Stationarity, Non Stationarity, Unit Roots and Spurious Regression

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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 of yt that
depends only on the difference apart in time
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Stationary time series
WHITE NOISE PROCESS Xt ut ut
IID(0, s2 )
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Stationary time seriesXt 0.5Xt-1 ut ut
IID(0, s2 )
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Many Economic Series Do not Conform to the
Assumptions of Classical Econometric Theory
Share Prices
Exchange Rate
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.
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Non-stationary time series

• RANDOM WALK
• Xt Xt-1 ut ut IID(0, s2 )
• Mean E(Xt) E(Xt-1)
  (mean is constant in t)
• X1 X0 u1 (take initial value X0)
• 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 IID(0, s2 )
• Xt X0 u1 u2 ut
• Variance Var(Xt) Var(X0) Var(u1) Var(ut)
• 0 s2 s2
• t s2
• (variance is not constant through time)

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Non-stationary time series Random WalkXt
Xt-1 ut ut IID(0, s2 )
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Constant covariance - use of correlogram

• Covariance between two values of Xt depends only
  on the difference apart in time for stationary
  series.
• Cov(Xt ,Xtk) ?(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 and 1997. (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 k increases.
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Stationary time series
First difference of GDP is stationary ?Yt
Yt - Yt-1 - Growth rate is reasonably constant
through time. Variance is also reasonably
constant through time
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Stationary time series UK GDP Growth (?
Yt) - correlogram

Sample autocorrelations decline towards zero as k
increases. Decline is rapid for stationary series.
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Non-stationary Time Series summary
Relationship between stationary and
non-stationary process AutoRegressive AR(1)
process Xt a ?Xt-1 ut ut IID(0, s2
) ? lt 1 stationary process -
process forgets past ? 1 non-stationary
process - process does not forget
past a 0 without drift a ? 0
with drift
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Stationary time series with driftXt a
0.5Xt-1 ut ut IID(0, s2 )
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Non-stationary time series Random Walk with
DriftXt a Xt-1 ut ut IID(0, s2 )
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Time Series Models summary
General Models AutoRegressive AR(1) process
without drift Xt ?Xt-1 ut ? lt 1
stationary process - process forgets
past ? 1 non-stationary process -
process does not forget past AutoRegressive
AR(k) process without drift Xt ?1Xt-1
?2Xt-2 ?3Xt-3 ?4Xt-4 ?kXt-k ut
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Spurious Regression Problem
yt yt-1 ut ut iid(0,s2) xt
xt-1 vt vt iid(0,s2) ut and vt are
serially and mutually uncorrelated yt ß0
ß1xt et since yt and xt are 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
with non-stationary time series. - if two series
are growing over time, they can be
correlated even if the increments in each series
are uncorrelated
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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,s2) xt xt-1 vt vt
iid(0,s2)
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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,s2)
xt xt-1 vt vt iid(0,s2)
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Spurious Regression Problem
Estimate regression using OLS in PcGive yt
ß0 ß1xt et based on two random walks yt
yt-1 ut ut iid(0,s2) xt xt-1 vt
vt iid(0,s2) 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 R2 0.325
F(1,496) 239.3 0.000 log-likelihood
-914.706 DW 0.0411 no.
of observations 498 no. of parameters
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Trend Deterministic or Stochastic?
The First
The Second
(with a2 lt1 and a3 gt0)
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This series has a deterministic trend (if a3 gt
0) Classical inference is valid (provided that
a2 is 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).
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This series is non-stationary - the trend is
stochastic Classical inference is not valid The
series is called difference stationary
Random Walk With Drift
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