### 商科代写|计量经济学代写Econometrics代考|Best 27

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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|Formal Tests for Serial Correlation

The Durbin-Watson (DW) test provides a formal test in which the null hypothesis is that the equation errors are serially uncorrelated and the alternative is that they follow a first-order autocorrelation process. This test was first introduced by Durbin and Watson in two papers published in Biometrika in 1950 and 1951 [Durbin1950] and [Durbin1951]. It is a standard part of the regression output for most econometrics packages. The DW test builds on a previous test developed by Von Neumann [VonNeumann1941] who developed a test for autocorrelation in a series of random variables with the null that the variables are independent random numbers. Unfortunately, this is not suitable when the series under examination comprises regression residuals, which are not independent by construction. Although Von Neumann’s statistic has a relatively simple distribution, that is, the standard normal distribution, Durbin and Watson showed that the distribution of their test statistic was necessarily more complex. The nature of the test statistic means that it is not possible to derive unique critical values for a test of the null of no autocorrelation against the alternative of first-order autocorrelation. However, they did demonstrate that the critical values for their test were bounded and were able to tabulate the bounds for small sample sizes.
The DW test is concerned with a specific form of serial correlation, that is, first-order autocorrelation but is arguably sensitive to other forms. Consider the following regression model with an error that follows an AR process of order one:
\begin{aligned} &Y_{t}=\beta X_{t}+u_{t} \ &u_{t}=\rho u_{t-1}+\varepsilon_{t} . \end{aligned}
Taking the residuals from an OLS regression of $Y$ on $X$, we can construct the test statistic
$$D W=\sum_{t=2}^{T}\left(\hat{u}{t}-\hat{u}{t-1}\right)^{2} / \sum_{t=1}^{T} \hat{u}_{t}^{2} .$$

## 商科代写|计量经济学代写Econometrics代考|DEALING WITH SERIAL CORRELATION

If serial correlation is present, then there are several ways to deal with the issue. Of course, the priority is to identify the nature, and hopefully the cause, of the serial correlation. If the root cause of the problem is the omission of a relevant variable from the model, then the natural solution is to include that variable. If it is determined that modeling of the serial correlation process is appropriate, then we have several different methods available for the estimation of such models by adjusting for the presence of serially correlation errors. It should be noted that mechanical adjustments, of the type we will describe in this section, are potentially dangerous. This process has been much criticized on the grounds that there is a risk that these methods disguise an underlying problem rather than dealing with it. McGuirk and Spanos [McGuirk2009] are particularly critical of mechanical adjustments to deal with autocorrelated arguments. In this paper, they show that unless we can assume that the regress and does not Granger-cause the regressors, adjusting for autocorrelation means that least squares yield biased and inconsistent estimates. However, these methods are still used and reported in applied work and it is therefore important that we consider how they work.

The first method we will consider is that of Cochrane-Orcutt estimation. This uses an iterative algorithm proposed by Cochrane and Orcutt [Cochrane1949] in which we use the structure of the problem to separate out the estimation of the behavioral parameters of the main equation from those of the AR process that describes the errors. Let us consider the case of an $\mathrm{AR}(1)$ error process as an example. Suppose we wish to estimate a model of the form (5.6). The two equations can be combined to give a single equation of the form
$$Y_{t}-\rho Y_{t-1}=\beta\left(X_{t}-\rho X_{t-1}\right)+\varepsilon_{t},$$
that is, an equation in “quasi-differences” of the data. If $\rho$ was known, then it would be straightforward to construct these quasi-differences and estimate the behavioral parameter $\beta$ by least squares. In the absence of such knowledge, we make a guess at $\rho$ and construct an estimate of $\beta$ on this basis. We then generate the residuals $\hat{u}{t}=Y{t}-\beta X_{t}$ on this basis and calculate an estimate of $\rho$ of the form $\hat{\rho}=\sum_{t=2}^{T} \hat{u}{t} \hat{u}{t-1} / \sum_{t=1}^{T} \hat{u}_{t}^{2}$. If, by some lucky chance, this estimate coincides with our assumption, then we stop. Otherwise, we use our estimate to recalculate the quasi-differences, reestimate $\beta$, and continue until our estimates of $\beta$ and $p$ converge. If a solution exists, then this provides a robust algorithm for estimation.

## 商科代写|计量经济学代写Econometrics代考|Formal Tests for Serial Correlation

Durbin-Watson (DW) 检验提供了一种形式检验，其中零假设是方程误差是序列不相关的，而另一种方法是它们 遵循一阶自相关过程。该测试由 Durbin 和 Watson 在 1950 年和 1951 年在 Biometrika 发表的两篇论文 [Durbin1950] 和 [Durbin1951] 中首次引入。它是大多数计量经济学软件包回归输出的标准部分。DW 测试建立 在 Von Neumann [VonNeumann1941] 开发的先前测试的基础上，该测试开发了一系列随机变量的自相关测 试，其中变量为独立随机数。不幸的是，当检查的序列包含回归残差时，这不适合，这些回归残差在构造上不是 独立的。尽管冯诺依曼的统计量具有相对简单的分布，即标准正态分布，但 Durbin 和 Watson 表明，他们的检 验统计量的分布必然更复杂。检验统计量的性质意味着不可能为无自相关的零点与一阶自相关的备选方案的检验 推导出唯一的临界值。然而，他们确实证明了他们测试的临界值是有界的，并且能够将小样本的界限制表。检验 统计量的性质意味着不可能为无自相关的零点与一阶自相关的备选方案的检验推导出唯一的临界值。然而，他们 确实证明了他们测试的临界值是有界的，并且能够将小样本的界限制表。检验统计量的性质意味着不可能为无自 相关的零点与一阶自相关的备选方案的检验推导出唯一的临界值。然而，他们确实证明了他们测试的临界值是有 界的，并且能够将小样本的界限制表。

DW 检验关注特定形式的序列相关，即一阶自相关，但可以说对其他形式敏感。考虑以下回归模型，其误差遵循 一阶 AR 过程:
$$Y_{t}=\beta X_{t}+u_{t} \quad u_{t}=\rho u_{t-1}+\varepsilon_{t} .$$

$$D W=\sum_{t=2}^{T}(\hat{u} t-\hat{u} t-1)^{2} / \sum_{t=1}^{T} \hat{u}_{t}^{2} .$$

## 商科代写|计量经济学代写Econometrics代考|DEALING WITH SERIAL CORRELATION

$$Y_{t}-\rho Y_{t-1}=\beta\left(X_{t}-\rho X_{t-1}\right)+\varepsilon_{t},$$

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