Find the spectral density of Xt =−0.7Xt−1+Zt −0.3Zt−1+0.7Zt−2, {Zt}~IID(0,2).
Find the spectral density functions of the following MA process: Xt = Zt + 0.5Zt−1 − 0.3Zt−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 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...
Suppose Zt = 2 + Xt -2Xt-1+Xt-2, where {Xt} is zero-mean stationary series with autocovariance function. Calculate the autocovariance of Zt
Consider the following AR(2) model: Xt – Xt–1 + + X4-2 = Zt, Z4 ~ WN(0,1). (a) Show that X+ is causal. (b) Find the first four coefficients (VO, ..., 43) of the MA(0) representation of Xt. (c) Find the pacf at lag 3, 233, of the AR(2) model.
Problem 3 Consider a random walk on the integers. Suppose we start from 0, and at each step, we either go left or right with probability 1/2, ie, Xo--0, and Xt+1 Xt+Zt, where Zt-1 with probability 1/2, and Zt1 with probability 1/2. What is the probability distribution of XT? What is E(X) and Var(XT)? Problem 3 Consider a random walk on the integers. Suppose we start from 0, and at each step, we either go left or right with probability...
Problem 3 Express growth rates for xt in terms of growth rates of Yt and zt. 1. It = y21 2. It = 4822- 3. It = 4ytzt
X~U(0,2) Y=3X-1 Find density of Y
(White noise is not necessarily i.i.d.). Suppose that {Wt} and {Zt} are independent and identically distributed (i.i.d.) sequences, also independent of each other, with P(Wt = 0) = P(Wt = 1) = 1/2 and P(Zt = −1) = P(Zt = 1) = 1/2. Define the time series Xt by Xt = . Show that {Xt} is white but not i.i.d. w (1 – W-1) ZŁ
X1,...,Xn are IID with N(0,2). a) Determine the mean and variance for (X (subscript 1)^2) b) Show sqrt(n) * [ log ( 1/n ∑(from i=1 to n) Xi2) − log(σ2 ) ] d → N(0, 2). We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Find the spectral density of the output signal ex(): Um u(0) uex(t) Reccomendation: Use the known spectral density of the rectangular pulse and apply properties of the Fourier transforms Find the spectral density of the output signal ex(): Um u(0) uex(t) Reccomendation: Use the known spectral density of the rectangular pulse and apply properties of the Fourier transforms