Properties of the estimates of the parameters of ARMA models.
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Transcript of Properties of the estimates of the parameters of ARMA models.
- Slide 1
- Properties of the estimates of the parameters of ARMA models
- Slide 2
- AR(1) models Comparison of Yule-Walker, Least Squares and Maximum Likelihood
- Slide 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
- Slide 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
- Slide 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
- Slide 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
- Slide 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
- Slide 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
- Slide 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
- Slide 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
- Slide 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
- Slide 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
- Slide 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
- Slide 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
- Slide 15
- AR(2) models Yule-Walker, Least squares and Maximum Likelihood for different N
- Slide 16
- N=20
- Slide 17
- a 1 = -1.8, a 2 = 0.9, N=20 Yule-Walker
- Slide 18
- a 1 = -1.8, a 2 = 0.9, N=20 Least Squares
- Slide 19
- a 1 = -1.8, a 2 = 0.9, N=20 Maximum Likelihood
- Slide 20
- a 1 = 0.05, a 2 = -0.9, N=20 Yule-Walker
- Slide 21
- a 1 = 0.05, a 2 = -0.9, N=20 Least Squares
- Slide 22
- a 1 = 0.05, a 2 = -0.9, N=20 Maximum Likelihood
- Slide 23
- N=100
- Slide 24
- a 1 = -1.8, a 2 = 0.9, N=100 Yule-Walker
- Slide 25
- a 1 = -1.8, a 2 = 0.9, N=100 Least Squares
- Slide 26
- a 1 = -1.8, a 2 = 0.9, N=100 Maximum Likelihood
- Slide 27
- N=1000
- Slide 28
- a 1 = 0.05, a 2 = -0.9, N=1000 Yule-Walker
- Slide 29
- a 1 = 0.05, a 2 = -0.9, N=1000 Least Squares
- Slide 30
- a 1 = 0.05, a 2 = -0.9, N=1000 Maximum Likelihood
- Slide 31
- AR(2) models Maximum Likelihood for different combinations of a 1, a 2
- Slide 32
- a 1 = -1, a 2 = 0.5, N=20
- Slide 33
- a 1 = -1, a 2 = 0.5, N=100
- Slide 34
- a 1 = -1, a 2 = 0.5, N=1000
- Slide 35
- a 1 = 1.3, a 2 = 0.8, N=20
- Slide 36
- a 1 = 1.3, a 2 = 0.8, N=100
- Slide 37
- a 1 = 1.3, a 2 = 0.8, N=1000
- Slide 38
- MA(1) models Conditional Likelihood for different b and N
- Slide 39
- b = 0.9
- Slide 40
- b = 0.6
- Slide 41
- b = -0.4
- Slide 42
- b = -0.9
- Slide 43
- b = -1 (not invertible, still stationary)
- Slide 44
- Here true model is MA(2) with 1 about 0.7. Estimated b is, on average, about 0.75 (corresponding 1 = 0.48)
- Slide 45
- ARMA(1,1) models Conditional Likelihood for different a, b and N
- Slide 46
- a = 0.8, b = 0.75
- Slide 47
- N=20
- Slide 48
- N=50
- Slide 49
- N=100
- Slide 50
- a = -0.7, b = -0.65
- Slide 51
- N=20
- Slide 52
- N=50
- Slide 53
- N=100
- Slide 54
- a = 0.8, b = -0.75 (practically a white noise)
- Slide 55
- N=20
- Slide 56
- N=50
- Slide 57
- N=100