VAR analysis in the presence of a Changing Correlation in the Structural Errors

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1 VAR analysis in he presence of a Changing Correlaion in he Srucural Errors By Sephen G. Hall Imperial College Business School Naional Insiue of Economic and Social research Absrac In his paper an exension o sandard VAR analysis has been proposed which allows us o invesigae he possibiliy ha he correlaion srucure of he shocks hiing he sysem is changing over ime. The proposal is o esimae a VAR in he usual way and hen apply Orhogonal GARCH analysis o calculae he condiional covariance marix for each period. Calculaing Impulse responses is hen sraighforward based on any of he sandard echniques for idenifying he srucural VAR. his proposal is illusraed using a four variable VAR for GDP for he UK, France, Germany and Ialy. This shows very clearly ha sandard diagnosics do no deec he changing correlaion srucure very well bu ha his change can have profound effecs on he esimaed impulse response and he associaed policy conclusions which migh be drawn from hem. JEL: C1, C2,C3 Keywords; VAR, srucural change, GARCH Corresponding Auhor: Sephen Hall, The Imperial College Business School, 53 Princes Gae, London SW7 2PG, UK. s.g.hall@ic.ac.uk Acknowledgemens: Financial Suppor from ESRC gran No L is graefully acknowledged.

2 Inroducion VAR analysis and he use of impulse responses has become one of he main ways o analyse reasonable size daa ses following he key paper of Sims(198). I has a number of aracions and also some disadvanages; he lack of a need for convenional heoreical resricions is one obvious aracion alhough he need o impose idenificaion o calculae he impulse responses parly offses his. The widely used orhogonalised impulse responses has he disadvanage ha he ordering of he variables changes he resuls. More recenly he proposal of Pesaran and Shin(????) o use generalised impulse responses has overcome his disadvanage alhough a he cos of making he inerpreaion of he resuls a lile less clear cu. In his paper I wish o raise a quesion, which has no so far been addressed. Wha is he impac on VAR analysis if he srucural errors have a changing correlaion srucure? This quesion arguably has considerable relevance o a number of areas of grea policy ineres. The convergence debae, which has been conduced in many conexs, may be characerised largely as one of a changing correlaion srucure of shocks. For example, he issue of he how similar shocks are across counries dominaes he quesion of forming a moneary union. Clearly he convergence debae is all abou his marix of shocks changing is covariance srucure. The issue o be addressed here hen is wha should be he consequence of a changing correlaion srucure for VAR analysis and how migh we approach he modelling problems in a racable way. Secion 2 of his paper ses ou he basic srucure of he VAR problem when he srucural error covariance marix is ime varying. Secion 3 discusses some problems in implemening his general framework and discusses a racable framework. Secion 4 hen gives a shor empirical example and secion 5 concludes. 2. The Srucure of VAR analysis Consider he Srucural VAR model p A x = A x 1,2,... T 1. i = 1 i i + Bw + e, = where x =(x 1,x 2,,x m ) is an mx1 vecor of joinly deermined variables, w is a qx1 vecor of exogenous or deerminisic componens and A i and B are mxm and mxq coefficien marices. I also, as usual, assume ha E(e )=, bu in addiion ha E(e e )= Ψ, ha is ha he covariance marix of he srucural shocks is ime varying. For simpliciy I also assume ha x are weakly saionary, alhough he analysis carries over o a coinegraed VAR quie easily. The sandard reduced form VAR represenaion of his model is x = p i = 1 Φ i x i + Ψw + = ε, 1,2,... T 2. Where of course i 1 Ai, 1 Φ = A Ψ = A B and ε = A e he covariance srucure of he reduced form errors will hen be given by E( ε ε ' ) course ime varying. 1 1 Ψ = Σ = A A 1 ', which is of 2

3 This model now raises wo quesions: How should we esimae such a sysem and how should we calculae impulse responses. 3. Esimaion In principal he specificaion of he likelihood funcion for (2) is quie sraighforward once we have specified a parameric form for he evoluion of Σ. A naural choice would be o specify a mulivariae GARCH process and a number of alernaive specificaions exis in he lieraure, Kraf and Engle(1982), Bollerslev, Engle and Wooldridge(1988), Hall Miles and Taylor( 199), Hall and Miles(1992), Engle and Kroner(1995). If we define he VECH operaor in he usual way as a sacked vecor of he lower riangle of a symmeric marix hen we can represen he sandard generalizaion of he univariae GARCH model as VECH ( Σ ) = C + A( L) VECH ( ε ε ' ) + B( L) VECH( Σ 1) 3. where C is an (N(N+1)/2) vecor and A i and B i are (N(N+1)/2)x(N(N+1)/2) marices. Esimaion of such a model is, in principle, quie sraighforward as he log likelihood is proporional o he following expression. l = T = 1 ln Σ 1 + ε ' Σ ε 4. However his general formulaion rapidly produces huge numbers of parameers as N rises (for jus 1 lag in A and B and a 5 variable sysem we generae 465 parameers o be esimaed) so for anyhing beyond he simples sysem his will almos cerainly be inracable. A second problem wih his sysem is ha wihou fairly complex resricions on he sysem he condiional covariance marix canno be guaraneed o be posiive semi definie. So much of he lieraure in his area has focused on rying o find a parameerizaion which is boh flexible enough o be useful and ye is also reasonably racable. One of he mos popular formulaions was firs proposed by Baba, Engle, Kraf and Kroner, someimes referred o as he BEKK(see Engle and Kroner(1993)) represenaion, his akes he following form Ω = C' C + q A' ε ε ' A + B' Ω B 5. i i i i i= 1 j= 1 p j j j This formulaion guaranees posiive semi definieness of he covariance marix almos surely and reduces he number of parameers considerably. However even his model can give rise o a very large number of parameers and furher simplificaions are ofen applied in erms of making A and B symmeric or diagonal. Orhogonal GARCH Any of he mulivariae GARCH models lised above are severely limied in he size of model, which is racable. Even a resriced BEKK model becomes largely unmanageable for a sysem above 4 or 5 variables. An alernaive approach, however which can be applied, poenially o a sysem of any size ress on he use of principal componens and is someimes referred o as orhogonal GARCH (see Ding(1994). Consider a se of n sochasic variables X, which have a covariance srucure V. Principal componens hen produces a se of n variables (P), which conain all he variaion of X bu are also orhogonal o each oher. The sandard principal componen represenaion can be wrien as follows. 3

4 n X i = µ + ω p i=1 n 6. i j= 1 ij j so if all n principal componens are used each x i can be exacly reproduced by weighing he principal componens ogeher wih he correc loading weighs. Now by simply aking he variance of boh sides of his equaion we can see ha VAR( X ) = V = W ( VAR( P)) W ' = WΨW ' 7. The advanage of his is of course ha as he principal componens are orhogonal Ψ will be a diagonal marix wih zeros on all non diagonal elemens. From applying principal componens we know W, we hen simply have o derive a se of univariae GARCH models o each principal componen o derive esimaes of he condiional variance a each poin in ime and apply he above formulae o derive an esimae of he complee covariance marix V. The condiional variance may be obained from any chosen procedure (GARCH, EGARCH or even an EWMA model of he squared errors) There are however wo furher issues here; i) as he principal componens are ordered by heir explanaory power we ofen find ha a subse of hem produces a very high degree of explanaory power. I may hen only be deemed necessary o use he firs k principal componens. I is even suggesed ha his helps o remove noise from he sysem as he minor principal componens may be reflecing pure random movemens. This can easily be done bu i inroduces an error erm ino he principal componens represenaion above and he resuling covariance marix may no longer be posiive definie. ii) Equaion (7) above is rue exacly for he average of he whole period he principal componens are calculaed for bu i does no necessarily hold a each poin in he sample. So his is really only delivering an approximaion. I may hen be useful o apply he procedure o a moving window of observaions so ha he W marix also effecively becomes ime varying, Yhap(23) has carried ou an exensive Mone Carlo analysis of his echnique and i seems o work well up o sample sizes of 5. The suggesion being proposed here is hen o esimae a sandard VAR as in (2), which gives consisen parameer esimaes even if he covariance srucure of he errors is ime varying, and hen o use he orhogonal GARCH model o generae he ime varying covariance marix of he esimaed residuals. This process is only limied by degrees of freedom consrains on he VAR in he usual way so ha any convenional VAR, which is racable, may also have he error covariance srucure decomposed in his way. The Impulse Response Funcion In general he impulse response funcion may be simply described following Koop e al(1996) as, I( n, δ, Ω 1) = E( x+ n ε = δ, Ω 1) E( x+ n Ω 1) 8. where Ω is he informaion se a ime and δ is a vecor of shocks applied a ime. Differen choices of he srucure of δ characerise differen schemes of idenificaion. The 4

5 dominan procedure is he orhogonalised residuals originally proposed by Sims, here we simply define P ' = Σ 9. P Using a Cholesky decomposiion where P is a lower riangular marix hen, defining p i as he ih row of P we may define he orhoganalised impulse response for he ih variable by seing. δ = p ' iσ 1. where σ is a suiable scaling facor, normally one sandard deviaion. (1) Will of course be dependen on he ordering of he variables in he decomposiion, (9), as is well known. An alernaive mehod is he generalised impulse response of Koop e al(1996) or Pesaran e al(1998), here raher han using an idenifying assumpion such as orhogonaliy hey simply ake he esimaed srucure of he covariance marix so ha if we ake s i o be he ih row of Σ hen he generalised impulse response is generaed by seing δ = s ' iσ The issue being considered here is no which alernaive is preferable bu ha whichever mehod is used he resuls will vary over ime if Σ is ime varying. Also i is obvious ha once an esimae of Σ is available he applicaion of any of hese idenificaion schemes is enirely sraighforward, even on a ime varying basis. 4. An Example In his secion a sandard VAR model of GDP for he UK, France, Ialy and Germany is firs esimaed in he usual way and hen he condiional correlaion srucure of he errors is assessed using he orhogonal GARCH echnique above. I hen illusrae he poenial change, which can occur in he impulse responses if hey are calculaed based on he condiional covariance marix a differen poins in ime. This is an ineresing applicaion given he imporance of he quesion of he correlaion of economic shocks o he formaion of a currency union wihin he sandard opimal currency area lieraure and hence he issue of he UK s membership of he European Moneary Union. Clearly we should be making he judgemen over enry based on an esimae of he curren condiional correlaion of errors no some average over he pas. And if convergence has been aking place we migh expec his o show up in he condiional correlaion srucure of he residuals. The saring place is herefore o esimae a sandard VAR for he log of he four GDP series, quarerly daa is used for he period 1973Q3-22Q2. A VAR(2) is chosen based on he Schwars crieria and a series of join F ess of excluding each lag (he probabiliy level of hese ess were lag 1=., lag 2=.69, lag 3=.78 and lag 4=.15). The following diagnosics confirm ha his VAR seems o be reasonably well specified. 5

6 Germany France UK Ialy Serial Corr ARCH Heero Heero-X Figures in each cell denoe probabiliy levels, hence a figure less han.5 denoes rejecion a he 5% criical value. Serial corr is a es of up o 5 h order serial correlaion, ARCH is a es of up o a fourh order ARCH process, Heero ess for heeroskedasiciy relaed o any of he lagged variables, Heero-X allows for cross producs of all he variables. Given he analysis laer i is paricularly ineresing o noe ha his VAR has very lile signs of ARCH or Heeroskedasiciy. In addiion i appears o be quie sable as recursive esimaion shows very lile sign of insabiliy (a recursive one period ahead chow es finds only 1 period ou of 32 ess which exceed he 1%criical value, he resuls are deailed in appendix A). We hen apply he Orhogonal GARCH model o he residuals of his VAR. The sandard correlaion marix of he residuals is presened below Germany France UK Ialy Germany France UK Ialy Here we see a picure of a relaively high correlaion beween France and Ialy and France and Germany bu uniformly low correlaions beween he UK and he oher hree counries. Bu of course if here is any sysemaic movemen over ime in hese correlaions hen his able will in effec simply be recording he average. The deails of he orhogonal GARCH esimaion will no be given in deail as here is no paricular inuiion o aach o he GARCH models for he principal componens. Given he lack of serial correlaion in he residuals he mean equaion for each principal componen is specified as a simple consan and a GARCH(1,1) model was specified for he condiional variance. This seemed o be adequae in all four esimaed equaions. The resuls of he Orhogonal GARCH process are given in Figures 1 and 2. Figure 1 shows he condiional correlaions for he UK wih Germany France and Ialy while figure 2 shows he correlaion BETWEEN Germany France and Ialy. The Broad picure is fairly simple, he correlaion beween Germany France and Ialy seems o be quie sable wih lile sign of any rend. The correlaion wih he UK however show quie srong rends especially wih France and Ialy, he 197 s and early 198s exhibis quie srong variaions in he correlaion wih average values below.2 and some values acually being negaive, by he las half of he 199s he correlaion was much more sable and averages over.4. There has however been lile rend change in he correlaion wih Germany. Clearly he change which has aken place in he correlaion srucure could profoundly affec he impulse responses of his sysem no maer wha form of idenificaion srucure we use o idenify hem. To illusrae his poin we presen he Generalised impulse response o a shock 6

7 o Ialy calculaed firs on he basis of he covariance marix calculaed for 1979Q3 (which exhibis he mos exreme negaive correlaion for he UK) and for 22Q1 which is fairly represenaive of he final years of he period. The conclusions of hese wo picures are very clear; in he firs char he UK is an obvious oulier and responds very differenly o he oher European counries. In figure 4 he UK is responding in a way, which is almos idenical o he oher counries. 5. Conclusion In his paper an exension o sandard VAR analysis has been proposed which allows us o invesigae he possibiliy ha he correlaion srucure of he shocks hiing he sysem is changing over ime. The proposal is o esimae a VAR in he usual way and hen apply Orhogonal GARCH analysis o calculae he condiional covariance marix for each period. Calculaing Impulse responses is hen sraighforward based on any of he sandard echniques for idenifying he srucural VAR. his proposal is illusraed using a four variable VAR for GDP for he UK, France, Germany and Ialy. This shows very clearly ha sandard diagnosics do no deec he changing correlaion srucure very well bu ha his change can have profound effecs on he esimaed impulse response and he associaed policy conclusions which migh be drawn from hem. 7

8 UK-G UK-F UK-I Figure 1: The Condiional Correlaion beween he shocks o he UK and Germany, France and Ialy G-F G-I F-I Figure 2: The condiional Correlaion beween Germany France and Ialy. 8

9 Germany France UK Ialy Figure 3: Generalised Impulse Response based on he covariance marix in 1979Q Germany France UK Ialy Figure 4: Generalised Impulse Response based on he covariance marix in 22Q1 9

10 1. 1up LGEY 1% Appendix A: Recursive chow ess for he VAR(2) 1. 1up LFRY 1% up LUKY 1% 1. 1up LITY 1% up CHOWs 1%

11 References Bollerslev T. Engle R.F. and Wooldridge J.M. (1988) A capial asse pricing model wih ime varying covariances Journal of Poliical economy, Ding Z. (1994) Time series analysis of Speculaive reurns PhD hesis UCSD Engle R.F. and Kroner K.F.(1995) Mulivariae simulaneous generalized ARCH, Economeric Theory 11(1) Hall S.G. Miles D.K. and Taylor M.P. (199) A Mulivariae GARCH in mean Esimaion of he Capial Asse Pricing Model, in Economic Modelling a he Bank of England, edied by K. Paerson and S.G.B. Henry, chapman and Hall, London. Hall S.G. and Miles D.K.(1992) An empirical sudy of recen rrends in world bond markes, Oxford Economic Papers 44, Koop G. Pesaran M.H. and Poer S.M. (1996) Impulse Response Analysis in Non-linear Mulivariae Models, Journal of Economerics, 74, pp Kraf and Engle(1982) Auoregressive Condiional Heeroskedasiciy in Muliple ime series Unpublished manuscrip, UCSD. Pesaran M.H. and Shin Y.(1998) Generalised Impulse Response Analysis in Linear Mulivariae Models. Economics Leers, Vol.58, pp Simms C. (198) Macroeconomics and realiy Economerica, 48, pp1-48 Yhap B (23) The analysis of Principal Componen GARCH models in Value-a-Risk calculaions London Universiy PhD hesis 11

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