![2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 mark] ii. Find the infinite](http://img.homeworklib.com/images/52acc692-9213-47cc-9143-c62d84eb8e60.png?x-oss-process=image/resize,w_560)
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2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 ...
Let Wt de a (Gaussian) white noise with variance σ 2 . Then, let Xt =...
Let Wt de a (Gaussian) white noise with variance σ 2 . Then, let
Xt = WtWt−1 + µ, where µ is a real constant. Determine the mean and
autocovariance of (Xt)? Is this process stationary?
Let W, de a (Gaussian) white noise with variance σ2. Then, let of where μ is a real constant. Determine the mean and (X)? Is this process stationary?
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. Let (et) be a zero mean white noise process with variance 1. Suppose that the...
2. Let (et) be a zero mean white noise process with variance 1. Suppose that the observed process is h ft + Xt where β is an unknown constant, and Xt-et- Explain why {X.) is stationary. Find its mean function μχ and autocorrelation function p for lk0,1,.. a. b. Show that {Yt3 is not stationary. C. Explain why w. = ▽h = h-K-1 is stationary. d. Calculate Var(Yt) Vt and Var(W) Vt . (Recall: Var(X+c)-Var(X) when c is a constant.)...
(A). Draw the Autocorrelaogram and Partial Autocorrelogram for a White Noise Time Series Process....
(A). Draw the Autocorrelaogram and Partial Autocorrelogram for a White Noise Time Series Process. (B). Assume that the optimal h-steps ahead forecast is noted as fth for a MA(1). Lets also assume that the optimal point forecast is a conditional expectation: Where Qt is the information set at time "t" and "h" is the forecast horizon. Now we can write the MA(1) process at time "t+1" as follows; Ü. What is the optimal one period ahead forecast, f,i? (ii). What...
2. Let [et be a zero mean white noise process with variance 0.25. Suppose that the...
2. Let [et be a zero mean white noise process with variance 0.25. Suppose that the observed process is k = et + 0.5e-2. a. Explain why {Yt) is stationary. b. Compute yo-V(Y.) c. Compute the autocorrelation pkY, kl-0,1,2,... for Y) d. Let Wt = 3 + 4t + h. i. Find the mean of {W) ii. Is W3 stationary? Why or why not? iii. Let Z Vw, W,- W,_1. Is {Z.1 stationary? Why or why not?
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|...
4.5 Consider the simple white noise process, Z= a. Discuss the consequence of overdifferencing by examining...
4.5 Consider the simple white noise process, Z= a. Discuss the consequence of overdifferencing by examining the ACF, PACF, and AR representation of the differ- enced series, W, = Z, - 2-1.
Consider the random walk model where {} is a white noise process with variance a) How...
Consider the random
walk model
where {}
is a white noise process with variance
a) How many parameters
does this model have?
b) calculate
and
c) Compute
for
d) Is this model
weakly stationary?
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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...
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).
Let Wt de a (Gaussian) white noise with variance σ 2 . Then, let
Xt = WtWt−1 + µ, where µ is a real constant. Determine the mean and
autocovariance of (Xt)? Is this process stationary?
Let W, de a (Gaussian) white noise with variance σ2. Then, let of where μ is a real constant. Determine the mean and (X)? Is this process stationary?
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. Let (et) be a zero mean white noise process with variance 1. Suppose that the observed process is h ft + Xt where β is an unknown constant, and Xt-et- Explain why {X.) is stationary. Find its mean function μχ and autocorrelation function p for lk0,1,.. a. b. Show that {Yt3 is not stationary. C. Explain why w. = ▽h = h-K-1 is stationary. d. Calculate Var(Yt) Vt and Var(W) Vt . (Recall: Var(X+c)-Var(X) when c is a constant.)...
(A). Draw the Autocorrelaogram and Partial Autocorrelogram for a White Noise Time Series Process. (B). Assume that the optimal h-steps ahead forecast is noted as fth for a MA(1). Lets also assume that the optimal point forecast is a conditional expectation: Where Qt is the information set at time "t" and "h" is the forecast horizon. Now we can write the MA(1) process at time "t+1" as follows; Ü. What is the optimal one period ahead forecast, f,i? (ii). What...
2. Let [et be a zero mean white noise process with variance 0.25. Suppose that the observed process is k = et + 0.5e-2. a. Explain why {Yt) is stationary. b. Compute yo-V(Y.) c. Compute the autocorrelation pkY, kl-0,1,2,... for Y) d. Let Wt = 3 + 4t + h. i. Find the mean of {W) ii. Is W3 stationary? Why or why not? iii. Let Z Vw, W,- W,_1. Is {Z.1 stationary? Why or why not?
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|...
4.5 Consider the simple white noise process, Z= a. Discuss the consequence of overdifferencing by examining the ACF, PACF, and AR representation of the differ- enced series, W, = Z, - 2-1.
Consider the random
walk model
where {}
is a white noise process with variance
a) How many parameters
does this model have?
b) calculate
and
c) Compute
for
d) Is this model
weakly stationary?
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imagevar(yt) We were unable to transcribe this imageWe were unable to transcribe this image
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...
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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