### 统计代写|应用时间序列分析代写applied time series analysis代考|Moments of a Random Variable

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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|应用时间序列分析代写applied time series anakysis代考|Moments of a Random Variable

The $\ell$-th moment of a continuous random variable $X$ is defined as
$$m_{\ell}^{\prime}=E\left(X^{\ell}\right)=\int_{-\infty}^{\infty} x^{\ell} f(x) d x$$
where ” $E “$ stands for expectation and $f(x)$ is the probability density function of $X$. The first moment is called the mean or expectation of $X$. It measures the central location of the distribution. We denote the mean of $X$ by $\mu_{x}$. The $\ell$-th central moment of $X$ is defined as
$$m_{\ell}=E\left[\left(X-\mu_{x}\right)^{\ell}\right]=\int_{-\infty}^{\infty}\left(x-\mu_{x}\right)^{\ell} f(x) d x$$
provided that the integral exists. The second central moment, denoted by $\sigma_{x}^{2}$, measures the variability of $X$ and is called the variance of $X$. The positive square root, $\sigma_{x}$, of variance is the standard deviation of $X$. The first two moments of a random variable uniquely determine a normal distribution. For other distributions, higher order moments are also of interest.

The third central moment measures the symmetry of $X$ with respect to its mean, whereas the 4th central moment measures the tail behavior of $X$. In statistics, skew ness and kurtosis, which are normalized 3 rd and 4 th central moments of $X$, are often used to summarize the extent of asymmetry and tail thickness. Specifically, the skewness and kurtosis of $X$ are defined as
$$S(x)=E\left[\frac{\left(X-\mu_{x}\right)^{3}}{\sigma_{x}^{3}}\right], \quad K(x)=E\left[\frac{\left(X-\mu_{x}\right)^{4}}{\sigma_{x}^{4}}\right]$$

The quantity $K(x)-3$ is called the excess kurtosis because $K(x)=3$ for a normal distribution. Thus, the excess kurtosis of a normal random variable is zero. A distribution with positive excess kurtosis is said to have heavy tails, implying that the distribution puts more mass on the tails of its support than a normal distribution does. In practice, this means that a random sample from such a distribution tends to contain more extreme values.

In application, skewness and kurtosis can be estimated by their sample counterparts. Let $\left{x_{1}, \ldots, x_{T}\right}$ be a random sample of $X$ with $T$ observations. The sample mean is
$$\hat{\mu}{x}=\frac{1}{T} \sum{t=1}^{T} x_{t}$$
the sample variance is
$$\hat{\sigma}{x}^{2}=\frac{1}{T-1} \sum{t=1}^{T}\left(x_{t}-\hat{\mu}{x}\right)^{2},$$ the sample skewness is $$\hat{S}(x)=\frac{1}{(T-1) \hat{\sigma}{x}^{3}} \sum_{t=1}^{T}\left(x_{t}-\hat{\mu}{x}\right)^{3},$$ and the sample kurtosis is $$\hat{K}(x)=\frac{1}{(T-1) \hat{\sigma}{x}^{4}} \sum_{t=1}^{T}\left(x_{t}-\hat{\mu}_{x}\right)^{4} .$$
Under normality assumption, $\hat{S}(x)$ and $\hat{K}(x)$ are distributed asymptotically as normal with zero mean and variances $6 / T$ and $24 / T$, respectively; see Snedecor and Cochran (1980, p. 78).

## 统计代写|应用时间序列分析代写applied time series anakysis代考| Distributions of Returns

The most general model for the log returns $\left{r_{i t} ; i=1, \ldots, N ; t=1, \ldots, T\right}$ is its joint distribution function:
$$F_{r}\left(r_{11}, \ldots, r_{N 1} ; r_{12}, \ldots, r_{N 2} ; \ldots ; r_{1 T}, \ldots, r_{N T} ; \boldsymbol{Y} ; \boldsymbol{\theta}\right)$$
where $\boldsymbol{Y}$ is a state vector consisting of variables that summarize the environment in which asset returns are determined and $\boldsymbol{\theta}$ is a vector of parameters that uniquely determine the distribution function $F_{r}(.)$. The probability distribution $F_{r}(.)$ governs the stochastic behavior of the returns $r_{i t}$ and $Y$. In many financial studies, the state

vector $\boldsymbol{Y}$ is treated as given and the main concern is the conditional distribution of $\left{r_{i t}\right}$ given $Y$. Empirical analysis of asset returns is then to estimate the unknown parameter $\boldsymbol{\theta}$ and to draw statistical inference about behavior of $\left{r_{i t}\right}$ given some past log returns.

The model in Eq. (1.14) is too general to be of practical value. However, it provides a general framework with respect to which an econometric model for asset returns $r_{i t}$ can be put in a proper perspective.

Some financial theories such as the Capital Asset Pricing Model (CAPM) of Sharpe (1964) focus on the joint distribution of $N$ returns at a single time index $t$ (i.e., the distribution of $\left{r_{1} t, \ldots, r_{N t}\right}$ ). Other theories emphasize the dynamic structure of individual asset returns (i.e., the distribution of $\left{r_{i 1}, \ldots, r_{i T}\right}$ for a given asset i). In this book, we focus on both. In the univariate analysis of Chapters 2 to 7 , our main concern is the joint distribution of $\left{r_{i t}\right}_{t=1}^{T}$ for asset $i$. To this end, it is useful to partition the joint distribution as
\begin{aligned} F\left(r_{i 1}, \ldots, r_{i T} ; \boldsymbol{\theta}\right) &=F\left(r_{i 1}\right) F\left(r_{i 2} \mid r_{1 t}\right) \cdots F\left(r_{i T} \mid r_{i, T-1}, \ldots, r_{i 1}\right) \ &=F\left(r_{i 1}\right) \prod_{t=2}^{T} F\left(r_{i t} \mid r_{i, t-1}, \ldots, r_{i 1}\right) \end{aligned}
This partition highlights the temporal dependencies of the log return $r_{i t}$. The main issue then is the specification of the conditional distribution $F\left(r_{i t} \mid r_{i, t-1,}\right)$-in particular, how the conditional distribution evolves over time. In finance, different distributional specifications lead to different theories. For instance, one version of the random-walk hypothesis is that the conditional distribution $F\left(r_{i t} \mid r_{i, t-1}, \ldots, r_{i 1}\right)$ is equal to the marginal distribution $F\left(r_{i t}\right)$. In this case, returns are temporally independent and, hence, not predictable.

It is customary to treat asset returns as continuous random variables, especially for index returns or stock returns calculated at a low frequency, and use their probability density functions. In this case, using the identity in Eq. (1.9), we can write the partition in Eq. (1.15) as
$$f\left(r_{i 1}, \ldots, r_{i T} ; \boldsymbol{\theta}\right)=f\left(r_{i 1} ; \boldsymbol{\theta}\right) \prod_{t=2}^{T} f\left(r_{i t} \mid r_{i, t-1}, \ldots, r_{i 1}, \boldsymbol{\theta}\right)$$

## MATLAB代写

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