1. Time Series Basics (2) Fin250f: Lecture 3.2 Fall 2005 Reading: Taylor, chapter 3.5-3.7, 3.9(skip 3.6.1)

2. Outline  Linear stochastic processes  Autoregressive process  Moving average process  Lag operator  Forecasting AR and MA’s  The ARMA(1,1)  Trend plus noise models  Bubble simulations

3. Linear Stochastic Processes  Linear models  Time series dependence  Common econometric frameworks  Engineering background

4. AR(1) Autoregressive Process, Order 1

5. AR(1) Properties

6. AR(m)

7. Moving Average Process of Order 1, MA(1)

8. MA(1) Properties

9. MA(m)

10. AR->MA

11. Lag Operator (L)

12. Using the Lag Operator

13. An important feature for L

14. MA -> AR

16. Forecasting the AR(1)

17. Forecasting the AR(1): Multiperiods

18. Forecasting an MA(1)

19. The ARMA(1,1): AR and MA parts

20. ARMA(1,1) with L

22. Forecasting 1 Period

23. ARMA(p,q)

24. Why ARMA(1,1)?  Small, but persistent ACF’s  Comparing the AR(1) and ARMA(1,1)

25. AR(1) ACF’s

26. ARMA(1,1) ACF’s

27. Adding an AR(1) to an MA(0) (Trend plus noise)

28. Why Is This Useful? (Taylor 3.6.2)  Returns follow a combination process  Sum of: Small, but very persistent trend Independent noise term

29. Trend Plus Noise

31. Parameter Example  A small   big  A = 0.02, 

32. Trend Plus Noise ACF

33. Temporary Pricing Errors Bubbles(3.6.1)

34. AR(1) Difference

35. Variance Ratio

36. Return Autocorrelations

37. An Example

38. Bubble Price Simulation

39. Return ACF

40. Outline  Linear stochastic processes  Autoregressive process  Moving average process  Lag operator  Forecasting AR and MA’s  The ARMA(1,1)  Trend plus noise models  Bubble simulations