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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
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