### 商科代写|计量经济学代写Econometrics代考|Best 22

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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|DETECTION OF SERIAL CORRELATION

Since serial correlation of the errors has been shown to have important implications for our interpretation of regression results, it becomes important to develop tests for whether it is present in the models we estimate. It will be helpful to have an example in mind as we develop such tests. Consider the following estimated consumption function for the US economy based on annual data from 1970 to $2019 .$
$\ln \left(C_{t}\right)=\underset{(0.0466)}{-0.2898}+\underset{(0.0052)}{1.0174} \ln \left(Y D_{t}\right)+\hat{u}{t}$ $R^{2}=0.9987 \quad \hat{\sigma}{u}=0.0154 \quad T=50$
$D W=0.5389$,
where $C$ is total consumers’ expenditure and $Y D$ is real personal disposable income. Both variables are measured in millions of dollars at 2012 prices.
On first inspection, this equation appears to have reasonable properties. The slope coefficient is statistically significantly different from zero and the coefficient of determination indicates a good fit. In this case, however, it is more interesting to test the null hypothesis that the slope coefficient is equal to 1 . This is because the slope coefficient in this relationship measures the income elasticity of consumption expenditure. However, this null hypothesis is also rejected by our model. The $t$ statistic for this test is $(1.0174-1) / 0.0052=3.35$ which leads us to reject the null at the $5 \%$ level. This conclusion is, however, may be unreliable if the errors in this model are serially correlated. Therefore, in order to assess the robustness of our estimates of the model parameters, we need to test for the presence of serial correlation in the residuals $\hat{u}$ from equation (5.18).

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

We can look for serial correlation informally by simply inspecting a plot of the residuals. If runs of positive or negative residuals are obvious, then this is a sign that serial correlation is present. For example, Figure $5.2$ shows the residuals for our consumption function equation and it is obvious that there are periods during which the residuals are consistently either positive or negative. This is a clear indication that the equation suffers from serial correlation. There are, however, forms of serial correlation, such as moving average errors, which are not so easily detected. Therefore, it is important to develop more formal tests as well as procedures for identifying the specific form of serial correlation that is relevant for this equation.

The correlogram 1 provides a more formal statistical method for the investigation of serial correlation. The correlogram is a table, or plot, of the sample autocorrelations of the regression residuals. The sample autocorrelations are defined in equation (5.19), and Figure $5.3$ gives both a table and a graph of the autocorrelations of the residuals from equation (5.18).
$$\hat{\rho}{k}=\sum{t=k+1}^{T} \hat{u}{t} \hat{u}{t-k} / \sum_{t=1}^{T} \hat{u}_{t^{2}}^{2} .$$

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

$$\hat{\rho} k=\sum t=k+1^{T} \hat{u} t \hat{u} t-k / \sum_{t=1}^{T} \hat{u}_{t^{2}}^{2} .$$

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