MATH461-HW2

MATH461-HW2 Vinh Pham 1.13 a) Since y t is a combination of independent normal random variables, the variance of y t is σ 2 w + θ 2 σ 2 w + σ 2 u for any t . For any y t , y s , the ACF or ρ ( t, s ) can be calculated as Cov ( Y t , Y s ) qqqqqqq V ar ( Y t ) × V ar ( Y s ) The numerator becomes Cov ( w t − θw t − 1 + µ t , w s − θw s − 1 + µ s ) . Clearly, if | s − t | ≥ 2 , then Cov ( y t , y s ) = 0 , which means that ρ ( t, s ) = 0 . Denote h = t − s , then ρ ( h ) = 0 if | h | ≥ 2 . If h = 0 , the ACF becomes 1 . If h = 1 , then ρ ( h ) = − θσ 2 w σ 2 w + θ 2 σ 2 w + σ 2 u . b) p xy ( s, t ) = γ xy ( s, t ) qqqqqqq γ x ( s, s ) × γ y ( t, t ) = E [( x s − E [ x s ])( y t − E [ y t ])] σ w qqqqqqq (1 + θ ) 2 σ 2 w + σ 2 u = E [ w s ( w t − θw t − 1 + u t )] σ w qqqqqqq (1 + θ ) 2 σ 2 w + σ 2 u Therefore, p xy ( s, t ) = 999 ????? ????? ===== ????? ????? ;;;; σ w √ (1+ θ ) 2 σ 2 w + σ 2 u if s = t − θσ w √ (1+ θ ) 2 σ 2 w + σ 2 u if s = t ± 1 0 otherwise Thus, p xy ( h ) = 999 ????? ????? ===== ????? ????? ;;;; σ w √ (1+ θ ) 2 σ 2 w + σ 2 u if h = 0 − θσ w √ (1+ θ ) 2 σ 2 w + σ 2 u if h ± 1 0 otherwise 1

c) The auto covariance and cross covariance of x and y depend only on the lag therefore, the series are jointly stationary. 1.20) a) sigma_squared = 1 n = 500 w = rnorm (n, 0 , sigma_squared) print ( acf (w, 20 , na.action = na.pass)) 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 Lag ACF Series w ## ## Autocorrelations of series 'w', by lag ## ## 0 1 2 3 4 5 6 7 8 9 10 ## 1.000 0.078 -0.055 -0.052 -0.083 0.083 -0.013 -0.005 -0.004 -0.048 0.069 ## 11 12 13 14 15 16 17 18 19 20 ## 0.020 -0.016 0.049 -0.066 -0.027 -0.005 0.019 0.042 -0.022 -0.005 2

The value that we obtained is somewhat similar to the theoretical values. They also both fluctuates between postive and negative values. Note that the ACF function does not equal to 0 as h > 0 but slowly converges to 0 b) sigma_squared = 1 n = 50 w = rnorm (n, 0 , sigma_squared) print ( acf (w, 20 , na.action = na.pass)) 0 5 10 15 20 -0.2 0.2 0.4 0.6 0.8 1.0 Lag ACF Series w ## ## Autocorrelations of series 'w', by lag ## ## 0 1 2 3 4 5 6 7 8 9 10 ## 1.000 -0.313 0.058 -0.076 -0.115 0.070 0.027 -0.026 -0.071 0.117 -0.123 ## 11 12 13 14 15 16 17 18 19 20 ## 0.251 -0.252 -0.087 0.027 0.061 0.129 0.073 -0.139 -0.104 0.132 There is more variance in the simulation, and also the values of the ACF when h > 0 is closer to zero in the case of n = 500 compared to n = 50 implies that if we increases sample size we will get the actual value of the ACF. 3