### 商科代写|计量经济学代写Econometrics代考|Find 2022

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 商科代写|计量经济学代写Econometrics代考|CAUSES OF SERIAL CORRELATION

Consider a regression equation of the standard form $Y_{t}=\beta X_{t}+u_{t}$. We assume a model in mean deviation form to simplify the notation and the discussion. The errors of this model are said to be serially correlated if $E\left(u_{t} u_{t-k}\right) \neq 0$ for some $k \neq 0$. A natural question is why the errors might be correlated in this way? For the moment, we will simply assume that this is an intrinsic property of the data. That is, we assume that shocks to the equation are not random drawings from a distribution but instead depend upon their own past values. An alternative would be to assume that correlation in the errors arises because the model is misspecified in some way. However, this would complicate much of the discussion and we will avoid this assumption for the moment, on the understanding that it will be relaxed later.

There are many different forms that serial correlation might take. For example, the errors might follow a first-order autoregressive (AR) process. This would mean that the error process could be described by an equation of the form $u_{t}=\rho u_{t-1}+\varepsilon_{t}$, where $\varepsilon_{t}$, is a truly random disturbance and $\rho \neq 0$. This is a very common and important case, but it is not the only form that serial correlation can take. An alternative is where the error term in the equation is an average over several time periods of the random disturbance $\varepsilon_{t}$. For example, we might have a first-order moving average process of the form $u_{t}=\varepsilon_{t}+\lambda \varepsilon_{t-1}$. Both error processes are said to be serially correlated but each produces different implications and problems for the modeler. However, in both cases, the problem of dealing with serial correlation is simplified because of the assumption that it is an intrinsic feature of the error themselves, that is, the problem is one of error dynamics. A more realistic conclusion might be that the errors are serially correlated because of some fundamental misspecification in the original equation.

## 商科代写|计量经济学代写Econometrics代考|CONSEQUENCES OF SERIAL CORRELATION

Now that we have established some of the reasons why serial correlation may arise in regression models, let us consider the implications for least squares regression analysis. Suppose we have a model in which the errors follow a first-order AR process as set out in (5.6)
\begin{aligned} &Y_{t}=\beta X_{t}+u_{t} \ &u_{t}=\rho u_{t-1}+\varepsilon_{t}, \end{aligned}
where $\varepsilon_{t}, t=1, \ldots, T$ are independent, identically distributed random disturbances with mean zero and constant variance. As we have seen, this is not the only possible type of serial correlation which may arise, but the results we derive for this model apply more generally to other forms of serial correlation.

The AR process defined in (5.6) can be written in moving average form. Using the method of backward substitution, we have
$$u_{t}=\varepsilon_{t}+\rho \varepsilon_{t-1}+\rho^{2} \varepsilon_{t-2} \ldots=\sum_{j=0}^{\infty} \rho^{j} \varepsilon_{t-j} .$$
This is an infinite moving average process. Providing $|\rho|<1$, then the sequence defined in (5.7) will converge, in the sense that it will have a finite variance. To see this note that
$$E\left(u_{t}^{2}\right)=\sum_{j=0}^{\infty} \rho^{2 j} E\left(\varepsilon_{t-j}^{2}\right)=\sum_{j=0}^{\mu_{j}} \rho^{2 j} \sigma_{\varepsilon}^{2}=\frac{\sigma_{\varepsilon}^{2}}{1-\rho^{2}}$$
Therefore, for the variance of the error term to be finite and positive, we need $|\rho|<1$. If this condition holds, then the process is said to be weakly stationary and it can be shown that a general feature of stationary, finite AR processes is that they can be written as infinite moving average processes. Moreover, since $E\left(\varepsilon_{t-j}\right)=0$ for all values of $j$, it follows that $E\left(u_{t}\right)=0$. This is a useful property because we have already seen that the expected value of the OLS estimator can be written as follows: $E(\hat{\beta})=\beta+\sum_{t=1}^{T} X_{t} E\left(u_{t}\right) / \sum_{t=1}^{T} X_{t}^{2}$. It therefore follows that $E(\hat{\beta})=\beta$ and that the OLS estimator is unbiased even when the errors are serially correlated.

## 商科代写|计量经济学代写Econometrics代考|CONSEQUENCES OF SERIAL CORRELATION

$$Y_{t}=\beta X_{t}+u_{t} \quad u_{t}=\rho u_{t-1}+\varepsilon_{t},$$

(5.6) 中定义的 AR 过程可以写成移动平均形式。使用向后替换的方法，我们有
$$u_{t}=\varepsilon_{t}+\rho \varepsilon_{t-1}+\rho^{2} \varepsilon_{t-2} \ldots=\sum_{j=0}^{\infty} \rho^{j} \varepsilon_{t-j} .$$

$$E\left(u_{t}^{2}\right)=\sum_{j=0}^{\infty} \rho^{2 j} E\left(\varepsilon_{t-j}^{2}\right)=\sum_{j=0}^{\mu_{j}} \rho^{2 j} \sigma_{\varepsilon}^{2}=\frac{\sigma_{\varepsilon}^{2}}{1-\rho^{2}}$$

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