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分类： EC212 Econometrics

经济代写|EC212 Econometrics

Posted on 2023年3月28日2023年3月28日 by statistics-lab

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）

问题 1.

1 What is autocorrelation? Which assumption of the CLRM is violated, and why?

Autocorrelation refers to the correlation of a variable with its own past values. In other words, autocorrelation occurs when the error terms in a regression model are not independent of each other, but instead are correlated with one another.

Autocorrelation violates the assumption of the classical linear regression model (CLRM) that the error terms are independently and identically distributed (i.i.d.). When autocorrelation is present in a regression model, it means that the error terms are not independent of each other, which violates the “independence” part of the i.i.d. assumption.

Autocorrelation can arise in time series data, where the observations are taken at regular intervals over time. For example, stock prices or GDP data are typically collected over time, and autocorrelation can occur if the error terms in a regression model are correlated with the previous time periods’ error terms.

Autocorrelation can also arise in cross-sectional data if there are unobserved factors that are correlated across observations. For example, if a regression model tries to explain the test scores of students in a class, and some students have a common unobserved factor such as the quality of the teacher, the error terms in the model could be correlated with each other.

When autocorrelation is present in a regression model, the estimated coefficients can still be unbiased, but the standard errors of the coefficients will be underestimated. This can lead to incorrect inferences about the significance of the coefficients and incorrect hypothesis testing. Therefore, it is important to detect and correct for autocorrelation in regression analysis.

问题 2.

Describe the steps of the DW test for autocorrelation. What are its disadvantages and which alternative tests can you suggest?

The Durbin-Watson (DW) test is a statistical test used to detect the presence of autocorrelation in the residuals of a regression model. The DW test statistic ranges from 0 to 4, with values close to 2 indicating no autocorrelation, values less than 2 indicating positive autocorrelation, and values greater than 2 indicating negative autocorrelation.

The steps of the DW test for autocorrelation are as follows:

Estimate the regression model and obtain the residuals.
Calculate the sum of squared differences between adjacent residuals, denoted as e(i) – e(i-1).
Calculate the sum of squared residuals, denoted as SSE.
Calculate the DW test statistic using the formula: DW = (sum of squared differences) / SSE.
Compare the calculated DW test statistic to the DW critical values, which depend on the sample size, the number of independent variables, and the desired significance level. If the calculated DW test statistic is greater than the upper critical value or less than the lower critical value, then there is evidence of autocorrelation.

The DW test has several disadvantages. First, it can only detect first-order autocorrelation, which means that it cannot detect higher-order autocorrelation. Second, it assumes that the error terms are normally distributed, which may not be true in practice. Third, it assumes that the regression model is correctly specified, which means that all relevant variables are included in the model and that the functional form is correctly specified.

Alternative tests for autocorrelation include the Breusch-Godfrey test, the Ljung-Box test, and the runs test. The Breusch-Godfrey test is a general test for higher-order autocorrelation, while the Ljung-Box test is a test for autocorrelation in time series data. The runs test is a non-parametric test for randomness in the residuals.

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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经济代写|EC212 Econometrics

Posted on 2023年3月18日2023年3月18日 by statistics-lab

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

PREREQUISITES

EC212 Econometrics HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

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.

问题 2.

(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

经济代写|EC212 Econometrics

Posted on 2023年3月15日2023年3月15日 by statistics-lab

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

PREREQUISITES

EC212 Econometrics HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

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.

a) Since income growth is not exogenous to consumption growth, we need to find an instrument that is correlated with income growth but not correlated with the error term in the consumption growth regression. One possible instrument for income growth could be lagged income growth, which is likely to be correlated with current income growth but not with the error term in the consumption growth regression. This is because lagged income growth is determined by past economic conditions and not affected by current consumption growth. Thus, using lagged income growth as an instrument would help us to obtain consistent estimates of the coefficient of income growth in the consumption growth regression.

b) To estimate the coefficients using the suggested IV estimator, we can use Matlab’s 2SLS (two-stage least squares) regression method. Here’s an example code that demonstrates how to use 2SLS to estimate the coefficients:

Note that we first define the variables y, x1, and x2 as before, and also create an instrument variable z using lagged income growth. We then specify the instruments and run the 2SLS regression using the ivregress function, which takes the instrument variables as the first input, the dependent variable y as the second input, and the endogenous variables x1 and x2 as the third input. The function returns the estimated coefficients, standard errors, and other statistics. Finally, we print the estimated coefficients using fprintf.

问题 2.

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.)

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.)

c) To calculate the standard errors of the coefficients in the IV regression, we can use the 2-step procedure:

Obtain the residuals from the first stage regression of the endogenous variable (income growth) on the instrument (lagged income growth) and the exogenous variable (interest rate).
Regress the dependent variable (consumption growth) on the instrumented endogenous variable and the exogenous variable, using the residuals from step 1 to correct for the endogeneity.

In Matlab, we can use the ivregress function from the Econometrics Toolbox to estimate the coefficients and standard errors using this 2-step procedure:

The output shows the coefficients and standard errors for the IV and OLS regressions:

We can see that the standard errors for the IV regression are generally larger than those for the OLS regression, which is expected given that the IV method involves additional sources of uncertainty.

d) To test if the coefficient to income growth is zero in the IV regression, we can use the t-test:The output shows the t-statistic and p-value for the test:

The p-value is greater than 0.05, so we cannot reject the null hypothesis that the coefficient to income growth is zero at the 5% significance level.

Textbooks

经济代写|EC212 Econometrics

Posted on 2023年3月10日2023年3月10日 by statistics-lab

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

PREREQUISITES

EC212 Econometrics HELP（EXAM HELP， ONLINE TUTOR）

问题 1.

(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.

问题 2.

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