1. Properties of the estimates of the parameters of ARMA models

2. AR(1) models Comparison of Yule-Walker, Least Squares and Maximum Likelihood

3. a=0.5, N=20, 50 simulations YW: average 0.44, st.dev. 0.183 LS: average 0.467, st.dev. 0.191 ML: average 0.463, st.dev. 0.19

4. a=0.5, N=100, 50 simulations YW: average 0.494, st.dev. 0.086 LS: average 0.498, st.dev. 0.086 ML: average 0.498, st.dev. 0.85

5. a=0.5, N=500, 50 simulations YW: average 0.495, st.dev. 0.039 LS: average 0.496, st.dev. 0.04 ML: average 0.496, st.dev. 0.04

6. a=0.5, N=1000, 50 simulations YW: average 0.499, st.dev. 0.027 LS: average 0.499, st.dev. 0.027 ML: average 0.499, st.dev. 0.0271

7. a=0.95, N=20, 50 simulations YW: average 0.837, st.dev. 0.118 LS: average 0.877, st.dev. 0.126 ML: average 0.882, st.dev. 0.117

8. a=0.95, N=100, 50 simulations YW: average 0.918, st.dev. 0.062 LS: average 0.929, st.dev. 0.06 ML: average 0.928, st.dev. 0.055

9. a=0.95, N=500, 50 simulations YW: average 0.942, st.dev. 0.015 LS: average 0.944, st.dev. 0.015 ML: average 0.945, st.dev. 0.015

10. a=0.95, N=1000, 50 simulations YW: average 0.948, st.dev. 0.011 LS: average 0.949, st.dev. 0.011 ML: average 0.949, st.dev. 0.011

11. a=-0.95, N=20, 50 simulations YW: average -0.796, st.dev. 0.165 LS: average -0.855, st.dev. 0.171 ML: average -0.841, st.dev. 0.164

12. a=-0.95, N=100, 50 simulations YW: average -0.926, st.dev. 0.037 LS: average -0.935, st.dev. 0.04 ML: average -0.932, st.dev. 0.038

13. a=-0.95, N=500, 50 simulations YW: average -0.938, st.dev. 0.019 LS: average -0.941, st.dev. 0.018 ML: average -0.941, st.dev. 0.018

14. a=-0.95, N=1000, 50 simulations YW: average -0.948, st.dev. 0.011 LS: average -0.949, st.dev. 0.012 ML: average -0.949, st.dev. 0.012

15. AR(2) models Yule-Walker, Least squares and Maximum Likelihood for different N

16. N=20

17. a 1 = -1.8, a 2 = 0.9, N=20 Yule-Walker

18. a 1 = -1.8, a 2 = 0.9, N=20 Least Squares

19. a 1 = -1.8, a 2 = 0.9, N=20 Maximum Likelihood

20. a 1 = 0.05, a 2 = -0.9, N=20 Yule-Walker

21. a 1 = 0.05, a 2 = -0.9, N=20 Least Squares

22. a 1 = 0.05, a 2 = -0.9, N=20 Maximum Likelihood

23. N=100

24. a 1 = -1.8, a 2 = 0.9, N=100 Yule-Walker

25. a 1 = -1.8, a 2 = 0.9, N=100 Least Squares

26. a 1 = -1.8, a 2 = 0.9, N=100 Maximum Likelihood

27. N=1000

28. a 1 = 0.05, a 2 = -0.9, N=1000 Yule-Walker

29. a 1 = 0.05, a 2 = -0.9, N=1000 Least Squares

30. a 1 = 0.05, a 2 = -0.9, N=1000 Maximum Likelihood

31. AR(2) models Maximum Likelihood for different combinations of a 1, a 2

32. a 1 = -1, a 2 = 0.5, N=20

33. a 1 = -1, a 2 = 0.5, N=100

34. a 1 = -1, a 2 = 0.5, N=1000

35. a 1 = 1.3, a 2 = 0.8, N=20

36. a 1 = 1.3, a 2 = 0.8, N=100

37. a 1 = 1.3, a 2 = 0.8, N=1000

38. MA(1) models Conditional Likelihood for different b and N

39. b = 0.9

40. b = 0.6

41. b = -0.4

42. b = -0.9

43. b = -1 (not invertible, still stationary)

44. Here true model is MA(2) with ρ 1 about 0.7. Estimated b is, on average, about 0.75 (corresponding ρ 1 = 0.48)

45. ARMA(1,1) models Conditional Likelihood for different a, b and N

46. a = 0.8, b = 0.75

47. N=20

48. N=50

49. N=100

50. a = -0.7, b = -0.65

51. N=20

52. N=50

53. N=100

54. a = 0.8, b = -0.75 (practically a white noise)

55. N=20

56. N=50

57. N=100