Appliedtimeseriesformacroeconomics
HildeC.BjørnlandLeifAndersThorsrud
# GyldendalNorskForlagAS2015 2.utgave,2.opplag2015
ISBN978-82-05-48089-6
Omslagsdesign:GyldendalAkademisk = HildeC.BjørnlandogLeifA.Thorsrud Layoutogsats:GammagrafiskAS(VegardBrekke)
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ToChristiane,EdvardandTrond
AinaandLars
Prefaceandacknowledgements
Macroeconomicsisaboutansweringquestionslike:
Whatwilltheinflationratebethecomingquarters?
Whatexplainseconomiccycleswhereboomsarefollowedbybusts?
Dooutputandconsumerpricesfalliftheinterestrateisincreased?
Haveallrecessionsbeenprecededbyanincreaseinthepriceofcrudeoil?
Thegoalofthisbookistogiveanintuitiveyetformalunderstandingofthebasictechniquesusedinappliedeconometricstoanswerquestionslikethese.Thefocusisontime serieseconometricswithapplicationsinmacroeconomicsandinternationalfinance. Throughoutthebookwewillcoverunivariateandmultivariatemodelsofstationaryand nonstationarytimeseries,includingstructuralvectorautoregression(SVAR)modelsand instrumentalvariable(IV)methods.Muchfocusisalsodevotedtothetopicofmacroeconomicforecasting,usingbothunivariateandmultivariatemodels.
Mostoftheeconometricmodelscoveredinthisbookcanbeestimatedusingordinary leastsquares.However,withvaryingdepth,wealsocovertheMaximumlikelihood estimatoranddifferentinstrumentalvariableestimatorslikeTwo-stageleastsquaredand GeneralizedMethodofMoments.
Thebookisdesignedprimarilyforuseintimeseriescoursesgiventomasterstudents ineconomicsandbusiness.However,thefirstpartsofthebookemphasisingunivariate methodscanbeusedincoursesgiventofinalyearunder-graduates,whiletheparts coveringforecastingandmultivariatemodelscanbeusedasasupplementtoareference bookusedincoursesgiventofirstyearPhDstudents.Wedonotintendthisbooktobea completereference.Formorerigoroustreatmentofmanyofthetopicsdiscussedwerefer thereadertoHamilton(1994)andLutkepohl(2005).
Oneoftheguidingprinciplesofthisbookisoperationality,andwehaveemphasized readyaccesstothecomputerprogramsusedinitspreparation.TheMATLABcodes, datafilesanddescriptionsareavailablefromthewebsitedevelopedforthisbook: http://www.timeseries.no.Ifthelinkdoesnotwork,pleasecontactoneoftheauthors.
TheframeworkofthebookgrewoutoflecturenotesdevelopedbyHildeC.Bjørnland formasterandPhDcoursesinappliedtimeserieseconometricsattheUniversityofOslo andBINorwegianBusinessSchool.WethankinparticularIdaWoldenBachefor contributingactivelytothedevelopmentofthesecoursesatanearlystage.
Ourworkhasbenefitedfromcommentsbymanyfriendsandcolleagues.Ourgrateful thanksgotoKnutAreAastveit,DragoBergholt,EleonoraGranziera,Steffen Grønneberg,AnneSofieJoreandBjørnNaugforhavingcommentedconstructivelyon ourwork.WewouldalsoliketothankQ.FarooqAkramforsharinghisdata.
WeowemuchtothepresentandpastnowcastingteamsofNorgesBank.1 Special thankstoFrancescoRavazzolo,ChristieSmithandShaunVaheyforinfluencingand shapingourperceptionofhowforecastingshouldbedone.
Wehavealsobenefitedfromcommentsbymanystudents,especiallythoseattending theMasterprogrammeattheBINorwegianBusinessSchool.SpecialthankstoStefanie FernandezforcommentsonanearlierdraftandtoStein-ErikHjørringforinvaluable editorialhelpinthefinalstage.
Planofthebook
Throughoutthecourseofthisbookwewillpresentthenecessarytoolstostudyeconomic timeseries,withampleapplicationstomacroeconomicsandfinance.Thebookis organizedasfollows.InChapter1westartbyreviewingsomeofthebasicconceptsand definitionsusedinstatisticalanalysis.Thatis,wereviewtheconceptofrandomnumbers, differencesbetweenapopulationandasample,andfinallytheordinaryleastsquares estimator(OLS).
Chapters2–5dealwithunivariatetimeseriesandmodels.Inparticular,Chapters2–3 provideuswiththenecessarytoolstodescribeandforecaststationarytimeseries.Most datainmacroeconomicsandfinancecanbedescribedastimeseries:asetofrepeated observationsovertimeofthesamevariable,suchasconsumerprices,grossdomestic product(GDP),stockprices,exchangerates,etc.Accordingly,tobeabletounderstand macroeconomicfluctuationsandfinancialmarketsweneedtolearnaboutthebasicdefinitionsandtoolsusedforunivariatetimeseriesanalysis.Animportantareaofapplication fortimeseriesdataandmodelsisforecasting.Tobeabletopredictorforecastthefuture withareasonabledegreeofaccuracyisfundamental,asmostdecisionstakentodayare basedonwhatwethinkwillhappeninthefuture.Wethereforedevotemuchemphasison macroeconomicforecasting.
Followingtheanalysisofstationarity,Chapters4and5introducenonstationarytime seriessothatwecananalysethetrendinthedataandthefluctuationsaroundthetrend (i.e.,thebusinesscycles).Wewillpresentandcompareseveraldetrendingmethodsused toextractthebusinesscycle,includingthepopularHodrick-Prescottandtheband-pass
1 TheviewsexpressedarethoseoftheauthorsanddonotnecessarilyreflectthoseofNorgesBank.
filter.Thelatterfilteroperatesonwhatisknownasthefrequencybandofatimeseries. Hence,thischapteralsodefinessomeofthebasicslinkingthetimedomaintothe frequencydomain.
Chapters6–9dealwithmultivariatemethodstostudyco-movementamongvariables. Manyoftheconceptsdiscussedtherewillbemultivariateextensionsofthetoolsand conceptsintroducedinChapter2.However,somearenewandrelateonlytomultivariate models,likethediscussionofsimultaneityandtheidentificationofstructuralshocks. WestartbyintroducingtheconceptofsimultaneityinChapter6andillustratethe usefulnessofinstrumentalvariable(IV)regressionfordealingwiththisproblem.TheIV andTwo-stageleastsquaresestimatorsaredescribed,andwealsogiveabriefintroductiontotheGeneralizedMethodofMoments(GMM)estimator.TheGMMmethod hasproventobeaveryefficientwaytoestimatetheparametersoftherationalexpectationmodelsthatarewidelyusedinmacroeconomics.
InChapter7weintroducevectorautoregressive(VAR)models,whichisusedthroughoutthenextthreechapters.Wefirstdiscussstabilitypropertiesandshowhowonecan derivethemovingaveragerepresentationoftheVAR.Thenwediscussissuesrelatedto specification,estimationandforecastingoftheVAR,andfinallyweintroduceanimportant conceptcalledGrangercausality.Chapter8isthemainchapteronvectorautoregression techniqueswherewedealwiththesimultaneityproblemfirstintroducedinChapter6. Weshowhowvariousmethodscanbeusedtoidentifythestructuralvectorautoregressive (SVAR)model,anddiscusshowonecanconstructandinterpretstructuralshocks,suchas monetarypolicy,aggregatedemandandsupplyshocks.Keyisthestudyofimpulse responsesandvariancedecompositions.
InChapter9weexplaintheconditionsandmethodsthatpermitustoworkwith nonstationarydatainamultivariatesettingdirectly.Suchconditionsarecommonlyreferred toascointegration.Weshowhowonecanusecointegrationasacommontoolforthe analysisoflong-runrelationshipsbetweenmacroeconomictimeseries.Asanexampleof apotentialcointegrationrelationshipderivedfromeconomictheory,wediscussthe theoryofpurchasingpowerparity.
Thefinalchapterofthebook,Chapter10,introducesandexplainstheso-calledstatespacerepresentation,theKalmanFilter,andhowtoestimatestate-spacemodelsusing Maximumlikelihoodtechniques.Togetherthestate-spacerepresentationandtheKalman Filtertoolsareusedinawidevarietyofsettingsintimeseriesanalysis.Theycanfor examplebeusedtoestimateunobservedeconomicvariables,likethetrendandthecycle, toestimatetheparametersoftimeseriesmodels,infermissingvalues,andforconditional forecasting.Assuch,thefilterappliestostationary,nonstationary,uni-andmultivariate modelsalike.
1.1.1Populationmoments ..................................23
1.1.2Samplemoments.....................................23
1.1.3Abriefreviewofprobability ............................26
1.2Ordinaryleastsquaresandassumptions..........................27
1.3MonteCarlosimulations .....................................36
1.3.1Biasvariancetrade-off................................39
Appendices
2.1Fundamentalsoftimeseries..................................48
2.1.1Whitenoise,differenceequationsandlagoperators...........48
2.1.2Conditionalandunconditionalmoments ....................52
2.1.3Stationarity,autocorrelationfunctionandergodicity...........54
2.1.4Multipliersandimpulseresponses........................55
2.2Movingaverage(MA)processes...............................56
2.2.1HigherorderMAprocesses .............................58
2.3Autoregressive(AR)processes................................59
2.3.1HigherorderARprocesses .............................64
2.3.2Autoregressivemovingaverage(ARMA)processes...........66 2.4Estimation...............................................67
2.4.1OLSandARmodels..................................67
2.4.2Lagselection.......................................68
2.4.3Structuralbreaks.....................................70
2.5Anexample:GDPandinflation...............................70
2.6Summary................................................76
Appendices
2.ASimulatingtimeseriesdata...................................77
3.1Forecastingwithautoregressiveprocesses .......................80
3.1.1Pointforecast......................................81
3.1.2Optimalmeansquarederrorprediction...................82
3.1.3Uncertainty........................................84
3.2Forecastevaluation........................................87
3.2.1Bias.............................................88
3.2.2Rootmeansquarederror(RMSE) .......................88
3.2.3Comparingdifferentforecasts..........................91
3.3Forecastingexperiment.....................................92
3.4Modelcombination ........................................95
3.4.1Linearopinionpoolandweights........................96
3.4.2Diversificationgainsintheory..........................96
3.4.3Diversificationgainsinpractice.........................99
3.5Forecastingwithothertimeseriesmodels.......................103
3.6Anexample:ForecastingNorwegianGDPgrowth.................104
3.7Summary...............................................107 Appendices
3.ADensityforecastingandevaluation............................108
3.A.1Evaluatingdensityforecasts. ...........................108
4Nonstationarity111
4.1Deterministictrends.......................................113
4.2Stochastictrends. .........................................114
4.2.1Randomwalk......................................114
4.2.2Randomwalkwithdrift..............................116
4.3Implicationsofnonstationarity...............................117
4.4Testsforunitroot.........................................118
4.4.1AugmentedDickey-Fuller(ADF)test....................118
4.4.2Unitrootversusstructuralbreak........................122
4.4.3KPSStest.........................................123
4.5Persistence..............................................124
4.6Summary...............................................127
5TrendandCycles129
5.1Statisticaldetrendingmethods................................132
5.2Deterministictrend........................................133
5.3Linearfilters............................................134
5.3.1Hodrick-Prescott(HP)filter............................135
5.3.2Bandpassfilters....................................137
5.4Beveridge-Nelson(BN)Decomposition.........................140
5.5Anexample:Stylizedfactsofbusinesscycles....................144
5.6Spuriouscycles..........................................149
5.7Summary...............................................150
Appendices
5.ASpectralanalysis..........................................151
5.A.1Linearfilters-Implications............................156
6Simultaneity159
6.1Simultaneousequationbias..................................160
6.2Instrumentalvariableregression(IV)...........................163
6.2.1Instrumentalvariables................................163
6.2.2Instrumentalvariablesestimators........................166
6.2.3ThegeneralIVregressionmodel........................169
6.3Identification............................................170
6.4GeneralizedMethodofMoments(GMM).......................173
6.4.1MethodofMoments:Intuition ..........................174
6.4.2GeneralizedMethodofMomentsEstimator................174
6.4.3GMMandOLSestimator.............................177
6.4.4GMMandIVestimator...............................177
6.4.5Anexample:TheConsumption-BasedAssetPricingModel....178
6.4.6Anexample:TheNewKeynesianPhillipsCurve............181
6.5Summary...............................................183
Appendices
6.AEstimatingthegeneralIVregressionmodel......................184
6.A.1Weakinstrumentsreconsidered.........................185
6.A.2Overidentificationtest................................185
6.BAnexample–Returntoeducation............................186
7Vectorautoregression(VAR)189
7.1TheVARmodel..........................................190
7.1.1Thecompanionform.................................191
7.1.2StabilityofVARs...................................192
7.1.3Woldrepresentation/decompositiontheorem................193
7.2MovingaveragerepresentationofaVAR.......................194
7.3EstimatingaVAR........................................199
7.3.1Choiceofvariablesandlags...........................200
7.3.2Anexample:AVARmodelforGDPandunemployment......201
7.4VARforecasts...........................................203
7.4.1Pointforecast......................................203
7.4.2MSE. ............................................204
7.4.3Uncertainty........................................205
7.4.4Forecastfailureinmacroeconomics......................206
7.5Grangercausality.........................................206
7.6Summary...............................................209
Appendices
7.AShortintroductiontolinearalgebra............................210
8Structuralvectorautoregression(SVAR)213
8.1Identification–Choleskydecomposition........................215
8.2AstructuralVARmodel ....................................217
8.2.1FromstructuraltoreducedformVAR ....................218
8.2.2Identificationofthestructuralmodel.....................219
8.3Theimpulseresponsefunction(IRF)...........................222
8.3.1Forecasterrorvariancedecomposition....................225
8.4Twoempiricalexamples....................................227
8.4.1Effectsofmonetarypolicyshocks.......................227
8.4.2Effectsofoilpriceshocks.............................232
8.5Alternativerestrictions .....................................234
8.5.1Long-runrestrictions.................................235
8.5.2Aspecialcaseoflong-runrestrictions ....................238
8.5.3Signrestrictions....................................239
8.6Anexample:Theexchangeratepuzzlerevisited ..................241
8.7LimitationoftheVARapproach ..............................245
8.8Summary...............................................247
9Cointegration249
9.1Spuriousregressions.......................................250
9.2Cointegrationdefined......................................254
9.3Testingforcointegration–singleequation......................255
9.3.1Engle-Grangerapproach..............................255
9.3.2Anexample:Consumptionandincome ...................257
9.4ErrorCorrectionModel(ECM)...............................259
9.5Cointegrationinamultivariatesetting ..........................262
9.5.1Vectorerrorcorrectionmodel(VECM) ...................263
9.5.2Testingforcointegration–theJohansenapproach...........265
9.5.3Identificationofthecointegratingrelations. ................266
9.5.4Cointegrationandcommontrends .......................268
9.5.5Weakexogeneity....................................269
9.6Anexample:Purchasingpowerparity(PPP).....................269
9.7Finalcomments:VARinlevelsordifferences....................273
9.8Summary...............................................275
10State-SpaceModelsandtheKalmanFilter277
10.1State-spacerepresentations..................................278
10.1.1AR-modelsinstate-space .............................279
10.1.2VARsinstate-space.................................280
10.1.3ARMAinstate-space ................................282
10.1.4Unobservedcomponentsmodelinstate-space..............284
10.2TheKalmanFilter........................................285
10.2.1Prediction.........................................286
10.2.2Updating.........................................287
10.2.3Smoothing........................................288
10.2.4Missingvalues,forecastingandchoosingstartingvalues......288
10.2.5KalmanFiltersummary...............................290
10.3Filteringexamples........................................292 10.4Parameterestimation......................................294
10.4.1Maximumlikelihood(ML)............................294
10.4.2TheKalmanFilterandMLestimation....................295 10.5Applications.............................................297
10.5.1EstimatinganARMA(1,1).............................297
10.5.2Anunobservedcomponentsmodel(UC)..................299
10.5.3Otherapplications...................................302 10.6Summary...............................................303