Time Series Econometrics 10 Vijayamohanan Pillai N

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1 Time Economerics 0 Vijayamohanan Pillai N Time Mehodology 7 November 03 Vijayamohan: CDS MPhil: Time 7 November 03 Vijayamohan: CDS MPhil: Time Time : Mehodology Three alernaive approaches: (): general o specific model (GETS): LSE Approach Hendry, Pagan and Sargan (984) and Hendry (987). (): vecor auoregressions model (VAR): Sims (980): dominan approach in he USA; and (3): vecor error correcion model (VECM): Follows he Granger Represenaion heorem (Engle and Granger 987) GETS Approach Suppose, he heory implies ha here is a relaionship beween consumpion (C ) and income (Y ) i.e., say C α 0 + α Y () Since his is an equilibrium relaionship, a dynamic adjusmen equaion can be searched by saring firs wih a very general and elaborae specificaion. This iniial general specificaion is ermed he general unresriced model (GUM). 3 /7/03 0:9:4 AM Vijayamohan: M Phil: Time 4 GETS Approach A good GUM for he consumpion equaion: C αc + β 0 Y + β Y + ε ; where Y (Y, Y,., Y k ), C (α) )C + β 0 Y + ( (β 0 +β )Y +ε ; or C β 0 Y + λ [C k Y ] +ε+ ; where k (β 0 +β )/ ( α) ) and λ (α) GETS Approach A good GUM for he consumpion equaion in general: n βci C i + β yi Y i + λ( C ky ) ε i m C + i Include enough lagged variables so ha here is no serial correlaion in he residuals of he GUM. Finally, a parsimonious version of equaion () is developed, by deleing he insignifican variables and imposing consrains on he esimaed coefficiens. GETS is hus a highly empirical approach. CDS MPhil FQ Vijayamohan 5 CDS MPhil FQ Vijayamohan 6

2 Consumpion Income Consumpion Inflaion Income Residual Analysis ACF PACF

3 Consumpion Income Inflaion 3 /7/03 0:9:4 AM Vijayamohan: M Phil: Time PcGive: Tes: Dynamic Analysis PcGive: Tes: Dynamic Analysis CDS MPhil FQ Vijayamohan 7 8 3

4 Residual Analysis GETS Approach Noe: he equilibrium heoreical consumpion relaionship can be recovered from equaion() n βci C i + β yi Y i + λ( C ky ) ε i m C + i () by imposing he equilibrium condiion ha all he ACF PACF changes in he variables are zero, i.e., from he erm in he las par in (), since in equilibrium, equaion () will be λ(c ky ) C* ky*. /7/03 0:9:4 AM Vijayamohan: M Phil: Time 0 GETS Approach The expression in he lagged level variables, λ(c ky ) in equaion () n m C β C + β Y + λ ( C ky ) + ε ci i i i yi i is known as he error correcion erm and models ha include i are known as he error correcion models (ECM). I implies ha deparures from he equilibrium posiion in he immediae pas period will be offse in he curren period by λ proporion. Noe ha λ should be negaive. /7/03 0:9:4 AM Vijayamohan: M Phil: Time Developed as an alernaive o he large scale economeric models based on he Cowles Commission approach. Sims (980) argued: The classificaion of variables ino endogenous and exogenous, The consrains implied by he radiional heory on he srucural parameers, and The dynamic adjusmen mechanisms used in he large scale models, are all arbirary and oo resricive. VAR Models Include all variables as endogenous. For a se of n ime series variables y (y,y,...,y a VAR model of order p (VAR(p)) can be wrien as: y VAR Models k )' Ay + Ay Apy p + ε VAR Models Consider a wo-variable VAR() wih k. y z bz + cy + cz by + cy + cz + ε y + ε wih ε i.i.d(, σ ) and cov( ε y, ε z ) 0 i ~ 0 ε i z where he A s i are (n x n) coefficien marices and,,..., )' ε ( ε ε ε k is an unobservable i.i.d. zero mean error erm. In marix form: b b y c z c c c y z ε y + ε z 3 4 4

5 b BX VAR Models b y c z c X c y c z Γ + ε Srucural VAR (SVAR) or he Primiive Sysem ε y + ε z VAR Models y a a y e + z a a z e These error erms are composies of he srucural innovaions from he primiive sysem. To normalize he LHS vecor, we need o muliply he equaion by inverse B: e B ε B ( b b ) b b B BX B Γ X + B ε X + e A X VAR in sandard form (unsrucured VAR: UVAR). e e ( bb ) b b ε y ε z 5 6 e e VAR Models ε y + bε z b ε + ε y E(e z i ) 0 b b E( ε y + bε z ) σ y + bσ z Var(e ) E(e ) ime independen, and he same is rue for he oher one. Bu covariances are no zero: Cov(e,e VAR Models ) E(e e ) E[( ε y + bε z )( ε z + bε y )] ( b y σ z + bσ ) 0 So he shocks in a sandard VAR are correlaed. The only way o remove he correlaion and make he covar 0 is o assume ha he conemporaneous effecs are zero: b b 0 y bz + cy + cz + ε y z by + cy + cz + ε z 7 8 VAR Models Same regressors for all equaions; so model esimaion is sraighforward: ML esimaor OLS esimaor for each equaion. This propery populariy of VAR models. Noe: All variables in he reduced form equaions are endogenous,, and hence he equaions can be seen as he basic ARDL formulaion of a simple VAR model. 9 VAR / GETS / VECM The simple VAR models do no idenify srucural coefficiens (hence Srucural VAR models ) nor do hey ake seriously he relevance of uni roo ess. In GETS, alhough here is some awareness of he uni roo characerisics of he variables, he crucial heoreical relaionship, in he error correcion par, is specified in he levels of he variables. In conras, VECM, like VAR, reas all variables as endogenous, bu limis he number of variables o hose relevan for y a z a a y a z a paricular heory. e + e 30 5

6 VECM This mehod developed by Johansen (988) is undoubedly he mos widely used mehod in applied work. Models using his approach are also known as Coinegraing VAR (CIVAR) models. VECM can be seen as scaled down (reduced form)var model in which he srucural coefficiens are idenified. The heoreical basis of VECM: Granger Represenaion Theorem. 3 VECM Granger Represenaion Theorem (Engle and Granger 987): If a se of variables are coinegraed,, hen here exiss a VARMA represenaion for hem and an error error-correcing mechanism (ECM); for example, If Y and X are boh I() and have consan means and are coinegraed,, hen here exiss an ECM,, (wih he equilibrium error U Y β X ), of he form: Y λ U + lagged{ Y, X } + θ (L)ε, Where θ (L) is a finie polynomial in lag operaor L and ε is a whie noise. 3 VECM In Y λ U + lagged{ Y, X } + θ (L)ε, The equilibraing error in he previous period, U, capures he adjusmen owards long-run equilibrium, and he expeced ve sign error would correc in he long-run run. Noe: No feedback assumed from Y o X. If feedback assumed from Y o X, an addiional equaion: VECM in Saa Saisics > Mulivariae ime series > Vecor errorcorrecion model (VECM) X λ U + lagged{ Y, X } + θ (L)ε, where ε is whie noise. Level variables 33 7 November 03 Vijayamohan: CDS M Phil: Time 7 34 VECM in Saa VECM in Saa provides informaion abou he sample, he model fi, and he idenificaion of he parameers in he coinegraing equaion. The main esimaion able he esimaes of he shor-run parameers, + heir sandard errors and confidence inervals

7 VECM in Saa VECM in Saa. predic ce, ce. line ce ime The second esimaion able he esimaes of he parameers in he coinegraing equaion, + heir sandard errors and confidence inervals. Prediced coinegraed equaion q 960q 970q 980q 990q Vijayamohan: ime CDS M Phil 03: Time 38 VECM Remember: VECM, like VAR, reas all variables as endogenous, bu limis he number of variables o hose relevan for a paricular heory. For example, wih wo variables: Y and X, wo endogenous variables, wo possible CVs; If our equaion of ineres is Y f(x ) only,, ha is, If No feedback assumed from Y o X, or X f(y ), we need o consider and esimae only one, firs, equaion, unlike in VAR. VECM Thus we need o es for direcion of feedback: From X o Y : wheher Y is exogenous o X ; From Y o X : wheher X is exogenous o Y ; Tha is, we need o es for exogeneiy of variables: This exogeneiy es is: Granger non- causaliy es (Granger 969): VECM: Granger non- causaliy es: an Exogeneiy es Consider he following equaions: Y Σα i Y i + Σβ i X i + e, () X Σγ i Y i + Σδ i X i + e, () where he summaions are for some lag lengh k, and e and e are independenly disribued whie noises. () hypohesises ha he curren value of Y is relaed o pas values of Y iself and hose of X, while () posulaes a similar behaviour for X. VECM: Granger non- causaliy es: an Exogeneiy es Y Σα i Y i + Σβ i X i + e, () We have he following implicaion: X does no Granger-cause Y if, and only if, β i 0, for all i, as a group; Thus he measure of linear feedback from X o Y is zero (Geweke 98). Tha is, he pas values of X do no help o predic Y. In his case, Y is exogenous wih (Engle e al. 983). respec o X 4 4 7

8 X Σγ i Y i + Σδ i X i + e, () Similarly, we have he following implicaion: Y does no Granger-cause X, if, and only if, γ i 0 for all i as a group; he measure of linear feedback from Y o X is zero. VECM: Granger non- causaliy es: an Exogeneiy es Tha is, he pas values of Y fail o help predic X. Here X is exogenous wih respec o Y. If he coefficiens, hen Y Σα i Y i + Σβ i X i + e, () X Σγ i Y i + Σδ i X i + e, () lagged erms here direcions. VECM: Granger non- causaliy es: an Exogeneiy es is causaliy have significan or feedback non-zero in boh Granger non- causaliy es in STATA Granger non- causaliy es in STATA In Saa, firs run a VAR model: Saisics > Mulivariae ime series > Vecor auoregression (VAR) Level variables 7 November 03 Vijayamohan: CDS M Phil: Time Granger non- causaliy es in STATA Granger non- causaliy es in STATA

9 In Saa Granger non- causaliy es in STATA Granger non- causaliy es in STATA H 0 : x does no Granger-cause y. Saisics > Mulivariae ime series > VAR diagnosics and ess > Granger causaliy ess H 0 : x does no Granger-cause y. Y X VECM: Granger non- causaliy es: an Exogeneiy es Granger causaliy : concerned wih only shor run forecasabiliy, while Coinegraion: concerned wih long run equilibrium; Error correcion model (ECM) (differen) conceps ogeher. brings he wo Suppose Y and X are boh I() series and hey are coinegraed such ha u Y βx is I(0). VECM: Granger non- causaliy es: an Exogeneiy es This coinegraed sysem can be wrien in erms of ECM as: y λ u + lagged{ y, x } + θ (L)ε, () x λ u + lagged{ y, x } + θ (L)ε, () where θ (L)ε and θ (L)ε are finie order moving averages and one of λ, λ VECM: Granger non- causaliy es: an Exogeneiy es y λ u + lagged{ y, x } + θ (L)ε, () x λ u + lagged{ y, x } + θ (L)ε, () VECM: Granger non- causaliy es: an Exogeneiy es y λ u + lagged{ y, x } + θ (L)ε, () x λ u + lagged{ y, x } + θ (L)ε, () In he ECM, he error correcion erm, U, Granger causes Y or X (or boh). As U iself is a funcion of Y and X, eiher X is Granger caused by Y or Y by X. Tha is, he coefficien of EC conains informaion on wheher he pas values of he variables affec affec he curren values of he variable under consideraion. This hen implies ha here mus be some Granger causaliy beween he wo series in order o induce hem owards equilibrium

10 Granger non- causaliy es: Causaliy - es? Firs suggesed by Wiener (956): More properly called Wiener-Granger non- causaliy es. Economiss (e.g., Zellner 979) and even philosophers (e.g., Holland 986) quesion he very erm causaliy : To mean cause-effec relaionship, when here is only emporal lead-lag relaionship? No causaliy bu precedence as suggesed by Edward Leamer. Unforunaely several sudies o infer cause-effec relaionship! 55 Granger non- causaliy es: Causaliy - es? Adrian Pagan (989) on Granger causaliy: There was a lo of high powered analysis of his opic, bu I came away from a reading of i wih he feeling ha i was one of he mos unforunae urnings for economerics in he las wo decades, and i has probably generaed more nonsense resuls han anyhing else during ha ime. Pagan, A.R. (989), '0 Years Afer: Economerics ,' in B. Corne and H. Tulkens (eds)., Conribuions o Operaions Research and Economerics, The XXh Anniversary of CORE, (Cambridge, Ma., MIT Press). 56 Consider a (linear) filer Y k Φ X k k Consider a (linear) filer Y k Φ X k k The impac muliplier is: Y X k response in he oupu a ime o a uni pulse in he inpu a ime k: one-period muliplier or ransien response or impulse response: Y Y k + X X k 7 November 03 Vijayamohan: CDS MPhil: Time 57 7 November 03 Vijayamohan: CDS MPhil: Time 58 The firs order Auorregresive process AR(): Given he linear filer k Impulse response funcion Y k Φ X k The ime pah of all he periodical impulse responses: ACF ρ k γ k / γ 0 Φ k, k 0,,,.. Impulse responses ACF, as k, since Φ < : Sign of saionariy. +Φ: direc convergence; negaive Φ: oscillaory convergence. Y Y + k X k X 7 November 03 Vijayamohan: CDS MPhil: Time 59 7 November 03 Vijayamohan: CDS MPhil: Time

11 In PcGive Firs run a VAR 6 6 SYS( ) Esimaing he sysem by OLS (using Daa.in7) The esimaion sample is: 953 (3) o 99 (3) URF equaion for: CONS Coefficien Sd.Error -value -prob CONS_ CONS_ INC_ INC_ INFLAT_ INFLAT_ Consan U sigma.877 RSS URF equaion for: INFLAT Coefficien Sd.Error -value -prob CONS_ CONS_ INC_ INC_ INFLAT_ INFLAT_ Consan U sigma RSS URF equaion for: INC Coefficien Sd.Error -value -prob CONS_ CONS_ INC_ INC_ INFLAT_ INFLAT_ Consan U Vecor Pormaneau(): Vecor AR -5 es: F(45,395) [0.748] Vecor Normaliy es: Chi^(6) [0.678] Vecor heero es: F(7,73).88 [0.064] Vecor heero-x es: F(6,695).38 [0.89] sigma RSS Quarerly daa Response of 953 () o 99 (3).0 Shock from 0.5 Consumpion 0.0 Shock from Income Shock from-.5 Inflaion -5.0 Consumpion Income Inflaion CONS (CONS eqn) INC (CONS eqn) INFLAT (CONS eqn) CONS (INC eqn) INC (INC eqn) 0.0 CONS (INFLAT eqn) INFLAT (INC eqn).0 0 INC (INFLAT eqn) INFLAT (INFLAT eqn) Consumpion Income Inflaion 6 cum CONS (CONS eqn) 4 cum INC (CONS eqn) 0.75 cum INFLAT (CONS eqn) Shock from 3 4 Consumpion 0.50 Shock from Income 5-50 Shock from Inflaion cum CONS (INC eqn) 0 cum CONS (INFLAT eqn) -50 Accumulaed Response of 5 0 cum INC (INC eqn) cum INFLAT (INC eqn) cum INC (INFLAT eqn) cum INFLAT (INFLAT eqn)

12 In Saa Firs run a VAR or VECM. var cons inc infla, noconsan lags(/) In Saa Firs run a VAR or VECM. var cons inc infla, noconsan lags(/). irf creae order, sep(0) se(myirf) (file myirf.irf creaed) (file myirf.irf now acive) (file myirf.irf updaed). irf graph oirf, impulse(inc) response(cons) order, inc, cons sep 95% CI orhogonalized irf Graphs by irfname, impulse variable, and response variable 7 7

13 . All he series are I(0) Simply model he daa in heir levels, using OLS esimaion.. All he series are inegraed of he same order (e.g., I()), bu no coinegraed. Jus difference (appropriaely)each series, and esimae a sandard regression model using OLS. 3. All he series are inegraed of he same order, and hey are coinegraed. esimae wo ypes of models: (i) An OLS regression model using he levels of he daa. he long-run equilibraing relaionship beween he variables. (ii) An error-correcion model (ECM), esimaed by OLS. he shor-run dynamics of he relaionship beween he variables Finally, Some of he variables in quesion may be saionary, some may be I() and here may be coinegraion among some of he I() variables. ARDL Auoregressive-Disribued Lag. in use for decades, bu in more recen imes provide a very valuable vehicle for esing for he presence of long-run relaionships beween economic ime-series In is basic form, an ARDL regression model : (MARMA model) y β 0 + β y β k y -p + α 0 x + α x - + α x α q x -q + ε where ε is a random "disurbance" erm. "auoregressive y is "explained (in par) by lagged values of iself. "disribued lag" successive lags of he explanaory variable "x". Pesaran MH and Shin Y An auoregressive disribued lag modelling approach o coinegraion analysis. Chaper in Economerics and Economic Theory in he 0h Cenury: The Ragnar Frisch Cenennial Symposium, Srom S (ed.). Cambridge Universiy Press: Cambridge. Someimes, he curren value of x is excluded

14 Bounds Tesing Approaches o he Analysis of Level Relaionships M. Hashem Pesaran, Yongcheol Shin and Richard J. Smih Journal of Applied Economerics 6: (00) a new approach o esing for he exisence of a relaionship beween variables in levels which is applicable irrespecive of wheher he underlying regressors are purely I(0), purely I() or muually coinegraed. Two ses of asympoic criical values : one when all regressors are purely I() and he oher if hey are all purely I(0). These wo ses of criical values provide criical value bounds for all classificaions of he regressors ino purely I(), purely I(0) or muually coinegraed. Accordingly, various bounds esing procedures are proposed The ARDL / Bounds Tesing mehodology of Pesaran and Shin (999) and Pesaran e al. (00) has a number of feaures ha many researchers feel give i some advanages over convenional coinegraion esing. For insance: I can be used wih a mixure of I(0) and I() daa. I involves jus a single-equaion se-up, making i simple o implemen and inerpre. Differen variables can be assigned differen lag-lenghs as hey ener he model. 8 8 A convenional ECM for coinegraed daa : y β 0 + Σ β i y -i + Σγ j x -j + Σδ k x -k + φz - + e ; The ranges of summaion : from o p, 0 o q, and 0 o q respecively. z, he "error-correcion erm", is he OLS residuals series from he long-run "coinegraing regression", Sep : use he ADF/PP/KPSS ess o check ha none of he series are I(). Sep : Formulae he following model: y β 0 + Σ β i y -i + Σγ j x -j + Σδ k x -k + θ 0 y - + θ x - + θ x - + e ; y α 0 + α x + α x + v

15 y β 0 + Σ β i y -i + Σγ j x -j + Σδ k x -k + θ 0 y - + θ x - + θ x - + e ; almos like a radiional ECM. The error-correcion erm, z - replaced wih he erms y -, x -, and x - from y α 0 + α x + α x + v he lagged residuals series would be z - (y - - a 0 - a x - - a x - ), where he a's are he OLS esimaes of he α's. "unresriced ECM", or an "unconsrained ECM". Pesaran e al. (00) call his a "condiional ECM". Sep 3: y β 0 + Σ β i y -i + Σγ j x -j + Σδ k x -k + θ 0 y - + θ x - + θ x - + e ; The ranges of summaion : from o p, 0 o q, and 0 o q respecively Maximum lags are deermined by using one or more of he "informaion crieria" : AIC, SC (BIC), HQ, ec. Remember: Schwarz (Bayes) crierion (SC) is a consisen model-selecor Sep 4: A key assumpion in he ARDL / Bounds Tesing mehodology of Pesaran e al. (00) : The errors of he equaion mus be serially independen. Sep 5: We have a model wih an auoregressive srucure, so we have o be sure ha he model is "dynamically sable". Once an apparenly suiable version of he equaion has been esimaed, use he LM es o es he null hypohesis ha he errors are serially independen, agains he alernaive hypohesis ha he errors are (eiher) AR(m) or MA(m), for m,, 3,... Check ha all he associaed wih he model Sep 6: Now perform he " " y β 0 + Σ β i y -i + Σγ j x -j + Σδ k x -k + θ 0 y - + θ x - + θ x - + e ; Do a "F-es" of he hypohesis, H 0 : θ 0 θ θ 0 ; agains he alernaive ha H 0 is no rue. Sep 6: "Bounds Tesing" y β 0 + Σ β i y -i + Σγ j x -j + Σδ k x -k + θ 0 y - + θ x - + θ x - + e ; As in convenional coinegraion esing, we're esing for beween he variables. This absence coincides wih zero coefficiens for y -, x - and x - in he equaion: H 0 : θ 0 θ θ 0 89 A rejecion of H 0 implies ha we have a long-run relaionship. 90 5

16 Exac criical values for he F-es no available for an arbirary mix of I(0) and I() variables. Pesaran e al. (00) supply bounds on he criical values for he asympoic disribuion of he F-saisic. lower and upper bounds on he criical values. he lower bound is based on he assumpion ha all of he variables are I(0), and he upper bound : all he variables are I(). If he compued F-saisic falls below he lower bound conclude : he variables are I(0), so no coinegraion possible, by definiion. If he F-saisic exceeds he upper bound, conclude : we have coinegraion. Finally, if he F-saisic falls beween he bounds, he es is inconclusive. 9 9 Sep 7: If he bounds es proves coinegraion, esimae he long-run equilibrium relaionship beween he variables: y α 0 + α x + α x + v ; as well as he usual ECM: y β 0 + Σ β i y -i + Σγ j x -j + Σδ k x -k + φz - + e ; where z - (y - -a 0 - a x - - a x - ), and he a's are he OLS esimaes of he α's. Sep 8: Exrac" long-run effecs from he unresriced ECM. y β 0 + Σ β i y -i + Σγ j x -j + Σδ k x -k + θ 0 y - + θ x - + θ x - + e ; a a long-run equilibrium, y 0, x x 0, he long-run coefficiens for x -(θ / θ 0 ) and x -(θ / θ ) CDS MPhil FQ Vijayamohan 96 6

17 CDS MPhil FQ Vijayamohan 97 7