Let Yt = β0 + β1t + Xt with {Xt} zero mean stationary with acf γk and β’s constant. Is {Yt}
stationary? Is ∇Yt?
Let Yt = β0 + β1t + Xt with {Xt} zero mean stationary with acf γk...
Yt = 5 − 2t + Xt, where {Xt} is stationary with mean 0 and autocovariance function γk. Now, let Wt = Yt − Yt−1. (a) Find the mean function for {Wt}. (b) Find the autocovariance function for {Wt}. (c) Is {Wt} stationary? Why or why not?
1. Let {Xt} be a stationary process with mean μt = E(Xt) = 0 and autocovariance function γX(k) = E(XtXt−k) - μ2 = E(XtXt+k) - μ2. De ne Yt = 5 + 2t + Xt. (a) Find E(Yt), the mean function for Yt. (b) Find γY (k), the autocovariance function for Yt in terms of γX (k). (c) Is Yt stationary? Explain. (d) De ne a new process Wt as Wt = Yt − Yt−1. Find E(Wt) and γW (k)....
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...
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.)...
Suppose Zt = 2 + Xt -2Xt-1+Xt-2,
where {Xt} is zero-mean stationary series with autocovariance
function.
Calculate the autocovariance of Zt
Consider a population linear regression model: Yt=β0 + β1Xt + ut Calculate: 1. Variance 2. Covariance of ut and Xt 3. β0 4. β1
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?
4. Let (Yi] be a stationary process with mean zero and let a, b and c be constants. Let st be a seasonal with period 4, that is, st-st+4, t-1, 2, . . . , and Xt = a + bt + ct2 + st + Y. (i) Let (ho, do )-min( (k, d)such that k > 0, d 0, and the proces s W t ▽k▽dX,-(1 B)a Find ko and do. For W, (with k = ko and d...
Let { be a zero-mean stationary process and let a and b be constants. (a) (5 points) If Xi a+bt+St+Yi, where St is a seasonal component with period 12, show that ▽12V is stationary and express its autocovariance function in terms of that of { (b) (5 points) If X1-(a + bt)Sİ + Y. where Sı is a seasonal component with period 12, show that Vi2 is stationary and express its autocovariance function in terms of that of {
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?