The Linear Regression Model with Autocorrelated Disturbances: Finite Sample Theory

(Reference – Greene, Chapter 13)

Consider the standard normal linear regression model

yt = Xt’β + εt , t = 1,…,T

where β is a kx1 constant vector, Xt is a kx1 strictly exogenous process and εt ~ i.i.d. N(0,σ2).

The OLS estimator is an unbiased, normally distributed, and efficient estimator of β.

We will begin by reviewing the problem of autocorrelated disturbances from the traditional finite sample theory approach.

Now suppose that we relax the independence assumption so that

ε ~ N(0, σ2Ω )

where ε is the Tx1 random vector [ε1 … εT]’ and Ω is a TxT p.d. symmetric matrix whose diagonal elements are equal to 1.

That is, Ωij = E(εiεj). We are assuming that ε’s are homoskedastic (so that the diagonal elements are all the same) but are allowing the off-diagonal elements to be non-zero to allow for serial correlation in the ε’s.

In this case, the OLS estimator of β is unbiased and normally distributed, but it is not an efficient estimator. In addition

.

So, hypothesis tests and confidence tests based on the OLS estimator will be invalid if the disturbances are autocorrelated and the simple variance formula is applied.

If Ω is known, then exact finite sample hypothesis tests and confidence intervals can be constructed from the OLS estimator by using the appropriate variance formula. However, if Ω is known it is straightforward to construct the GLS estimator, which is unbiased, normally distributed, and efficient.

Recall the GLS Estimator of β –

Another way to think about the GLS Estimator – Apply OLS to the transformed model:

, and CC’ = Ω-1

The transformed model is a classical normal linear regression model since the transformed X’s are strictly exogenous and transformed disturbances are normally distributed, zero-mean random variables with σ2I.

Valid hypothesis tests and intervals estimates for β can then be constructed by applying “OLS procedures” to the transformed model.

Suppose that we suspect that our disturbances may be serially correlated but, if they are, we don’t know the Ω. Then it is natural to consider testing the null hypothesis of no serial correlation. If we fail to reject the null, we can proceed under the assumption that the disturbances are not serially correlated. If we do reject the null, then we have to decide how to proceed. (More on this later.)

It would seem natural that to test for autocorrelated ε’s we should simply test for whether the OLS residuals, i.e., the , are serially correlated. However, in finite samples the OLS residuals will be serially correlated (and heteroskedastic) even if the true disturbances are independent (and homoskedastic)!

So, we will want to construct a test that accounts for the “normal amount” of serial correlation that appears in the OLS residuals under the null and see whether there is “too much” serial correlation in these residuals for the independence assumption to be plausible.

The Durbin-Watson (DW)Test :

The null hypothesis is that the ε’s are independently drawn N(0,σ2) random variables. The alternative hypothesis is that there is first-order (and, possibly higher) autocorrelation in the ε’s.

The DW Test Statistic –

where r1 is the first-order sample autocorrelation of the residuals.

If there is no first-order autocorrelation, we would expect the d statistic to be close to 2. Positive autocorrrelation would be evident in values of d less than 2 (bounded below by 0) and negative autocorrelation would be evident in values of d greater than 2 (bounded above by 4).

Under the null hypothesis of no first-order autocorrelation, the d statistic is drawn from the “Durbin-Watson distribution.”

The p-value of the test of H0:ρ1=0 vs. H1:ρ1>0 is

where d-hat is the sample value of d. So, for example, we reject H0 at the 5-percent test size if < d0.05.

The exact distribution of the DW statistic under H0 depends on T,k and X. Durbin and Watson derived lower and upper bounds for the percentiles of the distribution that do not depend on X. These are often available in econometric textbooks (e.g., Greene). Thus, for example, if T = 50 and k = 3 and any appropriate X matrix

1.46 d0.05 1.63

Then if < 1.46, reject H0 at the 5-percent size. If > 1.63 don’t reject H0, and if 1.46 1.63?

The exact p-value for given T,k, and X can be computed from a somewhat complicated formula. Most modern regression software will do these computations and provide you with the appropriate p-value so that the “inconclusive range” problem is no longer a serious problem.

For our purposes, the biggest problem with the DW test is that it requires the regressors to be strictly exogenous.

Question – What should we do if we find evidence of autocorrelation? FGLS? OLS with “correct” s.e.’s? Standard OLS?

Standard OLS is unbiased. But the unadjusted s.e.’s are incorrect leading to improperly sized confidence intervals and hypothesis tests.

Applying OLS but correcting the s.e.’s based on consistent estimates of the Ω matrix (or using “heteroskedasticity-autocorrelation-consistent” s.e.’s that don’t require knowing the exact form of the autocorrelation) provide the basis for asymptotically valid inference, but how well do these corrections work in finite samples?

FGLS is biased but consistent, asymptotically normal, and asymptotically efficient. How well does it work in finite samples?