autocorrelation from basicc econometrics

what is autocorrelation ? what happens if the error terms are correlated?

1. AUTOCORRELATION :WHATHAPPENSIF
THEERRORTERMSARECORRELATED?

2. •Inthischapterwetakeacriticallookatthefollowing
questions:
•1.Whatisthenatureofautocorrelation?
•2.Whatarethetheoreticalandpractical
consequencesofautocorrelation?
•3.Sincetheassumptionofnoautocorrelationrelates
totheunobservabledisturbancesu
t,howdoesone
knowthatthereisautocorrelationinanygiven
situation?Noticethatwenowusethesubscripttto
emphasizethatwearedealingwithtimeseriesdata.
•4.Howdoesoneremedytheproblemof
autocorrelation?

3. •THENATUREOFTHEPROBLEM
•Autocorrelationmaybedefinedas“correlationbetweenmembersofseriesof
observationsorderedintime[asintimeseriesdata]orspace[asincross-
sectionaldata].’’theCLRMassumesthat:
•E(u
iu
j)=0 i≠j (3.2.5)
•Putsimply,theclassicalmodelassumesthatthedisturbancetermrelating
toanyobservationisnotinfluencedbythedisturbancetermrelatingtoany
otherobservation.
•Forexample,ifwearedealingwithquarterlytimeseriesdatainvolvingthe
regressionofoutputonlaborandcapitalinputsandif,say,thereisalabor
strikeaffectingoutputinonequarter,thereisnoreasontobelievethatthis
disruptionwillbecarriedovertothenextquarter.Thatis,ifoutputislower
thisquarter,thereisnoreasontoexpectittobelowernextquarter.
Similarly,ifwearedealingwithcross-sectionaldatainvolvingthe
regressionoffamilyconsumptionexpenditureonfamilyincome,theeffectof
anincreaseofonefamily’sincomeonitsconsumptionexpenditureisnot
expectedtoaffecttheconsumptionexpenditureofanotherfamily.

4. •However,ifthereisautocorrelation,
•E(u
iu
j)≠0 i≠j (12.1.1)
•Inthissituation,thedisruptioncausedbyastrikethisquarter
mayverywellaffectoutputnextquarter,ortheincreasesin
theconsumptionexpenditureofonefamilymayverywell
promptanotherfamilytoincreaseitsconsumption
expenditure.
•InFigure12.1.Figure12.1atodshowsthatthereisa
discerniblepatternamongtheu’s.Figure12.1ashowsacyclical
pattern;Figure12.1bandcsuggestsanupwardordownward
lineartrendinthedisturbances;whereasFigure12.1dindicates
thatbothlinearandquadratictrendtermsarepresentinthe
disturbances.OnlyFigure12.1eindicatesnosystematicpattern,
supportingthenon-autocorrelationassumptionoftheclassical
linearregressionmodel.

6. •Whydoesserialcorrelationoccur?Thereareseveralreasons:
1.Inertia.Asiswellknown,timeseriessuchasGNP,priceindexes,production,
employment,andunemploymentexhibit(business)cycles.Startingatthe
bottomoftherecession,wheneconomicrecoverystarts,mostoftheseseries
startmovingupward.Inthisupswing,thevalueofaseriesatonepointin
timeisgreaterthanitspreviousvalue.Thusthereisamomentum’’built
intothem,anditcontinuesuntilsomethinghappens(e.g.,increasein
interestrateortaxesorboth)toslowthemdown.
2.SpecificationBias:ExcludedVariablesCase.Inempiricalanalysisthe
researcheroftenstartswithaplausibleregressionmodelthatmaynotbe
themost“perfect’’one.Forexample,theresearchermayplottheresiduals
ˆu
iobtainedfromthefittedregressionandmayobservepatternssuchasthose
showninFigure12.1atod.Theseresiduals(whichareproxiesforu
i)may
suggestthatsomevariablesthatwereoriginallycandidatesbutwerenot
includedinthemodelforavarietyofreasonsshouldbeincluded.Oftenthe
inclusionofsuchvariablesremovesthecorrelationpatternobservedamong
theresiduals.

7. •For example, suppose we have the following demand model:
•Y
t= β
1+ β
2X
2t+ β
3X
3t+ β
4X
4t+ u
t (12.1.2)
•where Y = quantity of beef demanded, X
2= price of beef, X
3= consumer
•income, X
4= price of poultry, and t = time. However, for some reason we run
•the following regression:
•Y
t= β
1+ β
2X
2t + β
3X
3t+ v
t (12.1.3)
•Now if (12.1.2) is the “correct’’ model or the “truth’’ or true relation,
running (12.1.3) is tantamount to letting v
t= β
4X
4t+ u
t. And to the extent the
•price of poultry affects the consumption of beef, the error or disturbance
term v will reflect a systematic pattern, thus creating (false) autocorrelation.
•A simple test of this would be to run both (12.1.2) and (12.1.3) and see
whether autocorrelation, if any, observed in model (12.1.3) disappears when
(12.1.2) is run.

8. 3.SpecificationBias:IncorrectFunctionalForm.Supposethe“true’’orcorrect
modelinacost-outputstudyisasfollows:
•Marginalcost
i=β
1+β
2output
i+β
3output
2
i+u
i (12.1.4)
•butwefitthefollowingmodel:
•Marginalcost
i=α
1+α
2output
i+v
i (12.1.5)
•Themarginalcostcurvecorrespondingtothe“true’’modelisshownin
Figure12.2alongwiththe“incorrect’’linearcostcurve.
•AsFigure12.2shows,betweenpointsAandBthelinearmarginalcostcurve
willconsistentlyoverestimatethetruemarginalcost,whereasbeyondthese
pointsitwillconsistentlyunderestimatethetruemarginalcost.Thisresult
istobeexpected,becausethedisturbancetermv
iis,infact,equaltooutput
2
+u
i,andhencewillcatchthesystematiceffectoftheoutput
2
termon
marginalcost.Inthiscase,v
iwillreflectautocorrelationbecauseoftheuseof
anincorrectfunctionalform.

10. 4.CobwebPhenomenon.Thesupplyofmanyagricultural
commoditiesreflectstheso-calledcobwebphenomenon,where
supplyreactstopricewithalagofonetimeperiodbecause
supplydecisionstaketimetoimplement(thegestationperiod).
Thus,atthebeginningofthisyear’splantingofcrops,farmers
areinfluencedbythepriceprevailinglastyear,sothattheir
supplyfunctionis
•Supply
t=β
1+β
2P
t−1+u
t (12.1.6)
•Supposeattheendofperiodt,priceP
tturnsouttobelower
thanP
t−1.Therefore,inperiod
t+1farmersmayverywelldecide
toproducelessthantheydidinperiodt.Obviously,inthis
situationthedisturbancesu
tarenotexpectedtoberandom
becauseifthefarmersoverproduceinyeart,theyarelikelyto
reducetheirproductionint+1,andsoon,leadingtoaCobweb
pattern.

11. 5. Lags. In a time series regression of consumption expenditure on
income, it is not uncommon to find that the consumption
expenditure in the current period depends, among other things,
on the consumption expenditure of the previous period. That
is,Consumption
t= β
1+ β
2income
t+ β
3consumption
t−1+ u
t
(12.1.7)
•A regression such as (12.1.7) is known as autoregression
because one of the explanatory variables is the lagged value of
the dependent variable. The rationale for a model such as
(12.1.7) is simple. Consumers do not change their consumption
habits readily for psychological, technological, or institutional
reasons. Now if we neglect the lagged term in (12.1.7), the
resulting error term will reflect a systematic pattern due to the
influence of lagged consumption on current consumption.

12. 6.“Manipulation’’ofData.Inempiricalanalysis,therawdataareoften
“manipulated.’’Forexample,intimeseriesregressionsinvolvingquarterly
data,suchdataareusuallyderivedfromthemonthlydatabysimplyadding
threemonthlyobservationsanddividingthesumby3.Thisaveraging
introducessmoothnessintothedatabydampeningthefluctuationsinthe
monthlydata.Therefore,thegraphplottingthequarterlydatalooksmuch
smootherthanthemonthlydata,andthissmoothnessmayitselflendtoa
systematicpatterninthedisturbances,therebyintroducingautocorrelation.
•Anothersourceofmanipulationisinterpolationorextrapolationofdata.For
example,theCensusofPopulationisconductedevery10yearsinthis
country,thelastbeingin2000andtheonebeforethatin1990.Nowifthere
isaneedtoobtaindataforsomeyearwithintheintercensusperiod1990–
2000,thecommonpracticeistointerpolateonthebasisofsomeadhoc
assumptions.Allsuchdata“massaging’’techniquesmightimposeuponthe
dataasystematicpatternthatmightnotexistintheoriginaldata.

13. 7.DataTransformation.considerthefollowingmodel:
•Y
t=β
1+β
2X
t+u
t (12.1.8)
•where,say,Y=consumptionexpenditureandX=income.Since(12.1.8)
holdstrueateverytimeperiod,itholdstruealsointheprevioustime
period,(t−1).So,wecanwrite(12.1.8)as:
•Y
t−1=β
1+β
2X
t−1+u
t−1 (12.1.9)
•Y
t−1,X
t−1,andu
t−1areknownasthelaggedvaluesofY,X,andu,respectively,
herelaggedbyoneperiod.Nowifwesubtract(12.1.9)from(12.1.8),we
obtain
•∆Y
t=β
2∆X
t+∆u
t (12.1.10)
•where∆,knownasthefirstdifferenceoperator,tellsustotakesuccessive
differencesofthevariablesinquestion.Thus,Y
t=(Y
t−Y
t−1),X
t=(X
t−X
t−1),
andu
t=(u
t−u
t−1).Forempiricalpurposes,wewrite(12.1.10)as
•Y
t=β
2X
t+v
t (12.1.11)
•wherev
t=u
t=(u
t−u
t−1).

14. •Equation(12.1.9)isknownasthelevelformandEq.(12.1.10)isknownas
the(first)differenceform.Bothformsareoftenusedinempiricalanalysis.
Forexample,ifin(12.1.9)YandXrepresentthelogarithmsofconsumption
expenditureandincome,thenin(12.1.10)YandXwillrepresentchangesin
thelogsofconsumptionexpenditureandincome.Butasweknow,achange
inthelogofavariableisarelativechange,orapercentagechange,ifthe
formerismultipliedby100.So,insteadofstudyingrelationshipsbetween
variablesinthelevelform,wemaybeinterestedintheirrelationshipsinthe
growthform.
•Nowiftheerrortermin(12.1.8)satisfiesthestandardOLSassumptions,
particularlytheassumptionofnoautocorrelation,itcanbeshownthatthe
errortermv
tin(12.1.11)isautocorrelated.Itmaybenotedherethatmodels
like(12.1.11)areknownasdynamicregressionmodels,thatis,models
involvinglaggedregressands.
•Thepointoftheprecedingexampleisthatsometimesautocorrelationmay
beinducedasaresultoftransformingtheoriginalmodel.

15. 8.Nonstationarity.WementionedinChapter1that,whiledealingwithtime
seriesdata,wemayhavetofindoutifagiventimeseriesisstationary.
•atimeseriesisstationaryifitscharacteristics(e.g.,mean,variance,and
covariance)aretimeinvariant;thatis,theydonotchangeovertime.Ifthatis
notthecase,wehaveanonstationarytimeseries.Inaregressionmodel
suchas
•Y
t=β
1+β
2X
t+u
t (12.1.8)
•itisquitepossiblethatbothYandXarenonstationaryandthereforethe
erroruisalsononstationary.Inthatcase,theerrortermwillexhibit
autocorrelation.
•Itshouldbenotedalsothatautocorrelationcanbepositive(Figure12.3a)as
wellasnegative,althoughmosteconomictimeseriesgenerallyexhibit
positiveautocorrelationbecausemostofthemeithermoveupwardor
downwardoverextendedtimeperiodsanddonotexhibitaconstantup-
and-downmovementsuchasthatshowninFigure12.3b.

17. OLS ESTIMATION IN THE PRESENCE OF AUTOCORRELATION
•WhathappenstotheOLSestimatorsandtheirvariancesifweintroduce
autocorrelationinthedisturbancesbyassumingthatE(u
tu
t+s)≠0(s≠0)but
retainalltheotherassumptionsoftheclassicalmodel?Werevertonce
againtothetwo-variableregressionmodeltoexplainthebasicideas
involved,namely,
•Y
t=β
1+β
2X
t+u
t.
•Tomakeanyheadway,wemustassumethemechanismthatgeneratesu
t,for
E(u
tu
t+s)≠0(s≠0)istoogeneralanassumptiontobeofanypracticaluse.
Asastartingpoint,orfirstapproximation,onecanassumethatthe
disturbance,orerror,termsaregeneratedbythefollowingmechanism.
•u
t=ρu
t−1+ε
t -1<ρ<1 (12.2.1)
•whereρ(=rho)isknownasthecoefficientofautocovarianceandwhereε
tis
thestochasticdisturbancetermsuchthatitsatisfiedthestandardOLS
assumptions,namely,
•E(εt)=0
•var(ε
t)=σ
2
ε (12.2.2)
•cov(ε
t,ε
t+s)=0s≠0

18. •Intheengineeringliterature,anerrortermwiththeprecedingpropertiesis
oftencalledawhitenoiseerrorterm.What(12.2.1)postulatesisthatthe
valueofthedisturbanceterminperiodtisequaltorhotimesitsvalueinthe
previousperiodplusapurelyrandomerrorterm.
•Thescheme(12.2.1)isknownasMarkovfirst-orderautoregressive
•scheme,orsimplyafirst-orderautoregressivescheme,usuallydenotedas
AR(1).Itisfirstorderbecauseu
tanditsimmediatepastvalueareinvolved;
thatis,themaximumlagis1.Ifthemodelwereu
t=ρ
1u
t−1+ρ
2u
t−2+ε
t,it
wouldbeanAR(2),orsecond-order,autoregressivescheme,andsoon.
•Inpassing,notethatrho,thecoefficientofautocovariancein(12.2.1),can
alsobeinterpretedasthefirst-ordercoefficientofautocorrelation,ormore
accurately,thecoefficientofautocorrelationatlag1.

19. •Given the AR(1) scheme, it can be shown that (see Appendix 12A, Section
•12A.2)
•Since ρ is a constant between −1 and +1, (12.2.3) shows that under theAR(1)
scheme, the variance of u
tis still homoscedastic, but u
tis correlatednot only
with its immediate past value but its values several periods in thepast. It is
critical to note that |ρ| < 1, that is, the absolute value of rho is lessthan one. If,
for example, rho is one, the variances and covarianceslistedabove are not
defined.

20. •If |ρ| < 1, we say that the AR(1) process given in(12.2.1) is stationary; that is,
the mean, variance, and covariance of u
tdo not change over time. If |ρ| is less
than one, then it is clear from (12.2.4) that the value of the covariance will
decline as we go into the distant past.
•One reason we use the AR(1) process is not only because of its simplicity
compared to higher-order AR schemes, but also because in many
applications it has proved to be quite useful. Additionally, a considerable
amount of theoretical and empirical work has been done on the AR(1)
scheme.
•Now return to our two-variable regression model: Y
t= β
1+ β
2X
t+ u
t. We
know from Chapter 3 that the OLS estimator of the slope coefficient is
•and its variance is given by
•where the small letters as usual denote deviation from the mean values.

21. •Now under the AR(1) scheme, it can be shown that the variance of this
estimator is:
•A comparison of (12.2.8) with (12.2.7) shows the former is equal to the latter
times a term that depends on ρ as well as the sample autocorrelations
between the values taken by the regressorX at various lags. And in general
we cannot foretell whether var(βˆ2) is less than or greater than var(βˆ2)AR1
[but see Eq. (12.4.1) below]. Of course, if rho is zero, the two formulas will
coincide, as they should (why?). Also, if the correlations among the
successive values of the regressorare very small, the usual OLS variance of
the slope estimator will not be seriously biased. But, as a general principle,
the two variances will not be the same.

22. •To give some idea about the difference between the variances given in
(12.2.7) and (12.2.8), assume that the regressorX also follows the first-order
autoregressive scheme with a coefficient of autocorrelation of r. Then it can
be shown that (12.2.8) reduces to:
•var(βˆ2)AR(1) = σ2x2 t 1 + rρ1 − rρ = var(βˆ2)OLS 1 + rρ1 − rρ (12.2.9)
•If, for example, r = 0.6 and ρ = 0.8, using (12.2.9) we can check that var
(βˆ2)AR1 = 2.8461 var(βˆ2)OLS. To put it another way, var(βˆ2)OLS = 1
2.8461var (βˆ2)AR1 = 0.3513 var(βˆ2)AR1 . That is, the usual OLS formula
[i.e., (12.2.7)] will underestimate the variance of (βˆ2)AR1 by about 65
percent. As you will realize, this answer is specific for the given values of r
and ρ. But the point of this exercise is to warn you that a blind application of
the usual OLS formulas to compute the variances and standard errors of
the OLS estimators could give seriously misleading results.

23. •Whatnowarethepropertiesofβˆ
2?βˆ
2is
stilllinearandunbiased.Isβˆ
2stillBLUE?
Unfortunately,itisnot;intheclassof
linearunbiasedestimators,itdoesnot
haveminimumvariance.

24. RELATIONSHIP BETWEEN WAGES AND PRODUCTIVITY IN THE BUSINESS
SECTOR OF THE UNITED STATES, 1959–1998
•Nowthatwehavediscussedtheconsequencesof
autocorrelation,theobviousquestionis,Howdowedetectit
andhowdowecorrectforit?
•Beforeweturntothesetopics,itisusefultoconsideraconcrete
example.Table12.4givesdataonindexesofrealcompensation
perhour(Y)andoutputperhour(X)inthebusinesssectorof
theU.S.economyfortheperiod1959–1998,thebaseofthe
indexesbeing1992=100.
•FirstplottingthedataonYandX,weobtainFigure12.7.Since
therelationshipbetweenrealcompensationandlabor
productivityisexpectedtobepositive,itisnotsurprisingthat
thetwovariablesarepositivelyrelated.Whatissurprisingis
thattherelationshipbetweenthetwoisalmostlinear,although
thereissomehintthatathighervaluesofproductivitythe
relationshipbetweenthetwomaybeslightlynonlinear.

27. •Therefore,wedecidedtoestimatealinearaswellasalog–linearmodel,
withthefollowingresults:
•Yˆ
t=29.5192+0.7136X
t
•se=(1.9423)(0.0241)
•t=(15.1977)(29.6066) (12.5.1)
•r2=0.9584d=0.1229ˆσ=2.6755
•wheredistheDurbin–Watsonstatistic,whichwillbediscussedshortly.
•lnY
t=1.5239+0.6716lnX
t
•se=(0.0762)(0.0175)
•t=(19.9945)(38.2892) (12.5.2)
•r2=0.9747 d=0.1542 ˆσ=0.0260

28. •Qualitatively,boththemodelsgivesimilarresults.Inbothcasesthe
estimatedcoefficientsare“highly”significant,asindicatedbythehight
values.
•Inthelinearmodel,iftheindexofproductivitygoesupbyaunit,on
average,theindexofcompensationgoesupbyabout0.71units.Inthelog–
linearmodel,theslopecoefficientbeingelasticity,wefindthatiftheindex
ofproductivitygoesupby1percent,onaverage,theindexofreal
compensationgoesupbyabout0.67percent.
•Howreliablearetheresultsgivenin(12.5.1)and(12.5.2)ifthereis
autocorrelation?Asstatedpreviously,ifthereisautocorrelation,the
estimatedstandarderrorsarebiased,asaresultofwhichtheestimatedtratios
areunreliable.Weobviouslyneedtofindoutifourdatasufferfrom
autocorrelation.Inthefollowingsectionwediscussseveralmethodsof
detectingautocorrelation.

29. DETECTING AUTOCORRELATION
•I.GraphicalMethod
•Recallthattheassumptionofnonautocorrelationoftheclassicalmodel
relatestothepopulationdisturbancesu
t,whicharenotdirectlyobservable.
Whatwehaveinsteadaretheirproxies,theresidualsˆu
t,whichcanbe
obtainedbytheusualOLSprocedure.Althoughtheˆutarenotthesame
thingasu
t,17veryoftenavisualexaminationoftheˆu’sgivesussomeclues
aboutthelikelypresenceofautocorrelationintheu’s.Actually,avisual
examinationofˆutor(ˆu2t)canprovideusefulinformationabout
autocorrelation,modelinadequacy,orspecificationbias.
•Therearevariouswaysofexaminingtheresiduals.Wecansimplyplot
themagainsttime,thetimesequenceplot,aswehavedoneinFigure12.8,
whichshowstheresidualsobtainedfromthewages–productivityregression
(12.5.1).ThevaluesoftheseresidualsaregiveninTable12.5alongwith
someotherdata.

32. •Toseethisdifferently,wecanplotˆu
tagainstˆ
ut−1,that
is,plottheresidualsattimetagainsttheirvalueattime
(t−1),akindofempiricaltestoftheAR(1)scheme.If
theresidualsarenonrandom,weshouldobtain
picturessimilartothoseshowninFigure12.3.This
plotforourwages–productivityregressionisasshown
inFigure12.9;theunderlyingdataaregivenin

34. •II.TheRunsTest
•IfwecarefullyexamineFigure12.8,wenoticeapeculiar
feature:Initially,wehaveseveralresidualsthatarenegative,
thenthereisaseriesofpositiveresiduals,andthenthereare
severalresidualsthatarenegative.Iftheseresidualswere
purelyrandom,couldweobservesuchapattern?Intuitively,it
seemsunlikely.Thisintuitioncanbecheckedbytheso-called
runstest,sometimesalsoknowastheGearytest,a
nonparametrictest.
•Toexplaintherunstest,letussimplynotedownthesigns(+or
−)oftheresidualsobtainedfromthewages–productivity
regression,whicharegiveninthefirstcolumnofTable12.5.
•(−−−−−−−−−)(+++++++++++++++++++++)(−−−−−−−−−−)
(12.6.1)

35. •Thus there are 9 negative residuals, followed by 21 positive residuals,
followed by 10 negative residuals, for a total of 40 observations.
•Wenowdefinearunasanuninterruptedsequenceofonesymbolor
attribute,suchas+or−.Wefurtherdefinethelengthofarunasthe
numberofelementsinit.Inthesequenceshownin(12.6.1),thereare3
runs:arunof9minuses(i.e.,oflength9),arunof21pluses(i.e.,oflength
21)andarunof10minuses(i.e.,oflength10).Forabettervisualeffect,we
havepresentedthevariousrunsinparentheses.
•Byexamininghowrunsbehaveinastrictlyrandomsequenceof
observations,onecanderiveatestofrandomnessofruns.Weaskthis
question:Arethe3runsobservedinourillustrativeexampleconsistingof
40observationstoomanyortoofewcomparedwiththenumberofruns
expectedinastrictlyrandomsequenceof40observations?Iftherearetoo
manyruns,itwouldmeanthatinourexampletheresidualschangesign
frequently,thusindicatingnegativeserialcorrelation(cf.Figure12.3b).
Similarly,iftherearetoofewruns,theymaysuggestpositive
autocorrelation,asinFigure12.3a.Apriori,then,Figure12.8wouldindicate
positivecorrelationintheresiduals.

36. •Now let
•N = total number of observations = N1 + N2
•N1 = number of + symbols (i.e., + residuals)
•N2 = number of − symbols (i.e., − residuals)
•R = number of runs
•Note: N = N1 + N2.
•If the null hypothesis of randomness is sustainable, following
the properties of the normal distribution, we should expect that
Prob [E(R) − 1.96σR ≤ R ≤ E(R) + 1.96σR] = 0.95 (12.6.3)

37. •Using the formulas given in (12.6.2), we obtain
•The 95% confidence interval for R in our example is thus:
[10.975 ±1.96(3.1134)] = (4.8728, 17.0722)

38. •Durbin–Watson d Test
•The most celebrated test for detecting serial correlation is that developed
•by statisticians Durbin and Watson. It is popularly known as the Durbin–
•Watson d statistic, which is defined as
•it is important to note the assumptions underlying the d statistic.
•1.Theregressionmodelincludestheinterceptterm.Ifitisnotpresent,asin
thecaseoftheregressionthroughtheorigin,itisessentialtorerunthe
regressionincludingtheintercepttermtoobtaintheRSS.
•2. The explanatory variables, the X’s, are nonstochastic, or fixed in repeated
sampling.

39. •3.Thedisturbancesutaregeneratedbythefirst-orderautoregressive
scheme:u
t=ρu
t−1+ε
t.Therefore,itcannotbeusedtodetecthigher-order
autoregressiveschemes.
•4. The error term u
tis assumed to be normally distributed.
•5.Theregressionmodeldoesnotincludethelaggedvalue(s)ofthe
dependentvariableasoneoftheexplanatoryvariables.Thus,thetestis
inapplicableinmodelsofthefollowingtype:
•Y
t= β
1+ β
2X
2t+ β
3X
3t+ ·· ·+β
kX
kt+ γY
t−1+ u
t (12.6.6)
where Yt−1 is the one period lagged value of Y.
•6.Therearenomissingobservationsinthedata.Thus,inourwages–
productivityregressionfortheperiod1959–1998,ifobservationsfor,say,
1978and1982weremissingforsomereason,thedstatisticmakesno
allowanceforsuchmissingobservations

40. •d≈2(1−ρˆ) (12.6.10)
•Butsince−1≤ρ≤1,(12.6.10)impliesthat
0≤d≤4 (12.6.11)
•Thesearetheboundsofd;anyestimatedd
valuemustliewithintheselimits.

42. •ThemechanicsoftheDurbin–Watsontestareasfollows,assumingthatthe
assumptionsunderlyingthetestarefulfilled:
•1.RuntheOLSregressionandobtaintheresiduals.
•2.Computedfrom(12.6.5).(Mostcomputerprogramsnowdothisroutinely.)
•3.Forthegivensamplesizeandgivennumberofexplanatoryvariables,
findoutthecriticaldLanddUvalues.
•4.NowfollowthedecisionrulesgiveninTable12.6.Foreaseofreference,
thesedecisionrulesarealsodepictedinFigure12.10.
•Toillustratethemechanics,letusreturntoourwages–productivity
regression.FromthedatagiveninTable12.5theestimateddvaluecanbe
showntobe0.1229,suggestingthatthereispositiveserialcorrelationinthe
residuals.FromtheDurbin–Watsontables,wefindthatfor40observations
andoneexplanatoryvariable,dL=1.44anddU=1.54atthe5percentlevel.
Sincethecomputeddof0.1229liesbelowdL,wecannotrejectthehypothesis
thatthereispositiveserialcorrelationsintheresiduals.