stationarity, non stationarity, unit roots and spurious regression
TRANSCRIPT
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Stationarity, Non Stationarity, Unit
Roots and Spurious Regression
Roger Perman
Applied Econometrics Lecture 11
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Stationary Time Series
Exhibits mean reversion in that it fluctuates around a constant
long run mean
Has a finite variance that is time invariant
Has a theoretical covariance between values ofytthat dependsonly on the difference apart in time
)( tyE)0()y(E)y(Var 2tt
)()y)(y(E)y,y(Cov tttt
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WHITE NOISE PROCESS
Xt= ut ut~ I I D(0, 2)
Stationary time series
White Noise
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
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Stationary time series
Xt= 0.5*Xt-1+ ut ut~ I I D(0, 2)
Stationary without drift
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
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0 50 100 150 200 250 300 350 400 450
.25
.5
.75
1
1.25
1.5
1.75
2
2.25
2.5
Many Economic Series Do not Conform to the
Assumptions of Classical Econometric Theory
Share Prices
0 50 100 150 200 250
.35
.4
.45
.5
.55
.6
.65
.7 Exchange Rate
1960 1965 1970 1975
8.7
8.8
8.9
9
9.1
9.2
9.3 Income
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Non Stationary Time Series
There is no long-run mean to which the series returns
and/or
The variance is time dependent and goes to infinity
as time approaches to infinity
Theoretical autocorrelations do not decay but, in finite
samples, the sample correlogram dies out slowly
The results of classical econometric theory
are derived under the assumption that variables of concern are stationary.
Standard techniques are largely invalid where data is non-stationary
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Non-stationary time series
UK GDP (Yt)
The level of GDP (Y) is not constant and the mean increases over time. Hence the level of
GDP is an example of a non-stationary time series.
GDP Level
0
20
40
60
80
100
120
1992Q
3
1993Q
2
1994Q
1
1994Q
4
1995Q
3
1996Q
2
1997Q
1
1997Q
4
1998Q
3
1999Q
2
2000Q
1
2000Q
4
2001Q
3
2002Q
2
2003Q
1
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Non-stationary time series
RANDOM WALK
Xt= Xt-1+ ut ut~ I I D(0, 2)
Mean: E(Xt) = E(Xt-1) (mean is constant in t)
X1= X0+ u1 (take initial valueX0)X2= X1+ u2= (X0+ u1) + u2
Xt= X0+ u1+ u2++ ut
E(Xt) = E(X0+ u1+ u2++ ut)(take expectations)
= E(X0)= constant
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Non-stationary time series
RANDOM WALK
Xt= Xt-1+ ut ut~ I I D(0, 2)
Xt= X0+ u1+ u2++ ut
Variance:Var(Xt) = Var(X0) + Var(u1) ++ Var(ut)= 0 + 2++ 2
=t 2
(variance is not constant through time)
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Non-stationary time series: Random Walk
Xt= Xt-1+ ut ut~ I I D(0, 2)
Random Walk
0
0.5
1
1.5
2
2.5
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Constant covariance - use of correlogram
Covariance between two values ofXtdepends only on the
difference apart in time for stationary series.
Cov(Xt,Xt+k) = (k) (covariance is constant in t)
(A) Correlation for 1980 and 1985 is the same as for 1990
and 1995. (i.e. t = 1980 and 1990, k = 5)
(B) Correlation for 1980 and 1987 is the same as for 1990 and1997. (i.e. t = 1980 and 1990, k = 7)
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Non-stationary time series
UK GDP (Yt) - correlogram
For non-stationary series the Autocorrelation Function (ACF) declines towards zero at a
slow rate as kincreases.
0 1 2 3 4 5 6 7
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
ACF-Y
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First difference of GDP is stationary
Yt=Yt-Yt-1- Growth rate is reasonably constant through time.Variance is also reasonably constant through time
Stationary time series
GDP Growth (YBEZ)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1992Q3
1993Q2
1994Q1
1994Q4
1995Q3
1996Q2
1997Q1
1997Q4
1998Q3
1999Q2
2000Q1
2000Q4
2001Q3
2002Q2
2003Q1
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Stationary time series
UK GDP Growth ( Yt) - correlogram
Sample autocorrelations decline towards zero as kincreases. Decline is rapid for stationary
series.
0 1 2 3 4 5 6 7
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
ACF-DY
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Relationship between stationary and non-stationary process
AutoRegressive AR(1) process
Xt= +Xt-1+ ut ut~ I I D(0, 2)
< 1 stationary process
- process forgets past = 1 non-stationary process
- process does not forget past
= 0 without drift
0 with drift
Non-stationary Time Series: summary
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Stationary time series with drift
Xt= + 0.5*Xt-1+ ut ut~ I I D(0, 2)
Stationary with Drift
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
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Non-stationary time series: Random Walk with Drift
Xt= + Xt-1+ ut ut~ I I D(0, 2)
Random Walk with Drift
0
2
4
6
8
10
12
Ti S i M d l
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General Models
AutoRegressive AR(1) process without dr if t
Xt= Xt-1+ ut
< 1 stationary process- process forgets past
= 1 non-stationary process
- process does not forget past
AutoRegressive AR(k) process without dr if t
Xt= 1Xt-1+ 2Xt-2+ 3Xt-3+ 4Xt-4++ kXt-k+ ut
Time Series Models: summary
S i R i P bl
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Spurious Regression Problem
yt= yt-1+ ut ut~ iid(0,2)
xt
= xt-1
+ vt
vt
~ iid(0,2)
utand vtare serially and mutually uncorrelated
yt=0+1xt+ t
since ytand xtare uncorrelated random walks we should expect R2
to tend to zero. However this is not the case.
Yule (1926): spurious correlation can persist in large samples withnon-stationary time series.
- if two series are growing over time, they can be correlated
even if the increments in each series are uncorrelated
S i R i P bl
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Spurious Regression Problem
Two random walks generated from Excel using RAND() command
hence independent
yt= yt-1+ ut ut~ iid(0,2)
xt= xt-1+ vt vt~ iid(0,2)
Two Random Walks
-4-2
0
2
4
6
8
10
12
14
1 40 79 118 157 196 235 274 313 352 391 430 469
RW1 RW2
S i R i P bl
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Spurious Regression Problem
Plot Correlogram using PcGive
(Tools, Graphics, choose graph, Time series ACF, Autocorrelation
Function)
yt= yt-1+ ut ut~ iid(0,2)
xt= xt-1+ vt vt~ iid(0,2)
0 5 10
0.25
0.50
0.75
1.00
ACF-RW1
0 5 10
0.25
0.50
0.75
1.00 ACF-RW2
S i R i P bl
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Spurious Regression Problem
Estimate regression using OLS in PcGive
yt=0+1xt+ t
based on two random walks
yt= yt-1+ ut ut~ iid(0,2)
xt= xt-1+ vt vt~ iid(0,2)
EQ( 1) Modelling RW1 by OLS (using lecture 2a.in7)The estimation sample is: 1 to 498
Coefficient t-value
Constant 3.147 25.8
RW2 -0.302 -15.5
sigma 1.522 RSS 1148.534
R^2 0.325 F(1,496) = 239.3 [0.000]**
log-likelihood -914.706 DW 0.0411
no. of observations 498 no. of parameters 2
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Trend: Deterministic or Stochastic?
0 50 100 150 200 250 300 350 400 450 500
100
200
0 50 100 150 200 250 300 350 400 450 500
100
200
0 5 10 15 20 25
.25
.5
.75
1
0 5 10 15 20 25
.25
.5
.75
1
The First
The Second
Y a Yt t t
1 1
Y a a Y a t t t
1 2 1 3
(with a2 0)
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Y a a Y a t t t
1 2 1 3
This series has a deterministic trend (if a3 > 0)
Classical inference is valid
(provided that a2is less than 1).
The series is transformed to a stationary series by
subtracting the deterministic trend from the left side
(and so the right side).
Y a t a a Y t t
3 1 2 1
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Y a Yt t t
1 1
This series is non-stationary - the trend is stochastic
Classical inference is not valid
The series is called difference stationary
Y Y at t t
1 1
Random Walk With Drift
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Parameter Set Description Properties
1 f b