How o Deal wih Srucural Breaks in Pracical Coinegraion Analysis Roselyne Joyeux * School of Economic and Financial Sudies Macquarie Universiy December 00 ABSTRACT In his noe we consider he reamen of srucural breaks in VAR models used o es for uni roos and coinegraion. We give pracical guidelines for he inclusion and he specificaion of inervenion dummies in hose models. JEL Classificaion Code: C3, C5, E43. Keywords: srucural break, dummy variable, coinegraion,var models. * Correspondence should be sen o: Dr Roselyne Joyeux School of Economic and Financial Sudies Macquarie Universiy Sydney 09 Ausralia e-mail: rjoyeux@efs.mq.edu.au
. Inroducion The empirical lieraure making use of uni roo and coinegraion ess has been growing over he las wo decades. The applicaion of hose ess is challenging for many reasons including he reamen of deerminisic erms (consan and rend) and srucural breaks. Franses (00) addresses he problem of how o deal wih inercep and rend in pracical coinegraion analysis. In his noe we use Franses (00) approach o consider he reamen of srucural breaks in VAR models used o ess for uni roos and coinegraion. In wha follows we assume ha srucural breaks occur a known break poins. There is a vas lieraure on srucural breaks and uni roo ess. If a series is saionary around a deerminisic rend wih a srucural break we are likely o accep he null of a uni roo even if we include a rend in he ADF regression. There is a similar loss of power in he uni roo ess if he series presen a shif in inercep. If he breaks are known he ADF es can be adjused by including dummy variables in he ADF regression (Perron (989, 990), Zivo and Andrews (99) among ohers). In his noe we show how inervenion dummies should be specified and included in VAR models o es for uni roos and coinegraion. Noe ha here is nohing new in his noe, he maerial is basically covered in he Johansen, Mosconi and Nielsen (000) paper (JMN (000) hereafer). This noe, however, provides a simple explanaion of he specificaion of inervenion dummies which is no presen in he laer paper. A survey of he applied lieraure using Johansen s es for coinegraion in a VAR seing would reveal ha inervenion dummies are usually inappropriaely specified. Indeed in empirical work i is ofen he case ha srucural breaks have o be accouned for. The inclusion of inervenion dummies should improve he normaliy properies of he esimaed residuals. This is, however, ofen no he case. The reason for his is ha he dummy variables are incorrecly specified. I is he aim of his noe o show how o specify and include inervenion dummies and o make accessible o applied economiss he laes developmen in he use of inervenion dummies when esing for coinegraion. In Secion we presen he resuls in he univariae case and in Secion 3 we generalize o he mulivariae case. Secion 4 concludes.
. Univariae Case In his secion we look a processes which can be modelled as auoregressive processes wih possibly a rend or an inercep shif a some poin in ime. Shif in Inercep Model In his secion we consider a univariae ime series y, =,,,T which has a shif in mean a ime T, < T < T, and can be described by: y - µ = φ (y - - µ ) ε when T and y (µ µ )= φ (y - (µ µ )) ε when > T where ε is a whie noise process. The parameer φ is assumed o be he same in all subsamples. The model above is formulaed condiionally on he firs observaions of each sub-sample: y and y T. When φ <, one can say ha y is araced by µ for T and by (µ µ ) for > T. This model can be rewrien as: y (µ µ D )= φ (y - (µ µ D - )) ε () where D = 0 if T and D = if > T. If we le φ = in equaion () we ge y = y - µ (D - D - ) ε () µ is no idenified when φ = bu he shif in mean µ is. We can rewrie () as: y ε or = ( φ )y ( φ ) µ µ ( D φ D ) y ) µ D ε = ( φ )y ( φ )( µ µ D (5) (6) If we le ρ = φ, (6) can be rewrien as: y = ρ y ρ( µ µ D ) µ D ε (7) 3
Since D = 0 if T or if > T, and D = if = T, he effec of D corresponding o he observaion y T is o render he associaed residual zero given he iniial value in he second sub-sample. The inclusion of D does no affec he asympoic disribuion of he saisic of he esimaed coefficien of y -, ˆρ, under he null of a uni roo. This represenaion also illusraes ha when esing for a uni roo he es regression should include boh he lagged inervenion dummy and he firs difference of he inervenion dummy, even hough under he null of ρ = 0 he lagged dummy disappears. Perron (990) and Perron and Vogelsang (99) abulae he asympoic disribuion of he saisic of he esimaed coefficien of y -, ˆρ, under he null of a uni roo. A beer es would be o es for he join significance of he coefficien of y -, he inercep and he lagged inervenion dummy in (7). In he mulivariae case he es considered in his noe is indeed a join es of he above hypoheses. Shif in Trend Model In his secion we consider a univariae ime series y, =,,,T which has a shif in mean and a shif in rend a ime T, < T < T, and can be described by: y - µ -δ = φ (y - - µ - δ (-)) ε when T and y (µ µ ) (δ δ )= φ (y - (µ µ ) (δ δ )(-)) ε when > T where ε is a whie noise process. As before he model above is formulaed condiionally on he firs observaions of each sub-sample: y and y T. This model can be rewrien as: y (µ µ D ) (δ δ D ) = φ [y - (µ µ D - ) (δ δ D - )(-)] ε (8) Alernaively (8) can be wrien as: y = ρ y [ ρ( µ µ D ) µ D φ ( δ δd )] [ δ D ρ ( δ δd )] ε (9) 4
The effec of D and D, corresponding o he observaion y, is o render he T associaed residual zero given he iniial value in he second sub-sample. There is, however, no poin in including boh and 0 oherwise, and δ rewrie (9) as: D D and D in (9) since µ D = µ if = T = δ (T ) for = T and 0 oherwise. We can hus y = ρ y ρδ ρδ D η η D κ0 D ε (0) where η = -ρ µ φ δ, η = -ρ µ φ δ and κ 0 = µ δ (T ). As for he shif in inercep only case his represenaion shows ha he es regression should include boh he lagged inervenion dummy and he firs difference of he inervenion dummy. I also shows ha he lagged inervenion dummy should be included boh in he inercep and he deerminisic rend variable, even hough under he null of ρ = 0 he lagged dummy disappears in he rend componen (bu no in he inercep par). So he pracical rule would be o include in he es regression he inercep, a lagged dummy inercep, a firs difference dummy inercep, he rend, and he lagged dummy imes he rend. Perron (989) abulaes he asympoic disribuion of he saisic of he esimaed coefficien of y -, ˆρ, under he null of a uni roo. Noe ha he rend and he lagged dummy imes he rend disappear under he null. A beer es of he null of a uni roo es is a join es of he join significance of he coefficiens of y -, he rend and he lagged dummy imes he rend in (0). Generalizaion o an AR(k) process In he case where he process follows an AR(k) model wih AR coefficiens φ,...,φ k equaion (0) becomes: k k k ρkδ ρk δd k η η D k Γi y i κi D i ε () 0 y = ρ y where ρ k = φ φ... φ k - and he model is formulaed condiionally on he firs k observaions of each sub-sample. This represenaion shows ha he es regression should include boh he inervenion dummy lagged k periods, he firs difference of he inervenion dummy and up o k- lags 5
of he firs difference of he inervenion dummy. I also shows ha he inervenion dummy lagged k periods should be included boh in he inercep and he deerminisic rend variable, even hough under he null of ρ = 0 he lagged dummy disappears in he rend componen (bu no in he inercep par). A uni roo es should be a join es of he join significance of he coefficiens of y -, he rend and he dummy lagged k periods imes he rend in (). Generalizaion o he case of more han one shif We allow for q samples periods, = T 0 < T < T <...< T q = T. The las observaion of he jh sample is T j and he firs period of he (j)h sample period is T j, j=,...,q. The model is formulaed condiionally on he firs k observaions of each sub-sample, for example for he jh sub-sample, dummy variables : D j, and D j, k = 0 for T j oherwise, T j y,..., yt k, Tj j for T j k T j k, = 0 oherwise, Correspondingly we define: for j =,...,q. We also define q- inervenion for j =,...,q. for = T j, I j, =. 0 oherwise, When q =, I j, is jus D j,. I j,-i is an indicaor variable for he ih observaion in he jh period. Equaion (), in he case of q periods becomes: y = ρ k y k q κ 0 j= ρ δ ρ j,i I k j, i ε k q δ j= j D j, k η q η D j= j j, k k Γ y i i () Our noaion for he inervenion dummies differ from JMN (000). In his laer paper D j, denoes an indicaor funcion for he las observaion in he j-h sample. 6
As before he effec of I j,,..., I j, k, corresponding o he observaions y,..., y k T j T j, is o render he respecive residuals zero given he iniial values in each period. In pracice we need o include he inervenion dummies for each sub-sample wih he appropriae lags as well as he dummies imes he rend and he indicaor variables for he break poins, again wih he appropriae lags. 3. Mulivariae Case The mos common mehod o es for he coinegraion rank is he maximum likelihood coinegraion es mehod developed by Johansen (988, 996). I is, however, he case ha he inclusion of inervenion dummies affecs he disribuion of coinegraion ess. JMN (000) generalize he likelihood-based coinegraion analysis developed by Johansen (988, 996) o he case where srucural breaks exis a known poins in ime. They show ha new asympoic ables are required. In his secion we show how o obain equaion (.6) of JMN (000) by expanding he resuls from Secion. In wha follows we assume ha we have a p-vecor process Y and ha wihou srucural breaks he model can be formulaed condiionally on he firs k observaions by: k Y = ΠY Π µ Γ Y ε i i (3) where ε,...,ε T are normal, independen and idenically disribued p vecors wih mean 0 and variance Ω. We also assume ha alhough some or all of he p ime series in Y may have a ime rend, none have a quadraic rend. The hypohesis of coinegraion can be reformulaed as a reduced rank problem of he Π marix, in which case Π = αβ, where α and β are (p r) full rank marices, and Y has a quadraic rend. If none of he p ime series displays a quadraic rend we need o assume ha Π = αγ, where γ is a ( r) full rank marix. If we now assume ha we have q- breaks (and q sub-samples), condiionally on he firs k observaions of each sub-sample he model can be rewrien as q equaions: Y k ( Π, Π j ) µ j Γi Y i ε Y = j =,...,q, where Π j and µ j are (p ) vecors. (4) 7
Under he null of coinegraion, we resric he rend o he coinegraing relaionships o exclude he possibiliy of quadraic rends in any ime series. This means ha Π j = αγ j. Insead of wriing q equaions we can define he following marices: D = (,...,D q, ), µ = (µ,...,µ q ), γ = (γ,...,γ q ) of dimensions (q ), (p q), (q r) respecively, and rewrie (4) in a form similar o (): β Y k q k Y α µ D k Γi Y i κ j,i I j, i ε γ D = k 0 j= where he dummy variables D j,, D j,-k and I j, are defined as in he previous secion, and he κ j,i are (p ).vecors. JMN (000) develop a maximum likelihood coinegraion es mehod based on he squared sample canonical correlaions, ˆλ i, of regressors: D -k, Y i and ( Y,D ) Y correced for he, i =,...,k-, I j,-i, i = 0,..., k-; j =,...,q. The likelihood raio es saisic for he hypohesis of a mos r coinegraing relaions is given by: p (5) LR = T log( λ ˆ i ) (6) r We consider nex hree cases:. none of he p ime series displays a rending paern, bu he coinegraing relaions have an inercep which can differ beween he sub-samples;. some or all of he ime series follow a rending paern in each sub-sample and he coinegraing relaions are rend saionary in each sub-sample; rend breaks are allowed boh in he coinegraing relaions and in he non-saionary series; 3. some or all of he ime series follow a rending paern in each sub-sample and he coinegraing relaions are saionary in each sub-sample (wih possibly a broken consan level); rend breaks are allowed only in he non-saionary series; Shif in Inercep Model: None of he p ime series have a deerminisic rend The only deerminisic componens in he model are he inerceps in he coinegraing relaions which can differ beween sub-samples. In ha case we have: 8
Π = Π =... Π q = 0, moreover µ is resriced o he coinegraing relaions. The inerpreaion is ha he coinegraing relaions have an aracor µ j which varies beween sub-samples. This model is denoed by H c (r) in JMN (000). β Y k k q Y α Γi Y i κ j,i I j, i ε ν D = k 0 j= where αν = µ. JMN (000) show ha he asympoic disribuion of he likelihood raio es is well approximaed by a Γ-disribuion. The reader is referred o secion 3.4 of JMN (000) for he compuaion of he criical values depending on he number of non-saionary relaions and he locaion of he break poins. (7) Some or all of he ime series follow a rending paern This model allows he individual ime series o have broken rends, while he coinegraing relaions may also broken rends. This model is denoed by H l (r) in JMN (000). I is he mos general case and is represened by equaion (5). The derivaion of he criical values for his model is also given in secion 3.4 of JMN (000). Some or all of he ime series follow a rending paern in each sub-sample and he coinegraing relaions are saionary in each sub-sample (wih possibly a broken consan level); rend breaks are allowed only in he non-saionary series This model is denoed H lc (r) in JMN (000). The asympoic disribuion of he likelihood raio es depends on nuisance parameers and canno easily be obained. k k q µ D k Γi Y i κ j,i I j, i ε (8) 0 j= Y = αβ' Y Uni Roo Tess In he firs wo cases, models (5) and (7), JMN (000) also show ha es for linear resricions on β, γ and ν are asympoically χ -disribued. Such ess are paricularly useful because hey make i possible o es wheher he individual ime series are rend saionary on each sub-sample. 9
4. Conclusion In he las decade applied economericians have usually reaed srucural breaks in VAR models in an ad hoc fashion. Inervenion dummies have been included wih lile care given o heir specificaion. In his noe we have considered hree models of ineres in applicaions and have given a deailed accoun of he specificaion and inclusion of inervenion dummies in hose cases. Saisical heory for hose cases has been developed in JMN (000). Alhough here is no new saisical heory in his noe, he discussion of he inclusion and specificaion of inervenion dummies should be useful o applied economiss. I is indeed ofen he case ha including dummies does no solve he non-normaliy problems of he residuals encounered in he esimaion of VAR and VECM models. The reason for his should now be clear. 0
References Franses, P.H. (00), How o Deal wih Inercep and Trend in Pracical Coinegraion Analysis, Applied Economics, 33, 577-579. Johansen, S., Moscow, R. and B. Nielsen (000), Coinegraion Analysis in he Presence of Srucural Breaks in he Deerminisic Trend, Economerics Journal, 3, 6-49. Perron, P. (989), The Grea Crash, he Oil Price Shock and he Uni Roo Hypohesis, Economeric, 57, 36-40. Perron, P. (990), Tesing for a Uni Roo in a Time Series wih a Changing Mean, Journal of Business and Economic Saisics, 8, 53-6. Perron, P. and T.J. Vogelsang (99), Nonsaionariy and Level Shifs wih a Applicaion o Purchasing Power Pariy, Journal of Business and Economic Saisics, 0, 30-30. Zivo, E. and D.W.K. Andrews (99), Furher Evidence on he Grea Crash, he Oil- Price Shock, and he Uni Roo Hypohesis, Journal of Business and Economic Saisics, 0, 5-70.