1. AUTOCORRELATED DATA

2. CALCULATIONS ON VARIANCES: SOME BASICS Let X and Y be random variables COV=0 if X and Y are independent.

3. WHAT IF COV(X i, X i+1 ) > 0? 1.We calculate an AVG by adding X’s 2.The VAR of the AVG is bigger by COV(X i, X i+1 ) 3.The formula for VAR assumes COV(X i, X i+1 ) =0 4.The formula underestimates VAR of the AVG 5.The formula for the width of the CI gives too small a width 6.The CI does not cover the true  with the advertized probability  7.Our conclusion has oversold accuracy

4. AUTOCORRELATED DATA Consider the formula, called the Auto- Regressive (Lag 1) Process

5. NORMAL(0, 1) INDEPENDENT

6. c=0.2

7. C=0.5

8. C=0.7

9. C=0.9

10. C=0.9, 200 sample

11. C=0.99

12. c=0.5

13. c=0.7

14. c=0.9

15. c=.99

16. The Test for Rank 1 Autocorrelation Ho:  (1) = 0 Ha:  (1) <> 0

17. STATISTICALLY SIGNIFICANT AUTOCORRELATION Lag 1 autocorrelation  (1) estimated by r(1) Normal Mean Variance

18. So the quantity z below is N(0, 1), and can be compared to critical values, and p-values can be computed… Simplifies when we are testing  (1) = 0 Remember that this is a classical “wrong-way” hypothesis test

19. Sample Results crho(1)zp-value 0.2000.1141.6160.053 0.2703.8900.000 0.1592.2510.012 0.1492.1050.018 0.3224.6900.000 0.2854.1210.000