Damodar GujaratiEconometrics by Example, second edition

CHAPTER 6

REGRESSION DIAGNOSTIC III: AUTOCORRELATION

AUTOCORRELATION

One of the assumptions of the classical linear regression (CLRM) is that the covariance between ui, the error term for observation i, and uj, the error term for observation j, is zero.

Reasons for autocorrelation include:The possible strong correlation between the shock in

time t with the shock in time t+1More common in time series data

Damodar GujaratiEconometrics by Example, second edition

CONSEQUENCES

If autocorrelation exists, several consequences ensue: The OLS estimators are still unbiased and consistent. They are still normally distributed in large samples. They are no longer efficient, meaning that they are no longer

BLUE. In most cases standard errors are underestimated. Thus, the hypothesis-testing procedure becomes suspect, since

the estimated standard errors may not be reliable, even asymptotically (i.e. in large samples).

Damodar GujaratiEconometrics by Example, second edition

DETECTION OF AUTOCORRELATION

Graphical methodPlot the values of the residuals, et, chronologically

If discernible pattern exists, autocorrelation likely a problem

Durbin-Watson test Breusch-Godfrey (BG) test

Damodar GujaratiEconometrics by Example, second edition

DURBIN-WATSON (d) TEST

The Durbin-Watson d statistic is defined as:

Damodar GujaratiEconometrics by Example, second edition

21

2

2

1

( )t n

t tt

t n

tt

e ed

e

DURBIN-WATSON (d) TEST ASSUMPTIONS

Assumptions are: 1. The regression model includes an intercept term. 2. The regressors are fixed in repeated sampling. 3. The error term follows the first-order autoregressive (AR1)

scheme:

where ρ (rho) is the coefficient of autocorrelation, a value between -1 and 1.

4. The error term is normally distributed. 5. The regressors do not include the lagged value(s) of the

dependent variable, Yt.Damodar GujaratiEconometrics by Example, second edition

1t t tu u v

DURBIN-WATSON (d) TEST (CONT.) Two critical values of the d statistic, dL and dU, called the lower and upper

limits, are established. The decision rules are as follows:

1. If d < dL, there probably is evidence of positive autocorrelation.

2. If d > dU, there probably is no evidence of positive autocorrelation.

3. If dL < d < dU, no definite conclusion about positive autocorrelation.

4. If dU < d < 4 - dU, probably there is no evidence of positive or negative autocorrelation.

5. If 4 - dU < d < 4 - dL, no definite conclusion about negative autocorrelation.

6. If 4 - dL < d < 4, there probably is evidence of negative autocorrelation.

d value always lies between 0 and 4. The closer it is to zero, the greater is the evidence of positive autocorrelation,

and the closer it is to 4, the greater is the evidence of negative autocorrelation. If d is about 2, there is no evidence of positive or negative (first) order autocorrelation.

Damodar GujaratiEconometrics by Example, second edition

BREUSCH-GODFREY (BG) TEST This test allows for: (1) Lagged values of the dependent variables to be included as

regressors (2) Higher-order autoregressive schemes, such as AR(2), AR(3), etc. (3) Moving average terms of the error term, such as ut-1, ut-2, etc.

The error term in the main equation follows the following AR(p) autoregressive structure:

The null hypothesis of no serial correlation is:

Damodar GujaratiEconometrics by Example, second edition

1 2 ... 0p

1 1 2 2 ...t t t p t p tu u u u v

BREUSCH-GODFREY (BG) TEST (CONT.) The BG test involves the following steps: Regress et, the residuals from our main regression, on the regressors in

the model and the p autoregressive terms given in the equation on the previous slide, and obtain R2 from this auxiliary regression.

If the sample size is large, BG have shown that: (n – p)R2 ~ X2p

That is, in large samples, (n – p) times R2 follows the chi-square distribution with p degrees of freedom.

 Rejection of the null hypothesis implies evidence of autocorrelation. As an alternative, we can use the F value obtained from the auxiliary

regression. This F value has (p , n-k-p) degrees of freedom in the numerator and

denominator, respectively, where k represents the number of parameters in the auxiliary regression (including the intercept term).

Damodar GujaratiEconometrics by Example, second edition

DURBIN ALTERNATIVE TEST OF AUTOCORRELATION

Alternative test that takes into account the lagged dependent variables

Provides a formal test of the null hypothesis of serially uncorrelated disturbances against the alternative of autocorrelation of order p

Post-estimation command in Stata:

estat durbinalt, lags(p)

Damodar GujaratiEconometrics by Example, second edition

REMEDIAL MEASURES

First-Difference Transformation If autocorrelation is of AR(1) type, we have: Assume ρ=1 and run first-difference model (taking first difference

of dependent variable and all regressors)

Generalized Transformation Estimate value of ρ through regression of residual on lagged

residual and use value to run transformed regression

Newey-West MethodGenerates HAC (heteroscedasticity and autocorrelation

consistent) standard errors Model Evaluation

Damodar GujaratiEconometrics by Example, second edition

1t t tu u v