Question

Let Xt = a + bt + Yt, where a and b are nonzero constants and Yt is a stationary time series. (i) Show that {Xt} is not stationary. (ii) Let ?Xt = Xt - Xt-1. Show that {?Xt} is stationary. (iii) Determine the autocovariances of {?Xt} in terms of those of {Yt}. (iv) Now suppose that {Yt} is a MA(1) process so that Yt = ?t + ??t-1 Write down an equation for ?Xt in terms of b, ?, ?t and B, the backward shift operator. (v) Show that Var(?Xt) ? Var(Yt).

          Let
Xt = a + bt + Yt,
where a and b are nonzero constants and Yt is a stationary time series.
(i) Show that {Xt} is not stationary.
(ii) Let ?Xt = Xt - Xt-1. Show that {?Xt} is stationary.
(iii) Determine the autocovariances of {?Xt} in terms of those of {Yt}.
(iv) Now suppose that {Yt} is a MA(1) process so that
Yt = ?t + ??t-1
Write down an equation for ?Xt in terms of b, ?, ?t and B, the backward shift operator.
(v) Show that Var(?Xt) ? Var(Yt).

Let
Xt = a + bt + Yt,
where a and b are nonzero constants and Yt is a stationary time series.
(i) Show that Xt is not stationary.
(ii) Let ?Xt = Xt - Xt-1. Show that ?Xt is stationary.
(iii) Determine the autocovariances of ?Xt in terms of those of Yt.
(iv) Now suppose that Yt is a MA(1) process so that
Yt = ?t + ??t-1
Write down an equation for ?Xt in terms of b, ?, ?t and B, the backward shift operator.
(v) Show that Var(?Xt) ? Var(Yt).

Added by Pamela V.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

11/14/2022

Step 1

{Xt} is not stationary: This is because the mean of {Xt} is a + bt, which is a function of time t. In a stationary time series, the mean should be constant over time. Therefore, {Xt} is not stationary. Show more…

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Let Xt = a + bt + Yt, where a and b are nonzero constants and Yt is a stationary time series. (i) Show that {Xt} is not stationary. (ii) Let ΔXt = Xt - Xt-1. Show that {ΔXt} is stationary. (iii) Determine the autocovariances of {ΔXt} in terms of those of {Yt}. (iv) Now suppose that {Yt} is a MA(1) process so that Yt = εt + βεt-1 Write down an equation for ΔXt in terms of b, β, εt and B, the backward shift operator. (v) Show that Var(ΔXt) ≥ Var(Yt).