difference sequence (mds) condition in (2). More general specifications on the error $u_{t}$ in the generating mechanism (1), such as those in Assumption 1 below, may be employed and are accommodated by allowing the regression lag order $p \rightarrow \infty$ as $T \rightarrow \infty$ in (2). Nonparametric adjustments for serial correlation may also be used, such as those developed in Phillips (1987) and Phillips and Perron (1988).

## 统计代写|金融统计代写financial statistics代考|The Recursive Evolving Algorithm

The recursive evolving algorithm of PSY enables real-time identification of bubbles and crises while allowing for the presence of multiple structural breaks within the sample period. Phillips et al. (2015a,b) show that this algorithm is superior to the forward expanding and rolling window algorithms in bubble identification, especially when the sample period contains multiple bubbles.

For the convenience of exposition, we use the standard “fraction of the total sample” notation for observations. Thus if $t=\lfloor T r\rfloor$ is the integer part of $T r$, then observation $t$ is represented fractionally as observation $r$ and then the total sample runs over values of $r$ from 0 to 1 . Suppose the observation of interest is $r^{\dagger}$. The PSY procedure calculates the ADF statistic recursively from a backward expanding sample sequence. Let $r_{1}$ and $r_{2}$ be the start and end points of the regression sample. The ADF statistic calculated from this sample is denoted by $A D F_{r_{1}}^{r_{2}}$. We fix the end point of all samples on the observation of interest such that $r_{2}=r^{\dagger}$ and allow the start point $r_{1}$ to vary within its feasible range, i.e., $\left[0, r^{\dagger}-r_{0}\right]$, where $r_{0}$ is the minimum window required to initiate the regression. The recommended setting of $r_{0}$ for practical implementation is $r_{0}=0.01$ $+1.8 / \sqrt{T}$. The PSY statistic is the supremum taken over the values of all the ADF statistics in the entire recursion, which is represented mathematically as
$$P S Y_{r^{\dagger}}\left(r_{0}\right)=\sup {r{1} \in\left[0, r^{\dagger}-r_{0}\right], r_{2}=r^{\dagger}}\left{A D F_{r_{1}}^{r_{2}}\right} .$$
The supremum enables the selection of the “optimal” starting point of the regression in the sense of providing the largest ADF statistic.

The PSY test can be conducted for each individual observation of interest ranging from $r_{0}$ to 1 , i.e., for $r^{\dagger} \in\left[r_{0}, 1\right]$. The recursive calculation evolves as the observation of interest moves forward and therefore the procedure is called a recursive evolving algorithm. See Fig. 1 for a graphical illustration of the algorithm. The corresponding PSY statistic sequence is $\left{P S Y_{r^{+}}(r 0)\right}_{r^{*} \in[r 0,1]}$.

## 统计代写|金融统计代写financial statistics代考|The Rationale

To illustrate the idea of bubble identification, consider the present value asset price formula
$$P_{t}=\sum_{i=0}^{\infty}\left(\frac{1}{1+r_{f}}\right)^{i} \mathbb{E}{t}\left(D{t+i}\right)+B_{t},$$
where $P_{t}$ is the price of the asset, $D_{t}$ is the payoff received from the asset, $r_{f}$ is the risk-free interest rate, $\mathbb{E}{t}(\cdot)$ is the conditional expectation operator given information to time $t$, and $B{t}$ is the bubble component. The bubble component satisfies the submartingale property (Diba and Grossman, 1988)
$$\mathbb{E}{t}\left(B{t+1}\right)=\left(1+r_{f}\right) B_{t} .$$
In the absence of a bubble, the degree of nonstationarity of the asset price is controlled entirely by the dividend series and hence is believed from empirical evidence to be at most $I(1)$. On the other hand, asset prices will be explosive in the presence of a bubble component in formula (7) whenever the initialization $B_{0}>0$ in (8).

Asset price dynamics over the expansionary phase of a bubble period may be modeled in terms of a mildly explosive process (Phillips et al., 2011; Phillips and Magdalinos, 2007; Phillips and Yu, 2009) of the form
$$\log P_{t}=\delta_{T} \log P_{t-1}+u_{t},$$
where the autoregressive coefficient $\delta_{T}=1+c T^{-\eta}$ mildly exceeds unity (with $c>0$ and $\eta \in(0,1))$ and yet still lies in its general vicinity. Detection of a bubble process in the data is therefore equivalent to distinguishing a martingale process of asset prices from a mildly explosive process. This can be achieved by the PSY procedure with null and alternative hypotheses specified as
\begin{aligned} &H_{0}: \mu=g_{T} \text { and } \rho=0, \ &H_{A}: \mu=0 \text { and } \rho>0 . \end{aligned}

## 统计代写|金融统计代写financial statistics代考|The Augmented Dickey–Fuller test

Δ是吨=μ+ρ是吨−1+∑j=1pφjΔ是吨−j+在吨,

iid 错误条件可以替换为 (2) 中的鞅差序列 (mds) 条件。有关错误的更一般规范在吨在生成机制 (1) 中，例如下面的假设 1 中的那些，可以通过允许回归滞后顺序来使用和适应p→∞作为吨→∞在 (2) 中。也可以使用序列相关的非参数调整，例如 Phillips (1987) 和 Phillips and Perron (1988) 开发的那些。

## 统计代写|金融统计代写financial statistics代考|The Recursive Evolving Algorithm

PSY 的递归演化算法能够实时识别泡沫和危机，同时允许在样本期间存在多个结构中断。菲利普斯等人。（2015a，b）表明该算法在气泡识别方面优于前向扩展和滚动窗口算法，尤其是在样本周期包含多个气泡时。

P S Y_{r^{\dagger}}\left(r_{0}\right)=\sup {r{1} \in\left[0, r^{\dagger}-r_{0}\right], r_{2}=r^{\dagger}}\left{A D F_{r_{1}}^{r_{2}}\right} 。P S Y_{r^{\dagger}}\left(r_{0}\right)=\sup {r{1} \in\left[0, r^{\dagger}-r_{0}\right], r_{2}=r^{\dagger}}\left{A D F_{r_{1}}^{r_{2}}\right} 。

PSY 测试可以针对每个感兴趣的观察进行，范围从r0为 1 ，即，对于r†∈[r0,1]. 递归计算随着感兴趣的观察向前移动而进化，因此该过程称为递归进化算法。有关该算法的图解说明，请参见图 1。对应的 PSY 统计序列为\left{P S Y_{r^{+}}(r 0)\right}_{r^{*} \in[r 0,1]}\left{P S Y_{r^{+}}(r 0)\right}_{r^{*} \in[r 0,1]}.

## 统计代写|金融统计代写financial statistics代考|The Rationale

H0:μ=G吨 和 ρ=0, H一种:μ=0 和 ρ>0.

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