![Exercise 2.31 Superposition [] Given two independent weakly stationary time series Xt and Yi) with autocovariance functions x(h) and y (h), show that Zt- Xt +Yt is also weakly stationary, with autocovariance function given by yz(h)-x(h)y(h).](http://img.homeworklib.com/questions/99802570-7252-11ea-8981-5beba1289e56.png?x-oss-process=image/resize,w_560)
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Exercise 2.31 Superposition [] Given two independent weakly stationary time series Xt and Yi) with autocovariance...
Suppose Zt = 2 + Xt -2Xt-1+Xt-2, where {Xt} is zero-mean stationary series with autocovariance function....
Suppose Zt = 2 + Xt -2Xt-1+Xt-2,
where {Xt} is zero-mean stationary series with autocovariance
function.
Calculate the autocovariance of Zt
You are given the three autoregressive time series models Xt 0.5Xt-1 + Z 4 Determine which...
You are given the three autoregressive time series models Xt 0.5Xt-1 + Z 4 Determine which of the following statements is true (A) {Xtj is stationary, {Ytj and {Utj are non-stationary (B) {Y;} is stationary, {X/} and {Ut} are non-stationary (C) {Ut} is stationary·(X} and {Yt} are non-stationary. (D) {X,) and {Yt) are stationary, Ut) is non-stationary (B) {Xt} and { are stationary, {E) is non-stationary
I. (5 points) Let {X, } be a stationary series with mean μ and autocovariance function...
I. (5 points) Let {X, } be a stationary series with mean μ and autocovariance function 7(), and icz Show Y is also stationary for a, ER, iE Z 2. (5 points) Let {Xi be the process Xi A cos(wt) Bsin(t),t 1,2, ., COS
Exercise 4: Heavy-tails (a) Consider the time series {Y determined by the equation 1t-1 where ao ...
Exercise 4: Heavy-tails (a) Consider the time series {Y determined by the equation 1t-1 where ao >0 and a20. Give a necessary condition for Y in this case find its mean and autocovariance function to be stationary, and (3 marks)
Exercise 4: Heavy-tails (a) Consider the time series {Y determined by the equation 1t-1 where ao >0 and a20. Give a necessary condition for Y in this case find its mean and autocovariance function to be stationary, and (3 marks)
2. Suppose that Ya ut where the ut are iid Normal with mean zero and variance σ2, but that you mistakenly think Yt is difference stationary. You therefore construct a new series a) Are the Xt i.i.d.?...
2. Suppose that Ya ut where the ut are iid Normal with mean zero and variance σ2, but that you mistakenly think Yt is difference stationary. You therefore construct a new series a) Are the Xt i.i.d.? Explain b) Is X stationary? Explain c) Calculate the mean, variance, and autocorrelation function of X d) How does the answer you obtained in (c) compare with the mean, variance and autocor- relation function of Y?
2. Suppose that Ya ut where the...
Time series question about independent stationary processes 2:23 Two processes |Z, and (Y are said to...
Time series question about independent stationary processes
2:23 Two processes |Z, and (Y are said to be independent if for any time points 11 Im and s1.8-2... Sn the random variables [Z, Z,. ..Z, are independent of the random variables [Ys,. Ys, ..., Y). Show that if IZ) and Y are inde- 23 mm pendent stationary proesses, then W-2,+ 7 is stationary.
It should be clear from the discussion that a strictly stationary, finite variance, time series is...
It should be clear from the discussion that a strictly stationary, finite variance, time series is also stationary. However, the converse may or may not be true, in general. A time series {Xti t 0,?1,?2, ) is said to be a Gaussian process, if the n-dimensional random vectors X = (Xti, Xt2, .. . , X.). for every collection of time points t1, t2,..., tn, and every positive integer n, have a Multivariate Normal distribution. } is a stationary Gaussian...
2. Consider the time series X, = 2 + 0.5t +0.8X1-1 + W, where W N(0.1). (a) (8 points) Calculate E(X2) Is this process weakly stationary? Give reasons for your answer. Hint: Find the mean function of...
2. Consider the time series X, = 2 + 0.5t +0.8X1-1 + W, where W N(0.1). (a) (8 points) Calculate E(X2) Is this process weakly stationary? Give reasons for your answer. Hint: Find the mean function of {X) and then substitute t = 20. (b) (3 points) Calculate Var(X20) Question 2 continues on the next page... Page 4 of 12 c)(4 points) Consider the first differences of the time series above, that is Is {%) a weakly stationary process. Prove...
4. Consider two functions x(h),y (h), Vh> 0 and cross-covariance function 7xy(h), Vh e Z (a)...
4. Consider two functions x(h),y (h), Vh> 0 and cross-covariance function 7xy(h), Vh e Z (a) (7 points) Find the Best Linear Predictor (BLP) of Y given Xt, and its Mean Square Prediction Error (MSPE), expressed in terms of x(h). (h), xy (h) jointly stationary time series (X, Y), with individual auto-covariance (b) (13 points) Find the BLP of Y given Xt, X-1,Y-1, and its MSPE, expressed in terms of 7x(h)(h), x.y (h). (Note: you don't need to solve the...
exercise 4 Exercises For each data set, plot the time series, and write the exact definition,...
exercise 4
Exercises For each data set, plot the time series, and write the exact definition, periodicity and units. Judge whether the underlying stochastic process may be first and sec- ond order weakly stationary. Explain your rationale. 77 a constant): a. LY, = b, Lc= nd d. LkY, for some integer k >0 5. The following table contains quarterly nominal GDP in U.S. (billions of dol- lars). Let Y, denote the GDP at time t and let y, In (Y)....
Suppose Zt = 2 + Xt -2Xt-1+Xt-2,
where {Xt} is zero-mean stationary series with autocovariance
function.
Calculate the autocovariance of Zt
You are given the three autoregressive time series models Xt 0.5Xt-1 + Z 4 Determine which of the following statements is true (A) {Xtj is stationary, {Ytj and {Utj are non-stationary (B) {Y;} is stationary, {X/} and {Ut} are non-stationary (C) {Ut} is stationary·(X} and {Yt} are non-stationary. (D) {X,) and {Yt) are stationary, Ut) is non-stationary (B) {Xt} and { are stationary, {E) is non-stationary
I. (5 points) Let {X, } be a stationary series with mean μ and autocovariance function 7(), and icz Show Y is also stationary for a, ER, iE Z 2. (5 points) Let {Xi be the process Xi A cos(wt) Bsin(t),t 1,2, ., COS
Exercise 4: Heavy-tails (a) Consider the time series {Y determined by the equation 1t-1 where ao >0 and a20. Give a necessary condition for Y in this case find its mean and autocovariance function to be stationary, and (3 marks)
Exercise 4: Heavy-tails (a) Consider the time series {Y determined by the equation 1t-1 where ao >0 and a20. Give a necessary condition for Y in this case find its mean and autocovariance function to be stationary, and (3 marks)
2. Suppose that Ya ut where the ut are iid Normal with mean zero and variance σ2, but that you mistakenly think Yt is difference stationary. You therefore construct a new series a) Are the Xt i.i.d.? Explain b) Is X stationary? Explain c) Calculate the mean, variance, and autocorrelation function of X d) How does the answer you obtained in (c) compare with the mean, variance and autocor- relation function of Y?
2. Suppose that Ya ut where the...
Time series question about independent stationary processes
2:23 Two processes |Z, and (Y are said to be independent if for any time points 11 Im and s1.8-2... Sn the random variables [Z, Z,. ..Z, are independent of the random variables [Ys,. Ys, ..., Y). Show that if IZ) and Y are inde- 23 mm pendent stationary proesses, then W-2,+ 7 is stationary.
It should be clear from the discussion that a strictly stationary, finite variance, time series is also stationary. However, the converse may or may not be true, in general. A time series {Xti t 0,?1,?2, ) is said to be a Gaussian process, if the n-dimensional random vectors X = (Xti, Xt2, .. . , X.). for every collection of time points t1, t2,..., tn, and every positive integer n, have a Multivariate Normal distribution. } is a stationary Gaussian...
2. Consider the time series X, = 2 + 0.5t +0.8X1-1 + W, where W N(0.1). (a) (8 points) Calculate E(X2) Is this process weakly stationary? Give reasons for your answer. Hint: Find the mean function of {X) and then substitute t = 20. (b) (3 points) Calculate Var(X20) Question 2 continues on the next page... Page 4 of 12 c)(4 points) Consider the first differences of the time series above, that is Is {%) a weakly stationary process. Prove...
4. Consider two functions x(h),y (h), Vh> 0 and cross-covariance function 7xy(h), Vh e Z (a) (7 points) Find the Best Linear Predictor (BLP) of Y given Xt, and its Mean Square Prediction Error (MSPE), expressed in terms of x(h). (h), xy (h) jointly stationary time series (X, Y), with individual auto-covariance (b) (13 points) Find the BLP of Y given Xt, X-1,Y-1, and its MSPE, expressed in terms of 7x(h)(h), x.y (h). (Note: you don't need to solve the...
exercise 4
Exercises For each data set, plot the time series, and write the exact definition, periodicity and units. Judge whether the underlying stochastic process may be first and sec- ond order weakly stationary. Explain your rationale. 77 a constant): a. LY, = b, Lc= nd d. LkY, for some integer k >0 5. The following table contains quarterly nominal GDP in U.S. (billions of dol- lars). Let Y, denote the GDP at time t and let y, In (Y)....
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