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AUTOCORRELATED DATA

AUTOCORRELATED DATA. CALCULATIONS ON VARIANCES: SOME BASICS. Let X and Y be random variables. COV=0 if X and Y are independent. WHAT IF COV(X i , X i+1 ) > 0?. We calculate an AVG by adding X’s The VAR of the AVG is bigger by COV(X i , X i+1 )

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AUTOCORRELATED DATA

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  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(Xi, Xi+1) > 0? • We calculate an AVG by adding X’s • The VAR of the AVG is bigger by COV(Xi, Xi+1) • The formula for VAR assumes COV(Xi, Xi+1) =0 • The formula underestimates VAR of the AVG • The formula for the width of the CI gives too small a width • The CI does not cover the true m with the advertized probability a • 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: r(1) = 0 Ha: r(1) <> 0

  17. STATISTICALLY SIGNIFICANT AUTOCORRELATION • Lag 1 autocorrelation r(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 r(1) = 0 Remember that this is a classical “wrong-way” hypothesis test

  19. Sample Results

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