Econometrics - Chapter 9

The Durbin-Watson d test is unusual

1. Econometricians almost never test the one-sided null hypothesis that there is negative serial correlation in the residuals because negative serial correlation is quite difficult to explain theoretically in economic or business analysis. Its existence usually means that impure serial correlation has been caused by some error of specification. 2. The Durbin-Watson test is sometimes inconclusive. Whereas previously explained decision rules always have had only "acceptance" regions and rejection regions, the Durbin-Watson test has a third possibility, called the inconclusive region. We do not recommend the application of a remedy for serial correlation if the Durbin-Watson test is inconclusive.

Problems with Generalized Least Squares estimates

1. Even though serial correlation causes no bias in the estmates of the Beta-hats, the GLS estimates usually are different from the OLS ones. 2. It turns out that GLS works well if ρ-hat is close to the actual ρ, but the GLS ρ-hat is biased in small samples. If ρ-hat is biased, then the biased ρ-hat introduces bais into the GLS estimates of the B-hats. Luckily, there is a remedy for serial correlation that avoids both of these problems: Newey-West standard errors.

Generalized Least Squares (GLS)

A method of ridding an equation of pure first-order serial correlation and in the process restoring the minimum variance property to its estimation. GLS starts with an equation that does not meet the Classical Assumptions (due in this case to the pure serial correlation in the error term) and transforms it into one that does meet those assumptions.

Negative serial correlation

A negative value of ρ implies that the error term has a tendency to switch signs from negative to positive and back again in consecutive observations. Negative pure serial correlation is much less likely than positive pure serial correlation. As a result, most econometricians analyzing pure serial correlation concern themselves primarily with positive serial correlation.

Positive serial correlation

A positive value for ρ implies that the error term tends to have the same sign from one time period to the next.

2. Serial correlation causes OLS to no longer be the minimum variance estimator (of all the linear unbiased estimators)

Although the violation of Classical Assumption IV causes no bias, it does affect the other main conclusion of the Gauss-Markov Theorem, that of minimum variance. In particular, we cannot prove that the distribution of the OLS Beta-hats is minimum variance (among the linear unbiased estimators) when Assumption IV is violated. OLS is more likely to misestimate the true beta in the face of serial correlation. On balance, the Beta-hats are still unbiased because overestimates are just as likely as underestimates, but these errors increase the variance of the distribution of the estimates, increasing the amount that any given estimate is likely to differ from the true beta.

Remedies for serial correlation

If you conclude that you have pure serial correlation, then the appropriate response is to consider the application of Generalized Least Squares or Newey-West standard errors.

Pure serial correlation

Occurs when Classical Assumption IV, which assumes uncorrelated observations of the error term, is violated in a correctly specified equation. If the expected value of the simple correlation coefficient between any two observations of the error term is not equal to zero, then the error term is said to be serial correlated. When econometricians use the term serial correlation without any modifier, they are referring to pure serial correlation.

Serial Correlation

Serial correlation implies that the value of the error term from one time period depends in some systematic way on the value of the error term in other time periods.

Consequences of serial correlation

Serial correlation is more likely to have internal symptoms; it affects the estimated equation in a way that is not easily observable from an examination of just the results themselves. The existence of serial correlation in the error term of an equation violates Classical Assumption IV, and the estimation of the equation with OLS has at least three consequences:

Impure serial correlation

Serial correlation that is caused by a specification error such as an omitted variable or an incorrect functional form. While pure serial correlation is caused by the underlying distribution of the error term of the true specification of an equation (which cannot be changed by the researcher), impure serial correlation is caused by a specification error that often can be corrected. If we omit a relevant variable or use the wrong functional form, then the portion of that omitted effect that cannot be represented by the included explanatory variables must be absorbed by the error term. The best remedy for impure serial correlation is to attempt to find the omitted variable (or at least a good proxy) or the correct functional form for the equation. Both the bias and the impure serial correlation will disappear if the specification error is corrected. Most econometricians try to make sure they have the best specification possible before they spend too much time worrying about pure serial correlation.

Cochrane-Orcutt method

The best known method to estimate GLS equations. A two step technique that first produces an estimate of ρ and then estimates the GLS equation using that ρ-hat.

First-order serial correlation

The most commonly assumed kind of serial correlation is first-order serial correlation, in which the current value of the error term is a function of the previous value of the error term.

First-order auto-correlation coefficient

The new symbol, ρ (rho, pronounced "row"), called the first-order auto-correlation coefficient, measures the functional relationship between the value of an observation of the error term and the value of the previous observation of the error term. The magnitude of ρ indicates the strength of the serial correlation in an equation. If ρ is zero, then there is no serial correlation (because Є would equal u, a classical error term).

Understanding Generalized Least Squares (GLS)

Unfortunately we can't use OLS to estimate a Generalized Least Squares model because GLS equations are inherently nonlinear in the coefficients. Since OLS requires that the equation be linear in the coefficients, we need a different estimation procedure.

3. Serial correlation causes the OLS estimates of SE (Beta hats) to be biased, leading to unreliable hypothesis testing

What will happen to hypothesis testing if OLS underestimates the SE(Beta-hats) and therefore overestimates t-scores? Well, the "too low" SE(beta-hats) will cause a "too high" t-score for a particular coefficient, and this will make it more likely that we will reject a null hypothesis when it is in fact true.

AR(1) method

estimates a GLS equation like Equation 9.18 by estimating beta0, beta1, and ρ simultaneously with iterative nonlinear regression techniques that are well beyond the scope of this chapter. The AR(1) method tends to produce the same coefficient estimates as Cochrane-Orcutt but with superior estimates of the standard errors, so we recommend the AR(1) approach as long as your software can support such nonlinear regression.