Residual Autocorrelation

The autocorrelation of a time series refers to the correlation of that time series with its past values. The k^th order autocorrelation is the autocorrelation between one time series observation and the value k periods before.

We cannot use the Durbin-Watson statistic to test for serial correlation in autoregressive models. This is because the test becomes invalid when the independent variables include past values of the dependent variable. Nevertheless, we can use the t-test to check whether the autocorrelations of the error terms are significantly different from zero. (Recall that serial correlation refers to the positive or negative correlation of the error terms.)

An accurately specified autoregressive model will have residual autocorrelations that do not vary significantly from zero.

To test whether an AR model is correctly specified, the following steps are followed:

• Estimate the autoregressive model and calculate the residuals;
• Calculate the autocorrelations of the residuals; and
• Perform a t-test to check if the autocorrelations are statistically different from 0.

$$\text{t}_{\text{statistic}}=\frac{\text{Residual Autocorrelation}}{\frac{1}{\sqrt{\text{T}}}}$$

i.e.

$$\text{t}=\frac{\rho_{\epsilon_{\text{t}}}, \epsilon_{\text{t}-\text{k}}}{\frac{1}{\sqrt{\text{T}}}}$$

Where:

• \(\frac{1}{\sqrt{\text{T}}}\)= Standard Error
• T= number of observations
• \(\rho_{\epsilon_{\text{t}}}, \epsilon_{\text{t}-\text{k}}\) = Correlation of the Error Term \(t\) with the k^th lagged error term.

Note: 

• The null hypothesis, \(\text{H}_{0}\) is that there is no autocorrelation.
• Reject the null hypothesis if \(\text{t}_{\text{statistic}}\) is greater than the critical value of t at the specified significance level and T-2 degrees of freedom. 

Rejection of the null hypothesis implies that the model is not correctly specified and should be modified. On the other hand, failure to reject the null hypothesis implies that the model is statistically valid.

Example: Testing an Autoregressive Model

Consider the following AR(1) model:

$$\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}\text{x}_{\text{t}-1}+\epsilon_{\text{t}}$$

The following table shows the autocorrelations of the residuals from the estimation of the model using a sample of 23 observations.

$$\small{\begin{array}{c|c|c} \textbf{Lag} & \textbf{Autocorrelation} & \textbf{t-statistic} \\ \hline1 & 0.3834 & 1.8388 \\ \hline 2 & 0.3483 & 1.6705 \\ \hline3 & 0.2897 & 1.3894 \\ \hline 4 & -0.1722 & -0.8259 \\ \hline 5 & -0.0725 & -0.3477 \\ \hline6 & -0.3952 & -1.8954 \\  \end{array}}$$

Check whether the model is correctly specified at the 5% level of significance.

$$\text{Standard error}= \frac{1}{\sqrt{T}}=\frac{1}{\sqrt{23}}=0.2085$$

We can compute the t-statistic for lag one as: 

$$\begin{align*}\text{t}_{\text{statistic}}&=\frac{\text{Residual Autocorrelation}}{\frac{1}{\sqrt{\text{T}}}}\\&=\frac{0.3834}{0.2085}=1.8388\end{align*}$$

The critical two-tail t-value using 21 degrees of freedom at the 5% level of significance is 2.08.

Notice that all the absolute values of the t-statistics corresponding to the residual autocorrelations are less than 2.08. This implies that none of them is statistically different from zero.

Therefore, we can conclude that the error terms from the AR(1) model are not serially correlated, and thus the model is correctly specified.

Question

Consider an AR(1) model relating to 100 observations with residual autocorrelations as presented in the following table:

$$\small{\begin{array}{c|c} \textbf{Lag} & \textbf{Autocorrelation} \\\hline1 & -0.0584 \\ \hline 2 & -0.0492 \\ \hline 3 & 0.0625 \\ \hline4 & 0.1455 \\  \end{array}}$$

At the 5% significance level, the most appropriate statement is that:

1. The AR(1) model is correctly specified.
2. The AR model is not correctly specified because the autocorrelations of the residuals for lag 1 are statistically different from 0.
3. The AR model is not correctly specified because the autocorrelations of the residuals for lag 4 are statistically different from 0.

Solution

The correct answer is A.

$$\text{Standard error}=\frac{1}{\sqrt{\text{T}}}=\frac{1}{\sqrt{100}}=0.10$$

We can compute the t-statistic for lag one as: 

$$\text{t}_{\text{statistic}}=\frac{\text{Residual autocorrelation}}{\frac{1}{\sqrt{\text{T}}}}$$

$$\text{t}_{\text{Statistic}}=\frac{-0.0584}{0.10}=-0.584$$

Repeating this for all the lags gives: 

$$\small{\begin{array}{c|c|c} \textbf{Lag} & \textbf{Autocorrelation} & \textbf{t-Statistic} \\ \hline 1 & -0.0584 & -0.584 \\ \hline2 & -0.0492 & -0.492 \\ \hline3 & 0.0625 & 0.625 \\ \hline 4 & 0.1455 & 1.455 \\  \end{array}}$$

The critical two-tail t-value using 98 degrees of freedom at the 5% level of significance is 1.98.

Notice that all the absolute values of the t-statistics corresponding to the residual autocorrelations are less than 1.98, which implies that none of them is statistically different from zero.

Therefore, we can conclude that the error terms from the AR(1) model are not serially correlated, implying that the model is correctly specified.

Reading 5: Time Series Analysis

LOS 5 (e) Explain how autocorrelations of the residuals can be used to test whether the autoregressive model fits the time series.