Lecture 6 Stationarity and Cointegration

8/17/2019 Lecture 6 Stationarity and Cointegration

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TOPIC 6:

STATIONARY AND

NONSTATIONARYVARIABLES

By:Assoc. Prof. Dr. Sallahuddi !assa

SEEQ5133 Applied Econometrics


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DE"INITION O"

STATIONARY A time series Yt is stationary if itsmean and variance are constant

over time, and if the covariancebetween two values from theseries depends only on the lengthof time seperating the two values,and not the actual times at whichthe variables are observed.

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DE"INITION O"STATIONARY

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Stationary and non-stationary variablesmust be distinguished.

Stationary is important because if the

series is non-stationary then all types ofthe typical results of the classicalregression analysis are not valid.

It is crucial for the properties of standardestimation and testing procedures

3


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SP#RIO#S RE$RESSION

Regressions with non-stationaryseries may have no meaning and aretherefore called ‘spurious regression

!"ranger # $ewbold, %&'(). *ecause both Yt and +t contain a

stochastic trend, the S estimator

tends to nd a signicant correlationbetween the two series, even if theyare completely unrelated

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SP#RIO#S RE$RESSION

/he results from this regression areli0ely to be characteri1ed by afairly high R2 statistic, highlycorrelated residuals, a signicantvalue for and a low 3urbin-4atson!34) statistics.

/herefore, the usual t 5 and F 5tests on the regression parametersmay be very misleading.

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#NIT ROOT TESTS

/here are many tests fordetermining whether a series isstationary or nonstationary.

A unit root test can be tested using6 3ic0ey-7uller !37) test, Augmented 3ic0ey-7uller !A37) test, 8hillips-8erron test and 9wiat0ows0i et al. test.

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Dic%&y "ull&r T&s'

/he 3ic0ey-7uller tests !37 tests)developed by 3avid 3ic0ey !%&':),4ayne 7uller !%&':), and 3avid

3ic0ey and 4ayne 7uller !%&'&,%&;%) are the most popular tests

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Dic%&y "ull&r T&s'

/he 37 test for unit root is actuallybased on the AR!%) process6


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Dic%&y "ull&r T&s'

8lot the time series of variable for selectingthe appropriate 37 testing procedure. If the series appears to be wandering or

>uctuating around a sample average of 1ero,use uctuating around a sample average which isnon1ero, use uctuating around a linear trend, use


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Dic%&y "ull&r T&s'

?ypothesis6 $on stationary

Stationary @se tau ! ) statistics

Reect ?o if

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1:0

= ρ H

11


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Dic%&y "ull&r T&s'


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A#$(ENTED D"

/he Augmented 3ic0ey-7uller!A37) !%&'&) regression tests for

the eCistence of unit root . /he variable eCpresses therst diDerence with k lag.

ECtension of 37 test allows forpossibility that the error term isautocorrelated.

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1−∆

t Y

t

Y


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A#$(ENTED D"

/he A37 test is referred to the t statisticsof coeFcient on the followingregression6

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∑=

−− +∆+=∆k

i

t it it t Y Y Y

1

1 ε β γ

∑=

−− +∆++=∆k

i

t it it t Y Y Y

1

1 ε β γ α

∑=

−− +∆+++=∆

k

i

t it it t Y Y t Y 1

1 ε β γ λ α


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COINTE$RATION

/he concept of cointegration was rstintroduced by "ranger !%&;%).

Elaborated further by6 Engle and "ranger !%&;') Engle and Yoo !%&;') 8hillips and uliaris !%&&G) Stoc0 and 4atson !%&;;) 8hillips !%&;:, %&;') Hohansen !%&;;, %&&%, %&&)


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COINTE$RATION

Engle and "ranger !%&;') denedcointegration as a state that if two !ormore) series are lin0ed to form an

eJuilibrium relationship spanning inthe long run.


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(AT!E(ATICAL APPROAC!O" COINTE$RATION


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Coi'&)ra'i) V&c'or

It is 0nown that there eCists at most onecointegrating vector !cv)

In the case k M 2, the number of cvs may

be 1ero or one !r M G,%) or uniJue cv. If we have more than two variables, there

is a possibility of having more than one cv.

4hen we have a set ofk

I!%) variables,there may eCist up to k – 1 linearlyindependent cv or

1−≤ k r


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Coi'&)ra'io i aBi*aria'& VAR et us consider a rst-order

!nonstationary) BAR for

. /hat is

/he matriC is given by

( )' t t t

X ,Y Y =

( )

−

−=Θ−=Π

1

11

2221

1211

θ θ

θ θ

Π

+

=

−

−

t

t

t

t

t

t

X Y

X Y

2

1

1

1

2221

1211

ε ε

θ θ θ θ


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Coi'&)ra'io i aBi*aria'& VAR If cointegrating vector6

/he matriC is given by

( )1211

1 θ θ β −='

Π

( ) ( )1211

11

21 1

1

1

θ θ θ

θ γβ −

−==Π '


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TESTIN$ "OR

COINTE$RATION


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Th& E)l&+$ra)&r ,E$-Aroach /est whether the nonstationary

variables, and , are cointegrated orspuriously related, we need to

eCamine the properties of theregression residuals for stationarity.

If error term ! ) is stationary, Yt and

+t are said to be cointegrated.

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t ε


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Th& E)l&+$ra)&rAroach Step %6 /est the variables for their order of

integration using 37 or A37. *oth variablesare integrated of the same order.

Step 26 Estimate the long run relationshipusing

Step =6


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Documents

Lecture 6 Stationarity and Cointegration