Time Series Summary

7/21/2019 Time Series Summary

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Univariate Models Autoregressive model (AR model) a linear regression of the current value of

the series against its previous values

Pure AR can be estimated with OLS

Moving average model (MA model) a linear regression of the current value of series against current and previous white noise error terms or random shocks

MA models can’t be estimated with OLS but using Ma!imum Likelihood

Autoregressive moving average model (ARMA model) a linear regression fthe current value of series against previous values of the series "ARcomponents) and current previous white noise error terms "MA component)

Autoregressive integrated moving average "AR#MA model)AR#MA lagged values of the dependent variable and$or random disturbance termas dependent variable% &ompared to ARMA AR#MA involves di'erencing thedata%

Stationarit( and modelso AR ma($ma( not be stationar( "unless prior conditions are imposed)o MA alwa(s stationar( white noise series *"+) ,-. var"+) , /0 cov "+t +t12) ,-

2% Stationarit( 3 the probabilit( laws governing the process do not change overtime

1 the process is in statistical e4uilibrium1 the process in free of a trend and seasonalit(

5 mean variance and autocorrelation structure do not change over time

&onse4uences of 6on1stationarit(2% 7iolation of &LRM assumption "OLS is not the best estimator an(more) in8nitevariance0% 9rong speci8cation:% Spurious regression



Strong stationarit( probabilit( distribution "e!pected value var) of thestochastic process is invariant under a shift in time "let’s sa( shifting it b( mperiods)5 means that the ;oint distribution does not depend on time1 does not impl( weak stationarit(

9eak stationarit( "covariance stationar() if its 1st and 2nd order moments areuna'ected b( a change of time origin- &ovariance and correlation are functions of onl( the time di'erences1 9hen e!pected value mean and covariance of the distribution randomvariables are constant for all points of time%

Spurious regression we fre4uentl( re;ect the null that b,- "i%e regressioncoe<cient of the e!planator( variable does not in=uence the dependentvariable) even when the null is correct1 strong "causal) relationship between variables that are not causall( related1 often it’s a conse4uence of regressing non1stationar( variables on each other

1 non1 stationar( residuals1 wrong distributional assumptions

Detection:1 ver( high R0

1 high t1values1highl( signi8cant results



Box Jenkins refers to a set of procedures for identif(ing and estimating time1series models within the class of AR#MA%

Steps2% Stationarit( checking0% Model identi8cation:% Parameter estimation>% ?iagnostic checking@% orecasting

Step 2% Stationarit( checkinga) Eye-ballingb) Inspection of autocorrelation function (correlograms)1 Statistical measures that indicate how a time series relates to itself over time%1 or stationar( processes autocorrelation between an( two observations onl(depends on the time lag h between them%1 Bhe time1series values y1 y2 C yn should be considered stationar( if itsvalues either cuts o fairl( 4uickl( or dies down fairl( 4uickl(



c) nit !oot "ests: not considered b( DE

#nclude A? Phillips1Perron test

&orrection1 Dierencing until stationarit( is achieved% *!amine the stationarit( again b(correlograms Fnit Root tests%- Level of integration , number of time a series should be di'erentiated tobecome stationar(1 #n practice 0nd di'erences should be enough

Step 0 Model identi8cation

Bhe model should be as parsimonious as possible1 including irrelevant lags in the model increases standard errors reduces the t1stat "making h(potheses testing more di<cult)- models that incorporate large numbers of lags do not tend to forecast well#



PAC are used to measure the degree of association between Gt and Gt1kwhen the e'ects of other time lags "2 0 : C k32) are removed%PA& identi8es the e!tent of the lag$ the order in an AR model%

Bo identif( MA lag structure we use A&

Step : Parameter estimation1 AR OLS while MA$ARMA using MLML: those value of the parameters$regression coe<cients for which the actuall(observed data are most likel( i%e% the values that ma!imiHe the likelihoodfunction L1 ML is a function of the intercept coe<cients and the variance of the error termgiven n observations (2 (0 C (n of the output variable

Step > ?iagnostic &hecking

Do!1Eenkins method does not automaticall( (ield a single IbestJ model% 9e needto compare themo Ttests for coe<cient signif% "add e!planator( power and should be included)o Residual anal!sis "the residuals should be uncorrelated)% #f the( are correlated

s(stematic movement in the output not accounted b( ARMA

$ortmanteau test (%&ung 'o) test for presence of serial correlation in theresidualsK- Residuals are independentl( distributed "i%e% an( observed correlations in thedata result from randomness of the sampling process)9e want the 6ull of L;ung Do!%

o "oodnesso#$t and model selection criteria



a) d&usted !2: the percentage of the variation of G around its mean that ise!plained b( the regression e4uation ad;usted for degrees of freedom% 9e needhigh values%b) I* and 'I* should be as small as possible%

Step @ orecasting

1 orecasts from ARMA model is not perfectl( accurate1 ualit( of forecasts diminishes with the amount of ;1steps1ahead into the future1 orecasts can be either in1sample or out-of-sample "better test for how themodel works since it uses data not included in the model estimation)% One ma(leave some observations at the end of the sample for this purpose1 Bhe model that 8ts the data best does not need to have the best forecastperformance

E+aluation: plot the forecast against the actual "observed) values1 direct comparison of patterns

1 comparison of turning points

,easures of orecast accuracy:2% Mean s4uared error "smaller value)0% Mean absolute error:% Mean average prediction error>% Bheil’s F statistic "forecast is compared to a benchmark model) the onesoutperforming the benchmark should be preferred

Pro%lems &it' BJM:2% Ad1hocism rel(ing on one’s ;udgement and e!perience

0% 6on 3 stationarit(a ) irst dierence 8lter does not work for trends structural breaks andcointegrationb) *ointegration: long run e4uilibrium e!ists between a set of variables% *venthrough individual variables ma( be non1stationar( it’s possible for linearcombinations to be stationar(%c) .imple de-trending of time series b( 8rst1di'erencing ma( lead to spuriousregression results when a long1run "cointegrating) relationship e!ist% Ma( bemodeled b( *&M%1 Bhe issue of cointegration arises onl( for AR and ARMA models that also includecontrol variables%

:% Seasonalit(>% Structural breaks one assumption is that data structure doesn’t change"constant coe<cients)% Otherwise parameters instabilit(%5 there is a sudden change in time series and$or relationship between time series



*how "est: when structural break is knownK- 6o structural break at given point in time1 it ARMA model for pre1 and post1 break data

1 Su<cient di'erence between models indicates the presence of a structuralbreak "large &how test and re;ection of null)1 Largest &how test stat indicates the actual break date "*ndogeneous break)or &how break date should be known% Dut breaks ma( not full( manifestthemselves at speci8c dates but graduall( ,

CUUM test "detects more graduall( changes)

A series of one1step ahead forecast errors is obtained b( running a series of regressions with the 8rst regression using the 8rst k observations "k , numberof parameters in original models) to generate a prediction of the dependentvariable at observation kN2 etc% "recursive estimations)% Bhe cumulated sum of

forecast errors is printed%K- is re;ected at @ if cumulated sum stra(s outside the con8dence band%

@% Other problems heteroskedasticit( and other speci8cation errors%

undamentals o# t'e BJM:

2% Parsimon( Model selection criteria0% Stationarit( and invertibilit( *!amination of correlogram

1 An MA process is said to be in+ertible if it can be converted into a stationar( ARprocess of in8nite order- MA processes are not invertible unless appropriate prior conditions are imposedon the parameters

◦Pure AR"p) alwa(s invertible◦Pure MA"4) ARMA"p4) ma($ ma( not be invertible

- /on-in+ertible models can not be estimated with the 'o-0enins method

:% ood 8t of data oodness of 8t measurements

>% 9hite noise appro!imation of residuals e!amination of residuals "1 stat)@% 6o coe<cient instabilit( "test for structural breaks "e! &how test)Q% ood out1of1sample forecasts e!amination of forecasting performance5 also look at t1test for coe<cient signi8cance



Topi*s +: ARC,- "ARC, models

AR&K "and AR&K) models heteroskedasticit( as a variance to be modeled1 simultaneous modeling of the mean and variance

ARC,: conditional variance of the disturbance of the output variable "8tteds4uared residuals) is se4uence is an AR process1 Signi8cant lags of the AR process , it’s order

Larger a2 more persistent change in (t stronger tendenc( to remain awa( fromits meanLarger 2 a more persistent shock in the + se4uence

Drawbacs: high persistence of variance , high order of AR&K1 Onl( positive coe<cients in the variance model

"ARC,: model of conditional variance as an ARMA process , it ma( be moreparsimonious



5 Make sure that2% no serial correlation "model of mean is appropriate)0% parsimonious:% if structural breas "high persistent conditional volatilit(): it can be capturedwith a dumm( variable indicating the structural break% Perform &how test

3ther models:2% #AR&K "AR has a unit root , cond% var acts like a Fnit Root process)0% BAR&K leverage e'ect bad "higher e'ect)$ good news1 model as(mmetricshocks:% *AR&K for as(mmetric shocks as well estimated coe<cients can benegative

"retl1# Inspection of summary statistics: mean is small. ma!imum and minimum 3large large standard deviation point to conditional heteroskedasticit(

2# 4raph inspection: tran4uilit( untilt,@- turbulence after that point ,AR&K ma( be ok

5# !egress 3%. with lags

>% $lot the correlograms and chec:1 the default lag lengths

1 presence of serial correlation5 residuals are not seriall( autocorrelated but their s4uares ma( follow an AR"p)or AR&K onl( A& is relevant. AR&K we check PA& as well

9hen both A& PA& die down 3 use AR&K

@% %, test for !*6 errors: OLS 1 tests1 AR&K#f LM1stat is large and signi8cant1 re;ect the null of 6O AR&K error1 model AR&K error"2)



7# itting (4)!*6: model of mean AR"2). model of variance is AR&K"2) orAR&K "-2)

8# 3btain residuals and suared residuals5 residuals no sign of serial correlation5 s4uared residuals no sign of volatilit(

1 7ariance is not 2 (et another ")AR&K ma( be more suitable but we need tokeep it parsimonious

9# orecasting: con8dence intervals become larger with larger forecast horiHon

Topi* .: Models &it' Trends

&omponents of Bime Series(t , trend N stationar( component N noise.tationary component: modeled with DE

/oise: AR&K$ AR&K

Brendsa# Deterministic Nnoise *"Gt) 3 *"Gt12) , b Bhe trending variable changes b( a constant amount each period "b)

Gt , b- Nb2t N +. t 3time trend + , white noise1 Bhe mean that grows around a 8!ed trend which is constant and independent

of time%1 Alwa(s revert to the trend in the long run "the e'ects of shocks is eliminated)Also known an trend1stationar( processes

b# .tochastic *"Gt) 3 *"Gt12) , a N vt. a, constant v , random amount Bhe trending variable changes by a random amount each period plus a constant "random but permanent shock to series)Also called ?i'erence stationar( or Fnit Root processes

c# !andom al ; noise: < t = < t-1 ; +the value of a time series at time t will be e4ual to the last period value plus astochastic component that is a white noise

1 #t is a mean1non1reverting process that can move awa( from the mean in apositive$ negative direction

1 7ariance evolves over time and goes to in8nit( as time goes to in8nit( , R9can’t be predicted

d# ! with a drift (;noise)



predicts that the value of a time series at time t will e4ual the last periodVs valueplus a constant or drift and a white noise term< t = a N< t-1 ; +1 both deterministic "constant) and stochastic trends#t does not revert to the long run mean and it’s variance depends on time



Removing Trends:a) ?i'erencing a series with unit root is transformed into a stationar( series%6umber of times a series is di'erences , order of integration #"d)% Fsuall( ma!#"0)%

Recall Bhe 8rst1di'erence of log1level data appro!imates the growth rate of thelog1level data

b) ?etrending a series with a deterministi* trend can be transformed into astationarH series b( removing the trend%1 Fse of OLS regression to identif( the trend component and then subtract thetrend component from t he original data series% 1Residuals from regression with trend, detrended time series

Dierence +s# Detrending1 ?etrending is not suitable for FR it does not eliminate the stochastic portion of

series1 ?i'erencing introduces the FR process into trend1stationar( series

/ete*ting Trends:2% *(e1balling a series should have constant mean variance and no visible LRtrend

0% #nspection of correlogram indicates how a time series is related to itself overtimeStationar( if A& cuts of$ dies down fairl( 4uickl(

:% t1test for Gt,b-N b2 Gt12N +t

K- b2,2 (non-stationary) c%v , 2%Q@Ka b2 W2 (stationary)Problem b2 ma( be downward biased especiall( in small samples "b$c use oflagged ()



9ewant BO

R*E*&B5

>% Fnit root test

a. Dickey Fuller test: for AR"2) process with a drift Gt , b- N b2 Gt12N +t

*stimate using OLS and e!amine the estimated b2K- b2 , 2 "non stationar()Ka b2 W 2 "stationar()

? suggest to subtract Gt12 from both sides Gt1Gt12, b-N b2 Gt12 1Gt12 N +tΔ Gt, b-N XGt12 N +t where X , b232

K- X , - "non1stationar( , FR e!ists)Ka X W - "stationar() , we want BO R*E*&B



5 9e test for R9 using 1test

b. ADF test : an e!tension that considers higher1order AR b( including lags ofthe dependent variablesinal lag should correspond to a parsimonious model1 too few residuals do not behave like white1noise "e! autocorrelation presence)1 too man( reduces the power of test to re;ect the null

.election of lag length:1 eneral to speci8c approach start with man( lags and pare down b( means of$t tests1 minimiHe A#&$ D#&

Etensions and $roblems:2% Structural Dreaks ? test is biased "null is accepted in pre1 and post1 break

periods)#t can lead to wrong decision of model 8tting%1 *(e1balling1 Yivot1 Andrews is a FR test that allows for single unknown break in trendand$or intercept1 #f breaks present include dumm( variables to account for breaks when testingFR

0% Power of Fnit Root Bests the probabilit( of correctl( re;ecting the nullh(pothesis%?$ A? often wrongl( indicate that a series contains a Fnit Root% #t has littlepower to distinguish between trend1stationar( and unit root processes%



1 Bhe closer is a2 to 2 lower the powerof the test

B(pe 2 Re;ect null when it is non1stationar( B(pe 0 Accept null when it is stationar(%

"ests with more power:

1 FRB that allow for structural breaks "e! Yivot Andrews)1 FRB that model more e<cientl( deterministic components that ma( include aFR process "e! Schmidt Phillips)1 Panel FRB#t ma( be useful to compare results of di'erent tests

*ointegration: when more than one variable is considered%1 ma( lead to spurious regression results when a long1run "cointegrating)relationship e!ist?etection prior tests for unit roots to test if series are non1stationar(1 &ointegration tests "e! *ngle 3 ranger)5 need to be corrected using error1correction1model

Remed(2% Fse FRB do determine order of integration0% Run cointegrating regression:% Appl( an appropriate FRB to residuals from this regression to test forcointegration>% #f cointegration is accepted use lagged residuals from the cointed regressionas an error correction term in *&M%

Topi* 0: Multie1uation Time eries Models

.ingle euation models: dependent variable is modeled using lags of itself 1AR%#t ma( contain MA to improve model 8t and parsimon(% #f involves di'erencing 3AR#MA%



1 #f heteroskedasticit( specif( the variance model ")AR&K while the meanremains an AR#MA model Fsed for forecasting1 Share little relationship with economic theor( include onl( IuninformativeJcomponents lagged ?7 time trends constants dumm( variables , potentialdisconnect between statistical modeling and economic theor(

,ulti-euation models: ma( still include an AR component constant trendsother dummies but also considers Itrul(J e!planator( variables%1 relation between economic theor( can be tested empiricall(1 polic( intervention anal(sis1 cause and e'ect anal(sis

'ene>ts: 1 reduced omitted variable bias1 improved goodness of 8t1 improved forecasting1 availabilit( of further tools of time1series anal(sis "e! causalit( anal(sis

variance decomposition)

B(pes23 2ntervention anal!sis 3 statistical techni4ue that generaliHes the univariatetime1series methodolog(1 time path of dependent variable in=uenced b( time path of an independentvariable1 6o feedback1 used for forecasting and various intervention anal(sis

H , intervention variable pure ;ump or di'erent forms

#t evaluates the initial and long run e'ects of a speci8c polic( Initial: denoted m( the magnitude sign and statistical signi8cance of H %ong !un Eect: e4ual to the long run mean after the intervention minus

value of the original "pre1intervention) mean

Disad+antages: 1 &an’t capture feedback1 ?i<cult to di'erentiate between dependent and independent variables1 "Presence of feedback) biases estimates b$c correlation of G with the error term

223 4AR models "vector autoregressive models) are used for multivariate timeseries% *ach variable is a linear function of past lags of itself and past lags of theother variables% So it contains more than one dependent variable% All variablesare endogenous%

i# *haracteristics? ssumptions:2% Stationarit( all parameters are stable0% Linearit( relationship between variables as linear regression



:% Reduced form 7AR models only past "lagged) variablesPrimitive 7AR "with contemporaneous terms) 1 algebraic transformation 1reduced 7AR>% S(mmetr( all variables enter 7AR with the same lag length@% inite1order 7AR number of lags is 8nite "data constraints)Q% orecast errors shocks can be given an economic interpretation modelingd(namic response of shocks and identi8cation of shocks@ error terms are homoscedastic and serially uncorrelated

orecasting: for macro variables future prices of securities1 orecast from unrestricted 7AR ma( be unreliable over parameteriHed modelsor insigni8cant coe<cients) , 6ear 7AR model with improved forecastperformance

ii# Identi>cation:a% Primitive$ structural 7AR "with contemporaneous independent variables) can’t

be estimated directl( b$c correlation between contemporaneous variables witherror terms- Bied to a particular theor(1 Detter interpretation of forecast

b% Reduced$ unrestricted 7AR "without contemporaneous i%v%) can be estimatedusing OLS1 Statistical description of data1 &ompatible with lots of theories1 ood short1term forecast

rom Reduced to Structural 7AR"2) *stimation of a reduced "unrestricted) 7AR"0) #mpose suitable theoretical restriction recursive s(stem coe<cientvariance s(mmetr( and other Long Run restriction

iii# Estimation: 7AR using OLS d%v% are identical for each e4uation in the 7AR s(stem

"s(mmetr( , same lags length) N homoscedastic and seriall( uncorrelatederrors

6ear17AR "e! SFR) allows for errors correlation across e4uations more e<cient when RKS are not identical across e4uations

So OLS ma( su'er from violations

i+# "ools:

2% #mpulse Responses response of ?7 to shocks to the error term where a shockis applied to each dependent variable and its e'ect are noted, assumes that error returns to Hero in subse4uent periods and all other errorsare Hero5 Measures how long and to what degree a shock to a given e4uation matters tothe variables in the s(stem



0% 7ariance ?ecomposition anal(He the relative importance of variables% #t givesinformation abut the relative importance of each shock to the variables in the7AR%1 Percentage of the forecast error variance attributable to each ?7 "like partialR0)

Ordering "placing variables in decreasing order to relative e!ogeneit() isimportant b$c1 7AR errors ma( not be statisticall( independent

ranger causalit( when past values of Z provide statisticall( signi8cantinformation about future values of G "Z ranger causes G)K- 6o ranger causalit(ISigni8cantJ 8ndings of 1test individual t1test ma( be biased due to collinearit(



Pro%lems:6o priori assumptions about e!o1$endogeneous variables% 7AR ma( not becomplete "omitted variable bias)

2% Lag length selection 3 incorrect lag ma( a'ect the variance decomposition0% Over 8tting Rule of Bhumb no more than > endog% variables use more data), Fse of #& for choosing IrightJ number of lags% MA#& is more conservativethan simple A#& "less danger of overparametriHation)

:% 6on1stationarit( 7AR tests are usuall( accompanied b( FRB

*ases and A! speci>cation:&ase 2% All variables are stationar( 1 7AR in levels "non1transformed data)&ase 0% 7ariables integrated of di'erent orders 1 7AR meaningless&ase :% Bhe same order of integration but 6OB cointegrated 17AR indi'erences "otherwise spurious regression)&ase > Bhe same order of integration and cointegrated 1 7AR with error*orre*tion model

"2) Fse FRB to determine order of integration"0) Run cointegrating regression":) Fse FRB to the residuals to test for cointegration">) #f cointegration is accepted use lagged residuals from the cointegratedregression as an error correction term in *&M

B# Other issues structural breaks outliers etc



Topi* 5: Cointegration and 46CM

&ointegration refers to a situation where there is a linear combination ofintegrated "i%e% non1stationar() time1series variables "within an econometricmodel) that is stationar(

&onse4uences of &ointegration2% .purious regressions "cointegrated variables are not alwa(s stationar()1 oftentimes the conse4uence of regressing non1stationar( variables on eachother

0% ,isspeci>cation error 7AR approach in di'erences rather than levels%A 7AR in di'erences ignores the information that the levels of the variablescannot move independentl( of each other , no Long Run e4uilibrium1 inferior forecasting performance1 ma( bias h(pothesis testing e! ranger causalit(1 can not di'erentiate between SR and LR d(namics

&hoosing the correct model > cases

46CM is a special case of 7AR% *rror correction term refers to the linearcombination of the 0 variables "in levels) that is stationar(%

4ranger !epresentation "heorem: if 0 series are cointegrated , SBdise4uilibrium can be e!pressed in the error correction form "it incorporates theLR information into the model)Error correction term: represents the speed to which the model returns toe4uilibrium following a shock% #t should be negativel( signed "N movement awa(from e4uilibrium)

*rror correction model , unrestricted 7AR*rror correction term , structural 7AR

Cointegration tests23 6ngle "ranger met'od: used when 0 variable s(stem are possible #"2)cointegrated%#t assesses if the residual of an estimated e4uilibrium is stationar(%

(1): $retest the considered +ariables for their order of integration variables needto have the same order of integration% 9e use A?

1 #f #"d) go to step 01 #f #"-) use levels



"0) *stimation of the LR relationship run OLS G, b-Nb2HtN +t 1 #f cointegrated strong linear relationship between these 0 due to a commontrend% Bhe residuals contain estimated deviations from the LR relationship

":) Saving of residuals from 0)

">) Best whether this residual has a Fnit Root using 6ngle "ranger: usedi'erent critical values than A?K- 6o cointegrationKa cointegration

%) Jo'ansen is more generall( applicable 3 tests for several #"#) time series andallows for more than one cointegrating relationship% #t is based on ma!imumlikelihood relationship and 0 statistics ma!imum eigenvalues and the trace1statistics%

K- for trace1statistics number of cointegrating vectors rW,[K- for eigenvalues r,[One can assess the number of cointegrating relationships to be modeled%

"2) Bheor( considering the statistical anal(sis and good data "0) Stationarit( Anal(sis e(e1balling A& correlograms unit root tests "lags

speci8cation power) ":) &ointegration Anal(sis test considering lag length speci8cation of

cointegration tests and di'erent choices ">) Model it choose the ade4uate multi1e4uation model > t(pes "@) ?iagnostic &hecks



1 Length selection for 7AR$ 7*&M1 Overall goodness of 8t1 1test for ;oint signi8cance "ranger causalit()1 B1test for individual regression coe<cients1 Possible elimination of Iinsigni8cantJ lags1 Sign and siHe of error correction term1 orecast performance and model comparison with #&1 Presence of multicollinearit( autocorrelation and other possible OLS

assumptions "Q) Anal(sis of 8ndings

1 Short run d(namics "regression coe<cients)1 Long1run d(namics "coe<cient associated with error correction term)1 ranger causalit( h(pothesis testing impulse response1 &onnection to theor(

Time Series Summary

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