1. CONSEQUENCES OF AUTOCORRELATION
The consequences of autocorrelation for OLS are similar to those of heteroscedasticity. In general, the regression coefficients remain unbiased, but OLS is inefficient because one can find an alternative regression technique that yields estimators with smaller variances. 1

2. CONSEQUENCES OF AUTOCORRELATION
The other main consequence is that autocorrelation causes the standard errors to be estimated wrongly, often being biased downwards. Finally, although in general OLS estimates are unbiased, there is an important special case where they are biased. 2

3. CONSEQUENCES OF AUTOCORRELATION
Unbiasedness is easily demonstrated, provided that Assumption C.7 is satisfied. In the case of the simple regression model shown, we have seen that the OLS estimator of the slope coefficient can be decomposed as the second line where the at are as defined in the third line. 3

4. CONSEQUENCES OF AUTOCORRELATION
Now, if Assumption C.7 is satisfied, at and ut are distributed independently and we can write the expectation of b2 as shown. At no point have we made any assumption concerning whether ut is, or is not, subject to autocorrelation. 4

5. CONSEQUENCES OF AUTOCORRELATION
All that we now require is E(ut) = 0 and this is easily demonstrated. 5

6. CONSEQUENCES OF AUTOCORRELATION
For example, in the case of AR(1) autocorrelation, lagging the process one time period, we have the second line. Substituting for ut–1 in the first equation, we obtain the third. 6

7. CONSEQUENCES OF AUTOCORRELATION
Continuing to lag and substitute, we can express ut in terms of current and lagged values of et with diminishing weights. Since, by definition, the expected value of each innovation is zero, the expected value of ut is zero. 7

8. CONSEQUENCES OF AUTOCORRELATION
For higher order AR autocorrelation, the demonstration is essentially similar. For moving average autocorrelation, the result is immediate. 8

9. CONSEQUENCES OF AUTOCORRELATION
For multiple regression analysis, the demonstration is the same, except that at is replaced by at*, where at* depends on all of the observations on all of the explanatory variables in the model. 9

10. CONSEQUENCES OF AUTOCORRELATION
We will not pursue analytically the other consequences of autocorrelation. Suffice to mention that the proof of the Gauss–Markov theorem, which guarantees the efficiency of the OLS estimators, does require no autocorrelation, as do the expressions for the standard errors. 10

11. CONSEQUENCES OF AUTOCORRELATION
Now we come to the special case where OLS yields inconsistent estimators if the disturbance term is subject to autocorrelation. 11

12. CONSEQUENCES OF AUTOCORRELATION
If the model specification includes a lagged dependent variable, OLS estimators are biased and inconsistent if the disturbance term is subject to autocorrelation. This will be demonstrated for AR(1) autocorrelation and an ADL(1,0) model with one X variable. 12

13. CONSEQUENCES OF AUTOCORRELATION
Lagging the ADL(1,0) model by one time period, we obtain the third line. Thus Yt–1 depends on ut–1. As a consequence of the AR(1) autocorrelation ut also depends on ut–1. 13

14. CONSEQUENCES OF AUTOCORRELATION
Hence we have a violation of part (1) of Assumption C.7. The explanatory variables, Yt–1, is not distributed independently of the disturbance term. As a consequence, OLS will yield inconsistent estimates. 14

15. CONSEQUENCES OF AUTOCORRELATION
This was described as a special case, but actually it is an important one. ADL models are frequently used in time series regressions and autocorrelation is a common problem. 15

16. Copyright Christopher Dougherty 2011.
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