Download Autocorrelation - Econometrics - Lecture Notes and more Study notes Econometrics and Mathematical Economics in PDF only on Docsity!
Chapter 08
Autocorrelation
As we have studied in the previous lectures, the desirable properties of OLS are conditional on the validity of assumption. Among these assumptions, one assumption is the autocorrelation. In this chapter we will study
- What is autocorrelation
- What happens if autocorrelation does not exist
- Tests for autocorrelation
- Remedies for the autocorrelation
What is autocorrelation?
Recall from the previous lecture that an assumption of OLS is
- The covariance of error term with another should be zero i.e. 𝑐𝑜𝑣�𝜀𝑖, 𝜀𝑗 � = 0 for all 𝑖 ≠ 𝑗 This property is called no auto-correlation
In the previous lecture we have shown two figures to clears the concept of e autocorrelation. However it is not always possible to judge the existence of e autocorrelation in such a straight forward way especially when we have multiple regressors. We need formal testing to investigate the incidence of e autocorrelation in a model.
The first fig shows the model where we do not find any pattern among the residuals this case of non auto-correlated error terms. The second figure shows that the residuals formulate a particular pattern around the regression. This is the autocorrelation.
What Happens If there is autocorrelation?
Autocorrelation may be result of one of the following problems
a. The non-linear relationship being modeled by OLS b. The Model has dynamic structure (lagged variables) but we have missed those variables c. There is an outlier in the data d. The variables are integrated
The incidence of autocorrelation means one of the above mentioned problems exist. For the first case, if non-linear relationship is being modeled by OLS, than the proper solution is introduce the nonlinear power in the model, if the variable are missing than we would need to include the lag terms in the model. There are some solutions for the problem of outlier however for the first two problems; one simple solution is to use a more general model.
Tests for autocorrelation
There are many tests for the autocorrelation. We will discuss the following two tests.
a. Durbin Watson statistics b. Breusch–Godfrey test
Durbin Watson statistics
DW statistics is routinely calculated in most of econometric softwares. The procedure for computing DW statistics is as follows:
- Estimate the regression model
- Calculate the residuals
- Durbin Watson statistics is given by following formula
𝐷𝑊 = ∑^ (𝑦𝑡^ −𝑦𝑡−1)
𝑇𝑡=2 2 ∑ 𝑇𝑡=2𝑦𝑡^2 =^
∑ 𝑇𝑡=2𝑦𝑡^2 +𝑦𝑡−1^2 −2𝑦𝑡 𝑦𝑡− ∑ 𝑇𝑡=2𝑦𝑡^2 =
Under the null hypothesis of no autocorrelation, the term 𝑦𝑡 𝑦𝑡−1 ≅ 0 and ∑ 𝑦𝑡^2 ≅ ∑ 𝑦𝑡−1^2 therefore
𝐷𝑊 ≅ 2 ∑ 𝑦𝑡
2 ∑ 𝑦𝑡^2 ≅^2
But if the null is not true, 𝑦𝑡 𝑦𝑡−1 ≠ 0 ; and the value of DW statistics will be far away from 2. It can take value between 0 and 4, with values greater than 2 implying negative autocorrelation and values smaller than 2 implying positive autocorrelation.
DW has been used as test for autocorrelation; however it has weak power properties. The detail of how DW should be used as test can be found in standard books like Gujrati. However, it would be more appropriate to use DW as a signal rather than a test. The values closer to 2 are signal for no autocorrelation whereas closer to 0 or 4 are signal for autocorrelation.
Cautions about DW statistics
