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