Time Series Econometrics

421 Pages·2016·6.145 MB·English

by  Klaus Neusser (auth.)

Springer Texts in Business and Economics
Klaus Neusser
Time Series
Econometrics
Springer Texts in Business and Economics
Moreinformationaboutthisseriesathttp://www.springer.com/series/10099

Klaus Neusser
Time Series Econometrics
123
KlausNeusser
Bern,Switzerland
ISSN2192-4333 ISSN2192-4341 (electronic)
SpringerTextsinBusinessandEconomics
ISBN978-3-319-32861-4 ISBN978-3-319-32862-1 (eBook)
DOI10.1007/978-3-319-32862-1
LibraryofCongressControlNumber:2016938514
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Preface
Overthepastdecades,timeseriesanalysishasexperiencedaproliferousincreaseof
applicationsineconomics,especiallyinmacroeconomicsandfinance.Todaythese
toolshavebecomeindispensabletoanyempiricallyworkingeconomist.Whereasin
thebeginningthetransferofknowledgeessentiallyflowedfromthenaturalsciences,
especially statistics and engineering, to economics, over the years theoretical and
applied techniques specifically designed for the nature of economic time series
and models have been developed. Thereby, the estimation and identification of
structuralvectorautoregressivemodels,theanalysisofintegratedandcointegrated
time series, andmodelsofvolatility havebeenextremelyfruitfulandfar-reaching
areas of research. With the award of the Nobel Prizes to Clive W. J. Granger and
RobertF. Engle III in 2003and to ThomasJ. Sargentand ChristopherA. Sims in
2011, the field has reached a certain degree of maturity. Thus, the idea suggests
itself to assemble the vast amount of material scattered over many papers into a
comprehensivetextbook.
Thebookisself-containedandaddresseseconomicsstudentswhohavealready
some prerequisite knowledge in econometrics. It is thus suited for advanced
bachelor,master’s,orbeginningPhDstudentsbutalsoforappliedresearchers.The
book tries to bring them in a position to be able to follow the rapidly growing
research literature and to implement these techniques on their own. Although the
book is trying to be rigorous in terms of concepts, definitions, and statements
of theorems, not all proofs are carried out. This is especially true for the more
technically and lengthy proofs for which the reader is referred to the pertinent
literature.
The book covers approximately a two-semester course in time series analysis
andisdividedintwoparts.Thefirstparttreatsunivariatetimeseries,inparticular
autoregressivemoving-averageprocesses.Most ofthe topicsare standardandcan
form the basis for a one-semester introductory time series course. This part also
contains a chapter on integrated processes and on models of volatility. The latter
topicscouldbeincludedinamoreadvancedcourse.Thesecondpartisdevotedto
multivariatetimeseriesanalysisandinparticulartovectorautoregressiveprocesses.
It can be taught independently of the first part. The identification, modeling, and
estimation of these processes form the core of the second part. A special chapter
treatstheestimation,testing,andinterpretationofcointegratedsystems. Thebook
alsocontainsachapterwithanintroductiontostatespacemodelsandtheKalman
v
vi Preface
filter. Whereasthe booksis almost exclusivelyconcernedwith linear systems, the
last chapter gives a perspectiveon some more recent developmentsin the context
ofnonlinearmodels.Ihaveincludedexercisesandworkedoutexamplestodeepen
the teachingand learning content.Finally, I have producedfive appendiceswhich
summarizeimportanttopicssuchascomplexnumbers,lineardifferenceequations,
andstochasticconvergence.
Astimeseriesanalysishasbecomeatremendouslygrowingfieldwithanactive
researchin manydirections,it goeswithoutsayingthatnotalltopicsreceivedthe
attentiontheydeservedandthatthereareareasnotcoveredatall.Thisisespecially
true for the recent advances made in nonlinear time series analysis and in the
applicationofBayesian techniques.These two topicsalonewould justify an extra
book.
The data manipulations and computations have been performed using the
software packages EVIEWS and MATLAB.1 Of course, there are other excellent
packages available. The data for the examples and additional information can
be downloaded from my home page www.neusser.ch. To maximize the learning
success, it is advised to replicate the examples and to perform similar exercises
with alternative data. Interesting macroeconomictime series can, for example, be
downloadedfromthefollowinghomepages:
Germany: www.bundesbank.de
Switzerland: www.snb.ch
UnitedKingdom: www.statistics.gov.uk
UnitedStates: research.stlouisfed.org/fred2
The book grew out of lectures which I had the occasion to give over the years
in Bern and other universities. Thus, it is a concern to thank the many students,
in particular Philip Letsch, who had to work through the manuscript and who
called my attention to obscurities and typos. I also want to thank my colleagues
andteachingassistantsAndreasBachmann,GregorBäurle,FabriceCollard,Sarah
Fischer,StephanLeist,SenadaNukic,KurtSchmidheiny,RetoTanner,andMartin
Wagner for reading the manuscript or part of it and for making many valuable
criticisms and comments. Specialthanksgo to my formercolleague and coauthor
Robert Kunst who meticulously read and commented on the manuscript. It goes
withoutsayingthatallerrorsandshortcomingsgotomyexpense.
Bern,Switzerland/Eggenburg,Austria KlausNeusser
February2016
1EVIEWSisaproductofIHSGlobalInc.MATLABisamatrix-orientedsoftwaredevelopedby
MathWorkswhichisideallysuitedforeconometricandtimeseriesapplications.
Contents
PartI UnivariateTimeSeriesAnalysis
1 Introduction................................................................. 3
1.1 SomeExamples ...................................................... 4
1.2 FormalDefinitions ................................................... 7
1.3 Stationarity ........................................................... 11
1.4 ConstructionofStochasticProcesses................................ 15
1.4.1 WhiteNoise................................................. 15
1.4.2 ConstructionofStochasticProcesses:SomeExamples .. 16
1.4.3 Moving-AverageProcessofOrderOne ................... 17
1.4.4 RandomWalk............................................... 19
1.4.5 ChangingMean............................................. 20
1.5 PropertiesoftheAutocovarianceFunction ......................... 20
1.5.1 AutocovarianceFunctionofMA(1)Processes............ 21
1.6 Exercises.............................................................. 22
2 ARMAModels.............................................................. 25
2.1 TheLagOperator..................................................... 26
2.2 SomeImportantSpecialCases ...................................... 27
2.2.1 Moving-AverageProcessofOrderq ...................... 27
2.2.2 FirstOrderAutoregressiveProcess........................ 29
2.3 CausalityandInvertibility ........................................... 32
2.4 ComputationofAutocovarianceFunction .......................... 38
2.4.1 FirstProcedure.............................................. 39
2.4.2 SecondProcedure........................................... 41
2.4.3 ThirdProcedure............................................. 43
2.5 Exercises.............................................................. 44
3 ForecastingStationaryProcesses ......................................... 45
3.1 LinearLeast-SquaresForecasts...................................... 45
3.1.1 ForecastingwithanAR(p)Process........................ 48
3.1.2 ForecastingwithMA(q)Processes ........................ 50
3.1.3 ForecastingfromtheInfinitePast.......................... 53
3.2 TheWoldDecompositionTheorem................................. 54
3.3 ExponentialSmoothing.............................................. 58
vii
viii Contents
3.4 Exercises.............................................................. 60
3.5 PartialAutocorrelation............................................... 61
3.5.1 Definition ................................................... 62
3.5.2 InterpretationofACFandPACF........................... 64
3.6 Exercises.............................................................. 65
4 EstimationofMeanandACF ............................................. 67
4.1 EstimationoftheMean .............................................. 67
4.2 EstimationofACF ................................................... 73
4.3 EstimationofPACF.................................................. 78
4.4 EstimationoftheLong-RunVariance............................... 79
4.4.1 AnExample................................................. 83
4.5 Exercises.............................................................. 85
5 EstimationofARMAModels ............................................. 87
5.1 TheYule-WalkerEstimator.......................................... 87
5.2 OLSEstimationofanAR(p)Model ................................ 91
5.3 EstimationofanARMA(p,q)Model................................ 94
5.4 EstimationoftheOrderspandq .................................... 99
5.5 ModelingaStochasticProcess ...................................... 102
5.6 ModelingRealGDPofSwitzerland................................. 103
6 SpectralAnalysisandLinearFilters..................................... 109
6.1 SpectralDensity...................................................... 110
6.2 SpectralDecompositionofaTimeSeries........................... 113
6.3 ThePeriodogramandtheEstimationofSpectralDensities........ 117
6.3.1 Non-ParametricEstimation ................................ 117
6.3.2 ParametricEstimation...................................... 121
6.4 LinearTime-InvariantFilters........................................ 122
6.5 SomeImportantFilters............................................... 127
6.5.1 ConstructionofLow-andHigh-PassFilters .............. 127
6.5.2 TheHodrick-PrescottFilter................................ 128
6.5.3 SeasonalFilters............................................. 130
6.5.4 UsingFilteredData......................................... 131
6.6 Exercises.............................................................. 132
7 IntegratedProcesses........................................................ 133
7.1 Definition,PropertiesandInterpretation............................ 133
7.1.1 Long-RunForecast ......................................... 135
7.1.2 VarianceofForecastError ................................. 136
7.1.3 ImpulseResponseFunction................................ 137
7.1.4 TheBeveridge-NelsonDecomposition.................... 138
7.2 Properties of the OLS Estimator in the Case
ofIntegratedVariables............................................... 141
7.3 Unit-RootTests....................................................... 145
7.3.1 Dickey-FullerTest.......................................... 147
7.3.2 Phillips-PerronTest......................................... 149
Contents ix
7.3.3 Unit-RootTest:TestingStrategy........................... 150
7.3.4 ExamplesofUnit-RootTests .............................. 152
7.4 GeneralizationsofUnit-RootTests.................................. 153
7.4.1 StructuralBreaksintheTrendFunction................... 153
7.4.2 TestingforStationarity..................................... 157
7.5 RegressionwithIntegratedVariables................................ 158
7.5.1 TheSpuriousRegressionProblem......................... 158
7.5.2 BivariateCointegration..................................... 159
7.5.3 RulestoDealwithIntegratedTimesSeries............... 162
8 ModelsofVolatility......................................................... 167
8.1 SpecificationandInterpretation ..................................... 168
8.1.1 ForecastingPropertiesofAR(1)-Models.................. 168
8.1.2 TheARCH(1)Model....................................... 169
8.1.3 GeneralModelsofVolatility............................... 173
8.1.4 TheGARCH(1,1)Model................................... 177
8.2 TestsforHeteroskedasticity ......................................... 183
8.2.1 AutocorrelationofQuadraticResiduals................... 183
8.2.2 Engle’sLagrange-MultiplierTest.......................... 184
8.3 EstimationofGARCH(p,q)Models................................. 184
8.3.1 Maximum-LikelihoodEstimation ......................... 184
8.3.2 MethodofMomentEstimation ............................ 187
8.4 Example:SwissMarketIndex(SMI) ............................... 188
PartII MultivariateTimeSeriesAnalysis
9 Introduction ................................................................ 197
10 DefinitionsandStationarity .............................................. 201
11 EstimationofCovarianceFunction....................................... 207
11.1 EstimatorsandAsymptoticDistributions........................... 207
11.2 TestingCross-CorrelationsofTimeSeries.......................... 209
11.3 SomeExamplesforIndependenceTests............................ 211
12 VARMAProcesses.......................................................... 215
12.1 TheVAR(1)Process ................................................. 216
12.2 RepresentationinCompanionForm................................. 218
12.3 CausalRepresentation................................................ 218
12.4 ComputationofCovarianceFunction............................... 221
13 EstimationofVARModels ................................................ 225
13.1 Introduction........................................................... 225
13.2 TheLeast-SquaresEstimator ........................................ 226
13.3 ProofsofAsymptoticNormality .................................... 231
13.4 TheYule-WalkerEstimator.......................................... 238