Statistics-lab™可以为您提供lse.ac.uk EC212 Econometrics计量经济学课程的代写代考和辅导服务！
EC212 Econometrics课程简介
The objective of this course is to provide the basic knowledge of econometrics that is essential equipment for any serious economist or social scientist. The course introduces statistical tools including regression analysis and its application using cross-sectional data.
The second week onwards will be focused on how various technical problems inherent in economic analysis, including heteroskedasticity, autocorrelation, and endogeneity should be handled. This section of the course will pay special attention to the application of the regression model to time-series data – both stationary and non-stationary.
Using the theories and their application in economics, you will participate in daily workshops to get hands-on experience implementing the various estimators and testing procedures in Stata using real-world data. As a result, you will consider how the theory can be applied to a wide range of questions of economic interest (For example, modelling long-term relationships between prices and exchange rates).
By the end of the course, you will be able to provide proof of the unbiasedness or biasedness and consistency or inconsistency of least squares, and instrumental variable estimators using simple models.
PREREQUISITES
It seems like you are describing a course in econometrics that aims to equip students with basic knowledge and skills in statistical analysis, with a focus on regression analysis and its application to cross-sectional and time-series data in economics. The course also covers various technical problems that can arise in econometric analysis, such as heteroskedasticity, autocorrelation, and endogeneity, and how to address them.
In addition to theoretical instruction, the course provides practical workshops to give students hands-on experience using statistical software (such as Stata) to implement various estimators and testing procedures on real-world data. The course aims to help students apply econometric theory to a wide range of economic questions, such as modelling long-term relationships between prices and exchange rates.
By the end of the course, students should be able to evaluate the unbiasedness or biasedness and consistency or inconsistency of least squares and instrumental variable estimators using simple models.
EC212 Econometrics HELP（EXAM HELP， ONLINE TUTOR）
- (For White robust standard errors)
a) Let
$$
X=\left(\begin{array}{cc}
x_{11} & x_{12} \
\vdots & \vdots \
x_{N 1} & x_{N 2}
\end{array}\right)
$$
and $e^{\prime}=e_1, \ldots, e_N$. In the White application, each column of $x$ will be $N$ observations of a regressor. Show that if the error terms are not autocorrelated and not correlated with $X$ and you set the terms with mean zero to zero (for the White variance estimator, we impose that the off-diagonal terms are 0$),$ then
$$
\left(X^{\prime} e\right)\left(X^{\prime} e\right)^{\prime}=\left(\begin{array}{cc}
\sum_{i=1}^N x_{1 i}^2 e_i^2 & \Sigma_{i=1}^N x_{1 i} x_{2 i} e_i^2 \
\Sigma_{i=1}^N x_{1 i} x_{2 i} e_i^2 & \sum_{i=1}^N x_{2 i}^2 e_i^2
\end{array}\right)
$$
(Each of these terms have mean different from 0 , if the columns of $X$ are not orthogonal, and if divided by $N$ they will satisfy a Law of Large Numbers under typical conditions. I did this part quickly in class, partly because it is better that you verify it yourself.)
To show that
\left(X^{\prime} e\right)\left(X^{\prime} e\right)^{\prime}=\left(\begin{array}{cc} \sum_{i=1}^N x_{1 i}^2 e_i^2 & \Sigma_{i=1}^N x_{1 i} x_{2 i} e_i^2 \\ \Sigma_{i=1}^N x_{1 i} x_{2 i} e_i^2 & \sum_{i=1}^N x_{2 i}^2 e_i^2 \end{array}\right),(X′e)(X′e)′=(∑i=1Nx1i2ei2Σi=1Nx1ix2iei2Σi=1Nx1ix2iei2∑i=1Nx2i2ei2),
we can expand the matrix multiplication using the definition of matrix multiplication: \begin{align*} \left(X^{\prime} e\right)\left(X^{\prime} e\right)^{\prime} &= \left(\begin{array}{c} \sum_{i=1}^N x_{1 i} e_i \ \sum_{i=1}^N x_{2 i} e_i \end{array}\right) \left(\sum_{i=1}^N e_i x_{1 i} \quad \sum_{i=1}^N e_i x_{2 i}\right) \ &= \left(\begin{array}{cc} \sum_{i=1}^N x_{1 i}^2 e_i^2 & \Sigma_{i=1}^N x_{1 i} x_{2 i} e_i^2 \ \Sigma_{i=1}^N x_{1 i} x_{2 i} e_i^2 & \sum_{i=1}^N x_{2 i}^2 e_i^2 \end{array}\right), \end{align*} where we have used the fact that the off-diagonal terms are zero by assumption (i.e., $e$ is not correlated with $X$), and that $e$ has zero mean (i.e., the terms with mean zero are set to zero).
Since the error terms are assumed to be uncorrelated, each diagonal term is the sum of squared errors multiplied by the squared values of the corresponding regressor. Since the error terms are also assumed to have mean zero and the terms with mean zero are set to zero, dividing by $N$ will give a Law of Large Numbers under typical conditions.
- Assume that random variables $y_i$ for $\mathrm{i}=1, \ldots, 20$ are independent with $E\left(y_i\right)=\alpha+\beta x_i, \operatorname{Var}\left(y_i\right)=$ $\sigma^2 x_i^2$, where $x_i=i$ and $\sigma^2=2$.
a) If you estimate $\alpha$ and $\beta$ by OLS, what is the variance of $\hat{\beta}$ ?
b) If you estimate $\alpha$ and $\beta$ by GLS, what is the variance of $\hat{\beta}$ ?
a) The OLS estimator of $\beta$ is given by:
\hat{\beta} = \frac{\sum_{i=1}^{n}(x_i – \bar{x})(y_i – \bar{y})}{\sum_{i=1}^{n}(x_i – \bar{x})^2},β^=∑i=1n(xi−xˉ)2∑i=1n(xi−xˉ)(yi−yˉ),
where $\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i$ and $\bar{y}=\frac{1}{n}\sum_{i=1}^{n}y_i$.
Substituting $y_i = \alpha+\beta x_i + \epsilon_i$, where $\epsilon_i \sim N(0,\sigma^2)$, we get:
\begin{aligned} \hat{\beta} &= \frac{\sum_{i=1}^{n}(x_i – \bar{x})[(\alpha+\beta x_i + \epsilon_i) – (\alpha+\beta \bar{x} + \frac{1}{n}\sum_{i=1}^{n}\epsilon_i)]}{\sum_{i=1}^{n}(x_i – \bar{x})^2} \\ &= \beta + \frac{\sum_{i=1}^{n}(x_i – \bar{x})\epsilon_i}{\sum_{i=1}^{n}(x_i – \bar{x})^2}. \end{aligned}β^=∑i=1n(xi−xˉ)2∑i=1n(xi−xˉ)[(α+βxi+ϵi)−(α+βxˉ+n1∑i=1nϵi)]=β+∑i=1n(xi−xˉ)2∑i=1n(xi−xˉ)ϵi.
The variance of $\hat{\beta}$ can be computed as:
\begin{aligned} \operatorname{Var}(\hat{\beta}) &= \operatorname{Var}\left(\frac{\sum_{i=1}^{n}(x_i – \bar{x})\epsilon_i}{\sum_{i=1}^{n}(x_i – \bar{x})^2}\right) \\ &= \frac{\sum_{i=1}^{n}(x_i – \bar{x})^2\operatorname{Var}(\epsilon_i)}{\left(\sum_{i=1}^{n}(x_i – \bar{x})^2\right)^2} \\ &= \frac{\sigma^2}{\sum_{i=1}^{n}(x_i – \bar{x})^2}. \end{aligned}Var(β^)=Var(∑i=1n(xi−xˉ)2∑i=1n(xi−xˉ)ϵi)=(∑i=1n(xi−xˉ)2)2∑i=1n(xi−xˉ)2Var(ϵi)=∑i=1n(xi−xˉ)2σ2.
Substituting $\sigma^2=2$ and $x_i=i$, we get:
\operatorname{Var}(\hat{\beta}) = \frac{2}{\sum_{i=1}^{20}(i-\bar{x})^2}.Var(β^)=∑i=120(i−xˉ)22.
b) To estimate $\alpha$ and $\beta$ by GLS, we need to first estimate the covariance matrix of the errors. The variance of $y_i$ is given by $\sigma^2 x_i^2$, so the covariance between $y_i$ and $y_j$ is:
\begin{aligned} \operatorname{Cov}(y_i, y_j) &= E[(y_i – E(y_i))(y_j – E(y_j))] \\ &= E[(\alpha+\beta x_i+\epsilon_i – \alpha – \beta x_i)(\alpha+\beta x_j+\epsilon_j – \alpha – \beta x_j)] \\ &= \beta^2(x_i^2+x_j^2) + \sigma^2 \delta_{ij}, \end{aligned}Cov(yi,yj)=E[(yi−E(yi))(yj−E(yj))]=E[(α+βxi+ϵi−α−βxi)(α+βxj+ϵj−α−βxj)]=β2(xi2+xj2)+σ2δij,
where $\delta_{ij}$ is the Kronecker delta (i.e., $\delta_{ij}=1$ if $i=j$ and $\delta_{ij}=0$ otherwise). Therefore, the covariance matrix of the errors is given
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
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