Topic 3 (Week 4)

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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)


(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)



(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





0

SE

Test statistic:


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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





0

SE

Test statistic:


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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-

Stationary24

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Criticism for DF/ADF/KPSS

• Small sample size ==> the power of test is

less.

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Topic 3 (Week 4)

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