The Durbins h test statistic
As been described above, the DW-test is made for the purpose of testing for first order autocorrelation. Furthermore, it assumes that none of the explanatory variables are lagged dependent variables which would be the case when estimating a dynamic model. When that is the case the DW-test has a tendency to be close to 2 even though the error terms are serially correlated. Hence, the DW-test should not be used with the following kind of model:
Fortunately there is an easy alternative to the DW-test that could be seen as a modified version of it and for that reason is called the Durbins h statistic. It is defined in the following way:
where DW is the standard DW-test, T the number of observations and Var (b1) the square of the standard error of the estimated parameter of the lagged dependent variable. The test statistic has been shown to be standard normally distributed under the null hypothesis of no autocorrelation, which means that the test value should be compared with a critical value from the standard normal table.
The presence of autocorrelation in models that include lagged dependent variables is even more affected than the standard model. When the error term is serially correlated in a dynamic model the estimated parameters are biased and inconsistent. It is therefore very important to correct for the problem before using the estimates for anything.
Assume that we have estimated the parameters of a dynamic model and received the following results with standard errors within parenthesis:
We use quarterly data over a 30 years period, which means that T=120. Since our model includes a lagged variable we lose one observation in the estimation. Using the information from the regression results, we may form the Durbins ^-statistic:
Using a one sided test at the 5 percent significance level we receive a critical value of 1.645. Since the test value is much larger than the critical value, we must conclude that our error terms are serially correlated.
The LM test is a more general test of autocorrelation compared to the two previous tests. Furthermore, it allows for a test of autocorrelation of higher order than one, and can be used even though lagged dependent variables are included in the model. However, the LM test is a large sample test which means that it should be treated as an approximation when using small samples, compared to the DW-test that could be seen as an exact test.
Estimate the parameters of your main model:
Create the residual term using the estimated parameters and lag it.
Extend your original model by including the estimated lagged residual in the specification:
Test the null hypothesis H0 : p = 0 using a simple f-test. If you reject the null hypothesis you can conclude that you have autocorrelation.
The equation given by (10.17) can be extended to include more lags of the residual terms in order to test for higher order of autocorrelation.
Assume that we have a time series model with two explanatory variables, and we suspect that the error term might be serially correlated of the second order.
Since the error term is unobserved we need to estimate the residuals for the model using the estimated parameters of the model:
Lag the estimated residuals from (10.19), re-specify equation (10.18) and receive:
We estimated the parameters of the extended version of the model given by (10.20) and received the following estimates with standard errors within parenthesis:
By investigating the significance of the coefficients of the two residual terms, we see that the first one is significantly different from zero, while the other is not. We therefore conclude that the residual term is serially correlated of the first order only.
Since we included lagged variables in the specification we lose some observations, and when including two estimated residuals as in (10.21) we loose two. If we have a large sample, losing two observations is not a big deal. However, if we only have 20 observations, loosing the first two observations might have an effect. One way to deal with this problem is to impose some values for e0 and e_l. Since the expected value of the residual terms equal zero, the missing observations could be replaced by zeros. Running the regression with and without the imposed values will give you an indication if the two missing observations are important. Other more advanced methods are available in the literature, but will not be discussed here.
Once we have found that our error term is serially correlated we need to correct for it before we can make any statistical inference on the population. As for the case of heteroskedasticity, we need to transform the involved variables and therefore use generalized least square. The transformation looks different dependent on the order of autocorrelation. We will therefore look at the two most frequently used error structures, AR(1) and AR(2), and show how it should be done for those two cases. After that it should be an easy task to generalize the transformation method for an autoregression of order n.
GLS when AR(1)
The transformation will be explained by an example. Assume that the objective is to transform the following model:
For simplicity reasons we use the specification of the simple regression model. However, the method can be generalized to any number of explanatory variables. The objective is to transform the autocorrelated u with something that is free from autocorrelation v. Assume that u is autoregressive of order one and given by:
If we substitute (10.23) into (10.22) we receive:
Form the following expression using (10.22):
Substitute (10.25) into (10.24) and rearrange:
Equation (10.26) is the transformed equation we are looking for. The error term of the original model is now replaced by v that is free from autocorrelation and we can estimate the regression equation using OLS. OLS, in combination with a variable transformation that results in a corrected error term, is what we call GLS.
GLS when AR(2)
The corresponding transformation in the AR(2) case is very similar. In this case our error term has the following shape:
With that in mind we can extend equation (10.26) in the following way:
The whole description above is based on the idea that the autocorrelation coefficient has been known. That is never the case and therefore it must be estimated. The estimated value is often received when we test for autocorrelation. In the Durbin Watson case the test statistic equal
This means that we can use the Durbin Watson test statistic to receive an estimate of the autocorrelation according to (10.29).
In case of higher order of autocorrelation the LM test should be applied. The coefficients in front of the lagged residual terms in (10.21) are estimates of the coefficients in (10.27). Those estimates could therefore be used when transforming the variables according to (10.28).
In the literature we will be able to find more advanced method to estimate the autocorrelation coefficient that could be used when applying GLS. However, statistical software, such as STATA and SPSS, will do most of the job for us. All we have to do is to specify the variables that we would like to have in our model.