TESTING NONLINEARITIES ETWEEN RAZILIAN EXCHANGE RATE AND INFLATION VOLATILITIES * Crisiane R. Albuquerque ** Marcelo S. Porugal *** Absrac Tere are few sudies direcly addressing excange rae and inflaion volailiies, and lack of consensus among em. However, is kind of sudy is necessary, especially because ey can elp moneary auoriies o know price beavior beer. Tis aricle analyses e relaion beween excange rae and inflaion volailiies using a bivariae GARCH model, and erefore modeling condiional volailiies, fac largely unexplored by e lieraure. We find a semi-concave relaion beween ose series, and is nonlineariy may explain eir apparenly disconnecion under a floaing excange rae sysem. Te aricle also sows a radiional ess, wi non-condiional volailiies, are no robus. Resumo Exisem poucos esudos analisando, direamene, a relação enre as volailidades cambial e da inflação, e pouco consenso enre os mesmos. Todavia, al análise é imporane, especialmene por auxiliar a auoridade moneária no conecimeno do comporameno de preços. Ese arigo analisa a relação enre as volailidades da axa de câmbio e da inflação empregando um modelo Garc bivariado e, porano, modelando as volailidades condicionais, fao não explorado pela lieraura. Enconramos uma relação semicôncava enre ais séries, e esa nãolinearidade pode explicar sua aparene desconexão em um regime de axas de câmbio fluuanes. O arigo ambém mosra que eses radicionais, com volailidades nãocondicionais, não são robusos. Keywords: Excange rae, inflaion, volailiy, Garc models Palavras cave: Taxa de câmbio, inflação, volailidade, modelos Garc JEL classificaion: E31, F41 Área ANPEC: Área 3 Macroeconomia, Economia Moneária, Finanças ** Researc Deparmen, anco Cenral do rasil. E-mail: crisiane.albuquerque@bcb.gov.br *** Professor of Economics, Universidade Federal do Rio Grande do Sul (UFRGS), and associae researcer of CNPq. Email: msp@ufrgs.br
1 1. Inroducion Te sudy of excange rae volailiy s effecs sould be imporan for moneary policy decisions, since iger volailiy means iger uncerainy, wic may affec inflaion expecaions, a crucial variable in moneary policy decisions. Aloug e lieraure abou e impac of excange rae volailiy on inflaion is no as exensive as e one available for is pass-roug o prices, some auors iglig suc relaion. Weer e impacs are significan or no remains conroversial: some auors defend e absence of connecion beween excange rae and macroeconomic variables volailiies, wile oers sae e opposie 1. According o e firs group, excange rae volailiy is no imporan o macroeconomic variables, since empirical evidence sows a subsanial increase in e former during floaing excange rae regimes, wile e laer did no presen a similar rise in eir volailiies. Te second group finds evidence of suc relaion, being eier posiive or negaive, in sudies conduced under differen aims and approaces. Tis paper ess e exisence of a relaion beween excange rae and inflaion volailiies for e razilian case, and our conclusions in is paper could be classified in e second group, relaed especially wi e findings of Dixi (1989) and Seabra (1996). y developing an opimizaion model for e firm, e firs auor sows a rade flows and prices would depend on invesmen made on a fuure basis and, consequenly, on bo expecaions and iger momens of e disribuions involved. In consequence, e macroeconomic environmen affecs e paern of price canges. Hence, no only e level of e devaluaion bu also e volailiy of e excange rae would affec is pass-roug o prices. Seabra (1996), on is urn, uses a model of ineremporal opimizaion wi asymmeric adjusmen coss and sows a e criical value a leads a firm o inves is a funcion of uncerainy. If uncerainy is ig, e opimal decision will be o wai before making a movemen (wai-and-see sraegy), even wi e excange rae a a level a makes invesmen profiable. Tis aiude impacs on aggregae supply and, erefore, on inflaion. Oer ineresing works are ose of Haussmann, Panizza and Sein (1), wo find a negaive and significan correlaion in eir ess beween pass-roug and measures of volailiy, and Smi (1999), were a reducion in inflaion volailiy as a resul of an increase in excange rae volailiy was found in approximaely 31% of e cases. Te welfare approac recalled by Gos, Gulde, Osry and Holger (1997) and by Suerland () are also wor menioning. Te former sow a inflaion volailiy is lower under floaing and inermediae excange rae regimes for counries wi low inflaion, wile e laer sow a e sign of e relaion beween excange rae and inflaion volailiies will depend on e model s parameers. In is paper, we adop a more sopisicaed economeric meodology an ose applied so far in lieraure: insead of consrucing exogenous volailiy series (by compuing e volailiy of subsamples or rolling windows) we apply a bivariae GARCH model, working wi condiional volailiy series. Te purpose of is procedure is o adop a measure no sensiive o individual selecion crieria. Apar from a, by modeling e condiional eeroskedasiciy of excange raes, i is also a more suiable economeric ecnique. One of e conribuions proposed by is paper is o verify weer excange rae volailiy as impacs srong enoug on inflaion so a e moneary auoriy sould monior i, an approac sill scarce, especially in razil. Te oer one is o sow a radiional ess are no robus for is ype of sudy and a Garc-ype models are more suiable for suc analysis. Te paper is divided ino six secions, including is inroducion. Secion inroduces e eoreical model a led o e economeric ess, wile daa is presened in secion 3. Te resuls obained by e use of radiional meods (i.e.: uncondiional variance series) are presened in secion 4. Secion 5 sows e resuls of e bivariae GARCH model, and secion 6 concludes. 1 For e firs group, see, for insance, Krugman (1998), Obsfeld and Rogoff (), axer and Sockman (1988), Flood and Rose (1995), Obsfeld and Rogoff (), Rogoff (1) and Duare and Sockman (). For e second, Calvo and Reinar (a, b), arkoulas, aum and Cavaglan (), Wei and Parsley (1995), Andersen (1997), Smi (1999), Engel and Rogers (1), Devereux and Engel (3) and Cen (4), Cen (4), arone-adesi and Yeung (199), leaney (1996), and leaney and Fielding (). Obsfeld and Rogoff () call e apparenly disconnecion beween e excange rae volailiy and macroeconomic fundamenals as e excange rae disconnec puzzle (ERDP).
. Te eoreical model We derive an equaion relaing inflaion and excange rae volailiies o es for e exisence of a significan relaion beween em. Te approac o acieve suc equaion is based on leaney and Fielding (), wi slig modificaions. Te governmen as a uiliy funcion Z, of e arro and Gordon (1983) ype, o be maximized. Z is given by equaion (1), wic represens e case were e governmen of a counry faces a rade-off beween price sabilizaion and oupu grow above is equilibrium level. Z =.5π.5b( y y * k) (1) Were π is inflaion, y is e oupu level and y* is poenial oupu. Te erm b > is incorporaed by e auors, meaning e relaive weig given o oupu, and k > represens e inflaionary bias of e governmen. Te presence of b and k comes from e assumpion a a governmen could evenually aribue a iger weig o oupu grow o e derimen of price sabiliy. Te resricion imposed by e auors upon funcion Z consiss of an expecaions-augmened Pillips Curve, including e excange rae. Here, we ave e firs difference o e model of leaney and Fielding () since we will focus no on e real bu on e nominal excange rae. Our resricion will be a Pillips Curve for an open economy, including bo e forward-looking and e backwardlooking erm, as described in equaion () below. e * ex π a π + a π + a ( y y ) + a ( p + a s + ε () = o 1 1 3 ) 4 were p ex e is e foreign price level, s, e nominal excange rae and π e inflaion expecaion beween period and period +1. We also assume e excange rae following a random walk, as in many parial equilibrium sudies. Tus, we ave s = s -1 + η η ~ N(, σ η) (3) applying () and (3) o (1), and obaining e firs-order condiion for e maximizaion of Z wi respec o π, we ave e ex π = βao π + βa1π 1 + βa3 p + βa4s 1 + βa4η + βε + K' (4) were β b and ' = βa k = a + b K Some assumpion also mus be made concerning e beavior of π e. We, en, consider a inflaionexpecaions are of e form: e π = π -1 + ν (5) Tus, subsiuing (5) in (4) we ge a ex E[ π ] = ( βao + βa1) π 1 + βa3 p + K' (6) Te erms ε, η and ν are independens, erefore, inflaion variance given by var( π ) = β a E( υ ) + β a4 E( η ) + β E( ε ) (7) u, from (), we ave a E(ε ) is e inflaion variance. Hence, var( π ) = µ E( υ ) + µ 1E( η ) (8) β were µ = a 4 and µ (1 β ) 1 = µ a a Inflaion variance is, erefore, a funcion of ν (e variance of e sock expeced in in relaion o -1 inflaion) and of η (variance of e excange rae process). Wi (8), we may es for a relaion beween volailiies and we aim o do a by using a mulivariae GARCH model. However, due o e small sample available from e beginning of e floaing excange rae sysem in razil, i.e., January, 1999 o Sepember, 4 e large number of erms o be esimaed does no allow us o esimae a mulivariae GARCH model wi ree variables. Aside from a, e inflaion expecaions researc publised by e Cenral ank of razil sared only on April,, reducing our sample even furer. Terefore, we will assume a e variance measured by ν is consan and, ence, equaion (8) becomes: Var(π) = µ + µ 1 var(η ),
3 were µ =µ + var(ν ) is e new consan. Aloug e assumpion a ν is consan is srong, we may consider i. Table 1 sows e resul of a regression of π e agains π -1 and a consan. If our ypoesis a e sock expeced o +1 in comparison wi -1 is, on average, consan, en e residuals of is equaion sould be omoskedasic. As we may see, we accep e null ypoesis of omoskedasiciy, wic suppors our assumpion a ν is consan. 3 e esides, if we compue v = π π 1 one can noice a almos e enire series is wiin e inerval of one sandard deviaion from e mean, as sown in Grap 1, wi e longes period in wic i was ouside a band being from December o April 3. Te daa for π refer o e average marke expecaions for IPCA 4 inflaion in mon +1 as in e e las business day of mon -1, and ey are publised by e Invesor Relaions Group (Gerin) from e Cenral ank of razil. 5 Table 1: Esimaion of Equaion π e = c + π -1 meod: OLS; sample: :3 o 4:1 Variable Coefficien Sandard deviaion -saisics p-value π -1.1593.54.9385.49 C.4463.555 8.351. MA(1).6576.45 5.745. R.4584 Durbin-Wason 1.8755 adjused R.4376 Wie Tes for omoskedasiciy: (p-value).4354 Grap 1 Evoluion of ν = π e - π -1.9.4 -.1 3-5- 7-9- - 1-1 3-1 5-1 7-1 9-1 -1 1-3- 5-7- 9- - 1-3 3-3 5-3 7-3 9-3 -3 1-4 3-4 5-4 7-4 9-4 -.6-1.1-1.6 v mean + 1 sandard deviaion mean - 1 sandard deviaion mean + sandard deviaions mean - sandard deviaions 3. Daa Our sample was compued on a monly basis, from 1999:1 o 4:9, and daa used in our esimaions were e following: a) Price Index: Exended Consumer Price Index (IPCA), consumer price index publised by e razilian Insiue of Geograpy and Saisics (IGE), 6 December/1993=1 and considered by e Cenral ank of razil as e reference index in e inflaion argeing regime; b) Excange Rae: Excange rae R$/US$, selling prices, monly average; c) Exernal Prices: Producer price index (PPI), publised by e ureau of Labour Saisics 7 (commodiies, final goods). d) GAP: oupu gap. I was compued by subracing e indusrial producion series publised by IGE (used as a proxy for monly GDP) from e rend obained by e Hodrick-Presco filer. All series were seasonally adjused by e X-1 meod and, aferwards, aken in logarims (ln). Nex, uni roo ess were performed. All series, excep for gap ave uni roos, as sown in Table A1 in 3 Equivalen ess o π and s from equaions and 3 acceped e alernaive ypoesis of eeroskedasiciy. 4 Index of consumer prices considered by e Cenral ank in e inflaion argeing. 5 p://www4.bcb.gov.br/?focuseries 6 p://www.ibge.gov.br 7 p://www.bls.gov/daa
4 e appendix I, and, erefore, ey were aken in firs differences. Te series in firs difference of Price Index, Excange Rae and Exernal Prices are encefor referred o as IPCA, E and PPI, respecively. 4. Tess wi uncondiional volailiy As a firs sep, we followed e main procedures found in lieraure and made ess using uncondiional volailiies. In suc cases, volailiy is more ofen compued by e sandard deviaion from e mean in small samples, or by e variance wiin em. Tese samples are given eier by spliing e series ino small subsamples or by adoping rolling windows 8. In is paper, we oped for ree differen meods o calculae e uncondiional volailiy series. Te firs one is consruced by compuing e sandard deviaion from e mean in rolling windows wi 4, 6, 8 and 1 observaions in eac window (series are compued as e firs difference of e naural logarim of e variable on a monly basis). Te second one considers e variances, insead of e sandard deviaion. Finally, we esed a VAR beween e price index (IPCA) and e excange rae (E) and analyzed e resuling variance decomposiion. 4.1. Rolling Windows wi sandard deviaions Te volailiies compued by e sandard deviaions are presened in Graps and 3, were E_i and IPCA_i are e volailiies of E and IPCA, respecively, wiin a window of size i. I is possible o noe a e series are sensiive o e size of e window. As Table A. sows, e uni roo es for IPCA _i is also affeced by window size: IPCA_4 is saionary and so is IPCA_6, aloug we rejec e presence of uni roos in e former a a level of significance of 1%. However, IPCA_8 and IPCA_1 ave uni roos. Since E_i is always saionary, we compued e firs differences of IPCA_8 and IPCA_1, named d_ IPCA_8 and d_ipca_1, respecively. Grap Variances of IPCA (sandard deviaions from e mean) - Rolling Windows 8,E-5 7,E-5 6,E-5 5,E-5 4,E-5 3,E-5,E-5 1,E-5,E+ 1999M1 1999M5 1999M9 M1 M5 M9 1M1 1M5 1M9 M1 M5 M9 3M1 3M5 3M9 4M1 4M5 4M9 Grap 3 Variances of E (sandard deviaions from e mean) - Rolling Windows,3,,,1,1, 1999M1 1999M5 1999M9 M1 M5 M9 1M1 1M5 1M9 M1 M5 M9 3M1 3M5 3M9 4M1 4M5 4M9 IPCA_4 IPCA_8 IPCA_6 IPCA_1 E_4 E_6 E_8 E_1 Te esimaion resuls also are very sensiive o window size, as i can be seen in ables A.3 o A.6 in appendix I 9. In e four-mon window, e lagged erms of a variable in is respecive equaion and e effec of inflaion variance on excange rae variance are considered o be saisically significan. Wi regard o e six-mon window, ere are significan cross-erms. However, e Wald es sows a e sum of e lagged coefficiens of E_6 in e IPCA_6 equaion is no saisically differen from zero, and 8 asourre and Carrera (4) aribue e few macroeconomic sudies abou volailiy o e lack of a paern o define or o measure volailiy. According o em, e use of rolling windows, insead of subsamples, as e advanage of reducing informaion loss (resulan from e reduced sample size). However, is procedure is also limied due o e difficuly in deermining e ideal number of observaions in a window. In addiion, i may imply a ig correlaion beween e compued series, wic may affec e qualiy of esimaors, and aler e rue relaion beween e volailiies. For insance, once e excange rae regime varies over ime, a cerain window may conain wo differen regimes. 9 Te number of lags in eac VAR was cosen by aking ino consideraion e informaion crieria, absence of residual auocorrelaion (LM es), absence of correlaion beween variables, and parsimony. In all models e dummy variable d_m - wic assumes e uniy value for November was included, since in all series ere is a peak in a mon, probably associaed wi e confidence crisis. Is inclusion allowed us o correc problems of residual auocorrelaion or correlaion beween e variables found in e model. For similar reasons, e dummy variables d1999 in e four-mon window and d3_m1 in e 1-mon window were included. Te laer assumes e uniy value for April and May 1999 (peak in E_4) wile e former equals e uniy value for Ocober 3 (peak in IPCA_1).
5 e same appens o e lagged coefficiens of IPCA_6 in e E_6 equaion. Only e dummy and firs lag of a variable are significan in e equaion. In e eig-mon window, only E_8(-1) in e equaion for E_8 is significan, wile only e dummy is significan in e D_IPCA_8 equaion. However, in is VAR, e correlaion beween IPCA_8 and E_8 equals.43, wic may jeopardize e OLS esimaion. Finally, e VAR beween d_ipca_1 and E_1 repors e coefficien of E_1(-1) as e only significan one in e E_1 equaion. E_1(-1), E_1(-6) and E_1(-7) are significan in e d_ipca_1 equaion and, according o e Wald es, eir sum is saisically differen from zero a a 1% level. In sum, e relaion beween ose wo endogenous variables is sensiive o window size. Depending on e size seleced, we may accep or rejec a e excange rae variance affecs inflaion variance and e oer way round, as well as accep or rejec a lagged values of inflaion variance will affec i. 4.. Rolling Windows wi variances Once again, we ave series a are very sensiive o window size, as sown in Graps 4 and 5 (pi and ei are e volailiy series for IPCA and E, respecively, compued as e variance of e sample inside e window). Concerning saionariy, e only difference from e sandard deviaion case is a e variance of IPCA in e six-mon window is no saionary (able A.7 in e appendix I). Hence, we ook e firs difference of p6, p8 and p1, and named em as dp6, dp8 and dp1, respecively. Grap 4 Variances of IPCA (variances) Rolling Windows Grap 5 Variances of E (variances) Rolling Windows,9,8,7,6,5,4,3,,1, 1999M1 1999M5 1999M9 M1 M5 M9 1M1 1M5 1M9 M1 M5 M9 3M1 3M5 3M9 4M1 4M5 4M9,16,14,1,1,8,6,4,, 1999M1 1999M5 1999M9 M1 M5 M9 1M1 1M5 1M9 M1 M5 M9 3M1 3M5 3M9 4M1 4M5 4M9 p4 p6 p8 p1 e4 e6 e8 e1 Tables A.8 o A. in e appendix I sow e resuls of e four esimaed VARs 1. For e fourmon window VAR, only e lagged erms of eac variable are significan and, differenly from e previous case, e volailiy of IPCA would no affec e excange rae volailiy. As for e six-mon window, conrary o wa was observed in e sandard deviaion case, e only significan erms are e dummy and e firs lag of e excange rae volailiy in is own equaion. In e eig-mon window, we do no find e correlaion problem we found before bu, again, e only erm a is significan is e6(- 1) in e equaion for e excange rae variance. Finally, e VAR beween dp1 and e1 indicaes e1(- 1) as e only significan variable in e equaion for e1. In e equaion for inflaion variance, e coefficiens for e1(-1) and e1(-) are significan and e Wald es sows a eir sum is saisically differen from zero a a 1% level. In sum, we noice a e resuls differ from e ones obained in e case wi sandard deviaions concerning uni roo ess, e number of lags in e VAR and e significance of some variances. None of e models sowed a inflaion volailiy is affeced by is lagged erm, differenly from wa appens o excange rae volailiy. Wen i comes o cross-erms, we find a excange rae volailiy is significan in explaining inflaion volailiy in e 1-mon windows. Hence, one can realize a resuls are sensiive no only o window size bu also o e meod cosen o compue volailiy. In addiion, since ere are lagged effecs in e case of excange rae variance, we reinforce e adequacy of invesigaing a GARCH-ype model. 1.D_M was included for e six-mon window case
6 4.3. Variance decomposiion in a VAR model Te las exercise performed in is secion was o es a VAR beween e price index and e excange rae and o analyze variance decomposiion. Since bo series ave uni roos, as sown in Table 1, we firs esed for e presence of coinegraion vecors. As sown in Table A.1 in e appendix I, e Trace and Eigenvalue ess do no accep e null ypoesis of presence of a coinegraion vecor. For is reason, we will es a VAR beween e firs differences of price index (IPCA) and excange raes (E). In e variance decomposiion facorizaion by Colesky meod, we cose E preceding IPCA, since we consider e former o be more exogenous an e laer. Te Granger es may be used o give furer suppor in e ordering decision (able A.13 in e appendix I. However, since e correlaion beween e residuals is low (-.17<. ) 1 e order does no ave significan effecs over e resuls. Table A.14 sows e VAR resuls, wile Table A.15 presens e variance decomposiion. y analyzing e variance decomposiion in able A.15, we find a abou 3% of e movemens in IPCA in +1 may be explained by socks in E in period. Tere are increasing accumulaed effecs over ime, and socks in E explain around 4% of e movemens in IPCA afer 1 mons. A sock in IPCA, in is urn, does no ave an immediae effec on e sequence of E, owever i as lagged effecs, aloug on a smaller scale. Graps 6 o 9 sow ese decomposiions over ime, as well as e inerval of ± sandard errors. We noice a socks o e variables ave posiive effecs on eir sequences, and apar from e impac of IPCA on E, ey are differen from zero. Terefore, we canno rule ou e ypoesis a socks o e excange rae represened by η in equaion (3) mig affec inflaion. Grap 6 - Percen E variance due o E 5 Grap 7 - Percen E variance due o IPCA 1 15 9 1 5 8-5 7 1 3 4 5 6 7 8 9 1-1 1 3 4 5 6 7 8 9 1 Grap 8 - Percen IPCA variance due o E 8 7 6 5 4 3 1-1 1 3 4 5 6 7 8 9 1 Grap 9 - Percen IPCA variance due o IPCA 1 9 8 7 6 5 4 3 1 3 4 5 6 7 8 9 1 ased on e resuls presened in is secion, we may infer a e radiional measures used o verify weer ere is a relaion beween e volailiies of excange rae and macroeconomic variables (sandard deviaions or variances in subsamples) yield resuls a are sensiive o e subsample size, leading us o accep or rejec e significance of e relaion according o e window size we are working wi. Te variance decomposiion, in is urn, indicaes a socks o e excange rae affec inflaion variance. Since volailiy is also a measure of uncerainy, is resul sounds more inuiive an some of ose presened before: if e excange rae affecs inflaion and as delayed effecs (incomplee excange rae pass-roug in e sor run), socks o a variable will affec e uncerainy abou fuure inflaion. esides, an adequae excange rae model mus consider e presence of condiional eeroskedasiciy, as illusraed in Table A.16 in e appendix. In is case, i is necessary o generae volailiy series for bo variables in e same way - ence, o consider condiional variance for bo - and Te conclusion is e same if we consider only one lag. 1 Enders (1995) suggess, as a rule-of-umb, a a correlaion beween residuals of e variables <. is no srong enoug o affec e resuls in e Colesky decomposiion.
7 no simply compare e variance series obained from a GARCH (p,q) model for e excange raes wi an exogenous measure of inflaion volailiy. Furermore, we sow a variance decomposiion repors a socks o e IPCA affec is variance, jus as well as some of e resuls obained in e rolling window procedure sow us a IPCA volailiy is affeced by is pas values, reinforcing e applicaion of e es for a bivariae GARCH model wi E and IPCA. 5. Tess wi condiional variance ivariae GARCH Tesing a GARCH model requires, firs, some assumpion abou e mean equaions. We considered, erefore, ree differen cases. Te firs one is consised of only lagged erms of eac variable; e second, of a Pillips Curve for e IPCA equaion (according o equaion in Secion ) and e lagged values for e excange rae; e ird, of e Pillips Curve for e IPCA and a random walk wi drif for e excange rae (equaion 3 in Secion ). According o uni roo ess previously performed, bo variables were considered in firs differences of eir logarims. Considering bo e cross-correlograms and OLS models, we cose e number of lags in e equaions for IPCA and excange rae. 13 Wi regard o variance specificaions, we esed five differen opions: diagonal-vec (ollerslev, Engle and Wooldridge, 1988), consan correlaion (CCORR, from ollerslev, 199), full parameerizaion (Vec), e EKK resricion (Engle and Kroner, 1993) and e dynamic condiional correlaion (DCC, from Engle, ). Only under e EKK resricion convergence was acieved, and we consider some reasons for a furer aead in is secion. Te general form of mean, variance and covariance equaions under e EKK model are: Mean equaions IPCA = δ + δ IPCA E = γ + γ E 1 1 1 1 + γ E + δ E + ε, 1 + δ E 3 + + δ GAP Variance and Covariance equaions 14 1 = c = c = c 1 + a ε + a ε 1 1, 1 1, 1 + a a ε + a a ε + a 1 1, 1 1 1, 1, 1 1 a + ( a 1, 1, 1 1 a ε ε ε + a 1 + a ε a 1 1 ) ε 4 + a ε, 1, 1 ε 1, 1, 1 + g + g + δ PPI 1 + a 5, 1, 1 a 1 + g ε 1, 1 + ε + g g 1 1, + g 1 1, 1 g 1, 1 1 g + g, 1 1 + g + ( g, 1, 1 In order o make e analysis clearer, we renamed e coefficiens above as: c = α ; a = α ; a a = α ; a = α ; g = α ; g g = α ; g = α c = β ; a 1 1 1 1 = β ; a a 1 = β ; a 1 3 3 = β ; g 1 4 4 1 = β ; g g 1 c1 = µ ; aa 1 = µ 1; a1a + a1a 1 = µ ; a1a = µ 3; g1g = µ 4; gg + g1g1 = µ 5; g g Hence, e variance and covariance equaions can be rewrien as: = α + α ε + α ε ε + α ε + α + α + α 1 = β = µ 1 + β ε 1 + µ ε 1 1, 1 1, 1 1, 1 + β ε + µ ε 1, 1, 1 ε 1, 1, 1 ε 1, 1, 1 3, 1 + β ε 3 + µ ε, 1 3, 1 4, 1 + β 4 + µ 4, 1 5 = β ; g 5, 1 1 + µ 6 = β 6 5 1, 1 + β 5 1, 1 5 1, 1 6, 1 + β 6 + µ, 1 6, 1 g 1 + g = µ 6 1 g 1 ) 1, 1 + g 1 g, 1 For eac case, differen simulaions were made canging e convergence crieria and e number of ieraions. Terefore, i is possible a, for eac case, we ended up wi more an one resul acieving convergence. Wen is occurred, e coice was made based on e following crieria: LM and e ARCH-LM ess (i.e. absence of residual serial correlaion and of arc-ype residuals), calculaion of e eigenvalues o assure a e condiion of covariance saionariy was respeced (see Engle and Kroner, 1993 for furer deails on condiions and ess), and, wen all e previous were respeced, we cose e resul a maximized, for e case considered, e likeliood funcion. Te final resuls are presened in able. y analyzing Table, we noice a e resuls for e mean equaions are quie similar, as well as e values in e variance equaion for cases (1) and (). Case (3) differs from e oer wo bu, since a model as ARCH residuals for e equaion of E and serial correlaion of residuals for bo mean equaions 15, i canno be considered as a good model. 13 Wen ey poined o differen number of lags, we esed e iges one. 14 Variance and covariance equaions are from Engle and Kroner (1993), equaion.3, pages 5 and 6, wiou suppressing e GARCH erms. 15 We could no find any model a removed e auocorrelaion in e mean equaion of e excange rae, wic was expeced since ere are no lagged erms in a equaion.
8 Table ivariae GARCH Resuls Monly daa from 1999:1 o 4:9 Variables Case 1 Case Case 3 a Funcion Value 548.5888337 558.78734 548.59755 Consan.839*.19733*.49* (.5596) (.5458) (.51313) IPCA -1.557154*.5784983*.68315183* (.7376155) (.69667) (.596744) E -1 -.386384*.638358* Equaion for IPCA (.9841) (.779833) E - - -.73484 -.9641 (.964137) (.7317) GAP - -.167364**.155899** (.97743) (.95768) PPI -1 -.9498593**.185839* (.5367548) (.5499651) Consan.669639.19489*.191751* (.4451) (.4367) (.53443) Equaion for E E -1.8948*.675556* - (.4656) (.154998) - E - -.75191** -.1677685 - Condiional variance of IPCA Condiional variance of E Covariance (.13597143) (.1694474) α α 1 + + + α + + + α 3 + + + α 4 α 5 α 6 β β 1 + + β + + β 3 + + + β 4 + + + β 5 + + + β 6 + + + µ + + - µ 1 - - µ - - - µ 3 - - - µ 4 µ 5 µ 6 Noes: (a) case presens residual auocorrelaion in bo mean equaions (LM es); residuals of ARCH-ype in e excange rae equaion; sandard deviaions in pareneses; * and ** denoe significance a 5% and 1%, respecively. Comparing e variance equaions in cases (1) and (), we see a e differences lie in e signs of g 1 and g, in e values of a, a and a 1 and in e significance of coefficiens µ 1, β 1 and β, a is, e impac of ε 1,-1 on e condiional variance of E (firs difference of e excange rae) and in e covariance and impac of ε 1,-1ε,-1 on e condiional covariance of E. However, i can be seen from Table 3 a e significance of µ 1, β 1 and β is e only significan difference beween bo cases. Te difference in e signs of g 1 and g does no affec e final resul because ese coefficiens are considered under ree siuaions: (i) squared values; (ii) muliplied by eac oer, (iii) muliplied by coefficiens a are saisically equal o zero. Te differences in a, a and a 1, in eir urn, fall wiin sandard deviaion boundaries, us, ey may no be considered o be significan. I is imporan o noice a for e inflaion equaion all cases provided e same signals and e same
9 significance (i.e. if saisically equal o or differen from zero). Terefore, our resuls for e response of IPCA o socks in E are robus. Table 3 - Esimaed Parameers in Variance and Covariance Equaions Monly daa from 1999:1 o 4:9 G -.141899 -.51387.39856* (.13487) (.99365) (.1496) g 1.81385.185598 -.743866* (.168344) (.1418979) (.156595) g 1 1.56518* -13.13891* 13.135566* (1.551793) (1.6838653) (1.988477) g.417613** -.43513845*.51845* (.3448881) (.1488) (.7538) a.749157*.443837* -.4443478* (.18945) (.1313784) (.17146475) a 1.59535*.53544* -.5393149* (.13164) (.7896) (.18349) a 1-3.4461647** -5.78533* 7.48698 (1.9813913) (.196985) (.434849) a -.46199166* -.5917669*.7865576* (.146473) (.144479) (.16694871) c -.16 -.8.6 (.76) (.455714) (.519676) c 1.834*.17565* -.189984* (.37541) (.3354) (.3171) c.366595.38599 -.46995 (.836578) (.9184) (.58) (.836578) (.9184) (.58) Te Wald Tes was performed o decide beween e cases considered. Te unresriced case a is, case () was preferred o e derimen of cases (1) and (3), as sown in Table 4. Hence, we will consider case () as our resuls from now on. Table 4 Wald Tes 16 Cases esed Observed χ saisic q Null ypoesis: Variables added in case () are no joinly significan Case (1) vs Case (). Rejec Case () vs.case (3) 1.3 Rejec Tables 5 o 8 sow e resuls of e Ljung-ox and LM ess for auo-correlaion of residuals, e Arc-LM es for Arc-ype residuals, e mulivariae Pormaneau es for cross-correlaion and e eigenvalue vecor 17 for case. As i can be seen from ese ables, e model esimaed in case respecs e condiions of no serial auocorrelaion or cross-correlaion of residuals, no Arc-ype residuals and is covariance saionary. Terefore, we can say a e dependence beween excange rae and inflaion volailiies was compleely capured by e bivariae-garc model. y analyzing e resuls of case, sown in e second column of Table, one can noice a e condiional variance of IPCA is affeced (saisically significan) by socks o e IPCA, E and socks common o bo. However, since α 1 and α 3 are square coefficiens, we canno deermine weer e effecs of IPCA and E socks ave a posiive or negaive sign, bu we can affirm a ey are saisically significan. Lagged variances and covariances, owever, do no play a significan role in explaining IPCA variance. 16 Wald Tes: -(l r -l u ) ~ χ q, were q is e number of added variables, l r and l u are e log-likeliood of e resriced and unresriced cases, respecively. Under H o, e added variables are no joinly significan. 17 For a brief explanaion of e eingevalue calculaion, see Appendix II.
1 As for e condiional variance of E, i is affeced by is lagged values and by lagged values of e condiional variance of IPCA e laer goes undeeced by almos all ess wi uncondiional variances aloug we also canno make asserions abou e sign. Socks common o bo variables (ε 1,-1 ε,-1 ) and in e covariance ave a posiive and significan sign. Grap 1 sows e esimaed condiional variances over ime. Ljung-ox Table 5 Lung-ox Tess for Residual Auocorrelaion E1 (residuals of inflaion Equaion) E (residuals of excange rae equaion) Q-Saisics Significance Level Q-Saisics Significance Level Q(1-).41.9491.934.7599 Q(-).8666.6484.151.977 Q(3-) 1.4555.696.1737.9817 Q(4-) 1.887.7577 1.716.866 Q(5-) 1.9857.85 1.7955.8766 Q(6-) 3.571.776.3153.8885 Q(7-) 3.6674.817.963.8935 Q(8-) 6.37.67 6.3.645 Q(9-) 6.5844.683 6.3149.78 Q(1-) 6.8781.7369 7.565.675 Q(-) 7.545.7536 1.845.346 Q(1-) 13.88.38 1.9173.3751 Qui-square criical level a 5% Table 6 LM and ARCH-LM Tess N*R values 18 Inflaion Equaion Excange Rae Equaion Lags LM es Arc-LM es LM es Arc-LM es 1 3.8415 1.197.98 3.355.4656 5.9915 4.993.154 4.145 3.63 3 7.8147 5.5838 6.5384.758 3.378 4 9.4877 6.76 7.8664 5.336 3.6771 5.75 7.8937 8.94 7.583 3.4554 6 1.5916 8.5463 8.9747 9.336 4.765 7 14.671 1.4 8.9349 13.569 4.779 8 15.573 14.68 9.1488 18.593 4.938 9 16.919 14.917 9.55 18.17 5.4148 1 18.37 14.8134 9.9457 18.35 5.354 19.6751 15.1475.377.7173 5.7317 1 1.61 19.79 9.759.341 1.397 Table 7 Mulivariae Pormaneau Tes for Cross-Correlaion 19 M Tes Saisics Significance Level 3 5.6954.337 5 1.948.536 7 16.5348.7389 1 5.13.8394 1 36.776.683 15 45.643.766 18 Te N*R value mus be < an e χ o accep e null ypoeses of no auocorrelaion and arc residuals. 19 H : ρ 1 = ρ = = ρ m = and H a : ρ i for some i Є {1,,m} (See TSAY,, for e mulivariae Pormaneau es for crosscorrelaion).
Condiional Variance of IPCA 1.E-4 1.E-4 8.E-5 6.E-5 4.E-5.E-5.E+ Table 8 Eigenvalue Vecor.663 -.45 y = -.77.637 Grap 1 Condiional Variances - IPCA and E..1.1.1.1.1... Condiional Variance of E 1-99 4-99 7-99 1-99 1-4- 7-1- 1-1 4-1 7-1 1-1 1-4- 7-1- 1-3 4-3 7-3 1-3 1-4 4-4 7-4 Condiional Variance of IPCA Condiional Variance of E Finally, resuls sow us a socks in e excange rae (E) and socks common o excange rae and IPCA ave negaive and significan effecs over e covariance beween e wo variables. Tis is an imporan resul in our model. I means a socks a affec e excange rae or e excange rae and IPCA simulaneously will cause a disconnecion of ese wo variables. Afer all, everying else e same, a reducion in e covariance means a reducion in e correlaion coefficien beween e variables. A firs, we considered a e lack of convergence for specificaions oer an e EKK model would resul from e small size of our sample (January, 1999 o Sepember, 4). However, is may be quesioned since e EKK specificaion as more parameers an some of e oer specificaions esed. Te negaive sign of socks in E over e condiional covariance (µ 1 < ) and e dispersion graps presened below (graps o 14) sugges a e sign of socks in E over e condiional variance of IPCA may no be e same all e ime. If is is rue, en we may ave a reason for e nonconvergence of specificaions a, insead of working wi squared erms (imposing e posiiviy of e marix), ry o find a sign for e relaion. In ese specificaions, if e signs of a coefficien in e equaions of excange rae s and inflaion s variances cange from posiive o negaive, ey will no converge o a final value, since e model will ave o esablis weer e coefficien is posiive or negaive. In e EKK specificaion, owever, is problem does no exis once i works wi square coefficiens. However, furer ess are necessary before we can make suc asserion. Graps o 14 are dispersion graps wi e condiional variances of E on e orizonal axis and of IPCA on e verical axis. Grap plos e enire sample and one can clearly see four ouliers in a grap, wic correspond o e period beween February and May 1999 (i.e. e firs mons afer e cange in e excange rae regime, caused by e 1999 crisis, and before e adopion of e inflaionargeing regime in June of a year). Hence, we excluded ese observaions and buil Grap 1. Again, five ouliers were removed o consruc Grap 13 (June 1999, November, December 1, December and January 3). Grap 14, in is urn, was buil using only e region wi e iges concenraion of observaions (57% of e sample). Te observaions removed from Grap 14 are, relaed o Grap : January o July 1999; November 1999 o January ; Marc, Augus, Ocober and November ; April 1; December 1 o Marc ; Ocober o Marc 3; May, July, Sepember and November 3 and Augus 4.
1 Grap Excange rae and inflaion volailiies (full sample).1.1 Condiional Variance of IPCA.8.6.4 Grap 14 Grap 13 Grap 1...4.6.8.1.1.14.16 Condiional Variance of E Grap 1 Excange rae and inflaion volailiies (reduced sample),3,5 Condiional Variance of IPCA,,15,1,5 Grap 13,5,15,5,35,45,55,65 Condiional Variance of E Grap 13 Excange rae and inflaion volailiies (reduced sample),14,1 Condiional Variance of IPCA,1,8,6,4 Grap 14,,5,1,15,,5,3 Condiional Variance of E Grap 14 Excange rae and inflaion volailiies (reduced sample),65,6 Condiional Variance of IPCA,55,5,45,4,35,3,7,9,,13,15,17,19 Condiional Variance of E
13 Grap 1 and, mainly, grap 13 sugges a semiconcave (if no concave) relaion beween e wo variables a sake (i.e. e condiional variances of excange rae and of IPCA). To illusrae e relaion, a rend was included in ose graps and in grap 14, and e adjused R of eac rend equaion was repored (see graps A.1 o A.3 in e appendix I). Te semiconcave relaion would imply a, aloug e response of inflaion volailiy o excange rae volailiy is posiive, e proporion in is variaion decreases as excange rae volailiy rises. If we consider grap 14, wic plos e region wi e iges concenraion of observaions, we find a clear concave relaion. Tis would mean a, afer a cerain poin, e posiive relaion beween volailiies becomes negaive, as opposed o e convex form observed in financial variables (e so-called smile of volailiy). Also, i is possible a i is reflecing e exisence of a regime swicing in e volailiies, since we are removing e exreme values of e sample. We need a longer sample o es if is beavior would be reproduced over ime. However, we can only use grap 14 o speculae abou ese possibiliies appening. Noneeless, i is a quesion o be answered in fuure researc, since some works of oer auors find, as we poined ou in e inroducion, e sign may cange according o e model s parameers. 6. Conclusions Te analysis presened in secions 4 and 5 sow a e use of uncondiional variances leads us o resuls a are sensiive o e cosen measure of volailiy, wic is based on subjecive crieria. Te mulivariae GARCH model, dealing direcly wi e effecs of condiional volailiies, finds a semiconcave relaion (differenly from e case for financial series, were is relaion as a convex form), saisically significan, beween excange rae and inflaion variances. Te resuls seem o be in line wi e inuiion obained from oer sudies, especially Dixi (1989) and Seabra (1996). Wen excange rae volailiy is very ig, increasing uncerainy, inflaion response may be reduced, leading o smaller effecs. Tis may explain wy some sudies o razil found a decrease in e sor-run pass-roug from excange raes o consumer prices afer e floaing regime. Te analysis based on e role played by uncerainy could also make a bridge beween e wo differen poins of view concerning e exisence of a relaion beween e volailiies of excange rae and inflaion. Te relaion would exis bu, under cerain condiions, e disconnecion beween e variables would be oo srong o be noiced. In periods of ig volailiy, agens will no respond wi e same inensiy as ey do in periods of sabiliy due o e lack of knowledge concerning e duraion of e movemens in e excange rae (weer emporary or permanen). Terefore, inflaion volailiy as smaller ampliude. On e oer and, wen excange rae volailiy is lower, inflaion would respond more promply 1. Tis disconnecion becomes clearer in e negaive sign found in e response of e condiional covariance o socks in e excange rae and would be reinforced if e sign reversion found in grap 14 is verified in fuure sudies. Te caveas of is paper basically lie in e small sample available for razil, since e floaing regime for excange raes aving sared only in 1999. ecause of a, we canno esablis wi cerainy weer e problems faced wi convergence were due o e sign insabiliy or o e small period involved. Noneeless, we end no o rely oo muc in e small sample explanaion, since ree ou of e oer four resricions esed diagonal VEC, CCORR and DCC ave less parameers o be esimaed. Noneeless, a large sample is essenial o corroborae e resuls. However, is aricle innovaes by (i) applying a mulivariae GARCH model, us, considering condiional variances o analyze e relaion beween volailiies, (ii) rying o esablis a relaion beween excange rae and inflaion volailiies and is possible implicaions for moneary policy and (iii) sowing a radiional ess performed wi exogenously consruced volailiy series are sensiive o e crieria cosen o consruc suc series and do no reveal relevan feaures of a relaion. 1 For insance, in an environmen wi fixed excange raes, e agens know a devaluaion is permanen. Terefore, facing a new level of e excange raes, ey need o adjus eir coss. Te same is no rue under a floaing sysem: agens ave coss o adjus eir prices o a new level of excange raes and coss o reurn o e original posiion if e (de)valuaion is no permanen.
References Andersen, T. M. (1997) Excange rae volailiy, nominal rigidiies and persisen deviaions from PPP, Journal of e Japanese and Inernaional Economies,, pp. 584-69 arkoulas, J. T.; aum, C.F.; Caglayan, M. () Excange rae effecs on e volume and variabiliy of rade flows, Journal of Inernaional Money and Finance, 1, pp. 481 496 arone-adesi, G.; Yeung,. (199) Price flexibiliy and oupu volailiy: e case for flexible excange rae, Journal of Inernaional Money and Finance, 9, pp, 76 98 arro, R.J., Gordon, D.. (1983) Rules, discreion, and repuaion in a model of moneary policy, Journal of Moneary Economics, 1, -1 asourre, Diego; Carrera, Jorge (4) Could e Excange Rae Regime Reduce Macroeconomic Volailiy? In: Lain American Meeing Of Te Economeric Sociey (LAMES), Saniago, 7-3, July, Analls, CD-ROM axer, M.; Sockman, A.C. (1988) usiness cycle and e excange rae sysem: some inernaional evidence, NER, working paper no. 689, Augus leaney, M. (1996) Macroeconomic sabiliy, invesmen and grow in developing counries, Journal of developmen economics, vol. 48, pp. 461-477 leaney, M.; Fielding, D. () Excange rae regimes, inflaion and oupu volailiy in developing counries, Journal of Developmen Economics, v. 68, pp. 33-45 ollerslev T., Engle, R. F. and Wooldridge, J. M. (1988) A Capial Asse Pricing Model wi Timevarying Covariances, Journal of Poliical Economy, v. 96, no. 1 ollerslev, Tim (199), Modelling e coerence in sor-run nominal excange raes, Review of Economics and Saisics, 7, pp. 498 55. Calvo, G.; Reinar, C. (a) Fear of Floaing, NER, working paper no. 7993, November (b) Fixing for your life, NER, working paper no. 86, November Cen, N. (4) Te beaviour of relaive prices in e European Union: A secoral analysis, European Economic Review, 48, pp. 157-186 Devereux, M..; Engel, C. (3) Moneary Policy in e Open Economy Revisied: Price Seing and Excange Rae Flexibiliy, Te Review of Economic Sudies, Ocober Dixi, A. (1989) Hyseresis, impor peneraion, and excange rae pass-roug, Te Quarerly Journal of Economics, CIV, May, pp. 5-8 Duare, M.; Sockman, A.C. () Commen on: Excange rae pass-roug, excange rae volailiy, and excange rae disconnec, Journal of Moneary Economics, v. 49, n. 5, July, 941-946 Enders, W. (1995) Applied economeric ime serie, Jon Wiley & Sons, Nova York Engel, C.; Rogers, J. H. (1) Deviaions from purcasing power pariy: causes and welfare coss, Journal of Inernaional Economics, 55, pp. 9-57 Engle, Rober F. () Dynamic condiional correlaion a simple class of mulivariae GARCH models, Journal of usiness and Economic Saisics, v., n.3, July Engle, R.F.; Kroner, K. F. (1993) Mulivariae Simulaneous Generalized ARCH, Universiy of California, San Diego, discussion paper 89-57R, June Flood, R.P.; Rose, A. K. (1995) Fixing excange raes a virual ques for fundamenals, Journal of Moneary Economics, 36, pp. 3 37 Gos, A. R.; Gulde, A.M.; Osry, J. D.; Holger, C. W. (1997) Does e nominal excange rae regime maer? NER, working paper no. 5874, January Hamilon, J. D. (1994) Time series Analysis, Prenice Hall, London Haussmann, R.; Panizza, U.; Sein, E. (1) Wy do counries floa e way ey floa? Journal of Developmen Economics, vol. 66, pp. 387 414 Krugman, P. (1988); Excange Rae Insabiliy, Cambridge, MA: Te MIT press Obsfeld, M; Rogoff, K. () Te six major puzzles in inernaional macroeconomics: is ere a common cause? NER, working paper no. 7777, July Rogoff, K. (1) Perspecives on excange rae volailiy in: Feldsein, M. (ed), Inernaional Capial Flows, Cicago: Universiy of Cicago Press, 441-453 14
15 Seabra, F. (1996) A relação eórica enre incereza cambial e invesimeno: os modelos neoclássico e de invesimeno irreversível, Políica e Planejameno Econômico, v. 6, n., pp. 183, Augus Smi, C. E. (1999) Excange rae variaion, commodiy price variaion and e implicaions for inernaional rade, Journal of Inernaional Money and Finance, 18, pp, 471 491 Suerland, A. () Incomplee pass-roug and e welfare effecs of excange rae variabiliy, CEPR, Discussion Paper No 3431 Tsay, Ruey S. () Analysis of Financial Time Series, Jon Wiley & Sons, Inc Wei, S.; Parsley, D. (1995) Purcasing power dispariy during e floaing rae period: excange rae volailiy, rade barriers and oer culpris, NER, working paper no. 53, February Appendix I Tables & Graps Table A.1 ADF Uni Roo Tes sample: :3 o 4:1 ADF es saisics firs Variable ADF es saisics Criical value a 5% difference of e variable Criical value a 5% Price Index -.174 (a) -3.4783-3.9417 (a) -3.4783 Excange Rae -1.797 (a) -3.4783-7.47513 (a) -3.4783 Exernal Prices -1.5419 (a) -3.4783-8.14687 (a) -3.4793 GAP -9.718 -.977 - - Noe: es performed wi (a) rend or inercep and (b) wiou rend. Table A. ADF Uni Roo Tes sd. dev. sample: :3 o 4:1 ADF es saisics firs Variable ADF es saisics Criical Value a 5% difference of variables Criical Value a 5% IPCA_4-3.4697-3.485 - - IPCA_6-1.854 (a) -1.9461 - - IPCA_8-1.346 (a) -1.9463-7.15-3.4865 IPCA_1 -.883 (a) -1.9465-5.9875-3.491 E_4 -.9597-3.4816 - - E_6-1.8574 (b) -.984 - - E_8-9.554 (b) -.91 - - E_1-7.551 (b) -.9136 - - Noe: es performed wi (a) rend or inercep and (b) wiou rend. Table A.3 VAR for four-mon windows Variables E_4 IPCA_4 Variables E_4 IPCA_4 E_4(-1).79-5.65E-5 D_M.5 7.E-5 (.449) (.4) (.1) (8.7E-6) [ 6.85] [-.143] [ 5.1319] [ 8.13] IPCA_4(-1) 1.86.76 D1999. 1.59E-6 (7.731) (.683) (.15) (1.4E-5) [.87] [ 1.5446] [ 7.8898] [.77] C.4.7E-6 R-squared.8856.7384 (.) (1.4E-6) Adj. R-squared.8779.71 [.99] [ 1.5965] F-saisic 6.581 4.3377 Noe: sandard deviaions beween pareneses; -saisics in brackes. Table A.4 VAR for six-mon windows Variables E_6 IPCA_6 Variables E_6 IPCA_6 Variables E_6 IPCA_6 E_6(-1).97.34 E_6(-6).76 -.1 IPCA_6(-5) -6.57 -.59 (.18) (.18) (.48) (.8) (7.889) (.148) [ 8.747] [ 1.9157] [ 1.4657] [-1.343] [-3.3858] [-.469] E_6(-).398 -.64 IPCA_6(-1).488.846 IPCA_6(-6) 19.654 -.141 (.1483) (.4) (7.85) (.9) (6.156) (.981) [.684] [-.797] [.689] [ 7.33] [ 3.197] [-.1434] E_6(-3).39 -. IPCA_6(-) 1.8361.1457 C. 1.86E-6 (.1548) (.5) (9.9 (.1449) (.1) (1.8E-6) [.1546] [-.77] [.] [ 1.61] [ 1.86] [ 1.46 E_6(-4) -.1345.43 IPCA_6(-3) -4.3719 -.71 D_M.8 4.59E-5 (.39) (.18) (8.765) (.1397) (.4) (6.8E-6) [-1.189] [.3814] [-.4989] [-.5159] [ 6.56] [ 6.7698] E_6(-5) -.8.5 IPCA_6(-4) 7.9487 -.954 R-squared.881.8588 (.874) (.19) (8.6463) (.1378) F-saisic 5.66.5897 [-1.1643] [.3891] [.9193] [-.69] Adj. R-squared.8459.8171
16 Table A.5 VAR for eig-mon windows Variables E_8 D_IPCA_8 Variables E_8 D_IPCA_8 E_8(-1).793.16 D_IPCA_8(-) 5.384 -.433 (.987) (.1) (9.1884) (.9) [ 8.398] [ 1.66] [.5856] [-.479] E_8(-).97 -.6 C. -1.79E-6 (.674) (.7) (.1) (1.1E-6) [.1439] [-.946] [ 1.87] [-1.63756] D_IPCA_8(-1) 1.8755.9867 D_M.15 4.1E-5 (9.15) (.93) (.5) (5.E-6) [.35] [ 1.9] [.83] [ 8.45] R-squared.71.5781 Adj. R-squared.6947.5383 F-saisic 7.3886 14.547 Table A.6 VAR for welve-mon windows Variables E_1 D_IPCA_1 Variables E_1 D_IPCA_1 Variables E_1 D_IPCA_1 E_1(-1) 1.166.66 E_1(-7) -.1345.166 D_IPCA_1(-6) 6.9544 -.856 (.1548) (.17) (.77) (.8) (8.41) (.874) [ 7.795] [ 3.949] [-.1849] [.49] [.8568] [-.9788] E_1(-) -.864 -.41 D_IPCA_1(-1) -7.3151.1543 D_IPCA_1(-7) -8.1538.164 (.7) (.9) (9.9846) (.176) (7.683) (.8) [-.3197] [-1.394] [-.736] [ 1.4338] [-1.717] [.5] E_1(-3) -.15.9 D_IPCA_1(-) 5.387.17 C 8.75E-5-1.55E-6 (.749) (.3) (9.8458) (.161) (9.6E-5) (1.E-6) [-.5499] [.39] [.5467] [ 1.81] [.999] [-1.4988] E_1(-4).197 -.8 D_IPCA_1(-3) -.8497 -.34 D_M -.391.64E-5 (.65) (.8) (1.516) (.3) (.3) (3.7E-6) [.7565] [-.994] [-1.1837] [-.318] [-1.1375] [ 7.175] E_1(-5) -.14.1 D_IPCA_1(-4) -.5193 -.49 D3_M1 -.18 1.46E-5 (.391) (.6) (1.1884) (.198) (.4) (3.8E-6) [-.466] [.7939] [-.51] [-1.465] [-5.76] [ 3.868] E_1(-6).18 -.4 D_IPCA_1(-5) 14.164.848 R-squared.9317.8543 (.1654) (.18) (9.135) (.993) Adj. R-squared.8986.7837 [.1315] [-.46] [ 1.543] [.854] F-saisic 8.165 1.931 Variable ADF es saisics Table A.7 ADF Uni Roo Tes variances sample: :3 o 4:1 Criical Value a ADF es saisics firs 5% difference of variables Criical Value a 5% p4-3.566 (b) -.969 - - p6 -.5156 (b) -.984-7.3599-3.484 p8 -.846 (a) -1.9463-7.3398-3.4865 p1 -.4167 (a) -1.9467-5.9193-3.49 e4-6.5715-3.4816 - - e6-5.8641 (b) -.984 - - e8-5.4361 (b) -.91 - - e1-4.5351 (b) -.9136 - - Noe: es performed wi (a) rend or inercep and (b) wiou rend Table A.8 VAR for four-mon windows Variable E4 p4 Variables E4 p4 E4(-1).655.53 C.88.8 (.65) (.665) (.3) (.4) [ 1.7898] [.3768] [.7471] [.13] P4(-1).791.797 R-squared.6643.4998 (.8394) (.933) Adj. R-squared.6534.4837 [.94] [ 7.594] F-saisic 61.333 3.976 Noe: Sd. deviaions in pareneses and -saisics in square brackes.
17 Table A.9 VAR for six-mon windows Variable E6 dp6 Variable E6 dp6 e6(-1).763.83 Dp6(-1).714.1 (.7) (.136) (.865) (.144) [ 6.7736] [.668] [.86] [.6] e6(-). -.16 Dp6(-).43334 -.1 (.93) (.19) (.8515) (.131) [.13] [-1.1549] [.59] [-1.1664] C.64 3.8E-5 D_M.998.56 (.4) (.3) (.77) (.9) [.665] [.1319] [ 3.8767] [ 6.19] R-squared.755.433 F-saisic 9.747 8.469 Adj. R-squared.76.3817 Table A.1 VAR for eig-mon windows Variable E8 dp8 Variables E8 dp8 e8(-1).8994.3 dp8(-1).669. (.77) (.151) (1.564) (.1351) [ 7.6418] [ 1.351] [.633] [.1639] e8(-) -.595 -.136 dp8(-).195 -.7 (.988) (.16) (1.548) (.1349) [-.61] [-1.74] [.186] [-.51] C.5 -. R-squared.767.338 (.5) (.3) Adj. R-squared.764 -.378 [.663] [-.6696] F-saisic 35.8896.47 Table A. VAR for welve-mon windows Variables E1 dp1 Variables E1 dp1 Variables E1 dp1 e1(-1).975.413 dp1(-1) -.946.1774 C.8 -.4 (.186) (.14) (1.1596) (.135) (.3) (.3) [ 8.976] [ 3.35] [-.78] [ 1.3391] [ 1.68] [-1.578] e1(-) -.543 -.9 dp1(-) -.1943.1777 R-squared.8348.45 (.4) (.6) (1.87) (.178) Adj. R-squared.816.1846 [-.5357] [-.557] [-.1736] [ 1.3851] F-saisic 63.1833 4.55 Table A.1 Coinegraion es beween excange rae and consumer price index Number of coinegraion vecors under Ho Eigenvalue Trace saisic Criical Value ( 5%) p-value ** Unresriced Coinegraion Rank Tes (Trace) None.148 8.319 15.495.438 A mos one.19 1.75 3.841.679 Unresriced Coinegraion Rank Tes (Maximum Eigenvalue) None.148 7.834 14.646.4793 A mos one.19 1.75 3.8415.679 ** MacKinnon-Haug-Micelis (1999) p-values Table A.13 Granger Causaliy Tes Null Hypoesis Number of Obs. F-saisic p-value E does no Granger-Cause IPCA 66 9.61 5.E-5 IPCA does no Granger-Cause E 1.4686.33 I is imporan o include as many lags as possible in variable x a may be significan over variable y. We esed an equaion wi 13 lags in bo variables and e iges significan lag of x over y was e ird lag of E over IPCA. In e Granger Causaliy es e null ypoesis a IPCA Granger-Causes E is rejeced bo wi 3 and wi 13 lags.
18 Table A.14 VAR beween E and IPCA Variables E IPCA Variables E IPCA E(-1).636.356 IPCA(-1).9185.6556 (.7) (.1) (1.735) (.63) [ 4.9597] [ 3.1] [.713] [ 5.639] E(-) -.68 -.37 IPCA(-) -.74 -.59 (.151) (.137) (1.5498) (.1415) [-.778] [-.669] [-1.7447] [-1.4555] E(-3).83. IPCA(-3).4593.145 (.139) (.3) (1.5498) (.1415) [.9547] [ 1.9463] [ 1.5868] [ 1.6] E(-4).87 -.51 IPCA(-4) -.5399.689 (.133) (.94) (1.154) (.15) [.87] [-.5394] [-.79] [.5988] C.165.18 D_M -.135.143 (.84) (.8) (.344) (.31) [ 1.9756] [.373] [-3.956] [ 4.568] R-squared.458.7168 Adj. R-squared.3696.674 F-saisic 5.1688 15.4665 Noe: Sd. deviaions in parenesis and -saisics in square brackes. Table A.15 Variance Decomposiion (Colesky ordering: E IPCA) Variance decomposiion of E: Variance decomposiion of IPCA: Period Sd. Error E IPCA Period Sd. Error E IPCA 1.36 1.. 1.8 3.385 96.9616.356 99.4964.536.34 7.31 9.8869 3.37 98.151 1.7849 3.37 15.7357 84.643 4.374 98.384 1.7616 4.4 7.31 7.997 5.383 97.469.531 5.4 3.387 67.6193 6.391 95.4898 4.51 6.43 36.3763 63.637 77.395 93.5715 6.485 7.44 39.664 6.7336 8.397 9.883 7.68 8.45 41.3734 58.666 9.398 9.73 7.798 9.46 4.343 57.6758 1.4 91.749 8.591 1.46 4.5519 57.4481.4 91.4451 8.5549.46 4.4477 57.553 1.43 91.399 8.68 1.46 4.399 57.761 Table A.16 - OLS Equaion for E Variable Coefficien Sandard Error -saisic p-value C.59.9.6378.558 AR(1).4341.964 4.541. R.351 LM Tes (1 lag) (a).878 Adjused R.35 ARCH-LM Tes (1 lag) 8.6673 (b) Noe: (a) null ypoesis of absence of auocorrelaion acceped also for iger number of lags; (b) null ypoesis of absence of ARCH residuals rejeced a 1%. Grap A.1 Excange rae and inflaion volailiies (reduced sample),3,5 Condiional Variance of IPCA,,15,1,5 y = -.37x +.64x - 3E-6 Adjused R =.6915 Grap A.,5,15,5,35,45,55,65 Condiional Variance of E
19,14 Grap A. Excange rae and inflaion volailiies (reduced sample),1 y = -1.1874x +.73x - E-6 R ajusado =.484 Condiional Variance of E,1,8,6,4 Grap 4,,5,1,15,,5,3 Condiional Variance of E,65 Grap A.3 Excange rae and inflaion volailiies (reduced sample),6 Condiional Variance of IPCA,55,5,45,4 y = -4.95x +.4x - 3E-6 Adjused R =.1339,35,3,7,9,,13,15,17,19 Condiional Variance of E Appendix II Diagnosic Tess for e i-garc Model Tis appendix brings a brief explanaion abou e es of covariance saionariy in e mulivariae Garc model under e EKK resricion. For deails, see ENGLE and KRONER (1993). In a bivariae GARCH (1,1), e condiional variance as e form: U u u u 1 c = c c 1 + 1 31 1 3 13 3 33 σ σ σ ( 1 1 ( 1 ( 1 ) A + ) A ) A Wiou exogenous variables, e EKK resricion as e form: * * ' * * * * ' *' * σ 1 1, 1σ, 1 1 A1 = C 1 C 1 * * * * * * 1 1 1 σ, 1 1 1 A + σ 1, 1 σ, 1σ 1, 1 31 A + A q K To assure e process is covariance saionary, e eigenvalues of A A A 1 3 A A A 13 3 33 u u u ( 1 ) 1 ( 1 ) ( 1 ) * * A A1 [ U ] A * 1 A * p K * * ( ik ik ) + i = 1 k = 1 i= 1 k = 1 ( A ik A ) mus be minor an e uniy, in absolue values. In oer words, we ave o calculae e eigenvalue of marix X below: (a a) + (g g) (a1 a) + (g1 g) X = (a1 a) + (g1 g) (a1 a1) + (g1 g1) (a a1) + (g g1) (a a) + (g g1) (a1 a1) + (g1 g1) (a1 a) + (g1 g) (a1 a) + (g1 g) (a1 a1) + (g1 g1) (a a) + (g g) (a1 a) + (g1 g) (a1 a1) + (g1 g1) (g1 g) + (g1 g1) (a a1) + (g g1) (a a) + (g g) Calculaing e eigenvalues of X using e coefficiens for case presened in able 3 along e ex, we find e vecor y of eigenvalues presened in able 8, were all absolue values are minor an one. Hence, e case cosen respec e condiion of covariance saionariy. ik