X; - X-1 = (H + y;) – (Hi-1 + yt–1) (2.27) = 6 + W; + Yt - Yt-1. It is easy to show z; = yt – Yt-1 is stationary using Chap. 1.1. That is, because stationary, Yt is Yz(h) = cov(z;+h, Zt) = cov(yt+h - Yt+h-1» Yt – Yt-1) %3D %3D = 2y, (h) – Yy(h + 1) – y,(h – 1) is independent of time; we leave it as an exercise (Problem 2.7) to show that x, – X1-1 in (2.27) is stationary. One advantage of differencing over detrending to remove trend is that no param- eters are estimated in the differencing operation. One disadvantage, however, is that differencing does not yield an estimate of the stationary process y; as can be seen in (2.27). If an estimate of y, is essential, then detrending may be more appropriate. If the goal is to coerce the data to stationarity, then differencing may be more appropri- ate. Differencing is also a viable tool if the trend is fixed, as in Example 2.4. That is, e.g., if µ; = Bo + Bị t in the model (2.24), differencing the data produces stationarity (see Problem 2.6):

X; - X-1 = (H + y;) – (Hi-1 + yt–1) (2.27) = 6 + W; + Yt - Yt-1. It is easy to show z; = yt – Yt-1 is stationary using Chap. 1.1. That is, because stationary, Yt is Yz(h) = cov(z;+h, Zt) = cov(yt+h - Yt+h-1» Yt – Yt-1) %3D %3D = 2y, (h) – Yy(h + 1) – y,(h – 1) is independent of time; we leave it as an exercise (Problem 2.7) to show that x, – X1-1 in (2.27) is stationary. One advantage of differencing over detrending to remove trend is that no param- eters are estimated in the differencing operation. One disadvantage, however, is that differencing does not yield an estimate of the stationary process y; as can be seen in (2.27). If an estimate of y, is essential, then detrending may be more appropriate. If the goal is to coerce the data to stationarity, then differencing may be more appropri- ate. Differencing is also a viable tool if the trend is fixed, as in Example 2.4. That is, e.g., if µ; = Bo + Bị t in the model (2.24), differencing the data produces stationarity (see Problem 2.6):

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter14: Discrete Dynamical Systems

Section14.3: Determining Stability

Problem 13E: Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii...

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