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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 -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...
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...
Onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process...
onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process? independent identically distributed random variables with variance 1. Is (Y How about the process ▽Yǐ = Yt-)t-1 ? Explain.
onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process? independent identically distributed random variables with variance 1. Is (Y How about the process ▽Yǐ = Yt-)t-1 ? Explain.
QUESTION4 (a) Let e be a zero-mean, unit-variance white noise process. Consider a process that begins...
QUESTION4 (a) Let e be a zero-mean, unit-variance white noise process. Consider a process that begins at time t = 0 and is defined recursively as follows. Let Y0 = ceo and Y1-CgY0-ei. Then let Y,-φ1Yt-it wt-1-et for t > ï as in an AR(2) process. Show that the process mean, E(Y.), is zero. (b) Suppose that (a is generated according to }.-10 e,-tet-+扣-1 with e,-N(0.) 0 Find the mean and covariance functions for (Y). Is (Y) stationary? Justify your...
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.
B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) proc...
B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) process, ao 0, α120, ai 1 > α1 + β1. Show that 0, A > 0 and 2IV2 t-1't-2 ..
B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) process, ao 0, α120, ai 1 > α1 + β1. Show that 0, A > 0...
If we fit the following model with n observations where i = 1,...,n and 's are...
If we fit the following model with n observations
where i = 1,...,n
and 's are
identically and independently distributed as a normal random
variable with a mean of zero and a variance ,
and all the x's are fixed.
How can I show that the point estimator of the mean response and
its residuals are statistically independent normal random
variables?
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
MA2500/18 8. Let X be a random variable and let 'f(r; θ) be its PDF where...
MA2500/18 8. Let X be a random variable and let 'f(r; θ) be its PDF where θ is an unknown scalar parameter. We wish to test the simple null hypothesis Ho: 0 against the simple alternative Hi : θ-64. (a) Define the simple likelihood ratio test (SLRT) of Ho against H (b) Show that the SLRT is a most powerful test of Ho against H. (c) Let Xi, X2.... , X be a random sample of observations from the Poisson(e)...
Consider the regression model where the εi are i.i.d. N(0,σ2) random variables, for i = 1,...
Consider the regression model
where the εi are i.i.d. N(0,σ2) random variables, for i = 1, 2, . .
. , n.
(a) (4 points) Show βˆ is normally distributed with mean β and
variance σ2 . 1 1SXX
Question 6 Consider the regression model y = Bo + B12 + 8 where the €, are i.i.d. N(0,0%) random variables, for i = 1,2, ..., n. (a) (4 points) Show B1 is normally distributed with mean B1 and variances
Consider the process Y.-μ + et-o, et-1-912 et-12, where {ed denotes a white-noise process with mean...
Consider the process Y.-μ + et-o, et-1-912 et-12, where {ed denotes a white-noise process with mean 0 and variance σ? > 0. Assume that et ls independent of Yt-1, Yt-2, Find the autocorrelation function for (Yt).
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...
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...
onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process? independent identically distributed random variables with variance 1. Is (Y How about the process ▽Yǐ = Yt-)t-1 ? Explain.
onsider the process Y, = Y + Σ|e, where Yo ~ (μ, σ2) and the e's are 0-mean, a stationary process? independent identically distributed random variables with variance 1. Is (Y How about the process ▽Yǐ = Yt-)t-1 ? Explain.
QUESTION4 (a) Let e be a zero-mean, unit-variance white noise process. Consider a process that begins at time t = 0 and is defined recursively as follows. Let Y0 = ceo and Y1-CgY0-ei. Then let Y,-φ1Yt-it wt-1-et for t > ï as in an AR(2) process. Show that the process mean, E(Y.), is zero. (b) Suppose that (a is generated according to }.-10 e,-tet-+扣-1 with e,-N(0.) 0 Find the mean and covariance functions for (Y). Is (Y) stationary? Justify your...
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.
B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) process, ao 0, α120, ai 1 > α1 + β1. Show that 0, A > 0 and 2IV2 t-1't-2 ..
B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) process, ao 0, α120, ai 1 > α1 + β1. Show that 0, A > 0...
If we fit the following model with n observations
where i = 1,...,n
and 's are
identically and independently distributed as a normal random
variable with a mean of zero and a variance ,
and all the x's are fixed.
How can I show that the point estimator of the mean response and
its residuals are statistically independent normal random
variables?
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
MA2500/18 8. Let X be a random variable and let 'f(r; θ) be its PDF where θ is an unknown scalar parameter. We wish to test the simple null hypothesis Ho: 0 against the simple alternative Hi : θ-64. (a) Define the simple likelihood ratio test (SLRT) of Ho against H (b) Show that the SLRT is a most powerful test of Ho against H. (c) Let Xi, X2.... , X be a random sample of observations from the Poisson(e)...
Consider the regression model
where the εi are i.i.d. N(0,σ2) random variables, for i = 1, 2, . .
. , n.
(a) (4 points) Show βˆ is normally distributed with mean β and
variance σ2 . 1 1SXX
Question 6 Consider the regression model y = Bo + B12 + 8 where the €, are i.i.d. N(0,0%) random variables, for i = 1,2, ..., n. (a) (4 points) Show B1 is normally distributed with mean B1 and variances
Consider the process Y.-μ + et-o, et-1-912 et-12, where {ed denotes a white-noise process with mean 0 and variance σ? > 0. Assume that et ls independent of Yt-1, Yt-2, Find the autocorrelation function for (Yt).
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