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4. Consider the ARMA (2, 3) process, I( 0.1%-1 +0.12%-2 + Ze + 0.3Zn-1-0.045-2-0.012Zt-3, where f...
time series analysis
where at is a white noise process with unit variance. It is known that the above process is overestimated.(a) Suggest a parismony model ARMA(2,1) for the above process.(b) Hence, determine the stationarity and invertibility of the process.(c) Find the mean, the variance and the first two lags of the autocovariance function of theprocess.(d) Find the first three lags of the autocorrelation function (ACF) for the process.(e) Find the first three lags of the partial autocorrelation coefficients.
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.
Question 2 (a) The following table gives the sample autocorrelation coefficients and partial auto...
Question 2 (a) The following table gives the sample autocorrelation coefficients and partial autocorrelation coefficients for a time series with 100 observations. 4 ,-0.55 -0.17 0.09 0.0.00.010.040.07 -0.55 | -0.4 0.29 | -0.22 -0.11- -0.13 -0.14 0,05 Suppose the sample mean of the time series is zero. Based on the above information, suggest an ARMA model for the data. Briefly explain your answer. (5 marks) (b) Let X, be a time series satisfying the following AR(2)model: X, = 0.3X,-1 +0.04X,-2...
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...
Question 2 (a) The following table gives the sample autocorrelation coefficients and partial autocorrelation coefficients for...
Question 2 (a) The following table gives the sample autocorrelation coefficients and partial autocorrelation coefficients for a time series with 100 observations. 4 ,-0.55 -0.17 0.09 0.0.00.010.040.07 -0.55 | -0.4 0.29 | -0.22 -0.11- -0.13 -0.14 0,05 Suppose the sample mean of the time series is zero. Based on the above information, suggest an ARMA model for the data. Briefly explain your answer. (5 marks) (b) Let X, be a time series satisfying the following AR(2)model: X, = 0.3X,-1 +0.04X,-2...
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...
2. FoRECASTING wITH MA PROCESSES. (i) How to check the invertibility of an MA(1) model? (ii) Suppose we use an MA model to model the process represented in Figure 2. Write down the model and find the...
2. FoRECASTING wITH MA PROCESSES. (i) How to check the invertibility of an MA(1) model? (ii) Suppose we use an MA model to model the process represented in Figure 2. Write down the model and find the estimates of the coefficients. (15 marks) 200 225 250 275 300 325 350 375 400 Sample: 1 2000 Included observations: 1999 Autocorrelation Partial Correlation AC PAC 10.498 0.498 2-0.042-0.386 -0.081 0.220 40.042 -0.183 5 0.003 0.157 6 0.013-0.127 7 0.019 0.131 8 0.013-0.112...
2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 ...
2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 mark] ii. Find the infinite moving average representation of X,i.e., find the scquence [6 marks] i. Explain why the process is stationary. (6) such that Xt = Σ b,2-j. iii. Calculate the mean and the autocovariance "Yo, γι and 72 of the process. 7 marks iv. Given 40 = 0.1 and Xo = 1.8, find the 2-step ahead forecast of the time series...
Consider the RC circuit shown below. Assume that R=(0.1)2 and C=(0.1)F 3. R i(t) y (t)...
Consider the RC circuit shown below. Assume that R=(0.1)2 and C=(0.1)F 3. R i(t) y (t) x(t) The input to this circuit is given as x(t) s(t)+ny (t), where the noise component of input, n(t), is a sample function realization of white noise process with an autocorrelation function given by Rpx(t) 8(T), and s (t) cos(6Tt) is the signal component of input. IS(fOI df, where S( a. Find the power of the signal component of input, Ps is the Fourier...
Consider the RC circuit shown below. Assume that R=(0.1)2 and C=(0.1)F 3. R i(t) y (t)...
Consider the RC circuit shown below. Assume that R=(0.1)2 and C=(0.1)F 3. R i(t) y (t) x(t) The input to this circuit is given as x(t) s(t)+ny (t), where the noise component of input, n(t), is a sample function realization of white noise process with an autocorrelation function given by Rpx(t) 8(T), and s (t) cos(6Tt) is the signal component of input. IS(fOI df, where S( a. Find the power of the signal component of input, Ps is the Fourier...
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.
Question 2 (a) The following table gives the sample autocorrelation coefficients and partial autocorrelation coefficients for a time series with 100 observations. 4 ,-0.55 -0.17 0.09 0.0.00.010.040.07 -0.55 | -0.4 0.29 | -0.22 -0.11- -0.13 -0.14 0,05 Suppose the sample mean of the time series is zero. Based on the above information, suggest an ARMA model for the data. Briefly explain your answer. (5 marks) (b) Let X, be a time series satisfying the following AR(2)model: X, = 0.3X,-1 +0.04X,-2...
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...
Question 2 (a) The following table gives the sample autocorrelation coefficients and partial autocorrelation coefficients for a time series with 100 observations. 4 ,-0.55 -0.17 0.09 0.0.00.010.040.07 -0.55 | -0.4 0.29 | -0.22 -0.11- -0.13 -0.14 0,05 Suppose the sample mean of the time series is zero. Based on the above information, suggest an ARMA model for the data. Briefly explain your answer. (5 marks) (b) Let X, be a time series satisfying the following AR(2)model: X, = 0.3X,-1 +0.04X,-2...
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...
2. FoRECASTING wITH MA PROCESSES. (i) How to check the invertibility of an MA(1) model? (ii) Suppose we use an MA model to model the process represented in Figure 2. Write down the model and find the estimates of the coefficients. (15 marks) 200 225 250 275 300 325 350 375 400 Sample: 1 2000 Included observations: 1999 Autocorrelation Partial Correlation AC PAC 10.498 0.498 2-0.042-0.386 -0.081 0.220 40.042 -0.183 5 0.003 0.157 6 0.013-0.127 7 0.019 0.131 8 0.013-0.112...
2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 mark] ii. Find the infinite moving average representation of X,i.e., find the scquence [6 marks] i. Explain why the process is stationary. (6) such that Xt = Σ b,2-j. iii. Calculate the mean and the autocovariance "Yo, γι and 72 of the process. 7 marks iv. Given 40 = 0.1 and Xo = 1.8, find the 2-step ahead forecast of the time series...
Consider the RC circuit shown below. Assume that R=(0.1)2 and C=(0.1)F 3. R i(t) y (t) x(t) The input to this circuit is given as x(t) s(t)+ny (t), where the noise component of input, n(t), is a sample function realization of white noise process with an autocorrelation function given by Rpx(t) 8(T), and s (t) cos(6Tt) is the signal component of input. IS(fOI df, where S( a. Find the power of the signal component of input, Ps is the Fourier...
Consider the RC circuit shown below. Assume that R=(0.1)2 and C=(0.1)F 3. R i(t) y (t) x(t) The input to this circuit is given as x(t) s(t)+ny (t), where the noise component of input, n(t), is a sample function realization of white noise process with an autocorrelation function given by Rpx(t) 8(T), and s (t) cos(6Tt) is the signal component of input. IS(fOI df, where S( a. Find the power of the signal component of input, Ps is the Fourier...
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