# 经济代写|计量经济学代写Econometrics代考|ECON3007

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

Violation of Assumption $3.3$ means that the disturbances are correlated, i.e., $E\left(u_i u_j\right)=\sigma_{i j} \neq 0$, for $i \neq j$, and $i, j=1,2, \ldots, n$. Since $u_i$ has zero mean, $E\left(u_i u_j\right)=\operatorname{cov}\left(u_i, u_j\right)$ and this is denoted by $\sigma_{i j}$. This correlation is more likely to occur in time-series than in cross-section studies. Consider estimating the consumption function of a random sample of households. An unexpected event, like a visit of family members, will increase the consumption of this household. However, this positive disturbance need not be correlated to the disturbances affecting consumption of other randomly drawn households. However, if we were estimating this consumption function using aggregate time-series data for the U.S., then it is very likely that a recession year affecting consumption negatively this year may have a carry over effect to the next few years. A shock to the economy like an oil embargo in 1973 is likely to affect the economy for several years. A labor strike this year may affect production for the next few years. Therefore, we will switch the $i$ and $j$ subscripts to $t$ and $s$ denoting time-series observations $t, s=1,2, \ldots, T$ and the sample size will be denoted by $T$ rather than $n$. This covariance term is symmetric, so that $\sigma_{12}=E\left(u_1 u_2\right)=E\left(u_2 u_1\right)=\sigma_{21}$. Hence, only $T(T-1) / 2$ distinct $\sigma_{t s}$ ‘s have to be estimated. For example, if $T=3$, then $\sigma_{12}, \sigma_{13}$, and $\sigma_{23}$ are the distinct covariance terms. However, it is hopeless to estimate $T(T-1) / 2$ covariances $\left(\sigma_{t s}\right)$ with only $T$ observations. Therefore, more structure on these $\sigma_{t s}$ ‘s needs to be imposed. A simple and popular assumption is that the $u_t$ ‘s follow a firstorder autoregressive process denoted by $\mathrm{AR}(1)$ :
$$u_t=\rho u_{t-1}+\epsilon_t \quad t=1,2, \ldots, T$$

where $\epsilon_t$ is $\operatorname{IID}\left(0, \sigma_\epsilon^2\right)$. It is autoregressive because $u_t$ is related to its lagged value $u_{t-1}$. One can also write (5.26), for period $t-1$, as
$$u_{t-1}=\rho u_{t-2}+\epsilon_{t-1}$$
and substitute $(5.27)$ in $(5.26)$ to get
$$u_t=\rho^2 u_{t-2}+\rho \epsilon_{t-1}+\epsilon_t$$
Note that the power of $\rho$ and the subscript of $u$ or $\epsilon$ always sum to $t$. By continuous substitution of this form, one ultimately gets
$$u_t=\rho^t u_0+\rho^{t-1} \epsilon_1+. .+\rho \epsilon_{t-1}+\epsilon_t$$

## 经济代写|计量经济学代写Econometrics代考|Consequences for OLS

How is the OLS estimator affected by the violation of the no serial correlation assumption among the disturbances? The OLS estimator is still unbiased and consistent since these properties rely on assumptions $3.1$ and $3.4$ and have nothing to do with assumption 3.3. For the simple linear regression, using (5.2), the variance of $\widehat{\beta}{O L S}$ is now \begin{aligned} \operatorname{var}\left(\widehat{\beta}{O L S}\right) &=\operatorname{var}\left(\sum_{t=1}^T w_t u_t\right)=\sum_{t=1}^T \sum_{s=1}^T w_t w_s \operatorname{cov}\left(u_t, u_s\right) \ &=\sigma_u^2 / \sum_{t=1}^T x_t^2+\sum_{t \neq s} \sum_t w_t w_s \rho^{|t-s|} \sigma_u^2 \end{aligned}
where $\operatorname{cov}\left(u_t, u_s\right)=\rho^{|t-s|} \sigma_u^2$ as explained in (5.33). Note that the first term in (5.34) is the usual variance of $\widehat{\beta}{\text {OLS }}$ under the classical case. The second term in (5.34) arises because of the correlation between the $u_t$ ‘s. Hence, the variance of OLS computed from a regression package, i.e., $s^2 / \sum{t=1}^T x_t^2$ is a wrong estimate of the variance of $\widehat{\beta}{O L S}$ for two reasons. First, it is using the wrong formula for the variance, i.e., $\sigma_u^2 / \sum{t=1}^T x_t^2$ rather than (5.34). The latter depends on $\rho$ through the extra term in (5.34). Second, one can show, see Problem 7, that $E\left(s^2\right) \neq \sigma_u^2$ and will involve $\rho$ as well as $\sigma_u^2$. Hence, $s^2$ is not unbiased for $\sigma_u^2$ and $s^2 / \sum_{t=1}^T x_t^2$ is a biased estimate of $\operatorname{var}\left(\widehat{\beta}{O L S}\right)$. The direction and magnitude of this bias depend on $\rho$ and the regressor. In fact, if $\rho$ is positive, and the $x_t$ ‘s are themselves positively autocorrelated, then $s^2 / \sum{t=1}^T x_t^2$ understates the true variance of $\widehat{\beta}{O L S}$. This means that the confidence interval for $\beta$ is tighter than it should be and the $t$-statistic for $H_0 ; \beta=0$ is overblown, see Problem 8 . As in the heteroskedastic case, but for completely different reasons, any inference based on $\operatorname{var}\left(\widehat{\beta}{O L S}\right)$ reported from the standard regression packages will be misleading if the $u_t$ ‘s are serially correlated.

# 计量经济学代考

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

$$u_t=\rho u_{t-1}+\epsilon_t \quad t=1,2, \ldots, T$$

$$u_{t-1}=\rho u_{t-2}+\epsilon_{t-1}$$

$$u_t=\rho^2 u_{t-2}+\rho \epsilon_{t-1}+\epsilon_t$$

$$u_t=\rho^t u_0+\rho^{t-1} \epsilon_1+\ldots+\rho \epsilon_{t-1}+\epsilon_t$$

## 经济代写|计量经济学代写Econometrics代考|Consequences for OLS

OLS 估计量如何受到干扰之间无序列相关假设的违反? OLS 估计量仍然是无偏且一致的，因为这些属性 依赖于假设 $3.1$ 和 $3.4$ 与假设 $3.3$ 无关。对于简单线性回归，使用 (5.2)，方差 $\widehat{\beta} O L S$ 就是现在
$$\operatorname{var}(\widehat{\beta} O L S)=\operatorname{var}\left(\sum_{t=1}^T w_t u_t\right)=\sum_{t=1}^T \sum_{s=1}^T w_t w_s \operatorname{cov}\left(u_t, u_s\right) \quad=\sigma_u^2 / \sum_{t=1}^T x_t^2+\sum_{t \neq s} \sum_t w_t w_s \rho^{|t-s|} \sigma_{u s}^2$$

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