Applied Econometrics 31 456

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Applied Econometrics 31 456

• This half of the course provides an introduction
  to stationary time series data.
• Nobel Prize for Economics in 2003 awarded to Rob
  Engle and Clive Granger, who highlighted the
  importance of stationarity in time series data.
• There are substantial implications for empirical
  modeling with time series data which is not
  stationary.
• Reading Thomas 13.1 Stationary and
  non-stationary stochastic processes

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Properties of Time Series I Stationary time
series
Xt is stationary if the series exhibits mean
reversion i.e. fluctuates around a constant long
run mean. Xt has finite variance which is not
dependent upon time. Covariance between two
values of Xt depends only on the difference apart
in time. E(Xt) µ (mean is constant in t)
Var(Xt) s2 (variance is constant in
t) Cov(Xt ,Xtk) ?(k) (covariance is constant
in t)
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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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Non-stationary time series
In contrast a non-stationary time series has
the following characteristics (1) Does not
have a long run mean which the series returns
(2) Variance is dependent upon time and goes to
infinity as the sample period approaches
infinity (3) Correlogram does not die out -
long memory
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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)

However, the level of a economic time series is
typically non-stationary. The level of GDP (Y) is
not constant and the mean increases over time.
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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