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经济代写|计量经济学代写Introduction to Econometrics代考|T-ARDL Model

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经济代写|计量经济学代写Introduction to Econometrics代考|T-ARDL Model

Finally, as an extension to the classical approach, we propose the T-ARDL model. ${ }^4$ The linear ARDL is a classical method used to capture persistence in time series data, and Pesaran et al. (2001) proposed a bounds test to detect cointegration based on the ARDL. An advantage of this method is its ability to determine the presence of cointegration without prior knowledge of the explanatory variables being stationary $(I(0))$ or non-stationary (I(1)). This is a useful feature in studies of bubbles, as economies often experience periods of tranquility and mild bubbles.
Pesaran et al. (2001) proposed five specifications of the ARDL with a different combination of deterministic terms. Here, we use the most popular model in financial research, with an unrestricted constant and no trend. For asset prices $(y)$, we can express this as
$$\Delta y_t=a+c y_{t-1}+\boldsymbol{b} \boldsymbol{x}{t-1}+\sum{i=1}^{p-1} \boldsymbol{d}i^{\prime} \Delta \boldsymbol{z}{t-i}+\boldsymbol{f}^{\prime} \Delta \boldsymbol{x}t+u_t$$ where $a, c, \boldsymbol{b}, \boldsymbol{d}$, and $\boldsymbol{f}$ are the parameters to estimate by the ordinary least squares (OLS) for time $(t=1, \ldots, T)$, and $u_t$ is the residual $\left(u_t \sim N\left(0, \sigma^2\right)\right)$. $\boldsymbol{x}$ is a matrix of explanatory variables and $z=[y, x]$. The appropriate lag length $(p)$ is determined such that it captures the data generating process of $y$. We can study the cointegrated relationship between $y$ and $\boldsymbol{x}$ by analyzing the time series properties of $c y{t-1}+\boldsymbol{b} \boldsymbol{x}_{t-1}$, known as the ECM. We can test the null hypothesis of no ECM $(c=0$ and $\boldsymbol{b}=\mathbf{0})$ by the $F$ test or $c=0$ the $t$ test.

经济代写|计量经济学代写Introduction to Econometrics代考|Data

We obtain quarterly data on housing price-to-rent ratios from the OECD for the Euro area, Japan, the UK, and the USA (Table 1). The maximum sample period is from 1968 to 2018 (nearly 200 observations for each country), and the base year of the data is 2015. Here, the Euro area consists of 15 countries: Greece, France, the Slovak Republic, Italy, Spain, Belgium, Luxembourg, Germany, Portugal, the Netherlands, Finland, Ireland, Austria, Slovenia, and Estonia. (Hereafter, we refer to the Euro area as a country for convenience.) Because financial bubbles are traditionally considered infrequent phenomena, we chose countries with more than 195 observations. In terms of the standard deviations of this ratio, the US housing market is most stable, and the Japanese market is most volatile.
We also obtain housing price indices from the OECD. Housing prices appear to have a positive relationship with the price-to-rent ratios for all countries and exhibit more stable movements compared to prices (Fig. 1). Furthermore, while there are some similarities in the ratios across four countries, they have a declining trend in Japan during the “Lost Decades” (i.e., after 1990). This trend indicates relatively higher inflation in rental properties than houses in Japan and is indeed attributable to the deflation in housing prices according to this figure. This result indicates a weak demand for house purchases during the weak economic conditions of this period. In contrast, there is an increasing trend in the UK ratio from the late 1990s, indicating a housing market boom.

经济代写|计量经济学代写Introduction to Econometrics代考|T-ARDL Model

Pesaran et al.(2001)提出了五种不同确定性术语组合的ARDL规范。在这里，我们使用金融研究中最流行的模型，不受限制的常数，没有趋势。对于资产价格$(y)$，我们可以表示为
$$\Delta y_t=a+c y_{t-1}+\boldsymbol{b} \boldsymbol{x}{t-1}+\sum{i=1}^{p-1} \boldsymbol{d}i^{\prime} \Delta \boldsymbol{z}{t-i}+\boldsymbol{f}^{\prime} \Delta \boldsymbol{x}t+u_t$$其中$a, c, \boldsymbol{b}, \boldsymbol{d}$、$\boldsymbol{f}$为对时间进行普通最小二乘估计的参数$(t=1, \ldots, T)$, $u_t$为残差$\left(u_t \sim N\left(0, \sigma^2\right)\right)$。$\boldsymbol{x}$是解释变量和$z=[y, x]$的矩阵。确定适当的延迟长度$(p)$，以便捕获$y$的数据生成过程。我们可以通过分析$c y{t-1}+\boldsymbol{b} \boldsymbol{x}_{t-1}$的时间序列特性(称为ECM)来研究$y$和$\boldsymbol{x}$之间的协整关系。我们可以通过$F$检验或$c=0$$t$检验来检验无ECM $(c=0$和$\boldsymbol{b}=\mathbf{0})$的零假设。

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。