1. STAT 497LECTURE NOTES 3 STATIONARY TIME SERIES PROCESSES (ARMA PROCESSES OR BOX-JENKINS PROCESSES)

2. AUTOREGRESSIVE PROCESSES • AR(p) PROCESS: or where

3. AR(p) PROCESS • Because the process is always invertible. • To be stationary, the roots of p(B)=0 must lie outside the unit circle. • The AR process is useful in describing situations in which the present value of a time series depends on its preceding values plus a random shock.

4. AR(1) PROCESS where atWN(0, ) • Always invertible. • To be stationary, the roots of (B)=1B=0 must lie outside the unit circle.

5. AR(1) PROCESS • OR using the characteristic equation, the roots of m=0 must lie inside the unit circle. B=1 |B|<|1| ||<1 STATIONARITY CONDITION

6. AR(1) PROCESS • This process sometimes called as the Markov process because the distribution of Yt given Yt-1,Yt-2,… is exactly the same as the distribution of Yt given Yt-1.

7. AR(1) PROCESS • PROCESS MEAN: 

8. AR(1) PROCESS • AUTOCOVARIANCE FUNCTION: K

9. AR(1) PROCESS

10. AR(1) PROCESS When ||<1, the process is stationary and the ACF decays exponentially.

11. AR(1) PROCESS • 0 <  < 1  All autocorrelations are positive. • 1 <  < 0  The sign of the autocorrelation shows an alternating pattern beginning a negative value.

12. AR(1) PROCESS • RSF: Using the geometric series

13. AR(1) PROCESS • RSF: By operator method _ We know that

14. AR(1) PROCESS • RSF: By recursion

15. THE SECOND ORDER AUTOREGRESSIVE PROCESS • AR(2) PROCESS: Consider the series satisfying where atWN(0, ).

16. AR(2) PROCESS • Always invertible. • Already in Inverted Form. • To be stationary, the roots of must lie outside the unit circle. OR the roots of the characteristic equation must lie inside the unit circle.

17. AR(2) PROCESS

18. AR(2) PROCESS • Considering both real and complex roots, we have the following stationary conditions for AR(2) process (see page 84 for the proof)

19. AR(2) PROCESS • THE AUTOCOVARIANCE FUNCTION: Assuming stationarity and that atis independent of Yt-k, we have

20. AR(2) PROCESS

21. AR(2) PROCESS

22. AR(2) PROCESS

23. AR(2) PROCESS

24. AR(2) PROCESS ACF: It is known as Yule-Walker Equations ACF shows an exponential decay or sinusoidal behavior.

25. AR(2) PROCESS • PACF: PACF cuts off after lag 2.

26. AR(2) PROCESS • RANDOM SHOCK FORM: Using the Operator Method

27. The p-th ORDER AUTOREGRESSIVE PROCESS: AR(p) PROCESS • Consider the process satisfying where atWN(0, ). provided that roots of all lie outside the unit circle

28. AR(p) PROCESS • ACF: Yule-Walker Equations • ACF: tails of as a mixture of exponential decay or damped sine wave (if some roots are complex). • PACF: cuts off after lag p.

29. MOVING AVERAGE PROCESSES • Suppose you win 1 TL if a fair coin shows a head and lose 1 TL if it shows tail. Denote the outcome on toss t by at. • The average winning on the last 4 tosses=average pay-off on the last tosses: MOVING AVERAGE PROCESS

30. MOVING AVERAGE PROCESSES • Consider the process satisfying

31. MOVING AVERAGE PROCESSES • Because , MA processes are always stationary. • Invertible if the roots of q(B)=0 all lie outside the unit circle. • It is a useful process to describe events producing an immediate effects that lasts for short period of time.

32. THE FIRST ORDER MOVING AVERAGE PROCESS_MA(1) PROCESS • Consider the process satisfying

33. MA(1) PROCESS • From autocovariance generating function

34. MA(1) PROCESS • ACF ACF cuts off after lag 1. General property of MA(1) processes: 2|k|<1

35. MA(1) PROCESS • PACF:

36. MA(1) PROCESS • Basic characteristic of MA(1) Process: • ACF cuts off after lag 1. • PACF tails of exponentially depending on the sign of . • Always stationary. • Invertible if the root of 1B=0 lie outside the unit circle or the root of the characteristic equation m=0 lie inside the unit circle.  INVERTIBILITY CONDITION: ||<1.

37. MA(1) PROCESS • It is already in RSF. • IF: 1= 2=2

38. MA(1) PROCESS • IF: By operator method

39. THE SECOND ORDER MOVING AVERAGE PROCESS_MA(2) PROCESS • Consider the moving average process of order 2:

40. MA(2) PROCESS • From autocovariance generating function

41. MA(2) PROCESS • ACF • ACF cuts off after lag 2. • PACF tails of exponentially or a damped sine waves depending on a sign and magnitude of parameters.

42. MA(2) PROCESS • Always stationary. • Invertible if the roots of all lie outside the unit circle. OR if the roots of all lie inside the unit circle.

43. MA(2) PROCESS • Invertibility condition for MA(2) process

44. MA(2) PROCESS • It is already in RSF form. • IF: Using the operator method:

45. The q-th ORDER MOVING PROCESS_ MA( q) PROCESS Consider the MA(q) process:

46. MA(q) PROCESS • The autocovariance function: • ACF:

47. THE AUTOREGRESSIVE MOVING AVERAGE PROCESSES_ARMA(p, q) PROCESSES • If we assume that the series is partly autoregressive and partly moving average, we obtain a mixed ARMA process.

48. ARMA(p, q) PROCESSES • For the process to be invertible, the roots of lie outside the unit circle. • For the process to be stationary, the roots of lie outside the unit circle. • Assuming that and share no common roots, Pure AR Representation: Pure MA Representation:

49. ARMA(p, q) PROCESSES • Autocovariance function • ACF • Like AR(p) process, it tails of after lag q. • PACF: Like MA(q), it tails of after lag p.

50. ARMA(1, 1) PROCESSES • The ARMA(1, 1) process can be written as • Stationary if ||<1. • Invertible if ||<1.