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AppliedEconometricWAIjTER ENDERSIowa StateUniversity

TimeSeries

JOHNTVILEY& SONS,NC.Brisbane Toronto Singaporeew York . Chichester.

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Chapter4TESTING ORTRENDS NDUNITROOTS

Ilnspectionof the autocorrelationunctionservesas a rough ndicatorof whetheratrend s presentn a series.A slowly decayingACF is indicativeof a largecharac-teristic root, true unit root process,or trend stationaryprocess.Formal testscanhelp determinewhetheror not a systemcontainsa trendand whether hat trend sdeterministicor stochastic.However, he existing estshave ittle powerto distin-guishbetweennearunit root and unit root processes.he aims of this chapterareto:l. Developand llustrate he Dickey-Fullerand augmented ickey-Fuller tests or

the presence f a unit root. These estscan also be used o helpdetect hepres-ence of a deterministic rend. Phillips-Perron ests,which entail less stringentrestrictions n the errorprocess, re llustrated.2. Consider ests or unit roots n the presence f structuralchange.Structuralchange ancomplicate he tests or trends;a policy regimechange an result na stncturalbreak hat makesan otherwise tationary eriesappear o be nonsta-tionary.3. Illustratea generalprocedureo determinewhetheror not a series ontains unitroot. Unit root tests are sensitive o the presence f deterministic egressors,suchas an intercept erm or a deterministic ime trend. As such, here s a so-phisticatedset of procedureshat can aid in the identificationprocess.Theseproceduresan be used f it is not known whatdeterministic lements repartofthe true data-generatingrocess.t is important o be wary of the results romsuch estssince l) they all have ow powerto discriminate etweena unit rootand nearunit rootprocess nd(2) they may haveusedan nappropriateetof de-terministic egressors.

ztl

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212 TestingforTrendsndUnitRoots1. UNITROOTPROCESSESAs shown n the ast chapter, hereare mportantdifferencesbetweenstationaryandnonstationaryime series.Shocks o a stationary ime seriesarenecessarilyempo-rary; over time, theeffectsof the shockswill dissipate nd he serieswill revert oits long-nrnmean evel.As such, ong-term orecasts f a stationary erieswill con-verge o the unconditionalmeanof the series. o aid n identification,we know thata covariance tationary eries:1. Exhibitsmeanreversion n that t fluctuatesarounda constantong-runmean.2. Hasa finite variance hat s time-invariant.3. Has a theoretical orrelogramhat diminishes s ag ength ncreases.

On theotherhand,a nonstationaryeriesnecessarily aspermanent omponents.The meanand/orvarianceof a nonstationary eriesare time-dependent.o aid inthe dentificationof anonstationaryeries,we know that:l. There s no long-runmean o which theseries eturns.2. Thevariances time-dependentndgoes o infinity as ime approachesnfinity.3. Theoreticalautocorrelationso not decaybut, in finite samples,he sample or-relogramdiesout slowly.

Although the propertiesof a samplecorrelogramare useful tools for detectingthe possiblepresence f unit roots, hemethod s necessarilymprecise.Whatmayappearas a unit root to oneobservermay appearasa stationary rocess o another.The problem s difficult because nearunit root processwill have he sameshapedACF as a unit root process.For example, he correlogramof a stationaryAR(l)process uch hat p(l) = 0.99will exhibit the typeof gradualdecay ndicativeof anonstationary rocess. o illustratesomeof the issuesnvolved,supposehat weknow a seriess generatedrom the ollowing first-orderprocess:r

l = A1l-1 I Cwhere{er} is generatedrom a white-noise rocess.

(4.1)

First, supposehat we wish to test he null hypothesishatat = O.Underthemaintained ull hypothesis f at=0,, w.ecanestimate 4'l) usingOLS' The factthate, s a white-nit. process nd la, I < I guaranteeshat the {y,} sequencess[ationaryand the estimateof a, is efficient.Calculating he standard rror of theestimate f a1, he researcheran usea t-test o determinewhetherar is signifi-cantly different rom zero.Thesituations quitedifferent f we want o test hehypothesisr = l. Now, un-der the null hypothesis,he {y,} sequences generatedy thenonstationaryrocess:

),=It,i= l (4.2)

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Unit Root ProcessesThus, f a, - l, the variancebecomes nfinitely largeas t increases. nder thenull hypothesis,t is inappropriateo useclassicalstatisticalmethods o estimteandperformsignificanceestson the coefficient r.lf the {yr} sequences gener-atedas n (4.2), t is simple o show hat he OLS estimate f (4.1)will yielda bi-asedestimate f ,. In Section8 of thepreviouschapter, t wasshown hat hefirst-

orderautocorrelationoeff,rcientn a randomwalk model sp r = [ ( r - l l r ] o - 5 < lSince heestimate f a, is directly related o thevalueof p,, the estimated alueof t is biased o be below ts true valueof unity. Theestimatedmodel will mimicthatof a stationaryAR(l) processwith a nearunit root. Hence, he usual -rcstcan-not beused o test hehypothesist= l.Figure4.1 shows he sampleconelogram or a simulated andomwalk process.One hundrednormally distributed andom deviateswereobtainedso as to mimicthe {er}sequence. ssumingo = 0, wecancalculatehe next 100valuesn the {yr}sequence S , = lrt * er.This particularcorrelograms characteristic f most sam-ple correlogramsonstructedrom nonstationary ata.The estimated alueof p, isclose o unity and the sampleautocorrelationsie out slowly. If we did not knowthe way in which the dataweregenerated,nspection f Figure4.1 might leadus ofalsely conclude hat the dataweregeneratedrom a stationaryprocess.With thisparticulardata, estimates f an AR(l) model with and without an interceptyield(sndarderrorsare n parentheses):

J,=0.9546y.-r , , R2= 0.86(0.030)), = 0.164 0.9247y,-, R2= 0.864(0.037)

(4.3)(4.4)

Examining 4.3),a careful researcher ould not be willing to dismiss he possi-bility of a unit root since he estimated alueof , is only 1.5133 tandard evia-tions from unity. We might correctly recognize hat under he null hypothesis f aunit root, the estimate f a, will be biasedbelow unity. If we knewthe true distrib-ution of a, under he null of a unit root, we couldperformsucha significance est.Of course, f we did not know the true data-generatingrocess,we might estimatethe modelwith an intercept. n (4.4), the estimateof a, is more than wo standarddeviations rom unity: (l - 0.9247)10.037 2.O35.However, t would be wrong touse his information o reject he null of a unit root. After all, the pointof this sec-tion hasbeen o indicate hat such -testsare nappropriate nder he null of a unitroot.Fortunately, ickeyandFuller 1979,1981) evised procedureo formally estfor the presence f a unit root. Their methodologys similar to that used n con-structing he data reported n Figure4.1. Suppose hat we generatedhousands fsuch andomwalk sequencesnd or eachwe calculated heestimated alueof ar.Although mostall of the estimateswouldbe close o unity, somewouldbe further

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A simulatedandomwalkProcess.

214 Testingor TrendsandUnit RootsFigure 4.11 0I6420

-2-4 50

(al

8 9 1 0(b )

from unity thanothers.n performing his experiment,DickeyandFuller found hatiri thepresence f an ntercept:Ninetypercent f the estimatedalues f a, are ess han2.58standard rrors romunity.Ninety-fivepercentof the estimated aluesof a, are ess han 2.89standard rrorsfrom unity.Ninety-nine ercent f the estimatedalues f a, are ess han3.51standardnorsfrom unity.2

100000

Correlogramf theprocess.

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Unit Root Processes 2L5The applicationof theseDickey-Fuller critical values o tests or unit roots sstraightforward.Supposewe did not know the true-datageneratingprocessandweretrying to ascertainwhether he data used n Figure 4.1 containeda unit root.Using theseDickey-Fullerstatistics,we would not reject he null of a unit root n(4.4).The estimated alueof a, is only 2.035standard eviations rom unity. In

fact, if the truevalueof a, doesequalunity, we should ind the estimated alue obe within 2.58standard eviations rom unity 90Vo f the ime.- Be aware hat stationaritynecessitatesl 1a, < 1. Thus, f the estimated alueof a, is close o -1, you shouldalsobe concerned boutnonstationarity.f we de-fine 1 - at - l, the equivalent esFiction s -2 < y < 0. In conductinga Dickey-Fullertest, f ispossible o check hat the estmated alueof yis greater han 2.3MonteCarloSimulationThe procedureDickeyandFuller (1979,1981)used o obtain heir critical values stypical of that found n the modern ime series iterature.Hypothesisestsconcem-ing the coefficientsof non-stationary ariablescannot be conductedusing tradi-tional f-testsor F-tests.The distributionsof the appropriate est statisticsare non-standardand cannotbe analytically evaluated.However,given the trivial costofcomputer ime, the non-standard istributionscan easilybe derivedusinga MonteCarlo simulation.The first step n the procedures to computergenerate setof randomnumbers(sometimes alledpseudo-randomumbers) rom a given distribution.Of course,the numbers annotbe entirely randomsinceall computeralgorithms ely on a de-terministicnumbergeneratingmechanism.However, he numbersaredrawnso asto mimic a randomprocesshaving somespecifieddistribution.Usually, he num-bersaredesignedo be normallydistributedand seriallyuncorrelated.he dea s touse hesenumbers o represent ne replicationof the entire{er} sequence.All major statisticalpackages avea built-in randomnumbergenerator. n inter-esting experiment s to useyour softwarepackage o draw a set of 100 randomnumbersandcheck or serialcorrelation.n almostall circumstances,hey will behighly correlated. n your own work, if you need o use serially uncorrelated um-bers, you can model the computergenerated umbersusing the Box Jenkinsmethodology. heresiduals houldapproximatewhite noise.The second tep s to specify he parametersnd nitial conditionsof the {y,} se-quence.Using theseparameters,nitial conditions,and random numbers, he {yr}can be constmcted.Note that the simulatedARCH processesn Figure3.9 andran-dom-walk process n Figure 4.1 were constructed n precisely his fashion.Similarly,Dickey andFuller(1979,1981)obined 100values or {e,}, std, = l,)o = 0, and calculated100values or {y,} according o (4.1).At thispoint, hepara-metersof interest suchas he estimate f a, or the n-sample ariance f yr) can beobtained.The beautyof the method s that all importantattributes of the constructed y,}sequence re known to the reseacher. or this reason,a Monte Carlosimulation soften referredto as an "experiment." The only problem is that the setof random

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TestingforTrends and Unit Rootsnumbersdrawn s just onepossibleoutcome.Obviously, he estimatesn (4.3)and(4.4) arc dependent n the values of the simulated{e,} sequence. ifferent out-comes or {e,} will yield differentvaluesof the simulated y,} sequence.This is why the Monte Carlo studiesperformmany replicationsof the processoutlined above.The third step s to replicatesteps and2 thousands f times.Thegoal is to ensure hat the statisticalpropertiesof the constructedy,} sequence rein accordwith the ruedistribution.Thus, or each eplication, heparametersf in-terestare abulatedandcritical values or confidencentervals)obtained.As such,the propertiesof your dataset can be compared o the propertiesof the simulateddatasothathypothesisestscan beperformed.This is the ustification or using heDickey-Fullercriticalvalues o test hehypothesis t - l.One imitation of a Monte Carloexperiment s that it is specific o the assump-tionsused o generatehe simulated ata. f you change he sample ize, nclude ordelete)an additionalparametern the datagenerating rocess, r usealternativeni-tial conditionsan entirelynew simulationneeds o be performed.Nevertheless,oushouldbe able to envisionmanyapplications f Monte Carlo simulations.As dis-cussed n Hendry, Neale, and Ericsson 1990), hey are particularlyuseful forstudying he small samplepropertiesof time-series ata.As you will seeshortly,Monte Carlo simulations re heworkhorse f unit root tests.Unit Roots n a Regresson odelThe unit root issuearisesquite naturally n the contextof the standardegressionmodel.Consider heregression quation:a

! t= ao+ a t 4+ e , (4.s)The assumptions f the classical egressionmodel necessitatehat both the {y,}and {2,} sequences e stationaryand the errors have a zero meanand finite vari-ance. n the presence f nonstationary ariables,heremight be whatGrangerandNewbold(1974)call a spurious regression.A spurious egression asa high R2,t-statistics hat appear o be significant,but the resultsare without any economicmeaning.The regression utput "looks good" because he least-squaresstimatesare not consistent nd the customary estsof statistical nferencedo not hold.GrangerandNewbold(1974)providea detailedexaminationof the consequencesof violating he stationarity ssumptiony generatingwo sequences,y,} and[2,],asrindependentandomwalks using he ormulas:

,, - Y,-t * t t (4.6)and

Z= Z-1* Er,where e' ande.,= white-noise rocessesndependent f eachother

(4.7)

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Unit Root Processes 217In theirMonte Carlo analysis,Grangerand Newboldgeneratedmanysuch sam-plesand or eachsampleestimated regressionn the form of (4.5).Since he {y,}and {2,} sequencesre independent f eachother, Equation 4.5) is necessarilymeaningless;ny relationshipbetween he two variabless spurious.Surprisingly,at the 57osignificance evel, they wereable o reject he null hypothesis r = 0 inapproximately 57oof the time. Moreover, he regressionssually hadveryhigh R2valuesand heestimatedesiduals xhibiteda highdegree f autocorrelation.To explain he Grangerand Newbold indings, note hat the regression quation(4.5) s necessari ly eaninglessf the residualseries e,} is nonstationary.Obviously, f the {e,} sequence asa stochasticrend,any error in periodt neverdecays, othat the deviation rom the model s permanent.t is hard o imagineat-tachingany importance o an economicmodel havingpermanent rrors.The sim-plestway to examine he properties f the {e,} sequences to abstractrom the n-tercepterm aoand rewrite(4.5)as

r = l r _ A l Z ,If z, and , Ne generated y (a.6)and(4.7),we can mpose he nitial conditions)o = ro = 0, sothat

t' , =ftr,i=l

I- orrrr,i= l

(4.8)

Clearly,the varianceof the errorbecomesnfinitely largeas t increases.More-over, he error hasapermanentomponentn that E&,*t = etfo all j 2 0. Hence,heassumptionsmbeddedn the usualhypothesis estsare violated,so thatany t-test,F-test,or R2valuesareunreliable.t is easy o seewhy theestimated esidualsroma spurious egressionwill exhibit a high degreeof autocorrelation. pdating 4.8),you shouldbe able o demonstratehat hetheoretical alueof the correlation oef-ficientbetween , ander+rgoes o unity as increases.The essence f theproblem s that 7fa, - 0, thedatagenerating rocessn (4.5)is y, = ao+ et.Given hat {y,} is integrated f orderone [i.e., (l)], it follows hat{e,} is I(l) under he null hypothesis. owever,he assumptionhat he error ermis a unit root processs inconsistentwith the distributional heory underlying heuse of OLS. This problemwill not disappearn largesamples. n fact, Phillips(1986)proves hat he arger hesample,he more ikely you are o falselyconcludethatc, * 0.Worksheet4.1 illustrates he problemof spurious egressions. he top twographsshow 100 ealizations fthe {yr} and {2,} sequencesenerated ccordingo(4.6)and(4.7).Although er,} and {e.,} aredrawn rom white-noise istributions,the realizations f thetwo sequencesre such haty,- is positiveandzroo egative.You can see hat the regression f lt oi z, captures he within-sampleendencyofthe sequenceso move n oppositedirections.The straight ine shown n the scatter

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ztt Testingfor Trendsand Unit Rootsplot is the OLS regressionine y, = {.31 - 0.462,.The correlation oefficientbe-tween{yr} and {e,} is 4.372. The residualsrom this regression avea unit root;as such, hecoefficients .31 and .46 are spurious.rl/orksheet4.2llustrateshesameproblemusing wo simulated andomwalkplusdrift sequences:, =0.2 + y,-,+ e' and4= 4.1* zr-r* e.r.Thedrift termsdominate, o hat or small valuesof t,it appearshaty, - -22,. As samplesize ncreases,owever, he cumulated um of

, , - r r - t * t r rConsider he two randomwalk processes:

Since he {gr} and {eo} sequencesre ndependent,heregression f y, on z, s spu-rious.Given the realizations f the randomdisturbances,t appears s f the two se-quences rerelated. n the scatterplot of y, againstz' you can see hat y, tends orise asz, decreases. he regtession quationof y, on z, will capture his tendency.The correlation oefficientbetween , andz,is 4.372 anda linear regression ieldsyr--4.462, - 0.31.However, he residuals rom the regression quationarenonsta-tionary.

+ Scatterplot- Regressionne

50t50t

el-2

-124 t-3

Z = Z - 1 i r ,

Scatter plot of y,and z,t+ *+ rf*

+ 'f +f++ f +++ +

f ++

{+

Regressionesiduals

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Unit RootProcessesthe errors(i.e., Ee,)will pull the relationship urther and further from -2.0. Thescatterplot of thetwo sequencesuggestshat heR2statisticwill be close o uni[;in fact, R2 s almost0.97.However,asyou can see n the lastgraphof Worksheet4.2, the residuals rom the regression quationare nonstationary.All departuresfrom this relationship renecessarilyermanent.The point is thattheeconometricianas o be very careful n workingwith non-stationary ariables.n termsof (4.5), hereare our caseso consider:CASE1Both {y,} and {e,} are stationary.Whenboth variablesare stationary,he classicalregressionmodel s appropriate.CASE2The {y,} and[,] sequencesre ntegratedf differentorders.Regressionquationsusingsuchvariables remeaningless-or example,eplace4.7)by the stationaryprocess ,= pz,-t+ .r,where p | < 1. Now (a.8) s replaced y t,= Ier; - Z1'er,-.Although he expressionp'e.,-, is convergent,he {er,} sequencetill containstrendcomponent.sCASE3The nonstationaryyr} and {2,} sequencesre ntegrated f thesameorder and heresidualsequenceontainsa stochasticrend.This is the case n which theregres-sion s spurious.The results rom suchspuriousegressionsremeaninglessn thatall errorsare perrnanent.n this case, t is oftenrecornmendedhat the regressionequationbeestimatedn frst differences.Considerhe irst differenceof (4.5):

Ly,= ar\,z,+ Le,Since t, zt, tde, each ontainunit roots, hefirst differenceof each s stationary.Hence, he usualasymptotic esultsapply.Of course,f oneof the trends s deter-ministicand heother s stochastic,irst-differencing ach s not appropriate.

CASE4The nonstationaryyr} and {zr} sequencesre ntegrated f thesameorderand heresidual equenes stationary.n this circumstance,y,} and {2,} are cointe-grated.A trivial exampleof a cointe$ratedystem ccurs f e., andyrareperfectlycorrelated.f er,= r' then (4.8)can be setequal o zero(which is stationary)bysettingat = l. To considera more nterestingexample,supposehat bothz, andy,are he randomwalk plusnoiseprocesses:

219

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tn Testingor Trendsand Unit Roots! ,= lL,* 9,4= l\ + ez,

where g, and.; trwhite-noise rocesses d , is therandomwalk process,=r-r * er.Notethat both [zr] and {yr} areunit rootprocesses,uty, - zt= rr . Sstationary.All of Chapter6 is devoted o the ssueof cointegrated ariables.For now, it issufficient o note hatpretestinghe variablesn a regressionor nonstationaritysextremelymportant.Estimatinga regressionn theform of (4.5) s meaninglessf

Consider he wo randomwalk plusdrift processes:

20& t o

0

0-5

zl-10

30 I t= O.2 * l _1 i ty

-to 50t

1000t

-150-

Again, the {e"r} and {er} sequencesre ndependent, othatthe regression f y, onz, is spurious.The scatterplot of y, againstz, stronglysuggestshat the two seriesarerelated. t is the deterministicime trend hat causeshe sustainedncreasen y,and sustained ecreasen 2,.The residuals rom theregression quation , = 12, +eraenonstationary.

20)r 1b

0-10

Scatter plot of y, and z,

z1

z t= -0 . 1 t z1 -1 * r

Regressionesiduals

- 1 5 - 1 0 -5

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Dickey-Fuller Tests 221Cases or 3 apply. f the variablesare cointegrated,he resultsof Chapter6 apply.The remainder f this chapterconsiders heformal testproceduresor the presenceof unit rootsand/ordeterministicime trends.

2. DICKEY-FULLERESTSThe last sectionoutlineda simpleprocedureo determinewhethear = I in themodel , = atltt + e,.Begin by subtracting ,-t from eachsideof the equationnorder o write he equivalentorm: L!,=W,-, * ,,whereT= at - 1.Of course,est-ing thehypothesis t = | is equivalento testing he hypothesis = 0. Dickey andFuller (1979)actuallyconsiderhreedifferentregression quationshat can be usedto test or thepresence f a unit root:

Yr=1Y,-t+e,Ly,= ao+ Yyr-l + Ly,= ao T!,-t + azt + et(4.e)(4.10)

(4 .1 l )The differencebetween hethree egressions oncernshepresence f thedeter-ministic elementsao and a2t.T\e first is a pure randomwalk model, the secondadds an interceptor drift term,and the third includesboth a drift and inear timetrend.The parameter f interestn all theregression quationss if T= 0, the {y,} se-quence ontainsa unit root. The test nvolvesestimatingone(or more)of the equa-tions aboveusing OLS in order to obtain the estimated alue of y and associatedstandarderror. Comparing he resulting f-statisticwith the appropriatevalue re-

ported n theDickey-Fullertablesallows heresearchero deterrninewhether o ac-ceptor reject henull hypothesis = 0.Recall hat n (4.3), heestimate f y, = dtJrt + rwassuch hatat=0.9546 witha standard rror of 0.030.Clearly, he OLS regressionn the form y,= T!,-t t c,will yield an estimateof l equal o 4.0454 with the samestandard rror of 0.030.Hence, he associated-statisticor thehypothesis = 0 is -1'5133(-0'0454/0'03-1.5133).The methodologys precisely hesame,egardless f which of thethree ormsofthe equationss estimated.However,be aware hat hecritical valuesof the f-statis-tics do dependon whetheran interceptand/or ime trend s included n theregres-sion equation.n their MonteCarlo study,Dickey andFuller (1979) ound thatthecritical values or y - 0 dependon the form of the regression ndsamplesize.Thestatistics abeled r, , and t" are the appropriatestatistics o use for Equations(4.9), 4.10), nd 4.11),espectively.Now, look at TableA at the endof this book.With 100observations,herearethreedifferent critical values or the f-statisticT = 0' For a regressionwithout theinterceptand rend erms ao= ar= 0), use hesectionabeled . With 100observa-tions, he criticalvalues or the f-statistic re 1.61, -1.95 and-2.60 at the 10, 5,

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Testing or Trends and Unit Rootsand 7o significanceevels, espectively. hus, n thehypothetical xamplewith y=-0.0454 anda standard rror of 0.03(so hat f = - I .5 33), t is notpossibleo rejectthe null of a unit root at conventional ignificanceevels.Note thatthe appropriatecritical valuesdependon samplesize.As in most hypothesisests, or any givenlevel of significance,he critical valuesof the t-statisticdecrease s sample ize n-creases.

Includingan ntercept ermbut not a trend erm(onlyaz= 0) necessitatesheuseof the critical values n the sectionabeled u. Estimating 4.4) in the form y, =aotT!,-t + ,necessai'ilyieldsa valueof yequal o (0.9247 l) = -0.0753with astandarderror of 0.037.The appropriate alculation or the tp statisticyields-0.0753/0.037= 1.035.If we read rom the appropriateow of Table A, with thesame100observations,he criticalvaluesare 2.58, -2.89, and 3.51 at the 10,5,and Vo significanceevels, espectively. gain, thenull of a unit root cannotbe re-jected at conventionalsignificance evels.Finally, with both intrceptand trend,use the critical values n the section abeledTr; now the critical valuesue -3.45and -4.04 at the 5 and l7o significanceevels,respectively. he equationwas notestimated sing a time trend; nspection f Figure4.1 indicates here s little reasonto includea deterministicrend n the estimating quation.As discussedn the next section, hesecritical valuesare unchangedf (4.9),(4.10),and 4.11)are eplaced y theautoregressiverocesses:6pLy,=Tlr-r*f i),-i*r e,i=2 *4^Ly t = ao * Tl a + /i|lt_i+l + e,i=2 p

y, = ao*^{!rt+a2t+)i},- i*r +e,

(4.t2)(4.13)

(4.r4)i=2The sameT,tu, and " statisticsare all used o test hehypotheses = 0. Dickeyand Fuller(1981)provide hreeadditional -statisticscalled0,, 0, md 0r) to testjoint hypothesesn the coefficients. /ith (4.10)or (4.13), he null hypothesis=ao= 0 is tested sing he0, statistic.ncludinga time trend n theregression-sothat(4.11) or (4.14) s estimated-the oint hypothesis o= T- a2= 0 is tested s-ing theQrstatistic nd he oint hypothesis= az= 0 is tested sing heQ3 tatistic.The 0,, Qr,andS, statisticsare constructedn exactly he sameway as ordinaryF-testsare:

Q =IRSSrestricted) RSSunrestricted)l/rRSS unrestricted)/( - /

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DickeY-Fuller Tests

r = numberof restrictionsT = numberof usableobservations/< = numberof parametersstimatedn theunrestrictedmodelHence,T - k =degreesof freedomn theunrestrictedmodel.comparingthe calculated alueof Q, o the appropriate aluereportedn DickeyandFuller (1981)allowsyou to determinehe significanceevelat whichthe re-striction is binding.The null hypothesiss that the dataare generated y the re-strictedmodeland the alternativehypothesiss that the dataare generated y theunrestrictedmodel. f the restrictions not binding,RSS(restricted)houldbe closeto RSS(unrestricted)nd Q,shouldbe small;hence, argevaluesof Q,suggestbindingrestrictionand ejecionof thenull hypothesis. hus, f thecalculated alueof 0, iI smallerthan thai reportedby Dickey and Fuller, you can accept he re-strictedmodel(i.e., you do not reject he null hypothesishat the restriction s notbinding).If the calclated alueof Q, s larger hanreportedby Dickey andFuller'you canreject he null hypothesis ndconclude hat the restrictions binding'Thecritical valuesof the hre$, statistics re eportedn TableC at theendof thistext'Finally, it is possible o testhypothesesoncerninghe significanceof the driftterm40and ime trendar. under the nult hypothesis = 0, the test or the presenceof the time trend n (4.14) s givenby the fp" statistic.Thus, his statistics the testaz= given that1= 0. To tesithehypothesis o= 0,-us: the1.."statisticf you esti-,ou," 1i'.1+)and he * statistic f yu estimate4.13).The completesetof teststa-tistics and their criti values or a samplesizeof 100 are summarizedn Table4 . t .

Tahte4.1 Summaryof theDickey-FullerTestsModel Hypothesis TestStatistic

Critical values or957oand99%oConfidence ntervalsLy,=ao*T) , - l +a+et

Ly,-- ao+ TY'-r+ ,

T = 04o= 0 given1= gaz=Ogiven = 0I = a z = O4 o = ^ t l a z = 0T = 04o= 0 given1- 0a s = l = 0T = 0

Tffo'"fp "0Qtfirtro.ttQ,I

-3.45and 4.M3.11 nd .78239 and3.536.49and8.734.88and6.50-2.89and 3.512.54and3.224.71and6.70-1.95and1.60LY,--YY'-t+ e,

Notes: Criticalvalues re df? sample izeof 100'

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TestingforTrendsandUnit RootsAn ExampleTo illustrate he useof thevarious eststatistics,DickeyandFuller(1981)usequar-terly valuesof the logarithmof theFederalReserveBoard's Production ndex overthe 1950:I o 1977:IVperiod o estimatehe ollowing threeequations:

Ly,=0.52+ 0.00120r 0.119y,- 0.498y,-r ,, RSS= 0'056448RSS= 0.063211RSS= 0.065966

whereRSS= residualsumof squares,ndstandard rors are n parentheses.To test henull hypothesishat the dataaregenerated y (a.17)againsthe alter-native hat (4.15) s the "tne" model,use heQ, statistic.DickeyandFullertest henull hypothesis o a2=T= 0 as ollows.Notethat heresidualsumsof squares fthe restrictedandunrestrictedmodelsare0.065966and0.056448and the null hy-pothesisentails hree estrictions.With110usableobservationsnd our estimatedparameters, he unrestrictedmodelcontains 106 degrees f freedom.Since0.056448/106 0.000533,heQ,statistics given by

02= (0.065966 0.056448y (0.000533) 5'95With 110observations,he critical valueof $2calculated y Dickey andFuller is5.59at rhe2.5Vo ignificanceevel.Hence, t is possibleo reject he null hypothe-sisof a randomwalk against he alternative tratthedatacontainan nterceptand/or

a unit root and/ora deterministic ime trend(i.e., rejectiEao- a, = T = 0 meansthat oneor moreof theseParametersoesnot equalzero).DickeyandFulleralsotestthenullhypothesisaz=T=0giventhealternativeof(4.15).Now if we view (4.16)as herestrictedmodeland 4.15)as heunrestrictedmodel, he0, statistics calculated s03= (0.063211 0'056448y (0.000533) 6'34

With I l0 observations,he critical value of 0 is 6.49 at the 57osignificancelevel and5.47at the \Vo significanceevel.TAt the lOToevel, hey reject he nullhypothesis.However,at the57o evel, hecalculated alueof S, is smaller han hecriticalValue; heydo not reject henull hypothesishat thedatacontaina unit rootand/ordeterministic imetrend.To comparewith the r test(i.e., hehypothesishatonly T= 0) note hat1"= -{.1 l910.033 -3.61.

whichrejectshe null of aunit rootat the 5Voevel.

(0.15) 0.00034) (0.033) (0.081)y,=0.0054 0.447LY--r',(0.002s) (0.083)),=0.51ly,-t,,(0.079)

(4.1s)(4.16)(4.t7)

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Extensions of the Dickey-Fuller Test3. EXTENSIONSFTHEDICKEY-FULLERESTNot all time-series rocessesan be well representedy the first-orderautoregres-siveprocess)l,= ao* T!,t + azt + e,. t is possible o use he Dickey-Fullertestsin higher-orderquationsuchas(4.12), 4.13),and(4.14).Considerhepth-orderautoregressiverocess

!, = Qo I atlrt I azl-z* az!rz + "' + ap-zltp+z* ao-r!t-n+r+ apyFp+ et (4.18)To bestunderstandhe methodology f the augmentedDickey-Fuller test, addand subtract plep+ro obtain:

It = ao* atlt-t * azJt-z* a,-t + '. ' + ar-zlrp+2+ (anr t apDt-p+t- ar\y,-*t + e,Next, add and subtract (an, * doDt-p+zto obtain

I t = ao t a tJ t * dzyrz * az! t-t + ... - (ao-, + ar)L! t-p*z - aoLy r-r*, t e,Continuingn this ashion,we get

pLy,= ao*^l !t-t+ Ip,^y,-i+l + ,i= 2where

(4.le)In (4.19), he coefficient f interests 1 if T = 0, theequations entirely n firstdifferences nd sohas a unit root.We cantest or thepresence f a unit root usingthe sameDickey-Fuller statisticsdiscussed bove.Again, the appropriate tatisticto use dependson the deterministiccomponentsncluded n the regression qua-tion. Without an interceptor trend,use he T statistic;with only the intercept,usethe tu statistic;and with both an nterceptand rend,use he 1r statistic. t is worth-while pointingout that the resultshere areperfectlyconsistentwith our studyof

differenceequationsn Chapter1. f the coefficientsof a differenceequation um oI, at leastonecharacteristicoot s unity.Here, f hi = l, Y= 0 and he system asa unit root.Note that the Dickey-Fuller tests assume hat the errors are independent ndhavea constant ariance. his raisesour importantproblems elated o the act thatwe do not know the true data-generating rocess.First, the true data-generatingprocessmay containboth autoregressivend moving average omponents.Weneed o know how to conduct he test f the orderof the moving average erms(if

( p \t = - l t - ! o , I I L l ' l\ i= t ./p9 i = \ a ,j=i \

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Testing or Trends and Unit Rootsany) is unknown.Second,we cannotproperlyestimateT and ts standard lror un-less all the autoregressiveermsare included n the estimatingequation.Clearly,the simpleregressionLy,= aor Tlt-t + e, s inadequateo this task f (4.18) s thetrue data-generatingrocess.However, he true orderof theautoregressiverocessis usuallyunknown o theresearcher,othat heproblem s to select heappropriatelag length.The third problem stems rom the fact that the Dickey-Fuller testcon-sidersonly a singleunit root.However,apth-orderautoregressionasp character-istic roots; f there alem

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Extensionsof the Dickey-Fuller Testincreased umberof lagsnecessitatesheestimationof additionalparametersndalossof degrees f freedom.The degrees f freedomdecrease ince he numberofpnametersstimated as ncreased nd becausehenumberof usableobservationshasdecreased.We lose oneobservationor eachadditionalag included n theau-toregression.) n theotherhand, oo few lagswill not appropriately apture heac-tualerrorprocess, othaty and ts standard rrorwill not bewell estimated.How does heresearcherelect he appropriateag ength n suchcircumstances?Oneapproachs to startwith a relatively ong lag lengthandparedown the modelby the usualr-rcstand/orF-tests.For example,onecouldestimateEquation 4.20)usinga lag engthof n*. If the -statistic n ag n* is insignificant t somespecifiedcritical value, eestimatehe regression singa lag lengthof r* - l. Repeat heprocess ntil the ag is significantlydifferent rom zero. n the pureautoregressivecase,sucha procedurewill yield the true ag lengthwith anasymptoticprobabilityof unity, provided hat he nitial choiceof lag length ncludes he true ength-Withseasonal ata, he processs a bit different.For example,usingquarterlydata,onecouldstartwith 3 yearsof lags n = l2).If the -statistic n ag 12 s insignificant tsomespecifiedcritical value and an F-test ndicates hat ags 9 to 12 arealso n-significant,move o lags I to 8. Repeatheprocess or lag 8 and ags 5 to 8 until areasonableag lengthhasbeendetermined.Oncea tentative ag lengthhas beendetermined, iagnosticcheckingshouldbeconducted. s always,plotting he residualss a most mportantdiagnosticool.There shouldnot appear o be any strongevidenceof structuralchangeor serialcorrelation.Moreover, he_eorrelogramf the residualsshouldappear o be whitenoise.The Ljung-Box Q-statisticshouldnot revealany significantautocorrelationsamong he residuals.t is inadvisableo use he alternative rocedure f beginningwith the mostparsimoniousmodeland keepadding agsuntil a significant ag isfound. In MonteCarlostudies,his procedures biased owardselectinga valueofn that s less han he truevalue.Multiple RootsDickey andPantula 1987)suggest simpleextension f the basicprocedurefmore thanone unit root is suspected.n essence,he methodologyentailsnothingmorethanperformingDickey-Fuller testson successiveifferences f {y'}- Whenexactlyone root is suspected,he Dickey-Fullerprocedures to estimatean equa-tion suchasAy,= ao*Ilrt * e,. nstead,f two rootsaresuspected,stimateheequation:

L 'y ,=ao+P,Y,- ,+e, (4.2r)Use the appropriate tatistic i.e.,t, tp, or t" depending n the deterministicele-mentsactually ncluded n the regression)o determinewhetherBt is significantlydifferent from zero.If you cannot eject he null hypothesishat Br = 0, concludethat the {y,} sequences I(2). If B1doesdiffer from zero,go on to determinewhether here s a singleunit root by estimating

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TestingforTrends and Unit Rootsy,= ao* p,Ay,-,+ ry,-, + e, (4.22)

Since here are not two unit roots,you should ind that pt and/orB, differ fromzero.Under he null hypothesis f a singleunit root, t < 0 andp, = 0; under he al-ternativehypothesis, yr} is stationary, o that pr and zare both negative.Thus,estimate 4.22)anduse he Dickey-Fuller critical values o test he null hypothesisr= 0. If you reject hisnull hypothesis, onclude hat {y,} is stationary.As a rule of thumb,economic eriesdo not need o be differencedmore han wotimes. However, n tlie odd case n which at most r unit roots are suspected,heprocedures to first estimate

L 'y ,=o+pr- I l t - t * tIf ^1,, is stationary, ou should ind that-2 < t < 0. If the Dickey-Fuller criticalvalues or p, are such hat t is notpossibleo reject henull of a unit root,you ac-cept thehypothesis hat {y,} contains unit roots. f you reject his null of exactly

r unit roots, he next steps to test or r - I rootsby estimatingL'y,=ao + pl^-r)r-r * rn2yr-, e,If both r and p, differ from zero, eject he null hypothesis f r - I unit roots.You can use he Dickey-Fullerstatistics o test he null of exactly r - I unit roots fthe t statistics or , and z le both statisticallydifferent rom zero. f you canre-ject thisnull, thenext step s to form

L'y,= do+ pr^-rlr-r * r!'-'y,-, + pr'-3y,-,+ e,As long as t is possibleo reject he null hypothesishat hevariousvaluesof the;arenonzero, ontinue oward he equation:

L'y,= ao+ pr^'-l !et t r!'-y,-, * rA'-'yr-r "' * Jr-r I , .Continue n this fashionuntil it is not possibleo reject he null of a unit root orthe y, series s shown o be stationary.Noticethat this procedures quitedifferentfrom the sequentialesting or successively reaternumbersof unit roots. t mightseem empting o test or a singleunit root and, f the null cannotbe rejected,go onto test or thepresence f a second nit root. In repeated amples,his method ends

to select oo few roots.SeasonalUnitRootsYou will recall hat the best-fittingmodel for the monthly Spanish ourism dataused n Chapter2had the form:

(l - t1(l - L)y,=(l + ,Xl + ,rL12e,

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Extensionsof the Dickey-Fuller TestTourist visits o Spainhavea unit rootandseasonalnit root.Sinceseasonalityis a key featureof manyeconomicseries,a sizable iteraturehasdevelopedo testfor seasonal nit roots.Before proceeding, ote that the first differenceof a sea-sonalunit root processwill not be stationary.To keepmatterssimple,supposehatthequarterlyobservationsf {yr} aregeneratedy

l - t = ! r - q * ,Here, he fourth differenceof {y,} is stationary; sing he notationof Chapter2,we canwrite, = er Given he nitial conditiono = )_r = ...= 0, thesolution orY, s:

It= t * r_ r_8 .. .so that

tl4 tl4l t - l t - t= I to , - I to , - ,j= 0 i= 0Hence, y, equals he differencebetween wo stochasticrends.Since he vari-anceof Ay, ncreases ithout imit as increases,he {y,} sequences not station-ary.However, he seasonal ifferenceof a unit rootprocessmay be stationary.Forexample,f {yr} is generatedy y, = lrt + r, he ourthdifference i.e.,oy,= , *r-r * t-z er-r) is stationary.However, he varianceof the fourth difference slargerthanthe varianceof the first difference.The point is that the Dickey-Fullerproceduremustbe modified n order o test or seasonal nit rootsanddistinguishbetween easonalersusnonseasonaloots.Thereare severalalternativeways to treat seasonalityn a nonstationary e-quence.The mostdirectmethodoccurswhenthe seasonal attern s purelydeter-ministic.For example,et Dp D2,and D, represent uarterlyseasonal ummy vari-ablessuch hat the valueof D, is unity in season andzerootherwise.Estimateheregression quation:

),,= 0o + lDt + ezDz + urD, +it, (4,23)where, is the regressionesidual, o that , canbe viewedas thedeseasonalizedvalueofy'Next,use hese egressionesidualso estimateheregression:

pi,= i,-r+f ,li_,*,.,i=2

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TestingforTrends and Unit RootsThe null hypothesisof a unit root (i'e', T = 0) canbe testedusing the Dickey-Fuller rp statistic.Rejecting he null hypothesiss equivalento acceptinghe alter-native hat ttre {y,} sequences stationary. he test s possibleasDickey,Bell, andMiller (19S6)show hat he imiting distribution or 1is not affectedby the removalof the deterministicseasonal omponents.f you want to includea time trend in(4.23),use he " statistic.Ifyou suspect seasonalnit root, t is necessaryo usean alternative rocedure.To keep he notationsimple,suppose ou havequarterlyobservations n the {y,}sequence ndwant to testfor the presence f a seasonal nit root. To explain hemethodology, ote that the polynomial(l - TLo) anbe factored,so that therearefour distinctcharacteristicoots:

(l - yLo)= (1 - y''oL)(l + TttoL1l iytt4 1| + iytt4L (4.24)If y, hasa seasonalnit root,T= l. Equation 4.24) s a bit restrictiven that tonly allows or a unit root at anannual requency.Hylleberget al. (1990)developa

clever technique hat allowsyou to test for unit rootsat various requencies; oucan testfor a unit root (i.e.,a root at a zero requency),unit root at a semiannualfrequency,or seasonal nit root.To understandheprocedure, uppose , is gener-atedbyA(L)y,--e,

where A(L) is a fourth-order olynomialsuch hat(l - arL)(l + arL)(l - a3iL)(l + aoiL)y,=e, (4.2s)

Now, f a, = a2= a3- a4= l' (4.25) s equivalento settingy= | in (4.24).Hence,f a, - az-- at = a*: l, there s a seasonalnit root.Consider omeof theotherpossible ases:GASE1lf a, = l, onehomogeneousolution o (4.25) s y, = ),-r. As such, he {yr} se-quence ends o repeat tself each andeveryperiod.This is the caseof a nonsea-sonalunitroot.CASE2 !I f a . r - l ,onehomogeneousso lu ti on to (4 .25) isy ,+ l t t=0 .In th i s ins tance, thesequenceends o replicatetself at 6-month ntervals,so that there s a semiannualunit root.Forexample,f y,= l, it follows haty,*, - -1 , !*2= *1, )r+ -1, !t*=I, etc.

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Extensions f the Dickey-FullerTest 231CASE3If eithera3or a4 s equal o unity, the {y,} sequenceasanannualcycle.For exam-ple, f at= l, a homogeneousolutiono (4.25) s y, = r)r_r.Thus,f y, - l, lr*r= i,!*2= i2= -1, )r+3 -i, and *= -i2 = l, so hat he sequenceeplicatestselfeveryfourthperiod.

To develop he test,view (4.25)as a functionof c, a2,a3, andao and takeaTaylor seriesapproximation f A(L) around he point et = e2 - a3 = e= l.Although he detailsof the expansion remessy, irst take hepartialderivative:A(L)I a,= (l - atL)(l + arL)(l - aiL)(l + aoiL)l a,= -(l + arL)(l - aJL)(l + aoiL)L

Evaluatinghis derivative t thepointet = a2- a3= a= | yields-L( l +Xl - Xl + iL) - -211+r ) ( l + I ] )=-L( l +L+ I +L3)

Next, formA(L)lar=(l - arL)(l + arL)(l - aiL)(l + aoLla,= (l - arL')(l- asiL)(l + aoiL)L

Evaluating t thepointat = a2= ar= d+= | yields l - L + I] - L3)L. tshouldnot taketoo long to convinceyourself hat evaluatingA(t)la3 and A(L)laoatthepointat = a2- a3= aq= | yields

A(L)lar= (1 - t])(t + iL)iL

A(L)lao=l - th(J- DrSinceA(L) evaluatedt et = e2= e3= dq= | is (l - L\,it is possibleo approxi-mate 4.25)by

0 - Lo) L(t + L + t + L3)(a,- l) + (t - L + I: - LtL(ar- t)- (l - L2)(l+ iL)L(ar- l) + (l - 2xl - iL)I-(a- ), = e,Define ,such haty,=(a- l) andnote hat l + iL)= i- L and l -iL)i=i+L; hence,

(l - Lo\y,=Tr(l + L + I3 + L3)y,_t Tz(l- L + I] - Lt)y,_,+ (l - t')[Tr( - L) -Tq(i + L)fy,-, + e,

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232 Testing or Trendsand Unit Rootsso that(l - Lo)y,= r(1+ L + L?+ L3)y,-,-T2fl - L+ I] - Lt)y,-,+ (1 - t')l(v,r- T)i (T,+ yo)Lfy,-,+ e, (4.26)

To purgethe imaginarynumbers rom (4.26),defineYsmd 1usuch thatZyt =-To- iy, and4q= -Te+ i1r. Hence, T, To)i y, andTz+ Tq=1u.Substitutingnto(4.26)yields0 - Ii)y,=Tr(l + L + t] + t])y,-, -Tz(1- L + I] - I])y,-'+ (1 - th$t-yuL)y,-, + e,

Thus, o implement heprocedure, se hefollowing steps:STEP : Form he ollowingvariables:

) r - r= ( l + L + L 2+ /3)y , - t=! . - t r ! rz* ! rz*Jr!a-t = Q - L + I] - I])y,-, = ),-r - J._z !-t - !rf3-t= 0 - I])y,-r = )-r - lrt so that 13,-z=Jez- ltaSTEP2: Estimate the regression:

(l - Lo)y,=Tr}r,-r - Tzlz,-t * Ts)r-r -Tolz,-z* e,You might want to modify the form of the equation by including an in-tercept, deterministic seasonaldummies, and a linear time trend. As in the

augmented form of the Dickey-Fuller test, lagged values of (1 - Lo)y,-,may also be included. Perform the appropriatediagnosticchecksto ensurethat the residuals from the regressionequation approximate a white-noiseprocess.STEp3: Form the r-statistic for the null hypothesis Yr = 0; the appropriatecriticalvaluesare reported n Hylleberg et al. (1990). If you do not reject the hy-pothesisTr = 0, conclude that a, = l, so there is a nonseasonal nit root.Next, form the t-test for the hypothesisy, = 0' If you do not reject the nullhypothesis,conclude that a,= l and there s a unit root with a semiannual

frequency. Finally, perform the F-test for the hypothesisTs= Te= 0. If thecalculatedvalue is less than the critical value reported n Hylleberg et al.(1990), conclude hat T5and/oryu is zero, so that there s a unit root withan annual frequency. Be aware that the three null hypothesesare not alter-natives;a seriesmay havenonseasonal,emiannual, nd annualunit roots.At the 5Vosignifrcanceevel, Hylleberg et al. (1990) report that the critical val-ues using 100observations re:

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Examplesof theAugmentedDickey-Fuller Test 233= 0

InterceptInterceptplusseasonalummiesInterceptplusseasonalummiesplus time trend

-2.88-2.95-3.533.086.576.60

4. EXAMPLES FTHEAUGMENTED ICKEY.FULLERTESTThe astchaptereviewedheevidenceeported y NelsonandPlosser1982)sug-gestingthat macroeconomic ariablesare differencestationary atherthan trendstationary.We arenow in aposition o considerheir formal testsof the hypothesis.For eachseriesunderstudy,NelsonandPlosser stimatedhe regression:

pLy,= ao+azt* )r-r + Ip,Ay,_r+i+ ,i=2The chosenag engthsare eported n the column abeledp in Table4.2.Thees-timatedvaluesas,a2,and are eported n columns3, 4, and5, respectively.

Table4.2 Nelsonand Plosser'sTests or Unit Rootsa 2 T T + 1

Tz=0

RealGNPNominalGNPIndustrialproductionUnemploymentrate

2 0.819(3.03)2 1.06(2.37)6 0.103(4.32)4 0.513(2.81)

0.006(3.03)0.006(2.34)0.007(2.44)-0.000(4.23)

-{.175(-2.ee)-0.101(-2.32)--0.165(-2.s3)4.294*(-3.55)

0.8250.8990.8350.706

Notes: l. p is the chosen ag length. Coefficientsdivided by their standarderrors are in parentheses.Thus, entries n parenthesesepresent he t-test for the null hypothesis hat a coefficient isequal to zero. Under the null of nonstationary,t is necessaryo use he Dickey-Fuller criti-cal values.At the 0.05 significance evel, the critical value for the r-statistic s -3.45.2. An asterisk *) denotessignificanceat the 0.05 level. For real and nominal GNP and indus-trial production, t is not possible o reject thenull T= 0 at the 0.05 level. Hence, he unem-ployment rateappearso be stationary.3. Theexpressionl+ I istheestimateof thepartialautocorrelationbetweeny,and),-r.

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Recall that the traditionalview of business yclesmaintains hat the GNP andproduction evelsaretrendstationary atherthandifferencestationary'An adherentof this view mustasserttraty is differentfrom zero; f 1= 0, the serieshasa unitroot and is differencestationary.Given the samplesizesusedby Nelson andPlosser 1982),at the 0.05 evel, the critical valueof the f-statistic or the null hy-pothesisT = 0 is -3.45. Thus,only if the estimated alueof 1 is more than 3.45standard eviationsrom zero, s it possible o reject he hypothesishat Y= 0' Ascan be seen rom inspectionof Table4.2' theestimated aluesof l for real GNP'nominal GNP, and industrialproductionare not statisticallydifferent from zero'only the unemploymentatehasanestimated alueof l that s significantlydiffer-ent rom zeoatthe0.05level.Unit RootsandPurchasing-PowerarityPurchasing-powerarity (PPP)s a simple elationshipinking nationalprice evelsandexchangeates.n iis simplestorm,PPPassertshat herateof currencydepre-ciation is approximately qual o the differencebetween he domestic

and foreigninflation rates.Ifp and * "noa" he ogarithmsof theu.S. and oreignprice evelsandethe ogarithmof ihe dollarpriceof foreignexchange, PP mplies

t=Pt- Pf + d,where d, representshedeviation rom PPP n period '

Testing or Trendsand Unit Roots

Figure 4.2 Realexchangeates'1 . 6

fi a | i l i l i l i l ' i l I ' i l [ i l I r ' r " r ' t " " " , " " " '" ' " 1973 1975 1977 1979 1981- Canad Germany - Japan(Jan.1973= 1.00)

1 . 4

1 . 2

0.8

1983 1985 1987 1989

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Examplesof theAugmentedDickey_Fuller Test Z3SIn appliedwork,P, ffidpf usuallyreferto nationalprice ndices n r relative o abaseyear,so that, refers o an ndexof thedomestic urrencypriceof foreignex-changeelative o a base ear.For example,f theU.S. nflation ate s l\Vo whitethe foreign nflation rate s lVo, thedollarpriceof foreignexchange hould all byapproximately Vo.The presence f the termd, allows or short-rundeviationsrom

PPP.Because f its simplicityand ntuitiveappeal,PPPhasbeenusedextensivelyntheoreticalmodelsof exchangeatedetermination.However,as n the well-knownDornbusch1976)"overshooting"model, ealeconomicshocks, uchasproductiv-ity or demand hocks, an cause ermanent eviationsrom PPP.For our purposes,the theoryof PPPservesasan excellentvehicle o illustratemany ime-seriesest-ing procedures.One testof long-runPPP s to determinewhetherd, is stationary.After all, if the deviations rom PPP arenonstationary i.e., if the deviationsarepermanentn nature),we can reject he theory.Notethat PPPdoesallow for persis-tent deviations;he autocorrelationsf the {d,} sequence eednot be zero.Onepopular estingprocedures to define he"real" exchangeate n period asr,= 9;+ PT P,

Long-runPPP s said o hold if the {r,} sequences stationary.For example,nEnders 1988), constructedeal exchangeates or threemajorU.S. tradingpart-ners:Germany,Canada, ndJapan.The datawere divided nto two periods:January1960 o April 1971(representinghe fixed exchange ate period)andJanuary1973 o November1986(representinghe flexible exchangeate period).Eachnation'sWholesalePrice ndex (!YPwas multipliedby an ndexof the U.S.dollar priceof the foreigncurrencyand thendividedby the U.S. WPI. The log ofthe constructed eriess the {rr} sequence. pdatedvaluesof therealexchangeatedataused n the study are n the file REAL.PRNcontainedon thedatadisk. As anexercise, ou shoulduse his dat o verify the results eportedbelow.A critical first step n any econometricanalysis s to visually inspect he data.The plotsof thethree ealexchangeateseriesduringthe lexibleexchangeatepe-riod areshown n Figure4.2.Eachseries eemso meandern a fashioncharacteris-tic of a randomwalk process.Noticethat there s little visualevidence f explosivebehavioror a deterministicime trend.ConsiderFigure4.3 that shows he autocor-relationfunctionof the Canadian eal rate n levels,part (a), and irst differences,part (b). This autocorrelation attern s typicalof all the seriesn theanalysis.Theautocorrelationunction shows ittle tendency o decay, wereas he autocorrela-tions of the first differencesdisplay the classicpatternof a stationaryseries. ngraph b),all autocorrelationswith thepossible xception f p,, that equals .1S)arenot statisticallydifferent rom zeroat theusualsignificanceevels.To formally test for the presence f a unit root in the real exchangeates,aug-mentedDickey-Fuller testsof the form given by (a.19) were conducted.The.re-gressionLr,= as * Tr,_t+ prrr_,+ r\rr_, + ... wasestimated ased n the ollow-ing considerations:

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236 TestingforTrendsndUnit RootsFigure 4.3 ACF of Canada'seal exchangeate.Levels.

0.40.2

(0.2)(0.4)

6 7lal

9 1 0 1 1 1 2( )

l. The theoryof PPPdoesnot allow for a deterministicime trendor multipleunitroots.Any such indingswould refute he theoryasposited.Althoughall the se-ries decline hroughoutheearly 1980sand all riseduringthemid to late 1980s,there s no apriori reason o expecta structural hange.Pretestinghe datausingthe Dickey-Pantula1987)strategyshowedno evidenceof multiple unit roots.Moreover, herewas no reason o entertain he notionof trend stationarity; heexpression 2twas col ncluded n theestimating quation.2. lnboth time periods,F-testsand he SBC ndicated hat p, throughprocouldbesetequal o zero.For GermanyandJapanduring heflexible rateperiod,pr wasstatisticallydifferent rom zero; n the other our instances, , could be setequalto zero. In spite of these indings,with monthly data t is always mportant oentertainhepossibilityof a lag lengthno shorter han 12 months.As such, estswere conducted sing he short ags selected y theF-testsand SBCand usingalag engthof l2 months.For the Canadian aseduringthe 1973 o 1986period, he r-statisticor thenullhypothesishaty = 0 is -1.42usingno lagsand 1.51 usingall 12lags.Given hecritical value of the Tp statistic, t is not possible o reject he null of a unit root inthe Canadian/U.S.eal exchangeateseries.Hence,PPP ails for thesewo nations.

In the 1960 o l97l period, he calculated alueof the -statistics -1.59; again, tis possibleo concludehatPPP ails.Table 4.3reports heresultsof all six estimations sing he short ag engthssug-gestedby the F-testsandSBC.Notice the following properties f the estimatedmodels:1. For all six models,t is not possibleo reject he null hypothesishat PPP ails.As can be seen rom the astcolumnof Table4.3, theabsolute alueof the -sta-

Firstdifferences.

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2.

Emmples of the Augmented Dickey-Fuller Testtistic for the null Y= 0 is nevermore han 1.59.The economic,interpretationsthatrealproductivityand/ordemandshockshavehada permanentnfluenceonrealexchangeates.As measured y the samplestandard eviation SD), real exchangeateswerefa r more volatile n the 1973 o 1986period han he 1960 o l97 l period.Moreover,as measured y the standard rrorof the estimate SEE), eal ex-change ate volatility is associated ith unpredictability. he SEEduring theflexible exchangeateperiod s severalhundred imes that of the fixed ratepe-riod. It seems easonableo conclude hat the change n the exchange ateregime(i.e., the end of Bretton-Woods)affected he volatility of the real ex-change ate.Care must be taken o keep he appropriate ull hypothesisn mind. Under thenull of a unit root,classical estproceduresre nappropriate ndwe resort o thestatistics abulatedby Dickey and Fuller.However,classical estprocedures(which assume tationary ariables)are appropriate nder he null that the realratesare stationary. hus, he following possibilityarises.Supposehat he f-sta-tistic in the Canadian asehappenedo be -2.16 insteadof -1.42. Using theDickey-Fullercritical values, ou wouldnot reject he null of a unit root;hence,you could conclude hat PPP ails. However, under the null of stationarity(wherewe canuseclassicalprocedures), is more han wo standard eviationsfrom zeroandyou would concludePPPholds since heusual -testbecomes p-plicable.This apparentdilemma commonlyoccurswhen analyzingserieswith rootsclose o unity in absolute alue.Unit root testsdo not have muchpower n dis-criminatingbetween haracteristicootsclose o unity andactualunit roots.Thedilemma s only apparent ince he two null hypotheses requitedifferent. t isperfectlyconsistent o maintaina null that PPPholds and not be able o reject anull that PPP ails! Notice that this dilemmadoesnot actuallyarise or any ofthe series eported n Table 4.3; for each, t is not possible o reject a null ofT= 0 at conventionalignificanceevels.Looking at someof the diagnostic tatistics,we see hat all the F-statisticsndi-cate hat t is appropriateo excludeags 2 (or 3) through 12 rom the regressionequation.To reinforce heuseof short ags,notice hat he first-ordercorrelationcoefficientof theresidualsp) is low and he Durbin-Watsonstatisticclose o 2.It is interestinghat all thepointestimates f the characteristicoots ndicate hatreal exchange ates are convergent.To obtain the characteristicoots,rewritethe estimated quations n the autoregressiveorm rt = ao+ a{Ft ot rt = ao+atrt-t + ezrFz.For the four AR(l) models, hepointestimates f theslopecoef-ficientsare all less hanunity. n thepost-Bretton-W'oodseriod 1973-1986),the point estimatesof the characteristic oots of Japan'ssecond-order rocessare0.931and0.319; or Germany,he rootsare0.964and0.256.However,hisis preciselywhatwe would expect f PPP ails; under he null of a unit root, weknow thaty is biaseddownward.

3.

4.

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Tabte4.3 Real ExchangeRateEstimation

t973-1986CanadaJapanGermany196L]_t971CanadaJapanGermany

4.022(0.015s)4.047(0.074)4.027(0.076)-0.031(0.019)-0.030(0.028)-0.016(0.012)

1

0.978r.251.22

42

4.297-0.247

Mean

0.9690.9700.984

1.051.01l . l

plDw0.0591.88

-0.0072.Ot-0.0142.004-0.1072.2r0.0461.980.038t.93

r.020.9801.0r

0.r940.2260.858

5.471 . 1 6r0.42.8120.683.710.0140.0040.0r70.0050.0260.004

Hozy=g

0.4340.3300.097

t = -1 .42

t = 4.635t = 4.280

t = -I.59t = -1.04t = -I.23

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Statistical Tables 419STATISTICALABLESTahleA Empirical CumulativeDistribution of tProbability of a SmallerValueSampleSize 0.0f 0.025 0.05 0.10 0.90 0.9s 0.975 0.99No Constantor Time (ao= az- O) -1.9s-1.95-1.95-1.95-1.95-1.95

-1.60-1.61-1. l-r.62-r.62-r.62Lp1.62-2.60-2.58-2.57-2.57-2.57

ft -3.24-3.18-3.15-3.13-3.13-3.12

o.920.910.900.890.890.894.37{.404.424.42-0.434.44- 1 . 1 4-1 . 19-t.22-r.23-1,.24-r.25

1.33r .311.29r.29r.281.280.00-0.03

-0.05-o.0-0.07-0.07

2550100

2550r002so300Constant az--O)-3.75-3.58

-3.51250 -3.46500 -3.M -3.43Constant time

-2.66 -2.26-2.62 -2.25-2.60 -2.24-2.58 -2.23-2.58 -2.23-2.58 -2.23

-4.38-4.154.M-3.99-3.98-3.96

-3.33 -3.00-3.22 -2.93-3.r7 -2.89-3.r4 -2.88-3.13 -2.87-3.r2 -2.86-3.95 -3.60-3.80 -3.50-3.73 -3.45-3.69 -3.43-3.68 -3.42-3.66 -3.4r

1.70 2.161.66 2.08r.& 2.031.63 2.0r1.62 2.N1.62 2.000.34 0.720.29 0.660.26 0.630.24 0.624.24 0.610.23 0.60

2550r00250500

-{.80 -0.50 -0.15-o.87 --0.58 4.24{.90 4.62 {.284.92 -0.4 -0.31-{.93 -0.65 4.324.94 -{.66 -0.33Source:This tablewas constructed y David A. Dickey using MonteCarlo methods.Standard rrorsofthe estimatesary,but mostare ess han0.20.The able s reproducedromWayneFuller. ntroductionto StatisticalTime eries.NewYork JohnWiley). 1976.