经济代写|EC212 Econometrics
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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)
- Computer question (continuation of previous homeworks). In Matlab, regress real per capita U.S. data consumption growth on income growth and the interest rate using the posted dataset. (This is the what you did in homework 1.)
a) You are told that income growth is not exogenous to consumption growth, but lagged income growth is. Suggest a suitable IV estimator. (Just words here.)
b) In Matlab, estimate the coefficients using your suggested IV estimator.
c) Calculate the standard errors of the coefficients and compare to the estimated standard errors from an OLS regression.
d) Based on the IV estimation, test if the coefficient to income growth is zero.
(NOTE: I ask you to use the lagged variable here, because you already have it. It is, or was, rather common to lagged variables used as instruments without much discussion and that is very often not a good idea. So do not take the setup of this problem as a suggestion for doing good empirical economics.)
a) Since income growth is not exogenous to consumption growth, we need to find an instrument that is correlated with income growth but uncorrelated with consumption growth. One possible instrument could be lagged income growth. This is because income growth in the current period could be influenced by consumption growth in the previous period, but lagged income growth should not be affected by current consumption growth. Therefore, lagged income growth can be a suitable IV estimator for income growth.
b) To estimate the coefficients using the suggested IV estimator, we can use the 2SLS (two-stage least squares) method in Matlab. We first regress the endogenous variable (income growth) on the instrument (lagged income growth) and any other exogenous variables (in this case, the interest rate). Then we obtain the predicted values of the endogenous variable from this first-stage regression and use them as the instrument in the second-stage regression of consumption growth on the predicted values of income growth and the interest rate.
c) To calculate the standard errors of the coefficients, we can use the usual formula for the standard errors in 2SLS. We can compare these standard errors to the estimated standard errors from an OLS regression to see if there is any difference. In general, the standard errors from 2SLS should be larger than those from OLS because 2SLS takes into account the additional uncertainty from the first-stage regression.
d) To test if the coefficient to income growth is zero, we can use the standard t-test based on the estimated coefficient and its standard error. We can compare this test to the corresponding t-test from an OLS regression to see if there is any difference. If the coefficient is statistically significant in the 2SLS regression but not in the OLS regression, this would suggest that there is endogeneity bias in the OLS regression and that the 2SLS regression provides a better estimate of the causal effect of income growth on consumption growth.
- (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
$$
\left(X^{\prime} e\right)\left(X^{\prime} e\right)^{\prime}=\left(\begin{array}{cc}
\sum_{i=1}^N x_{1 i}^2 e_i^2 & \sum_{i=1}^N x_{1 i} x_{2 i} e_i^2 \
\Sigma_{i=1}^N x_{1 i} x_{2 i} e_i^2 & \sum_{i=1}^N x_{2 i}^2 e_i^2
\end{array}\right) .
$$
(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.)
We have
\begin{aligned} \left(X^{\prime} e\right)\left(X^{\prime} e\right)^{\prime} &=\left(\begin{array}{ccc} \sum_{i=1}^N x_{1 i} e_i & \ldots & \sum_{i=1}^N x_{N 1} e_i \\ \vdots & \ddots & \vdots \\ \sum_{i=1}^N x_{1 i} e_i & \ldots & \sum_{i=1}^N x_{N 2} e_i \end{array}\right)\left(\begin{array}{ccc} \sum_{i=1}^N x_{1 i} e_i & \ldots & \sum_{i=1}^N x_{N 1} e_i \\ \vdots & \ddots & \vdots \\ \sum_{i=1}^N x_{1 i} e_i & \ldots & \sum_{i=1}^N x_{N 2} e_i \end{array}\right) \\ &=\left(\begin{array}{cc} \sum_{i=1}^N x_{1 i}^2 e_i^2 & \sum_{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{aligned}(X′e)(X′e)′=⎝⎛∑i=1Nx1iei⋮∑i=1Nx1iei…⋱…∑i=1NxN1ei⋮∑i=1NxN2ei⎠⎞⎝⎛∑i=1Nx1iei⋮∑i=1Nx1iei…⋱…∑i=1NxN1ei⋮∑i=1NxN2ei⎠⎞=(∑i=1Nx1i2ei2Σi=1Nx1ix2iei2∑i=1Nx1ix2iei2∑i=1Nx2i2ei2).
We know that $\operatorname{Cov}(e_i,e_j) = 0$ for $i \neq j$, and also $\operatorname{Cov}(e_i, x_{jk}) = 0$ for all $i, j, k$. Therefore, we have
\begin{aligned} \operatorname{E}\left(\sum_{i=1}^N x_{1 i} e_i \sum_{i=1}^N x_{2 i} e_i\right) &=\operatorname{E}\left[\sum_{i=1}^N \sum_{j=1}^N x_{1 i} x_{2 j} e_i e_j\right] \\ &=\sum_{i=1}^N \sum_{j=1}^N x_{1 i} x_{2 j} \operatorname{E}\left(e_i e_j\right) \\ &=\sum_{i=1}^N x_{1 i} x_{2 i} \operatorname{E}\left(e_i^2\right) \\ &=\Sigma_{i=1}^N x_{1 i} x_{2 i} \operatorname{E}\left(e_i^2\right), \end{aligned}E(i=1∑Nx1ieii=1∑Nx2iei)=E[i=1∑Nj=1∑Nx1ix2jeiej]=i=1∑Nj=1∑Nx1ix2jE(eiej)=i=1∑Nx1ix2iE(ei2)=Σi=1Nx1ix2iE(ei2),
which gives us the desired result. Note that since we have assumed mean zero errors, the diagonal terms are just the sum of squares of each element in the $X$ matrix times the squared errors.
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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