商科代写|计量经济学代写Econometrics代考|A New Approach to Investigate Cyclical Dependence in Economic Time Series

statistics-lab™ 为您的留学生涯保驾护航 在代写计量经济学Econometrics方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写计量经济学Econometrics代写方面经验极为丰富，各种代写计量经济学Econometrics相关的作业也就用不着说。

• 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

商科代写|计量经济学代写Econometrics代考|Harmonic Regression Models and Laplace Periodograms

For a given time series $\left{Y_{t}\right}$ of length $n$ and its frequency $\omega \in(0, \pi)$, the ordinary periodogram is defined as
$$G_{n}(\omega):=\frac{1}{n}\left|\sum_{t=1}^{n} Y_{t} \exp (-i t \omega)\right|^{2}$$

In the above equation, if $\omega=2 \pi k / n$, where $k$ is a certain integer, it can also be expressed as
$$G_{n}(\omega)=\frac{1}{4} n\left|\tilde{\boldsymbol{\beta}}{n}(\omega)\right|^{2}=\frac{1}{4} n \tilde{\boldsymbol{\beta}}{n}^{\prime}(\omega) \tilde{\boldsymbol{\beta}}{n}(\omega),$$ where $|\cdot|$ denotes the Euclidian norm, and $\tilde{\boldsymbol{\beta}}{n}(\omega)$ denotes the least squares estimator in the linear model with regressors $\boldsymbol{x}{t}(\omega)=[\cos (\omega t), \sin (\omega t)]^{\prime}$, corresponding to an $L{2}$-projection of the observed series onto the harmonic basis, which are obtained as the solution of the following equation.
$$\left{\tilde{\lambda}{n}(\omega), \tilde{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left(Y_{t}-\lambda-\boldsymbol{x}{t}^{\prime}(\omega) \boldsymbol{\beta}\right)^{2}$$ When the OLS criterion is replaced by the least absolute deviation (LAD) criterion in the harmonic regression, the LAD coefficient $\ddot{\boldsymbol{\beta}}{n}(\omega)$ is obtained as follows:
$$\left{\ddot{\lambda}{n}(\omega), \ddot{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left|Y_{t}-\lambda-\boldsymbol{x}{t}^{\prime}(\omega) \boldsymbol{\beta}\right|$$ By using $\ddot{\boldsymbol{\beta}}{n}(\omega), \mathrm{Li}(2008)$ has defined the Laplace periodogram as
$$L_{n}(\omega):=\frac{1}{4} n\left|\ddot{\boldsymbol{\beta}}{n}(\omega)\right|^{2}$$ Therefore, both $G{n}(\omega)$ and $L_{n}(\omega)$ are obtained by the squared norm (or sum of squares) of harmonic regression coefficients multiplied by some constant terms. In particular, the Laplace periodogram inherits the robustness properties of linear LAD regression. Just as the OLS estimator is used to characterize the sample mean, the LAD estimator applied captures the behavior of the observation around the median $(0.5$ quantile)
$\mathrm{Li}(2008)$ has derived the asymptotic normality and useful related theorems of the Laplace periodogram, which are very useful to consider asymptotic behaviors of several periodograms. His results are based on the concept of zero-crossings.

Definition (Stationarity in zero-crossings) The lagged zero-crossing rate of a random process $\left{\varepsilon_{t}\right}$ between $t$ and $s$ is defined as $\gamma_{t s}:=P\left(\varepsilon_{t} \varepsilon_{s}<0\right)$, and $\left{\varepsilon_{t}\right}$ is called to be stationary in zero-crossings if and only if $\gamma_{t s}$ depends only on $t-s$, that is, $\gamma_{t s}=\gamma_{t-s}$ for all $t$ and $s . \gamma_{\tau}$ is called as the lag-zero-crossing rate of $\left{\varepsilon_{t}\right}$ and $S(\omega):=\sum_{\tau=-\infty}^{\infty}\left(1-2 \gamma_{\tau}\right) \cos (\omega \tau)$ is called as the zero-crossing spectrum of $\left{\varepsilon_{t}\right}$.

商科代写|计量经济学代写Econometrics代考|Sample and Smoothed Laplace Periodogram

Define the following new variable of interest called a quantile crossing indicator:
$$V_{t}(\tau, q(\tau))=\tau-I\left{Y_{t}<q(\tau)\right}$$
If the distribution function of $Y_{t}$ is continuous and increasing at
$$q(\tau):=\inf {y: P(Y \leq y)}$$
$V_{t}(\tau)$ is bounded, stationary and mean zero random variable. Using Koenker and Basset’s approach, we define an estimate of $V_{t}(\tau)$ as follows:
$$\widehat{V}{t}(\tau)=V{t}\left(\tau, \hat{q}_{n}(\tau)\right)$$

where $\hat{q}{n}(\tau)=\operatorname{argmin}{q \in \mathbb{R}} \sum_{t=1}^{n} \rho_{\tau}\left(Y_{t}-y\right), \rho_{\tau}(x)=x{\tau-I(x<0)} . \hat{q}{n}(\tau)$ is the estimate of the $\tau$ th quantile. The $\tau$ th quantile periodogram is given by $$Q{n, \tau}(\omega):=\frac{1}{2 \pi}\left|\frac{1}{\sqrt{n}} \sum_{t=1}^{n} \widehat{V}{t}(\tau) \mathrm{e}^{-i t \omega}\right|=\frac{1}{2 \pi} \sum{|j|<n} \hat{r}{n, \tau}(j) \cos (\omega j)$$ where $i^{2}=1$ and $\hat{r}{u, r}(j)=\frac{1}{n} \sum_{t}^{n}|j|+1 \widehat{V}{l}(\tau) \widehat{V}{t-|/|}(\tau), \quad|j| \propto n . Q_{n, r}(\omega)$ is an unbiased estimate of the $\tau$ th spectral density, but is not consistent. A consistent estimator is obtained by smoothing the periodogram using kernel functions (all the results below are taken from Hagemann 2013).
We obtained a smoothed $\tau$ th quantile periodogram as
$$\widehat{Q}{n, \tau}(\omega)=\frac{1}{2 \pi} \sum{|j|<n} \lambda\left(j / b_{n}\right) \hat{r}{n, \tau}(j) \cos (\omega j)$$ $\lambda\left(j / b{n}\right)$ is a lag window and $b_{n}$ is a bandwidth parameter. It is known from the literature on spectral analysis that an optimal lag window leading a non-negative periodogram is the so-called quadratic spectral window defined as
$$\lambda_{Q S}(x)=\frac{25}{12 \pi^{2} x^{2}}\left{\frac{\sin \left(\frac{6 \pi x}{5}\right)}{\frac{6 \pi x}{5}}-\cos \left(\frac{6 \pi x}{5}\right)\right}$$

商科代写|计量经济学代写Econometrics代考|Copula-Based Periodogram and Rank-Based Laplace Periodogram

Laplace periodograms can be used to estimate copula spectra density kernels. We briefly present the methodology here since copula models have become widely used in economics and finance (see Patton 2012 for a review of theory and empirical estimation). One important advantage of copulas is that they do not require any distributional assumption, such as for instance the existence of finite moments.
Let us consider again a strictly stationary time series $\left{Y_{t}\right}_{t \in \mathbb{Z}}$ and its marginal distribution function $F$. In the traditional approach, the spectral density kernels are defined associated with autocovariance kernels of the series. To capture more general features of pairs of $Y_{t}$ and $Y_{t-k}$, the clipped processes $\left(I\left{Y_{t} \leq q\right}\right){t \in \mathbb{Z}}$ and $\left(I\left{U{t} \leq \tau\right}\right){t \in \mathbb{Z}}$, where $U{t}:=F\left(Y_{t}\right)$ are introduced; then, the spectral density kernels are defined associated with covariance kernels of these clipped processes, which are shown below.
$$\gamma_{k}\left(q_{1}, q_{2}\right):=\operatorname{Cov}\left(I\left{Y_{t} \leq q_{1}\right}, I\left{Y_{t-k} \leq q_{2}\right}\right), \quad q_{1}, q_{2} \in \overline{\mathbb{R}}, k \in \mathbb{Z}$$

where $I{\cdot}$ denotes the indicator function and $\overline{\mathbb{R}}:=\mathbb{R} \cup{-\infty, \infty}$ the extended real line. The definition descriked above is the I añlace cross-covariance. The copula cross-covariance is
$$\gamma_{k}^{U}\left(\tau_{1}, \tau_{2}\right):=\operatorname{Cov}\left(I\left{U_{t} \leq \tau_{1}\right}, I\left{U_{t-k} \leq \tau_{2}\right}\right), \quad \tau_{1}, \tau_{2} \in[0,1], k \in \mathbb{Z}$$
By using the Laplace cross-covariance and the copula cross-covariance, researchers can consider more general dependence structures of $Y_{t}$ and $Y_{t-k}$ that traditional covariance-based methods unable to deal with, such as time-irreversibility, tail dependence, varying conditional skewness or kurtosis, and so on, though various extensions and revisions have been proposed in the $L_{2}$-periodograms (Kleiner et al. 1979; Klüppelberg and Mikosch 1994; Mikosch 1998; Katkovnik 1998; Hong 1999; Hill and McCloskey 2014).

Under summability conditions on $\gamma_{k}$ and $\gamma_{k}^{U}$, the population spectral densities are defined as follows.
$$\begin{gathered} f_{q_{1}, q_{2}}(\omega):=\frac{1}{2 \pi} \sum_{k=-\infty}^{\infty} \gamma_{k}\left(q_{1}, q_{2}\right) \mathrm{e}^{-i k \omega}, \quad q_{1}, q_{2} \in \overline{\mathbb{R}}, \omega \in \mathbb{R}, \ f_{q_{t_{1}}, q_{\tau_{2}}}(\omega):=\frac{1}{2 \pi} \sum_{k=-\infty}^{\infty} \gamma_{k}^{U}\left(\tau_{1}, \tau_{2}\right) \mathrm{e}^{-i k \omega}, \quad \tau_{1}, \tau_{2} \in[0,1], \omega \in \mathbb{R}, \end{gathered}$$

商科代写|计量经济学代写Econometrics代考|Harmonic Regression Models and Laplace Periodograms

Gn(ω):=1n|∑吨=1n是吨经验⁡(−一世吨ω)|2

Gn(ω)=14n|b~n(ω)|2=14nb~n′(ω)b~n(ω),在哪里|⋅|表示欧几里得范数，并且b~n(ω)表示带有回归量的线性模型中的最小二乘估计量X吨(ω)=[因⁡(ω吨),罪⁡(ω吨)]′，对应于大号2-将观察到的序列投影到谐波基础上，作为以下方程的解获得。

\left{\tilde{\lambda}{n}(\omega), \tilde{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \ mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left(Y_{t}-\lambda-\boldsymbol{x}{t}^ {\prime}(\omega) \boldsymbol{\beta}\right)^{2}\left{\tilde{\lambda}{n}(\omega), \tilde{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \ mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left(Y_{t}-\lambda-\boldsymbol{x}{t}^ {\prime}(\omega) \boldsymbol{\beta}\right)^{2}当OLS准则被调和回归中的最小绝对偏差（LAD）准则代替时，LAD系数b¨n(ω)得到如下：

\left{\ddot{\lambda}{n}(\omega), \ddot{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \ mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left|Y_{t}-\lambda-\boldsymbol{x}{t}^ {\prime}(\omega) \boldsymbol{\beta}\right|\left{\ddot{\lambda}{n}(\omega), \ddot{\boldsymbol{\beta}}{n}(\omega)\right}:=\operatorname{argmin}{\lambda \in \ mathbb{R}, \beta \in \mathbb{R}^{2}} \sum{t=1}^{n}\left|Y_{t}-\lambda-\boldsymbol{x}{t}^ {\prime}(\omega) \boldsymbol{\beta}\right|通过使用b¨n(ω),大号一世(2008)已将拉普拉斯周期图定义为

商科代写|计量经济学代写Econometrics代考|Sample and Smoothed Laplace Periodogram

V_{t}(\tau,q(\tau))=\tau-I\left{Y_{t}<q(\tau)\right}V_{t}(\tau,q(\tau))=\tau-I\left{Y_{t}<q(\tau)\right}

q(τ):=信息是:磷(是≤是)

\lambda_{Q S}(x)=\frac{25}{12 \pi^{2} x^{2}}\left{\frac{\sin \left(\frac{6 \pi x}{5} \right)}{\frac{6 \pi x}{5}}-\cos \left(\frac{6 \pi x}{5}\right)\right}\lambda_{Q S}(x)=\frac{25}{12 \pi^{2} x^{2}}\left{\frac{\sin \left(\frac{6 \pi x}{5} \right)}{\frac{6 \pi x}{5}}-\cos \left(\frac{6 \pi x}{5}\right)\right}

商科代写|计量经济学代写Econometrics代考|Copula-Based Periodogram and Rank-Based Laplace Periodogram

\gamma_{k}\left(q_{1}, q_{2}\right):=\operatorname{Cov}\left(I\left{Y_{t} \leq q_{1}\right}, I\左{Y_{tk} \leq q_{2}\right}\right), \quad q_{1}, q_{2} \in \overline{\mathbb{R}}, k \in \mathbb{Z}\gamma_{k}\left(q_{1}, q_{2}\right):=\operatorname{Cov}\left(I\left{Y_{t} \leq q_{1}\right}, I\左{Y_{tk} \leq q_{2}\right}\right), \quad q_{1}, q_{2} \in \overline{\mathbb{R}}, k \in \mathbb{Z}

\gamma_{k}^{U}\left(\tau_{1}, \tau_{2}\right):=\operatorname{Cov}\left(I\left{U_{t} \leq \tau_{1 }\right}, I\left{U_{tk} \leq \tau_{2}\right}\right), \quad \tau_{1}, \tau_{2} \in[0,1], k \在 \mathbb{Z}\gamma_{k}^{U}\left(\tau_{1}, \tau_{2}\right):=\operatorname{Cov}\left(I\left{U_{t} \leq \tau_{1 }\right}, I\left{U_{tk} \leq \tau_{2}\right}\right), \quad \tau_{1}, \tau_{2} \in[0,1], k \在 \mathbb{Z}

Fq1,q2(ω):=12圆周率∑ķ=−∞∞Cķ(q1,q2)和−一世ķω,q1,q2∈R¯,ω∈R, Fq吨1,qτ2(ω):=12圆周率∑ķ=−∞∞Cķ在(τ1,τ2)和−一世ķω,τ1,τ2∈[0,1],ω∈R,

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。