Lecture 15 Autocorrelation in Time Series
Stationarity If the time series is normally distributed then strictly stationary and weekly stationary time series are considered equivalent.
Autocorrelation The autocorrelation (Box and Jenkins, 1976) function can be used for the following two purposes: 1. To detect non-randomness in data. 2. To identify an appropriate time series model if the data are not random.
Definition Given measurements, Y1, Y2, ..., YN at time X1, X2, ..., XN, the lag k autocorrelation function is defined as − − = − Although the time variable, X, is not used in the formula for autocorrelation, the assumption is that the observations are equi-spaced.
Autocorrelation is a correlation coefficient. However, instead of correlation between two different variables, the correlation is between two values of the same variable at times Xi and Xi+k .
When the autocorrelation is used to detect non-randomness, it is usually only the first (lag 1) autocorrelation that is of interest. When the autocorrelation is used to identify an appropriate time series model, the autocorrelations are usually plotted for many lags.
Sample Output Dataplot generated the following autocorrelation output using the data set: The lag-one autocorrelation coefficient of the 200 observations = -0.3073 The computed value of the constant a = -0.3073 lag autocorrelation 12. -0.55 25. -0.66 38. -0.64 0. 1.00 13. 0.73 26. 0.42 39. 0.08 1. -0.31 14. 0.07 27. 0.39 40. 0.58 2. -0.74 15. -0.76 28. -0.65 41. -0.45 3. 0.77 16. 0.40 29. 0.03 42. -0.28 4. 0.21 17. 0.48 30. 0.63 43. 0.62 5. -0.90 18. -0.70 31. -0.42 44. -0.10 6. 0.38 19. -0.03 32. -0.36 45. -0.55 7. 0.63 20. 0.70 33. 0.64 46. 0.45 8. -0.77 21. -0.41 34. -0.05 47. 0.25 9. -0.12 22. -0.43 35. -0.60 48. -0.61 10. 0.82 23. 0.67 36. 0.43 49. 0.14 11. -0.40 24. 0.00 37. 0.32
Questions The autocorrelation function can be used to answer the following questions 1. Was this sample data set generated from a random process? 2. Would a non-linear or time series model be a more appropriate model for these data than a simple constant plus error model? Instructor: Azmat Nafees Time Series and Forecasting 30-Oct-10 1
Importance Randomness is one of the key assumptions in determining if a univariate statistical process is in control. If the assumptions of constant location and scale, randomness, and fixed distribution are reasonable, then the univariate process can be modeled as: