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Consider ARMA(2.1) model X4 - X:-1 +62X-2 = 2+ 2-1. When the process is stationary and...
2. Consider an ARMA(1,1) process, X4 = 0.5X:-1 +0+ - 0.25a4-1, where az is white noise...
2. Consider an ARMA(1,1) process, X4 = 0.5X:-1 +0+ - 0.25a4-1, where az is white noise with zero mean and unit variance. (a) Is the model stationary? Explain your answer briefly. (b) Is the model invertible? Explain your answer briefly. (c) Find the infinite moving-average representation of Xt. Namely, find b; such that X =< 0;&–; j=0 (d) Evaluate the first three lags of the ACF and PACF.
2. Consider an ARM A(2,2) model h" ф.xt-1-фг%2 :: at + at-NI 1at-1 + 20.-2, a. Under what condition, the above A...
2. Consider an ARM A(2,2) model h" ф.xt-1-фг%2 :: at + at-NI 1at-1 + 20.-2, a. Under what condition, the above ARMA(2,2) model is causal/stationary. b. Under what condition, the above ARMA(2,2) model is invertible. State the reason for us to consider an invertible ARMA model. Suppose that xt is causal, i.e. C. Calculate , j = 1,2,3,4,5,6. d. Suppose that x is invertible, i.e. Calculate nj, j-1,2,3,4,5,6.
2. Consider an ARM A(2,2) model h" ф.xt-1-фг%2 :: at + at-NI...
QUESTION 3 (a) Consider the ARMA (1, 1) process -Bat-1-where o and θ are model parame-...
QUESTION 3 (a) Consider the ARMA (1, 1) process -Bat-1-where o and θ are model parame- are independent and identically distributed random variables with mean 0 z, oz,-1 ters, and a1, a2, and variance σ (i) Show that the variance of the process is γ,- (ii) Using (i) or otherwise, show that the autocorrelation function (ACF) of the process is: ifk=0. (b) Let Y be an AR(2) process of the special form Y-2Y-2e (i) Find the range of values of...
Q1. (10 points total) Consider an ARMA(1,1) model X4 = 0.9X-1 + Z+ +0.527-1, {Z4}~ IID...
Q1. (10 points total) Consider an ARMA(1,1) model X4 = 0.9X-1 + Z+ +0.527-1, {Z4}~ IID N(0,1). 1. (2 points) (i) Generate n = 200 observations from this ARMA model. (ii) Find the maximum likelihood (ML) estimates of the three parameters o, e, and o2. 2. (8 points) Repeat (i) and (ii) in part 1 nine more times using all different seed numbers. Compare the estimates to their true values. Are the average of 10 estimates for each parameter close...
4. (Forecasting an ARMA(2,2) process) Consider the ARMA(2,2) process: y = 0,8-1 + 0,8-2 + 4...
4. (Forecasting an ARMA(2,2) process) Consider the ARMA(2,2) process: y = 0,8-1 + 0,8-2 + 4 + 0,8-1 +0,4-2 a. Verify that the optimal 1-step ahead forecast made at time T is YT+1,0,+ $y- + 0,4 +0,9-1 b. Verify that the optimal 2-step ahead forecast made at time is YT.27 - $,$t1,7 * 0,81 +0, and express it purely in terms of elements of the time-T information set e. Verify that the optimal 3-step ahead forecast made at time is...
Consider the following ARMA(1,1) Process: Describe the ACF and the PACF plot for the process. Is...
Consider the following ARMA(1,1) Process: Describe the ACF and the PACF plot for the process. Is the process stationary or invertible? (Assume that Also, )
If you model a time series Yt using a stationary ARMA process with a nonzero constant...
If you model a time series Yt using a stationary ARMA process with a nonzero constant (µ unequal to 0) and use it to forecast future values of Yt, then as you forecast further and further into the future, the confidence interval widths for your forecasts will (a) continue to increase and eventually reach arbitrarily large values. (b) gradually decay to zero. (c) cutoff to zero after some lag. (d) converge to a non-zero limiting value.
Consider the following AR(2) model: Xt – Xt–1 + + X4-2 = Zt, Z4 ~ WN(0,1)....
Consider the following AR(2) model: Xt – Xt–1 + + X4-2 = Zt, Z4 ~ WN(0,1). (a) Show that X+ is causal. (b) Find the first four coefficients (VO, ..., 43) of the MA(0) representation of Xt. (c) Find the pacf at lag 3, 233, of the AR(2) model.
QUESTION 3 (a) Consider the ARMA(1, 1) process Zt-oZt_itat-θ4-1 :Where φ and θ are model parame-...
QUESTION 3 (a) Consider the ARMA(1, 1) process Zt-oZt_itat-θ4-1 :Where φ and θ are model parame- ters, and a, a are independent and identically distributed random variables with mean 0 and variance σ 1-1.4. (i) Show that the variance of the process is γ,- (i) Using () or otherwise, show that the autocorrelation function (ACF) of the process is: if k 0,
Consider the ARMA(2,1) model 2+ = 0.624-1 -0.092-2 + at – 0.204-1, a4~WN(0,1) Find the AR...
Consider the ARMA(2,1) model 2+ = 0.624-1 -0.092-2 + at – 0.204-1, a4~WN(0,1) Find the AR representation of {Z}.
2. Consider an ARMA(1,1) process, X4 = 0.5X:-1 +0+ - 0.25a4-1, where az is white noise with zero mean and unit variance. (a) Is the model stationary? Explain your answer briefly. (b) Is the model invertible? Explain your answer briefly. (c) Find the infinite moving-average representation of Xt. Namely, find b; such that X =< 0;&–; j=0 (d) Evaluate the first three lags of the ACF and PACF.
2. Consider an ARM A(2,2) model h" ф.xt-1-фг%2 :: at + at-NI 1at-1 + 20.-2, a. Under what condition, the above ARMA(2,2) model is causal/stationary. b. Under what condition, the above ARMA(2,2) model is invertible. State the reason for us to consider an invertible ARMA model. Suppose that xt is causal, i.e. C. Calculate , j = 1,2,3,4,5,6. d. Suppose that x is invertible, i.e. Calculate nj, j-1,2,3,4,5,6.
2. Consider an ARM A(2,2) model h" ф.xt-1-фг%2 :: at + at-NI...
QUESTION 3 (a) Consider the ARMA (1, 1) process -Bat-1-where o and θ are model parame- are independent and identically distributed random variables with mean 0 z, oz,-1 ters, and a1, a2, and variance σ (i) Show that the variance of the process is γ,- (ii) Using (i) or otherwise, show that the autocorrelation function (ACF) of the process is: ifk=0. (b) Let Y be an AR(2) process of the special form Y-2Y-2e (i) Find the range of values of...
Q1. (10 points total) Consider an ARMA(1,1) model X4 = 0.9X-1 + Z+ +0.527-1, {Z4}~ IID N(0,1). 1. (2 points) (i) Generate n = 200 observations from this ARMA model. (ii) Find the maximum likelihood (ML) estimates of the three parameters o, e, and o2. 2. (8 points) Repeat (i) and (ii) in part 1 nine more times using all different seed numbers. Compare the estimates to their true values. Are the average of 10 estimates for each parameter close...
4. (Forecasting an ARMA(2,2) process) Consider the ARMA(2,2) process: y = 0,8-1 + 0,8-2 + 4 + 0,8-1 +0,4-2 a. Verify that the optimal 1-step ahead forecast made at time T is YT+1,0,+ $y- + 0,4 +0,9-1 b. Verify that the optimal 2-step ahead forecast made at time is YT.27 - $,$t1,7 * 0,81 +0, and express it purely in terms of elements of the time-T information set e. Verify that the optimal 3-step ahead forecast made at time is...
Consider the following AR(2) model: Xt – Xt–1 + + X4-2 = Zt, Z4 ~ WN(0,1). (a) Show that X+ is causal. (b) Find the first four coefficients (VO, ..., 43) of the MA(0) representation of Xt. (c) Find the pacf at lag 3, 233, of the AR(2) model.
QUESTION 3 (a) Consider the ARMA(1, 1) process Zt-oZt_itat-θ4-1 :Where φ and θ are model parame- ters, and a, a are independent and identically distributed random variables with mean 0 and variance σ 1-1.4. (i) Show that the variance of the process is γ,- (i) Using () or otherwise, show that the autocorrelation function (ACF) of the process is: if k 0,
Consider the ARMA(2,1) model 2+ = 0.624-1 -0.092-2 + at – 0.204-1, a4~WN(0,1) Find the AR representation of {Z}.
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