1. Economics 310 Lecture 15 Autocorrelation

2. Correlation between members of series of observations order in time or space. For our classic model, we have E(  i  j )  0 for i  j. Some times use the term serial correlation for autocorrelation.

3. Autocorrelation Model

4. Rho = 0.9

5. Plot error and error lagged

6. Rho = -0.9

7. Plot of mu and mu lagged

8. Rho = 0.0 (Pure Random Error)

9. Plot of mu and mu lagged

10. Causes of Autocorrelation Inertia Specification bias: excluded variables case Specification bias: incorrect functional form Cobweb Phenomenon Lags Manipulation of data interpolation extrapolation

11. Models of Autocorrelation

12. OLS Estimation with AR(1) Error

13. OLS Estimation Disregarding Autocorrelation The residual variance is likely to underestimate the true variance. R 2 is likely to be overestimated. Estimate of variance of b 2 is likely to underestimate the true variance of b 2. t and F tests are no longer valid.

14. Variance estimate is biased.

15. Methods of Detecting Autocorrelation Graphic Method Runs Test Durbin-Watson d test Breusch-Godfrey test of higher-order autocorrelation

16. Durbin-Watson d Test

17. Durbin-Watson d Test Assumptions Regression model includes an intercept The explanatory variables, the X’s are nonstochastic. The disturbances are generated by a AR(1) process. Model includes no lagged values of dependent variable. There are no missing observations.

18. Durbin-Watson d statistic

19. Distribution Durbin-Watson Statistic dLdL dUdU d 0 24

20. Decision Regions Durbin-Watson d d 4 2 0 Reject H 0 * Evidence of negative auto- correlation Zone of indecision Zone of indecision Reject H 0 Evidence of positive auto-correlation Do not reject H 0 or H 0 * or both. H 0 : no positive autocorrelation H 0 * : no negative autocorrelaton dLdL dUdU 4- d u 4- d L

21. Durbin-Watson Decision Rules

22. Bid-Ask Spread an Example The spread between the bid price for US currency and the ask price for US currency in the Brazilian blackmarket is function of opportunity cost of holding currency and the risk of holding currency. Opportunity cost is interest rate risk is the rate of variability in exchange rate

23. Example Durbin-Watson |_Ols spread interest sigma / dw resid=e DURBIN-WATSON STATISTIC = 1.51549 DURBIN-WATSON P-VALUE = 0.019933 R-SQUARE = 0.6124 R-SQUARE ADJUSTED = 0.5983 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.57694 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.75956 SUM OF SQUARED ERRORS-SSE= 31.732 MEAN OF DEPENDENT VARIABLE = 3.4959 LOG OF THE LIKELIHOOD FUNCTION = -64.8077 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 55 DF P-VALUE CORR. COEFFICIENT AT MEANS INTEREST 0.19908 0.4677E-01 4.256 0.000 0.498 0.4563 0.3327 SIGMA 0.39287 0.1021 3.847 0.000 0.460 0.4124 0.2798 CONSTANT 1.3547 0.2507 5.404 0.000 0.589 0.0000 0.3875