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UnitRoots

Anautoregressiveprocess

hasaunitrootif

ThesimplestcaseistheAR(1)model

or

tt eyLa =)(

0)1( =a

tt eyL = )1(

ttt eyy+=

1

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ExamplesofRandomWalks

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RandomWalkwithDrift

AR(1)withnonzerointerceptandunitroot

ThisissameasTrendplusrandomwalk

ttt eyy ++= 1

ttt

t

ttt

eCC

tT

CTy

+=

=

+=

1

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Examples )1,0(~1.0

1

Ne

eyy

t

ttt ++=

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OptimalForecastsinLevels

RandomWalk

RandomWalkwithdrift

ttt yy =+ |1ttht yy =+ |

ttht yy +=+ |

ttht yhy +=+ |

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OptimalForecastsinChanges

Takedifferences(growthratesif y inlogs)

Optimalforecast:Randomwalk

Optimalforecast:Randomwalkwithdrift

1== tttt yyyz

0|=

+ thtz

hz tht =+ |

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ForecastErrors

Bybacksubstitution

Sotheforecasterrorfromanhstepforecastis

Whichhasvariance

Thustheforecastvarianceislinearin h

11

1

++

+++=

+=

ththt

ttt

eeyeyy

L

11 ++++ tht ee L

222 h=++L

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Forecastintervals

Theforecastintervalsareproportionaltothe

forecaststandarddeviation

Thustheforecastintervalsfanoutwiththe

squarerootoftheforecasthorizon h

hh =2

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Example:RandomWalk

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GeneralCase

If y hasaunitroot,transformbydifferencing

Thiseliminatestheunitroot,soz isstationary.

Makeforecastsof z

Forecastgrowthratesinsteadoflevels

1== tttt yyyz

tt

tt

ezLb

LLbLa

eyLa

=

=

=

)(

)1)(()(

)(

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Forecastinglevelsfromgrowthrates

Ifyouhaveaforecastforagrowthrate,you

alsohaveaforecastforthelevel Ifthecurrentlevelis253,andtheforecasted

growthis2.3%,theforecastedlevelis259 Ifa90%forecastintervalforthegrowthis

[1%,4%],the90%intervalforthelevelis

[256,263]

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EstimationwithUnitRoots

Ifaserieshasaunitroot,itisnonstationary,

sothemeanandvariancearechangingovertime.

Classicalestimationtheorydoesnotapply However,leastsquaresestimationisstill

consistent

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ConsistentEstimation

Ifthetrueprocessis

AndyouestimateanAR(1)

Thenthecoefficientestimateswillconvergein

probabilitytothetruevalues(0and1)asTgetslarge

ttt eyy += 1

ttt eyy 1 ++=

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Exampleonsimulateddata

N=50

N=200

N=400

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Modelwithdrift

Ifthetruthis

AndyouestimateanAR(1)withtrend

Thenthecoefficientestimatesconvergein

probabilitytothetruevalues(,0,1) Itisimportanttoincludethetimetrendinthis

case.

ttt eyy ++= 1

ttt eyty 1 +++=

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Examplewithsimulateddatawithdrift

N=50

N=200

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NonStandardDistribution

Aproblemisthatthesamplingdistributionof

theleastsquaresestimatesandtratiosarenotnormalwhenthereisaunitroot

Criticalvaluesquitedifferentthanconventional

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Densityoftratio

NonNormal

Negativebias

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TestingforaUnitRoot

Nullhypothesis:

Thereisaunitroot

InAR(1)

Coefficientonlaggedvariableis1 InAR(k)

Sumofcoefficientsis1

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AR(1)Model

Estimate

Orequivalently

Testfor =1sameastestfor =0. Teststatisticistratioonlaggedy

ttt eyy 1 ++=

1

1

=

++=

ttt eyy

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AR(k+1)model

Estimate

Testfor =0

CalledADFtest

AugmentedDickeyFuller

(TestwithoutextralagsiscalledDickeyFuller,test

withextralagscalledAugmentedDickeyFuller)

tktkttt eyyyy

111 +++++= L

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TheoryofUnitRootTesting

WayneFuller(IowaState)

DavidDickey(NCSU) DevelopedDFandADFtest

PeterPhillips(Yale) Extendedthedistribution

theory

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STATAADFtest

dfuller t3,lags(12) ThisimplementsaADFtestwith12lagsof

differenceddata

EquivalenttoanAR(13)

Alternatively

reg d.t3L.t3L(1/12).d.t3

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Example:3monthTbill

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Example:3monthTbill

Thepvalueisnotsignificant

Equivalently,thestatisticof2isnotsmallerthanthe10%criticalvalue

Donotrejectaunitrootfor3monthTBill

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Alternatively

ThetforL1.t3is 2

Ignorereportedpvalue,comparewithtable

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InterestRateSpread

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ADFtestforSpread

Thetestof4.8issmallerthanthecriticalvalue

Thepvalueof.0001ismuchsmallerthan0.05

Werejectthehypothesisofaunitroot

Wefindevidencethatthespreadisstationary

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TestingforaunitRootwithTrend

Iftheserieshasatrend

Againtestfor =0. dfuller y,trendlags(2)

tktkttt eyytyy

111 ++++++= L

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Example:Log(RGDP)

ADFwith2lags

Thepvalueisnotsignificant. Wedonotrejectthehypothesisofaunitroot

Consistentwithforecastinggrowthrates,notlevels.

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UnitRootTestsinPractice

Examineyourdata.

Isittrended?

Doesitappearstationary?

Ifitmaybenonstationary,applyADFtest

Includetimetrendiftrended

Iftestrejectshypothesisofaunitroot Theevidenceisthattheseriesisstationary

Ifthetestfailstoreject

Theevidenceisnotconclusive Manyusersthentreattheseriesasifithasaunitroot

Differencethedata,forecastchangesorgrowthrates

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SpuriousRegression

One problemcausedbyunitrootsisthatit

caninducespuriouscorrelationamongtimeseries CliveGrangerandPaulNewbold (1974)

Observedthephenomenon

PaulNewbold aUWPhD(1970)

PeterPhillips(1987) Inventedthetheory

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SpuriousRegression

Supposeyouhavetwoindependenttime

series yt and xt

Supposeyouregress yt on xt

Sincetheyareindependent,youshouldexpectazerocoefficienton xt andan

insignificanttstatistics,right?

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Example

TwoindependentRandomWalks

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Regressionof y on x

Xhasanestimatedcoefficientof.6

Atstaitsitc of18!Highlysignificant!

Butxandyareindependent!

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SpuriousRegression

Thisisnotanaccident

Ithappenswheneveryouregressarandomwalkonanother.

Traditionalimplication:

Dontregresslevelsonlevels

Firstdifferenceyourdata

Evenbetter Makesureyourdynamicspecificationiscorrect

Includelagsofyourdependentvariable

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DynamicRegression

Regressyonlaggedy,plusx

Now x hasinsignificanttstatistic,andmuch

smallercoefficientestimate Coefficientestimateonlaggedyiscloseto1.

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Message

Ifyourdatamighthaveaunitroot

TryanADFtest Considerforecastingdifferencesorgrowthrates

Alwaysincludelaggeddependentvariablewhen

seriesishighlycorrelated