Homework Answers
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?
1. Let {Xt} be a stationary process with mean μt = E(Xt) = 0 and autocovariance...
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)....
Consider the following model on a return series rt=t+ at +0.25at-1, where at riid N(0,02), t = 1, ... ,T. (a) What are...
Consider the following model on a return series rt=t+ at +0.25at-1, where at riid N(0,02), t = 1, ... ,T. (a) What are the mean function and autocovariance function for this return series? Is this return series {rt} weakly stationary? Justify your answer. (b) Consider first differences of the return series above, that is, consider wt = Vrt=rt – Pt-1. What are the mean function and autocovariance function for this time series? Is this time series {wt} weakly stationary? Justify...
Yt = 5 − 2t + Xt, where {Xt} is stationary with mean 0 and autocovariance...
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?
2.6 Consider a process consisting of a linear trend with an additive noise term consisting of...
2.6 Consider a process consisting of a linear trend with an additive noise term consisting of independent random variables wt with zero means and variances o that is where Bo, B1 are fixed constants (a) Prove t is nonstationary. (b) Prove that the first difference series Vxt finding its mean and autocovariance function Xt t-s stationary by
Consider the time series of Xt = Xt−1 + Wt, whereWt are i.i.d and Wt ∼...
Consider the time series of Xt = Xt−1 + Wt, whereWt are i.i.d
and Wt ∼ N (0, σ2 ) and X0 = 0. Let X¯ = 1 n Pn i=1 Xi . Derive the
general form for var(X¯). (Hint: Pn i=1 i 2 = n(n+1)(2n+1) 6 )
Consider the time series of X-Xi-1+Wi, whereW are i.i.d and W N(0, σ2) and Xo 0. Let X = n Σ, Derive the general form for var(X). (Hint: Σ i- n(n+1)(2n+1))
(White noise is not necessarily i.i.d.). Suppose that {Wt} and {Zt} are independent and identically distributed...
(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
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...
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.
Consider the time series Consider the time series of Xt Xt-1 + Wt, where Wt ~...
Consider the time series
Consider the time series of Xt Xt-1 + Wt, where Wt ~ N(0, σ2) and Xi. Derive the general form for var(X). (Hint: i=1 5 ? -n(n+1)(2n+1)
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?
Consider the following model on a return series rt=t+ at +0.25at-1, where at riid N(0,02), t = 1, ... ,T. (a) What are the mean function and autocovariance function for this return series? Is this return series {rt} weakly stationary? Justify your answer. (b) Consider first differences of the return series above, that is, consider wt = Vrt=rt – Pt-1. What are the mean function and autocovariance function for this time series? Is this time series {wt} weakly stationary? Justify...
2.6 Consider a process consisting of a linear trend with an additive noise term consisting of independent random variables wt with zero means and variances o that is where Bo, B1 are fixed constants (a) Prove t is nonstationary. (b) Prove that the first difference series Vxt finding its mean and autocovariance function Xt t-s stationary by
Consider the time series of Xt = Xt−1 + Wt, whereWt are i.i.d
and Wt ∼ N (0, σ2 ) and X0 = 0. Let X¯ = 1 n Pn i=1 Xi . Derive the
general form for var(X¯). (Hint: Pn i=1 i 2 = n(n+1)(2n+1) 6 )
Consider the time series of X-Xi-1+Wi, whereW are i.i.d and W N(0, σ2) and Xo 0. Let X = n Σ, Derive the general form for var(X). (Hint: Σ i- n(n+1)(2n+1))
(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
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...
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.
Consider the time series
Consider the time series of Xt Xt-1 + Wt, where Wt ~ N(0, σ2) and Xi. Derive the general form for var(X). (Hint: i=1 5 ? -n(n+1)(2n+1)
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