### 商科代写|计量经济学代写Econometrics代考|ECON 7204

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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

## 商科代写|计量经济学代写Econometrics代考|Estimation Method

Since we have some unobserved variables, the starting point is to write the model using a state-space representation. The parameters and unknown variables are then estimated using Kalman filter method. The state-space representation of the model is as follows:
\begin{aligned} X_{t} &=A X_{t-1}+Z_{t}+F_{t} W_{t}: \text { state equation } \ Y_{t} &=\mu_{t}+C_{t}^{\prime} X_{t}+V_{t}: \text { measurement equation } \end{aligned}
$X_{t}$ is the vector of $k_{1}$ state variables (unobserved), $Y_{t}$ is the vector of $k_{2}$ observed variables, A is a $k_{1} \times k_{1}$ matrix, $Z_{I}$ is a $k_{1} \times 1$ vector of deterministic terms, $W_{t}$ is a $r_{1} \times 1$ vector of residuals, $F_{t}$ is a $k_{1} \times r_{1}$ matrix, $\mu_{t}$ is the product of a $k_{2} \times n e x p l$ matrix of coefficients by a vector of nexpl explanatory variables. $C_{t}$ is a matrix of dimension $k_{2} \times k_{1}$ and $V_{t}$ is a vector of $r_{2}$ residual terms.
To estimate the model, we adopt a sequential approach based on five steps.
Step 1: we estimate the model with three state variables: $y_{t}^{}, y_{t-1}^{}, y_{t-2}^{*}$ and three observed variables $y_{t}, \pi_{t}, \tilde{u}_{t}$.

The influence of the import prices and oil prices on inflation in Eq. (9) is assumed to be measured by the average impact over two lagged quarters. In Eq. (11) $u_{t}^{*}$ is assumed to be a constant (exogenous). In the IS curve, the financial variables are assumed to be exogenous and the influence of world demand is captured by the average of the impacts of current and previous quarters $\left(D_{c}^{p_{c}}(L)=0.5 a_{31}\left(L+L^{2}\right.\right.$, $\left.D_{s}^{p_{x}}(L)=0.5 a_{32}\left(L+L^{2}\right), D_{w}^{p_{w}}(L)=0.5(1+L)\right)$. Moreover, we do not consider the influence of the interest rate. Finally potential growth rate is assumed to be a constant (exogenous)

Denoting $\Theta$ the set of parameter to estimate, we, therefore, search for $\hat{\Theta}, \hat{X}{t}$ that minimizes the following loss function: $$\sum{t=1}^{T}\left{\frac{1}{\sigma_{\tilde{y}}^{2}}\left(y_{t}-y_{t}^{*}\right)^{2}+\frac{1}{\sigma_{y *}^{2}}\left(\epsilon_{t}^{y *}\right)^{2}+\frac{1}{\sigma_{\pi}^{2}}\left(\epsilon_{t}^{\pi}\right)^{2}+\frac{1}{\sigma_{\tilde{u}}^{2}}\left(\epsilon^{\tilde{u}}\right)^{2}\right}$$
The estimates depends upon the values of the following weights (scaling factors): $\lambda_{1}=\frac{\sigma_{y}^{2}}{\sigma_{y y}^{2}}, \lambda_{2}=\frac{\sigma_{i}^{2}}{\sigma_{\pi}^{2}}, \lambda_{3}=\frac{\sigma_{y}^{2}}{\sigma_{v}^{2}}$. As has been evidenced in the literature, there is no clear guidance about the choice of particular values for these scaling factors. Even the conventional value of 1600 usually chosen for $\lambda_{1}$ would not be appropriate here as the cyclical properties of the output-gap depend upon all the scaling factors. We leave those scaling be estimated by the data. This first-step estimation, with constant $g$ aims at generating some first guess values for the output-gap and the coefficients of the Phillips, IS and Okun equations.

## 商科代写|计量经济学代写Econometrics代考|The Theoretical Concept of Rational Bubbles

The main objective of this study is to review statistical methods, but it is still important to understand the underlying economic theory of financial bubbles in order to specify a composition of statistical models. As discussed, recent research casts doubt on the rationality of investors, but many economic analyses maintain this assumption and it is often explained using the present value model (PVM). The rationality assumption prevails in academic research largely for convenience; it is easier to model rational behaviors than irrational ones. Survey data on investors’ expectations are the best source of information about investors’ expectations, but in the absence of survey data for a comprehensive number of countries and infrequent dissemination of survey data, we also maintain the rationality assumption.

According to the PVM, rational bubbles are defined as sizable and persistent deviations from economic fundamentals and follow a non-stationary process in a statistical sense. Based on the definition of asset returns or returns on real estate $\left(r_{t+1}=\left(P_{t+1}+D_{t+1}\right) / P_{t}-1\right)$, the PVM suggests that the contemporaneous prices $\left(P_{t}\right)$ will be determined by the expected value of future economic fundamentals $(D)$ and prices:
$$P_{t}=E_{t}\left[\frac{P_{t+1}+D_{t+1}}{1+r_{t+1}}\right]$$
where $t$ denotes time $(t=1, \ldots, T)$ and $E$ is an expectation operator. $D$ is an economic fundamental, such as dividend payments in equity analyses or rental costs in housing analyses. Solving Eq. (1) forwardly and using the law of iterated expectations, we can obtain Eq. (2):
$$P_{t}=E_{t}\left[\sum_{h=0}^{\infty}\left(\prod_{k=0}^{h}\left(\frac{1}{1+r_{t+k}}\right)\right) D_{t+h}+\lim {h \rightarrow \infty} \prod{k=0}^{h}\left(\frac{1}{1+r_{t+k}}\right) P_{t+h}\right]$$ When $P$ and $D$ are cointegrated, and when the transversality condition holds (i.e., $\left.E_{t}\left[\lim {h \rightarrow \infty} \prod{k=0}^{h}\left(\frac{1}{1+r_{i+k}}\right) P_{t+h}\right] \rightarrow 0\right)$, then the result shows no evidence of bubbles. Therefore, in this case, asset prices tend to move along with the economic fundamentals. On the other hand, when these conditions do not hold, then the results indicate evidence of bubbles. For this reason, prior studies frequently investigated rational bubbles using integration methods.

## 商科代写|计量经济学代写Econometrics代考|Econometric Methods

Econometricians proposed many statistical methods, with statistical hypotheses that seem designed to be suitable from their perspectives. Consequently, some approaches were developed to look for tranquil periods, while others investigate financial bubbles. Unlike previous studies, we make a clear distinction between statistical approaches to determine tranquil and bubble periods. This distinction is important since differences in the statistical hypotheses can explain the different results from these two approaches. In this section, we will clarify these two approaches using popular statistical specifications in studies of bubbles.

To investigate the theoretical model and predictions in Eq. (2) quantitatively, previous studies often focused on a single market utilizing time series methods. These statistical methods are one-tailed tests, but can be broadly categorized into left- and right-tailed approaches according to their alternative hypotheses. The lefttailed test is classical and is designed to look for cointegration between prices and economic fundamentals, and thus revealing tranquil periods. As an extension, we also propose a nonlinear approach that can be classified into a group of left-tailed tests. On the other hand, the right-tailed test, which has become popular, is an approach to identify explosive bubbles.

## 商科代写|计量经济学代写Econometrics代考|Estimation Method

X吨=一个X吨−1+从吨+F吨在吨: 状态方程  是吨=μ吨+C吨′X吨+在吨: 测量方程
X吨是向量ķ1状态变量（未观察到），是吨是向量ķ2观察到的变量，A 是ķ1×ķ1矩阵，从我是一个ķ1×1确定性术语的向量，在吨是一个r1×1残差向量，F吨是一个ķ1×r1矩阵，μ吨是一个产品ķ2×n和Xpl由 nexpl 解释变量向量组成的系数矩阵。C吨是一个维度矩阵ķ2×ķ1和在吨是一个向量r2剩余条款。

\sum{t=1}^{T}\left{\frac{1}{\sigma_{\tilde{y}}^{2}}\left(y_{t}-y_{t}^{*} \right)^{2}+\frac{1}{\sigma_{y *}^{2}}\left(\epsilon_{t}^{y *}\right)^{2}+\frac{1 }{\sigma_{\pi}^{2}}\left(\epsilon_{t}^{\pi}\right)^{2}+\frac{1}{\sigma_{\tilde{u}}^ {2}}\left(\epsilon^{\tilde{u}}\right)^{2}\right}\sum{t=1}^{T}\left{\frac{1}{\sigma_{\tilde{y}}^{2}}\left(y_{t}-y_{t}^{*} \right)^{2}+\frac{1}{\sigma_{y *}^{2}}\left(\epsilon_{t}^{y *}\right)^{2}+\frac{1 }{\sigma_{\pi}^{2}}\left(\epsilon_{t}^{\pi}\right)^{2}+\frac{1}{\sigma_{\tilde{u}}^ {2}}\left(\epsilon^{\tilde{u}}\right)^{2}\right}

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