3. Let {X} be an autoregressive process of order one, usually written as AR(1). (a) Write down an equation defining X₁ in terms of an autoregression coefficient a and a white noise process {} with variance σ². Explain what the phrase "{} is a white noise process with variance o?" means. (b) Derive expressions for the variance 70 and the autocorrelation function Pk, k 0,1,. of the {X} in terms of o2 and a. Use these expressions to suggest an estimate of a in terms of the sample autocor- relations {k}. (c) Suppose that only every second value of X is observed, resulting in a time series Y X2, t = 1, 2,.... Show that {Y} forms an AR(1) process. Find its autoregression coefficient, say d', and the variance of the underlying white noise process, in terms of a and o². (d) Given a time series data set X1, ..., X256 with sample mean = 9.23 and sample autocorrelations ₁ = -0.6, 2 = 0.36, 3 = -0.22, p = 0.13, 5 = -0.08, estimate the autoregression coefficients a and a' of {X} and {Y}.
3. Let {X} be an autoregressive process of order one, usually written as AR(1). (a) Write down an equation defining X₁ in terms of an autoregression coefficient a and a white noise process {} with variance σ². Explain what the phrase "{} is a white noise process with variance o?" means. (b) Derive expressions for the variance 70 and the autocorrelation function Pk, k 0,1,. of the {X} in terms of o2 and a. Use these expressions to suggest an estimate of a in terms of the sample autocor- relations {k}. (c) Suppose that only every second value of X is observed, resulting in a time series Y X2, t = 1, 2,.... Show that {Y} forms an AR(1) process. Find its autoregression coefficient, say d', and the variance of the underlying white noise process, in terms of a and o². (d) Given a time series data set X1, ..., X256 with sample mean = 9.23 and sample autocorrelations ₁ = -0.6, 2 = 0.36, 3 = -0.22, p = 0.13, 5 = -0.08, estimate the autoregression coefficients a and a' of {X} and {Y}.
Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015
1st Edition
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:HOUGHTON MIFFLIN HARCOURT
Chapter4: Writing Linear Equations
Section: Chapter Questions
Problem 12CR
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Transcribed Image Text:3. Let {X} be an autoregressive process of order one, usually written as AR(1).
(a) Write down an equation defining X₁ in terms of an autoregression coefficient a
and a white noise process {} with variance σ².
Explain what the phrase "{} is a white noise process with variance o?" means.
(b) Derive expressions for the variance 70 and the autocorrelation function Pk, k
0,1,. of the {X} in terms of o2 and a.
Use these expressions to suggest an estimate of a in terms of the sample autocor-
relations {k}.
(c) Suppose that only every second value of X is observed, resulting in a time series
Y X2, t = 1, 2,....
Show that {Y} forms an AR(1) process. Find its autoregression coefficient, say
d', and the variance of the underlying white noise process, in terms of a and o².
(d) Given a time series data set X1, ..., X256 with sample mean = 9.23 and sample
autocorrelations ₁ = -0.6, 2 = 0.36, 3 = -0.22, p = 0.13, 5 = -0.08,
estimate the autoregression coefficients a and a' of {X} and {Y}.
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