topic 3 (week 4)
TRANSCRIPT
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Week 4
Topic 3 :
Testing for Trends and
Unit Roots
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Assumption of Stationarity
• A stationary time series exhibits mean
reversion in that it fluctuates around a
constant long run mean.
• Absence of unit root implies that the series
has a finite variance which do not depend
on time (crucial for economic forecasting),
and that the effects of shocks dissipate overtime.
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Assumption of StationarityStrict Stationary
• Constant mean
• Constant variance
• Constant covariance
– Cov(yt,yt-1) = Cov(yt-1,yt-2) =…Cov(yt-i,yt-i-1)
– Cov(yt,yt-2) = Cov(yt-1,yt-3) = … Cov(yt-I, yt-i-2)
– Cov(yt,yt-3) = Cov(yt-1,yt-4) = … Cov(yt-I, yt-i-3)
(All Cov (yt,yt-i) are constant)
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Assumption of StationarityWeak Stationary
• Constant mean
• Constant variance
• Constant covariance
– Cov(yt,yt-1) = Cov(yt-1,yt-2) =…Cov(yt-i,yt-i-1)
– Cov(yt,yt-2) = Cov(yt-1,yt-3) = … Cov(yt-I, yt-i-2)
– Cov(yt,yt-3) = Cov(yt-1,yt-4) = … Cov(yt-I, yt-i-3)
(At least one of Cov (yt,yt-i) is not constant)
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Stationarity
This series has non-stationary movement because
its mean and variance are not constant across
time.
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Non-Stationarity• A series can strongly influence its behaviour
and properties - e.g. persistence of shocks willbe infinite for non-stationary series.
• Non-stationary series have no tendency to
return to a long-run path. The variance of theseries is time-dependent and goes infinity astime approaches infinity, which results in seriesproblem in forecasting.
• Presence of trends – Deterministic Trend
– Stochastic trend
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Characteristic of Time Series Data
• There are two models which have beenfrequently used to characterize non-stationarity(recall stochastic trend & deterministic trend inTopic 1):
a. the random walk model with drift:
yt = α + yt-1 + ut
b. the deterministic trend process:
yt = µ + φt + ut
where ut is IID in both cases.
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There are FIVE types of univariate models
• White Noise (WN)==> It must be stationary
• Moving Average (MA)
==>It can be stationary or non-stationary
• Autoregressive (AR)
==> It can be stationary or non-stationary
• Autoregressive Moving Average (ARMA)
==> It must be stationary• Autoregressive Integrated Moving Average(ARIMA)
==> It must be stationary
Characteristic of Time Series Data
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Why stationary is importantSpurious regressions
• If two variables are trending over time, aregression of one on the other could have a highR square even if the two are totally unrelated.
• If the variables in the regression model are notstationary, then it can be proved that the standardassumptions for asymptotic analysis will not bevalid.
• In other words, the usual “t-ratios” will not follow at-distribution, so we cannot validly undertakehypothesis tests about the regression parameters.
==> R square > Durbin-Watson test statistic
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The Impact of Shocks
• The AR (1) could be generalized to threecases:
yt = α + βyt-1 + ut
When β > 1, yt is an explosive process
When β = 1, yt is a unit root process
(non-stationary process)
When β < 1, yt is a stationary process
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The Impact of Shocks
• Typically, the explosive case is ignoredand we use β = 1 to characterize thenon-stationarity because
– β > 1 does not describe many data seriesin economics and finance.
– β > 1 has an intuitively unappealingproperty: shocks to the system are not
only persistent through time, they arepropagated so that a given shock will havean increasingly large inf luence.
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Unit Root Test
– Dickey-Fuller (1979)
==> Parametric testing
– Augmented Dickey-Fuller (1981)
==> Parametric testing
– Phillips-Perron (1988)
==> Non-parametric testing
Note: We make sure there is no structural break
in a series across time.
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t t t Y Y
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Dickey-Fuller (DF) Unit Root Test:
Model with constant andwith trend:
Model with constant and
without trend:
Graphical: Given that Yt has not trend , so we should use model
as below to conduct the unit root test.
Graphical: Given that Yt has trend , so we should use model as
below to conduct the unit root test.
- This test was developed by Dickey and Fuller (1979).
t t t Y t Y
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DF Unit Root Test
Decision rule: We reject Ho is test statistic is less than
critical value. Other-wise, do not reject Ho.
Ho: Yt is non-stationarity (Yt has unit root),
H1: Yt is stationarity (Yt has no unit root),0
0
SE
Test statistic:
Critical value: It can be obtained from the t statisticaltable that has been modified by Dickeyand Fuller. Later, the distribution ofmodified t is expanded by Mackinnon.
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Limitations of DF test
• DF regression model does not taken
dynamic effect into account.
==> error in the model does not longer to
have normal distribution or white noiseprocess
==> autocorrelation problem
==> hypothesis results will be invalid
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Note:
The optimal lag length for unit root test model is based on the minimum AIC
or SIC, where the autocorrelation problem does not exist in both models.
t it
k
i
it t Y Y Y
1
1
Augmented Dickey-Fuller (ADF) Unit Root Test:
Model with constant andwith trend:
Model with constant and
without trend:
Graphical: Given that Yt has not trend , so we should use model
as below to conduct the unit root test.
Graphical: Given that Yt has trend , so we should use model as
below to conduct the unit root test.
- This test was further developed by Dickey and Fuller (1981).
t it
k
i
it t Y Y t Y
1
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ADF Unit Root Test
Decision rule: We reject Ho is test statistic is less than
critical value. Other-wise, do not reject Ho.
Ho: Yt is non-stationarity (Yt has unit root),
H1: Yt is stationarity (Yt has no unit root),0
0
SE
Test statistic:
Critical value: It can be obtained from the t statisticaltable that has been modified by Dickeyand Fuller. Later, the distribution ofmodified t is expanded by Mackinnon.
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Phill ips-Perron (PP) Unit Root Test:
• This test was developed by Phillips andPerron (1988).
• It is deal with the autocorrelation problem in
DF test.
• It is a non-parametric test (ranking) with no
assumptions are required (waste some
information) ==> only for small sample size
t t t Y
nt Y
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PP Unit Root Test
Decision rule: We reject Ho is test statistic is less than
critical value. Other-wise, do not reject Ho.
Ho: Yt is non-stationarity (Yt has unit root),
H1: Yt is stationarity (Yt has no unit root),0
0
SE
Test statistic:
Critical value: It can be obtained from the t statisticaltable that has been modified by Dickeyand Fuller. Later, the distribution ofmodified t is expanded by Mackinnon.
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Stationary Test
- Kwiatkowski, Phillips, Schmidt and Shin (1992)==> Parametric testing
– Note: We make sure there is no structural break
exists in a series across time.
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Kwiatkowski –Phillips –Schmidt –Shin
(KPSS) Stationary Test
- This test was developed by Kwiatkowski, Phillips,Schmidt and Shin (1992).
- KPSS test is intended to complement unit root
tests, such as the DF, ADF and PP tests.- By testing both the unit root hypothesis and thestationarity hypothesis, one can distinguish seriesthat appear to be stationary, series that appear to
have a unit root, and series for which the data (orthe tests) are not sufficiently informative to be surewhether they are stationary or integrated.
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KPSS test
- Please refer to reference as follows for
extra information:
Kwiatkowski D., P. C. B. Phillips, P.Schmidt, and Y. Shin (1992): Testing the
Null Hypothesis of Stationarity against
the Alternative of a Unit Root. Journal ofEconometrics 54, 159 –178.
Ho: Yt is stationarity
H1: Yt is non- stationarity
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Comparison
Unit Root Test Stationary TestConclusionHo: Yt has a unit
root
Ho: Yt is
stationary
Reject Ho Do not reject Ho Stationary
Reject Ho Reject Ho Inconclusive
Do not reject Ho Do not reject Ho Inconclusive
Do not reject Ho Reject Ho Non-
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Criticism for DF/ADF/KPSS
• Small sample size ==> the power of test is
less.
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