# 经济代写|计量经济学代写Econometrics代考|EC5216

## 经济代写|计量经济学代写Econometrics代考|Normality of the Disturbances

If the disturbances are not normal, OLS is still BLUE provided assumptions 3.1-3.4 still hold. Normality made the OLS estimators minimum variance unbiased MVU, and these OLS estimators turn out to be identical to the MLE. Normality allowed the derivation of the distribution of these estimators, and this in turn allowed testing of hypotheses using the $t$ and $F$-tests considered in Chap. 4. If the disturbances are not normal, yet the sample size is large, one can still use the normal distribution for the OLS estimates asymptotically by relying on the Central Limit Theorem, see Theil (1978). Theil’s proof is for the case of fixed $X$ ‘s in repeated samples, zero mean, and constant variance on the disturbances. A simple asymptotic test for the normality assumption is given by Jarque and Bera (1987). This is based on the fact that the normal distribution has a skewness measure of zero and a kurtosis of 3 . Skewness (or lack of symmetry) is measured by
$$S=\frac{\left[E(X-\mu)^3\right]^2}{\left[E(X-\mu)^2\right]^3}=\frac{\text { Square of the } 3 r d \text { moment about the mean }}{\text { Cube of the variance }}$$
Kurtosis (a measure of flatness) is measured by
$$\kappa=\frac{F(X-\mu)^4}{\left[E(X-\mu)^2\right]^2}=\frac{4 \text { th moment ahout the mean }}{\text { Square of the variance }}$$
For the normal distribution $S=0$ and $\kappa=3$. Hence, the Jarque-Bera (JB) statistic is given by
$$J B=n\left[\frac{S^2}{6}+\frac{(\kappa-3)^2}{24}\right]$$
where $S$ represents skewness and $\kappa$ represents kurtosis of the OLS residuals. This statistic is asymptotically distributed as $\chi^2$ with two degrees of freedom under $H_0$. Rejecting $H_0$ rejects normality of the disturbances but does not offer an alternative distribution. In this sense, the test is non-constructive. In addition, not rejecting $\mathrm{H}_0$ does not necessarily mean that the distribution of the disturbances is normal; it only means we do not reject that the distribution of the disturbances is symmetric and has a kurtosis of 3. See the empirical example in Sect. $5.3$ for an illustration. The Jarque-Bera test is part of the standard output using EViews.

## 经济代写|计量经济学代写Econometrics代考|Heteroskedasticity

Violation of Assumption $3.2$ means that the disturbances have a varying variance, i.e., $E\left(u_i^2\right)=\sigma_i^2, i=1,2, \ldots, n$. First, we study the effect of this violation on the OLS estimator. For the simple linear regression, it is obvious that $\widehat{\beta}{O L S}$ given in Eq. (3.5) is still unbiased and consistent because these properties depend upon Assumptions $3.1$ and 3.4, and not Assumption 3.2. However, the variance of $\widehat{\beta}{O L S}$ is now different
$$\operatorname{var}\left(\widehat{\beta}{O L S}\right)=\operatorname{var}\left(\sum{i=1}^n w_i u_i\right)=\sum_{i=1}^n w_i^2 \sigma_i^2=\sum_{i=1}^n x_i^2 \sigma_i^2 /\left(\sum_{i=1}^n x_i^2\right)^2$$
where the second equality follows from assumption $3.3$ and the fact that $\operatorname{var}\left(\mathrm{u}i\right)$ is now $\sigma_i^2$. Note that if $\sigma_i^2=\sigma^2$. this reverts back to $\sigma^2 / \sum{i=1}^n x_i^2$, the usual formula for $\operatorname{var}\left(\widehat{\beta}{O L S}\right)$ under homoskedasticity. Furthermore, one can show that $E\left(s^2\right)$ will involve all of the $\sigma_i^2$,s and not one common $\sigma^2$, see Problem 1. This means that the regression package reporting $s^2 / \sum{i=1}^n x_i^2$ as the estimate of the variance of $\widehat{\beta}_{O L S}$ is committing two errors. One, it is not using the right formula for the variance, i.e.

Eq. (5.9). Second, it is using $s^2$ to estimate a common $\sigma^2$ when in fact the $\sigma_i^2$ ‘s are different. The bias from using $s^2 / \sum_{i=1}^n x_i^2$ as an estimate of $\operatorname{var}\left(\widehat{\beta}{O L S}\right)$ will depend upon the nature of the heteroskedasticity and the regressor. In fact, if $\sigma_i^2$ is positively related to $x_i^2$, one can show that $s^2 / \sum{i=1}^n x_i^2$ understates the true variance and hence the $t$-statistic reported for $\beta=0$ is overblown, and the confidence interval for $\beta$ is tighter than it is supposed to be, see Problem 2. This means that the $t$-statistic in this case is biased toward rejecting $H_0 ; \beta=0$, i.e., showing significance of the regression slope coefficient, when it may not be significant.

The OLS estimator of $\beta$ is linear unbiased and consistent, but is it still BLUE when Assumption $3.2$ is violated? In order to answer this question, we note that if $\operatorname{var}\left(u_i\right)=\sigma_i^2$, one can divide $u_i$ by $\sigma_i / \sigma$, and the resulting $u_i^=\sigma u_i / \sigma_i$ will have a constant variance $\sigma^2$. It is easy to show that $u^$ satisfies all the classical assumptions of Chap. 3 including homoskedasticity. The regression model becomes
$$\sigma Y_i / \sigma_i=\alpha \sigma / \sigma_i+\beta \sigma X_i / \sigma_i+u_i^*$$
and OLS on this model (5.10) is BLUE. The OLS normal equations on (5.10) are
\begin{aligned} &\sum_{i=1}^n\left(Y_i / \sigma_i^2\right)=\alpha \sum_{i=1}^n\left(1 / \sigma_i^2\right)+\beta \sum_{i=1}^n\left(X_i / \sigma_i^2\right) \ &\sum_{i=1}^n\left(Y_i X_i / \sigma_i^2\right)=\alpha \sum_{i=1}^n\left(X_i / \sigma_i^2\right)+\beta \sum_{i=1}^n\left(X_i^2 / \sigma_i^2\right) \end{aligned}

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Normality of the Disturbances

$$S=\frac{\left[E(X-\mu)^3\right]^2}{\left[E(X-\mu)^2\right]^3}=\frac{\text { Square of the } 3 r d \text { moment about the mean }}{\text { Cube of the variance }}$$

$$\kappa=\frac{F(X-\mu)^4}{\left[E(X-\mu)^2\right]^2}=\frac{4 \text { th moment ahout the mean }}{\text { Square of the variance }}$$

$$J B=n\left[\frac{S^2}{6}+\frac{(\kappa-3)^2}{24}\right]$$

## 经济代写|计量经济学代写Econometrics代考|Heteroskedasticity

$$\operatorname{var}(\widehat{\beta} O L S)=\operatorname{var}\left(\sum i=1^n w_i u_i\right)=\sum_{i=1}^n w_i^2 \sigma_i^2=\sum_{i=1}^n x_i^2 \sigma_i^2 /\left(\sum_{i=1}^n x_i^2\right)^2$$

OLS估计量 $\beta$ 是线性无偏且一致的，但是当假设时它仍然是蓝色的 $3.2$ 被侵犯? 为了回答这个问题，我们 注意到如果 $\operatorname{var}\left(u_i\right)=\sigma_i^2$, 可以分 $u_i$ 经过 $\sigma_i / \sigma$ ， 以及由此产生的 $u_i^{=} \sigma u_i / \sigma_i$ 会有一个常量方差 $\sigma^2$. 很容 易证明在^满足第 1 章的所有经典假设。3 包括同方差性。回归模型变为
$$\sigma Y_i / \sigma_i=\alpha \sigma / \sigma_i+\beta \sigma X_i / \sigma_i+u_i^*$$

$$\sum_{i=1}^n\left(Y_i / \sigma_i^2\right)=\alpha \sum_{i=1}^n\left(1 / \sigma_i^2\right)+\beta \sum_{i=1}^n\left(X_i / \sigma_i^2\right) \quad \sum_{i=1}^n\left(Y_i X_i / \sigma_i^2\right)=\alpha \sum_{i=1}^n\left(X_i / \sigma_i^2\right)+\beta \sum_{i=1}^n\left(X_i^2 / \sigma_i^2\right)$$

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