Deterministic or Stochastic Trend? -...
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Deterministic or Stochastic Trend? Let us consider two of the simplest versions: Deterministic trend (DT) : y t = β t + t Stochastic trend (ST) : y t = β + y t -1 + t , where t is white noise with variance σ 2 (= 1, for simplicity) and y 0 = 0 (also for simplicity). It is easy to see that E DT (y t )= E ST (y t )= β t but V DT (y t )=1 and V DT (y t )= t . Expectation with respect to all information up to time t = 0. 1
Transcript of Deterministic or Stochastic Trend? -...
Deterministic or Stochastic Trend?
Let us consider two of the simplest versions:
Deterministic trend (DT) : yt = βt + εt
Stochastic trend (ST) : yt = β + yt−1 + εt ,
where εt is white noise with variance σ2 (= 1, for simplicity) andy0 = 0 (also for simplicity).
It is easy to see that
EDT (yt) = EST (yt) = βt
butVDT (yt) = 1 and VDT (yt) = t.
Expectation with respect to all information up to time t = 0.1
Simulating DT and ST time series
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0 y(t) = 0.5*t + rby(t) = 0.5 + y(t−1) + rb
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How to model y1t and y2t?Even with n = 100 one can argue that the trend of {y2t} “looks”more deterministic than the trend of {y1t}.
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Y1Y2
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Model y1t and y2t with deterministic trendsEven after removing a determinist trend from y1t , the residuals stillbehave like a random walk. On the other hand, y2t is definitelytrend-stationary.
Modeling y1 with DT
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y1
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Time
Res
idua
ls
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−6
−4
−2
02
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Noise doesn't look white
0 5 10 15 20
0.0
0.2
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Lag
AC
F
FAC+FACP => random walk
5 10 15 20
0.0
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Lag
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tial A
CF
Modeling y2 with DT
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y2
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−2
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Noise looks white
0 5 10 15 20
0.0
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Lag
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FAC+FACP => white noise
5 10 15 20
−0.
15−
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Lag
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tial A
CF
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Model y1t and y2t with stochastic trendsAfter fitting a random walk plus drift for y1t , the residuals behavelike a white noise, so y1t is difference-stationary.
y1 : random walk + drift
Time
y1
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Time
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ls
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Noise looks white
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−0.
20.
00.
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40.
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81.
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Lag
AC
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FAC+FACP => white noise
5 10 15 20
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20−
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0.00
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tial A
CF
y2 : random walk + drift
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y1
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Noise doesn't look white
0 5 10 15 20
−0.
50.
00.
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Lag
AC
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FAC+FACP => not white noise
5 10 15 20
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4−
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tial A
CF
Fitting a random walk plus drift for y2t (which is trend-stationary),induces an MA(1) behavior in the residuals. 5
If yt is trend stationary,
yt = βt + εt
thenyt−1 = β(t − 1) + εt−1
and∆yt = β + vt
where vt = εt − εt−1, such that E (vt) = 0, V (vt) = 2 and
Cov(vt , vt−1) = Cov(εt − εt−1, εt−1 − εt−2) = −V (εt) = −1
and Cov(vt , vt−h) = 0, for h > 1. Therefore, the 1st orderautocorrelation is
ρ(1) =Cov(vt , vt−1)
V (vt)= −0.5.
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Summary
If yt is trend-stationary:
I Stochastic trend fit: residuals with MA(1) behavior.
I Deterministic trend fit: residuals are white noise.
If yt is difference-stationary:
I Stochastic trend fit: residuals are white noise.
I Deterministic trend fit: residuals are random walk.
Lesson: ALWAYS check the residuals!
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