Unit Roots and Smooth Transitions: A Replication

Munich Personal RePEc Archive

Unit Roots and Smooth Transitions: A

Replication

Kulaksizoglu, Tamer

4 February 2015

A Repliation

Tamer Kulaksizoglu

∗

February 4,2015

Abstrat

ThispaperrepliatesLeybourneetal.(1998),whoproposea Dikey-Fullertypetestforunitrootthatismostappropriatewhenthereisreason tosuspetthepossibilityofdeterminististruturalhangeintheseries. Wendthatourrepliatedresultsarequitesimilartotheauthors'results. WealsomaketheOxsoureodeavailable.

Keywords: Dikey-Fullertest;Integratedproess;Nonlineartrend;Strutural hange

JEL lassiation: C12;C15

1 Introdution

ThealternativehypothesisofthestandardaugmentedDikey-Fuller(ADF)tests isstationarityaroundaxed meanoralineartrend. However,another possi-bilityisstationarityaroundalineartrendwith aninstantaneous break

1 . Ley-bourne et al. (1998) broaden this lass of alternatives to allow for a smooth transitionbetweentrendsbydevelopingDikey-Fullertypetestsandexploring theirproperties. Inthis paper, wetryto repliatetheirresults. Setion 2 dis-usses the smooth transition regressionmodels. Setion 3 disussesthe tests. Setion4presentstherepliatedresults. Setion 5onludesthepaper.

2 Smooth Transition Regression Models

Leybourneet al.(1998)onsider thefollowingthreelogisti smoothtransition regression(STR)models

ModelA:

y

t

=

α

1

+

α

2

S

t

(

γ, τ

) +

v

t

ModelB:

y

t

=

α

1

+

α

2

S

t

(

γ, τ

) +

β

1

t

+

v

t

ModelC:

y

t

=

α

1

+

α

2

S

t

(

γ, τ

) +

β

1

t

+

β

2

tS

t

(

γ, τ

) +

v

t

∗

where

y

t

istheseriestobetestedforunitroot,

t

isatimetrend,and

v

t

isa zero-mean I(0) proess.

S

t

(

γ, τ

)

is the logisti transition funtion, whih ontrols thetransitionbetweenregimes

S

t

(

γ, τ

) = [1 + exp

{−

γ

(

t

−

τ T

)

}

]

−

1

where

γ >

0

,

t

= 1

, . . . , T

,and

T

isthesamplesize.

τ

isthetransitionmidpoint fration,whih is also alled the loationparameter. It deteminesthe timing ofthetransitionmidpoint.

γ

isthespeedoftransitionparameter,whihisalso alled the slope orsmoothness parameterin the literature. It determines the speedofthetransitionbetweenregimes.

Themodelsrepresentdeterminististrutural hange. Inorder tosee this, rewriteModelC,whihneststheothermodels,as

y

t

= (

α

1

+

α

2

S

t

(

γ, τ

)) + (

β

1

+

β

2

S

t

(

γ, τ

))

t

+

v

t

Therstpartontherighthand-sideoftheequationisthetime-varyinginterept andtheseondpartisthetime-varyingslopewithrespettotimetrend. How-ever,the model isalso ableto miminobreak when

γ

= 0

and instantaneous struturalbreakwhen

γ

→ ∞

.

3 Unit Root Tests

Theauthorsonsider thefollowinghypotheses

H

0

:

y

t

=

µ

t

µ

t

=

µ

t

−

1

+

ε

t

µ

0

=

ψ

H

1

:

ModelA,ModelB,orModelC and

H

0

:

y

t

=

µ

t

µ

t

=

κ

+

µ

t

−

1

+

ε

t

µ

0

=

ψ

H

1

:

ModelBorModel C

where

ε

t

is zero-mean

I

(0)

proess. Notie that the null hypothesis is unit root but the alternativehypothesis is trend-stationarity. Theypropose atest statistithat anbealulatedinatwo-stepproedure.

STEP1. Estimatethemodelswithnonlinearleastsquares(NLS)andobtain theresiduals

2

ModelA:

v

ˆ

t

=

y

t

−

α

ˆ

1

−

α

ˆ

2

S

t

(ˆ

γ,

ˆ

τ

)

ModelB:

ˆ

v

t

=

y

t

−

α

ˆ

1

−

α

ˆ

2

S

t

(ˆ

γ,

ˆ

τ

)

−

β

ˆ

1

t

ModelC:

ˆ

v

t

=

y

t

−

α

ˆ

1

−

α

ˆ

2

S

t

(ˆ

γ,

ˆ

τ

)

−

β

ˆ

1

t

−

β

ˆ

2

tS

t

(ˆ

γ,

τ

ˆ

)

2

∆ˆ

v

t

= ˆ

ρ

v

ˆ

t

−

1

+

k

X

i=1

ˆ

δ

i

∆ˆ

v

t

−

i

+ ˆ

η

t

TheADF teststatistiis the

t

-ratioof theparameter

ρ

ˆ

. Thesummation part aountsfor anystationary dynamisin

ε

t

.

The test statistisaredenoted by

s

α

,

s

α(β)

,and

s

αβ

iftheresidualsomefromModelA,ModelB,andModelC, respetively.

EstimationofthesmoothtransitionregressionmodelsaredisussedinGranger andTeräsvirta(1993),Teräsvirta(1998),andTeräsvirta(2004). Ingeneral,the estimationis notaneasy task. However,observing that themodelsare linear in

(

α

1

, α

2

, β

1

, β

2

)

one

γ

and

τ

are known, Leybourneet al.(1998) propose a onentrated NLSestimation

3

. ForModel C,theobjetivefuntion isset to be

SSR

=

T

X

t=1

y

t

−

π

ˆ

′

x

ˆ

t

2

where

ˆ

π

=

h

ˆ

α

1

,

β

ˆ

1

,

α

ˆ

2

,

β

ˆ

2

i

′

=

T

X

t=1

ˆ

x

t

ˆ

x

′

t

!

−

1

T

X

t=1

ˆ

x

t

y

t

and

ˆ

x

t

= ˆ

x

t

(ˆ

γ,

τ

ˆ

) =

{

1

, t, S

t

(ˆ

γ,

τ

ˆ

)

, tS

t

(ˆ

γ,

τ

ˆ

)

}

The estimated values

ˆ

γ

and

τ

ˆ

are obtained from the optimization algorithm. TheyusetheBFGSoptimizationalgorithmin theOPMUMlibraryofGAUSS 3.1. We use the same optimization algorithm implemented in the MaxBFGS funtion in Ox. Theadvantageof this proedure is that the parameterspae thatneedstobesearhedoverisonlytwo-dimensionalinsteadoffour through six.

TheNLSestimatesof

γ

and

τ

donothavelosed-formsolutions. That'swhy it is diult to establish theanalytial relationshipbetween

v

ˆ

t

and

y

t

. Thus the null asymptoti distributions of the test statistis

s

α

,

s

α(β)

, and

s

αβ

are noteasily tratable byanalytialmeans. Instead, the authors ondut Monte Carlosimulationexperiments to approximate the null distribution of the test statistis.

Beforepresentingtheresults,wewouldliketomentionafewdetailsinregard tothealulations. Firstofall,itisonventionaltodividetheargumentofthe exponentialfuntioninthelogistitransitionfuntionbythestandarddeviation ofthetransitionvariabletomakeitsale-free

4

. Sinethis isnotmentionedin thepaper,wedonotapplythesaling. Seond,itisalsoonventionaltousea

3

Maugeri(2014)warnsthatthisproedureneedstobeusedwithautionasitmayyield biasedandinonsistentestimates,espeiallywhenfaedwithsmallsamples.

4

gridsearhto ndreasonablestartingvaluesfor

γ

and

τ

5

. However,following theauthors,weset thestartingvaluesfor

γ

and

τ

to 1and0.5, respetively

6 . Finally,astheauthorssuggested,weusetheMoore-Penrosegeneralizedinverse toestimatethe

π

ˆ

vetorwhentheomponentsof

ˆ

x

t

beomeslinearlydependent due tothe fat that theoptimization algorithm seletsvaluesof

γ

ˆ

and

τ

ˆ

that make

S

t

(ˆ

γ,

ˆ

τ

)

onstantforall

t

7 .

4 Experiments and Appliation

Inthissetion,wepresenttheresultsofourrepliations. Sinetherepliations inlude generating random numbers, exat repliation results should not be expeted. Instead,wefousontheproximityof thenumbersandonrmation ofthequalitativeresults. Theexperimentsexaminethenite samplesize and powerharateristisofthetests. Morespeially,ExperimentIgeneratesthe ritialvalues,ExperimentII generatesthesizesofthetests,andExperiments III andIVinvestigatethepowerpropertiesofthetests.

4.1 Experiment 1

TherstexperimentinthispaperrepliatesTableIinLeybourneetal.(1998), whih showsthe ritialvaluesof thetest statistis. Thoseritial valueswill beused latertoalulate theempirialsizesandpowersofthetests. Thenull DGPispurerandomwalk

y

t

=

µ

t

,

µ

t

=

µ

t

−

1

+

ε

t

,

µ

0

= 0

,

ε

t

∼

N ID

(0

,

1)

Asapreautionagainstpossibleonvergenefailures,wesetthenumberof repli-ationsfor the null distributionto 25,000 but used therst 20,000 in ritial valuealulations. Wealsoset

k

= 0

intheADFtests. Table1showsthe repli-atedritialvalues

8

ofthenulldistributionsoftheteststatistis

s

α

,

s

α(β)

,

and

s

αβ

at0.10,0.05,and0.01signianelevelsfor varioussamplesizes. Overall, therepliatedritialvaluesarequiteloseto theauthors'ritalvalues. The minimumabsolute diereneis0.001andthemaximumis0.151.

5

SeeTeräsvirta(2004),page228.Thisgridsearhisalsousedintheeonometrisoftware JMulTi,whihanbedownloadedattheURLhttp://www.jmulti.de/.

6

Theauthors laimthat thesolutions atonvergene werenot found tobesensitiveto thesehoies.

7

They suggest applyingthe standard ADFtest when thishappens and also when the onvergeneisveryslow.

8

Leybourne et al. (1998) hek the robustness of the simulated ritial values by also generatingrandomwalkswithinnovationsdrawnfrom

χ

2

(1)

s

α

s

α(β)

s

αβ

T 0.10 0.05 0.01 0.10 0.05 0.01 0.10 0.05 0.01 25 -4.308 -4.737 -5.717 -5.066 -5.529 -6.554 -5.624 -6.098 -7.218 50 -4.039 -4.405 -5.122 -4.634 -4.982 -5.768 -5.025 -5.396 -6.107 100 -3.922 -4.235 -4.841 -4.468 -4.775 -5.389 -4.774 -5.098 -5.742 200 -3.858 -4.174 -4.749 -4.373 -4.676 -5.265 -4.657 -4.978 -5.586 500 -3.834 -4.125 -4.716 -4.326 -4.610 -5.124 -4.582 -4.854 -5.381 Note: Nominalsizes0.10,0.05,and0.01.

Table2: EmpirialSizesoftheTest

s

α

forARIMA(1,1,0)Proesses T=100 T=200

ϕ

k

0.10 0.05 0.01 0.10 0.05 0.01 0.0 0 0.112 0.060 0.015 0.107 0.051 0.015 0.0 1 0.102 0.058 0.014 0.099 0.051 0.010 0.0 4 0.078 0.037 0.008 0.090 0.044 0.008 -0.4 0 0.602 0.485 0.300 0.631 0.530 0.331 -0.4 1 0.100 0.048 0.008 0.104 0.056 0.011 -0.4 4 0.068 0.034 0.005 0.093 0.046 0.007 0.4 0 0.008 0.004 0.001 0.006 0.004 0.002 0.4 1 0.094 0.045 0.009 0.093 0.047 0.011 0.4 4 0.079 0.046 0.009 0.093 0.047 0.006 -0.8 0 0.982 0.971 0.931 0.991 0.983 0.959 -0.8 1 0.094 0.047 0.011 0.103 0.055 0.014 -0.8 4 0.069 0.029 0.004 0.082 0.040 0.013 0.8 0 0.017 0.012 0.006 0.027 0.018 0.006 0.8 1 0.100 0.054 0.016 0.104 0.057 0.012 0.8 4 0.074 0.039 0.008 0.089 0.039 0.009 Note: Nominalsizes0.10,0.05,and0.01.

4.2 Experiment 2

The seond experiment repliates TableII, whih showsthe empirial size of thetest

s

α

. ThenullDGPisamoregeneralI(1)proess

y

t

=

µ

t

,

∆

µ

t

=

ϕ

∆

µ

t

−

1

+

ε

t

,

µ

0

= 0

,

ε

t

∼

N ID

(0

,

1)

T=100 T=200

s

α

τ

τ

s

α

τ

τ

ϕ

k

0.10 0.05 0.10 0.05 0.10 0.05 0.10 0.05 0.9 0 0.167 0.090 0.312 0.172 0.526 0.337 0.798 0.627 0.9 4 0.078 0.038 0.203 0.107 0.318 0.166 0.609 0.432 0.8 0 0.552 0.359 0.815 0.644 0.993 0.961 1.000 0.999 0.8 4 0.198 0.104 0.468 0.300 0.771 0.578 0.961 0.881 0.7 0 0.894 0.765 0.991 0.952 1.000 1.000 1.000 1.000 0.7 4 0.329 0.192 0.671 0.481 0.953 0.872 0.996 0.986 Note: Nominalsizes0.10and0.05.

4.3 Experiment 3

Thethirdexperiment repliatesTableIII,whih showstheempirialpowerof thetest

s

α

. ThenullDGPisastationaryAR(1)proess

y

t

=

µ

t

,

µ

t

=

ϕµ

t

−

1

+

ε

t

,

µ

0

= 0

,

ε

t

∼

N ID

(0

,

1)

where

ϕ <

1

. Thenumberof repliations is2,500 but weused the rst2,000 in the table. By way of omparison, the powers of the standard ADF test with trend and interept, denoted by

τ

τ

(a naturalompetitor to

s

α

), is also inluded

9

. One again, our repliated values are lose to the author's. The maximumabsolutediereneis0.049.

4.4 Experiment 4

ThefourthexperimentrepliatesTableIV,whihalsoshowstheempirialpower ofthetest

s

α

. WhentheDGPisastationaryAR(1)proess,thestandardADF test hasmore powerthan the

s

α

test. The fourth experimentgenerates data fromModelAwithrst-orderautoregressiveinnovationsandondutsthesame tests. TheDGP is

y

t

= 1 + 10

S

t

(

γ, τ

) +

µ

t

,

µ

t

= 0

.

8

µ

t

−

1

+

ε

t

where

µ

0

= 0

and

ε

t

∼

N ID

(0

,

1)

. The number of repliations is 2,600 but we used the rst 2,000 in the table. The experiment is repeated for various valuesof

γ

and

τ

. Table4showsthe resultsofthepowersimulations. Again, ourrepliatedvaluesareverylosetotheauthor's. Wendthatthemaximum absolutediereneis0.09.

4.5 Appliation

Leybourneetal.(1998)applythemostgeneralformoftheirtestproedure

s

αβ

and the standard ADF test

τ

τ

to the U.S. data set rst analyzed by Nelson

9

T=100 T=200

s

α

τ

τ

s

α

τ

τ

T

k

s

αβ

τ

τ

USSeries

RealGNP 62 2 -4.31 -2.94 NominalGNP 62 1 -3.18 -2.32 PerapitarealGNP 62 1 -4.46 -3.05 Industrialprodution 111 7 -4.57 -2.67 Employment 81 1 -3.70 -3.12 Unemployment 81 1 -4.55 -3.92 GNPdeator 82 1 -3.40 -2.52 Consumerpries 111 5 -3.26 -2.37 Wages 71 1 -3.45 -2.52 Realwages 71 1 -4.19 -3.05 Moneystok 82 1 -3.42 -3.08 Veloity 102 0 -3.28 -1.66 Bondyield 71 6 -5.95 -0.19 SP500 100 1 -5.26 -2.65

andPlosser(1982). Thedatasetontains14annualmaroeonomiserieswith thenumbersofobservationsranging from 62to 111. Theypresentthe results in Table V, whih is repliated in Table5

10

. As an beseen from the table, exeptfor the unemploymentseries, weget very loseresults. Themaximum absolutedierenefortheunemploymentseries is0.86for

s

αβ

and0.66for

τ

τ

. The maximum absolute dierene for the series exluding the unemployment seriesis0.15for forthe

s

αβ

testand0.13forthe

τ

τ

test. Thedierenein the resultsfor the standardADF test

τ

τ

is a bit puzzling sinethe test is pretty standard. The reasons for these dierenes might be a slightly dierent data set

11

,softwareissue,orjusttypo.

5 Conlusion

In this paper, we tried to repliate Leybourne et al. (1998), who propose an auxiliary test for the augmented Dikey-Fuller test that is most appropriate whenthereisreasontosuspetthepossibilityofstruturalhangeintheseries. With afew exeptions,wendthat our repliatedresultsare quitesimilar to the authors' results, whih is atestimony to the authors' areful eonometri analysis. We also make the Ox soure ode available to allow for others to double-hekourresults.

10

Theyalso applythesametests toU.K.onsolsand industrialprodution data butwe don'thavethesedatasetssoweannotrepliatetheirresults.

11

Granger,C. W. J. and Teräsvirta,T. (1993). Modelling Nonlinear Eonomi Relationships. OxfordUniversityPress.

Leybourne, S., Newbold, P., and Vougas, D. (1998). Unit roots and smooth transitions. Journalof Time SeriesAnalysis,19(1):8397.

Maugeri,N. (2014). Somepitfalls in smooth transition models estimation: A MonteCarlostudy. Computational Eonomis,44(3):339378.

Nelson,C.R. and Plosser, C.I.(1982). Trendsand randomwalksin maroe-onomitime series: Someevidene and impliations. Journal of Monetary Eonomis,10:139162.

Perron, P. (1989). The Great Crash, the oil prie shok and the unit root hypothesis. Eonometria,57:13611401.

Perron,P.(1990).Testingforaunitrootinatimeserieswithahangingmean. JournalofBusinessandEonomi Statistis,8:153162.

Teräsvirta,T.(1998). Handbook of Applied Eonomi Statistis,hapter Mod-elling Eonomi Relationships with Smooth Transition Regressions, pages 229246. Dekker,NewYork.

Teräsvirta,T.(2004).AppliedTimeSeriesEonometris,hapterSmooth Tran-sition Regression Modelling, pages 222242. Cambridge University Press, Cambridge.