Properties of the estimates of the parameters of ARMA models.

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Presentation transcript:

Properties of the estimates of the parameters of ARMA models

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

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

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

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

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

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

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

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

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

a=-0.95, N=20, 50 simulations YW: average , st.dev LS: average , st.dev ML: average , st.dev

a=-0.95, N=100, 50 simulations YW: average , st.dev LS: average , st.dev ML: average , st.dev

a=-0.95, N=500, 50 simulations YW: average , st.dev LS: average , st.dev ML: average , st.dev

a=-0.95, N=1000, 50 simulations YW: average , st.dev LS: average , st.dev ML: average , st.dev

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

N=20

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

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

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

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

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

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

N=100

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

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

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

N=1000

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

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

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

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

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

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

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

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

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

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

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

b = 0.9

b = 0.6

b = -0.4

b = -0.9

b = -1 (not invertible, still stationary)

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)

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

a = 0.8, b = 0.75

N=20

N=50

N=100

a = -0.7, b = -0.65

N=20

N=50

N=100

a = 0.8, b = (practically a white noise)

N=20

N=50

N=100