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 the
process.
(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.
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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...
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 fZ) is a white noise process with unit variance. It is known that the above process is overestimated [4 marks (b) Hence, determine the stationarity and invertibility of the process. [4 marks (c) Find the first three lags of the autocorrelation function (ACF) for the process. [12 marks) (5 marks] (a) Suggest a parsimonious model for the above process. (d) Find the first three lags...
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...
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 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...
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,
Dear tutor, Please could you help me with these questions. Please kindly give brief explanations ...
Dear tutor,
Please could you help me with these questions. Please kindly
give brief explanations to the answers.
Thank you.
4. Which of the following sets of characteristics would usually best describe an autoregressive process of order 3 (i.e. an AR(3)? (a) A slowly decaying acf, and a pacf with 3 significant spikes (b) A slowly decaying pacf and an acf with 3 significant spikes (c) A slowly decaying acf and pacf (d) An acf and a pacf with 3...
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...
For each of the following first construct an example and then show that it has the...
For each of the following first construct an example and then
show that it has the correct properties: (a) (Xt) with constant
mean but has a variance that is a function of time. (b) (Wt) white
noise process that is not strongly stationary. (c) (Zt) is
nonstationary process with an autocovariance function such that
γ(t, t) = σ 2 for all t. (d) (Vt) is nonstationary with an
autocovariance function such that γ(t, t+ h) = 0 for all |h|...
(b) Jan has collected a monthly time series consisting of 167 observations on the differenced log Yen/SAU exchange rate and plotted the sample ACF for the series (see below). Use the QLB(2) statistic...
(b) Jan has collected a monthly time series consisting of 167 observations on the differenced log Yen/SAU exchange rate and plotted the sample ACF for the series (see below). Use the QLB(2) statistic to test whether or not the series is a white-noise process at the 5% level of significance. Write down your hypotheses, decision rule and conclusion. [3 marks] Lag-1.00 -0.60-0.20 0.20 0.60 ACF 1.00 I 0.001073 0.121080 1 -0.084039 1 -0.052353 I -0.054732 1 -0.095489 0 . 044049...
: Assume Yt is a time series process and Et is a white noise process with...
: Assume Yt is a time series process and Et is a white noise process with mean zero and constant variance. (a). Write an equation for AR(4) process. (b). Write an equation for AR(5) process. (c). Write an equation for MA(3) process. (d). Write down an equation for MA(2) process. (e). Write an equation for ARMA (4,2) process. (f). Do more research and write an equation for ARIMA (4,0,2) proce
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 fZ) is a white noise process with unit variance. It is known that the above process is overestimated [4 marks (b) Hence, determine the stationarity and invertibility of the process. [4 marks (c) Find the first three lags of the autocorrelation function (ACF) for the process. [12 marks) (5 marks] (a) Suggest a parsimonious model for the above process. (d) Find the first three lags...
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...
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 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,
Dear tutor,
Please could you help me with these questions. Please kindly
give brief explanations to the answers.
Thank you.
4. Which of the following sets of characteristics would usually best describe an autoregressive process of order 3 (i.e. an AR(3)? (a) A slowly decaying acf, and a pacf with 3 significant spikes (b) A slowly decaying pacf and an acf with 3 significant spikes (c) A slowly decaying acf and pacf (d) An acf and a pacf with 3...
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...
For each of the following first construct an example and then
show that it has the correct properties: (a) (Xt) with constant
mean but has a variance that is a function of time. (b) (Wt) white
noise process that is not strongly stationary. (c) (Zt) is
nonstationary process with an autocovariance function such that
γ(t, t) = σ 2 for all t. (d) (Vt) is nonstationary with an
autocovariance function such that γ(t, t+ h) = 0 for all |h|...
(b) Jan has collected a monthly time series consisting of 167 observations on the differenced log Yen/SAU exchange rate and plotted the sample ACF for the series (see below). Use the QLB(2) statistic to test whether or not the series is a white-noise process at the 5% level of significance. Write down your hypotheses, decision rule and conclusion. [3 marks] Lag-1.00 -0.60-0.20 0.20 0.60 ACF 1.00 I 0.001073 0.121080 1 -0.084039 1 -0.052353 I -0.054732 1 -0.095489 0 . 044049...
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