Texos para Discussão PPGE/UFRGS Programa de Pós-Graduação em Economia Universidade Federal do Rio Grande do Sul ENDOGENEITY AND NONLINEARITIES IN CENTRAL BANK OF BRAZIL S REACTION FUNCTIONS: AN INVERSE QUANTILE REGRESSION APPROACH Gabriela Bezerra de Medeiros Marcelo Savino Porugal Edilean Kleber da Silva Bejarano Aragón Nº 2015/01 (hp://www.ufrgs.br/ppge/exos-para-discussao.asp) Poro Alegre/RS/Brasil 1
ENDOGENEITY AND NONLINEARITIES IN CENTRAL BANK OF BRAZIL S REACTION FUNCTIONS: AN INVERSE QUANTILE REGRESSION APPROACH Gabriela Bezerra de Medeiros Marcelo Savino Porugal Edilean Kleber da Silva Bejarano Aragón Absrac: In his work, we seek o invesigae nonlineariies in he reacion funcion of he Cenral Bank of Brazil by esimaing quanile regressions. As he moneary policy rule has endogenous regressors, we followed he procedures suggesed by Wolers (J Macroecon 34:342-361, 2012) and he mehod of inverse quanile regression, proposed by Chernozhukov and Hansen (Economerica 73:245 261, 2005). This mehod enabled us o deec nonlineariies in he Cenral Bank of Brazil s reacion funcion wihou he need o make specific assumpions abou he facors ha deermine hese nonlineariies. In paricular, we observed ha: i) he response of he ineres rae o he curren and expeced inflaion was, in general, sronger in he upper ail of he condiional ineres rae disribuion; ii) he response o he oupu gap showed a growing and significan rend in he lower ail of he condiional Selic rae disribuion; iii) he response of he Cenral Bank of Brazil o he real exchange rae was posiive and higher in he upper ail of he condiional Selic rae disribuion. Keywords Moneary policy rules Quanile regression Endogenous regressors Cenral Bank of Brazil. JEL Classificaion C32 E52 E58 1 Inroducion The inflaion-argeing regime was adoped by he Cenral Bank of Brazil (CBB) in July 1999. This decision was aken six monhs afer he ransiion from an exchange rae band sysem o a floaing sysem. Owing o exchange rae overshooing and o he rise in inflaion and in inflaion expecaions, he Brazilian governmen aimed o implemen a policy regime ha was insiuionally commied o mainaining price sabiliy and providing a new nominal exchange rae anchor for inflaion. For he analysis of he CBB s moneary policy decisions in he inflaionargeing regime, several papers have esimaed he Taylor (1993) rule or he forward- G. B. de Medeiros Pos Graduae Program in Economics (PPGE), Universidade Federal do Rio Grande do Sul (UFRGS), Av. João Pessoa, 52 sala 33B - 3 andar, Cenro, Poro Alegre/RS, Brazil. CEP 90040-000. e-mail: grabriela.bm@homail.com M. S. Porugal Graduae Program in Economics and in Business Adminisraion of Universidade Federal do Rio Grande do Sul (PPGE and PPGA/UFRGS) and CNPq researcher. Av. João Pessoa, 52 sala 33B - 3 andar, Cenro, Poro Alegre/RS, Brazil. CEP 90040-000. e-mail: msp@ufrgs.com E. K. da S. B. Aragón Deparmen of Economics and Graduae Program in Economics of Universidade Federal da Paraíba (PPGE/UFPB), Jardim Cidade Universiária, João Pessoa/PB, Brazil. CEP 58.051-900. e-mail: edilean@homail.com 2
looking reacion funcion proposed by Clarida e al. (2000). 1 For insance, Minella e al. (2003) and Minella & Souza-Sobrinho (2013) esimaed a forward-looking reacion funcion and observed ha he CBB srongly reaced o inflaion expecaions. Mello & Moccero (2009) uilized coinegraion analysis and M-GARCH model esimaions o check for he presence of long-erm relaionships beween he moneary policy ineres rae (Selic rae), inflaion expecaions, and inflaion arge, and o verify he presence of volailiy spillovers beween inflaion expecaions and moneary policy. For Brazil, he resuls gahered by hese auhors revealed here exis long-erm relaionships beween he ineres rae, expeced inflaion, and inflaion arge, and ha higher volailiy in moneary policy increases he volailiy of he expeced inflaion. Sanches-Fung (2011) esimaed reacion funcions for he CBB in a daa-rich environmen. Sanches-Fung s (2011) evidence poins ou ha he CBB adjused he Selic ineres rae according o he Taylor principle, bu ha i did no reac sysemaically o he exchange rae behavior. An imporan assumpion of he papers menioned above is ha ineres rae rules are linear funcions relaive o he variables describing economic condiions. By conras, he economic lieraure has come up wih numerous reasons why he moneary auhoriy responds nonlinearly o inflaion and/or o he oupu gap. Nobay & Peel (2000), Schaling (2004) and Dolado e al. (2005) demonsrae ha an opimal nonlinear moneary rule emerges when he cenral bank has a quadraic loss funcion and he Phillips curve is nonlinear. Bec e al. (2002), Nobay & Peel (2003), Dolado e al. (2004), Surico (2007) and Cukierman & Muscaelli (2008) show nonlineariies in he opimal moneary rule may arise if he moneary auhoriy s preferences are asymmeric in relaion o inflaion and/or o he oupu gap. By assessing an opimal moneary policy in an economy where he cenral bank is uncerain over he Phillips curve slope, Tillmann (2011) evidences ha he ineres rae adjusmen is nonlinear. Lasly, he zero lower bound for he nominal ineres rae can promp he cenral bank o respond nonlinearly o he inflaion rae (Kao & Nishiyama, 2005; Adam & Billi, 2006). 2 For Brazil, sudies on nonlineariies in he moneary policy rule assess specific feaures of he CBB s asymmeric reacion. For example, Aragón & Porugal (2010), Sá & Porugal (2011) and Aragón & Medeiros (2013) reveal ha he Brazilian moneary auhoriy had an asymmeric preference for an above-arge inflaion in he inflaionargeing regime. Moura & Carvalho (2010) find empirical evidence of nonlineariies in he reacion funcion ha corroboraes he CBB s asymmeric preference concerning inflaion. Lopes & Aragón (2014) describe ha he nonlineariy in he ineres rae rule sems from ime-varying asymmeric preferences raher han from possible nonlineariies in he Phillips curve. Schiffino e al. (2013) show ha he nonnegaiviy consrain on he Selic ineres rae may affec he calibraion of he CBB s preferences, implying nonlineariies in he opimal moneary rule. Aragón and Medeiros (2014) esimae a reacion funcion whose parameers vary over ime and conclude ha he reacion of he Selic rae o inflaion varies remarkably hroughou he period, showing a downrend during he inflaion-argeing regime. Unlike he afore-menioned sudies, he presen paper seeks o verify nonlineariies in he CBB s reacion funcion by quanile regression esimaion. An imporan advanage of his approach over convenional mehods (e.g., leas ordinary 1 According o he moneary rule proposed by Taylor (1993), he cenral bank changes he nominal ineres rae in response o deviaions of he curren inflaion from he inflaion arge and o he curren oupu gap. In urn, he policy rule formulaed by Clarida e al. (2000) assumes he moneary auhoriy adjuss he ineres rae based on expeced fuure inflaion raes and on he oupu gap. 2 Kao & Nishiyama (2005) and Adam & Billi (2006) argue ha, close o he zero bound, he cenral bank responds more srongly o a decrease in inflaion rae in order o minimize he likelihood of deflaion. 3
squares (OLS) and insrumenal variables (IV)) is ha i allows esimaing he Selic ineres rae rule across differen quaniles of he condiional ineres rae disribuion and no only in he condiional mean of his variable. This permis deecing nonlineariies in he CBB s reacion funcion wihou having o make specific inferences abou he causal facors of hese nonlineariies. Thus, as nonlineariy is deermined by he daa, he quanile regression mehod allows comparing he esimaes of he moneary rule parameers obained for he quaniles of he condiional ineres rae disribuion wih he mean from he linear reacion funcion. Empirically, we used inverse quanile regression (IVQR), proposed by Chernozhukov & Hansen (2005, 2006), o esimae he CBB s quanile reacion funcion parameers during he inflaion-argeing regime. This mehod was chosen because of he presence of endogenous regressors (inflaion rae and oupu gap) in he ineres rae rule. Some auhors, such as Chevaparakul e al. (2009), Wolers (2012), and Chevaparakul & Paez-Farrell (2014), esimae he reacion funcion by quanile regression. To add he presence of endogeneiy, Chevaparakul e al. (2009) and Chevaparakul & Paez-Farrell (2014) apply he wo-sage quanile regression (2SQR) mehod, while Wolers (2012) uses IVQR. 3 Noe ha IVQR is a good alernaive o he 2SQR mehod because: i) i yields consisen and unbiased esimaes of all parameers in he model; and ii) he esimaes are consisen even when endogenous regressors change he disribuion of he dependen variable (Wolers, 2012). 4 The IVQR esimaion resuls for he CBB s reacion funcion can be summarized as follows. While condiional mean esimaions showed an insignifican response of he Selic rae o he curren inflaion gap, quanile regression resuls indicaed ha he CBB s shor-erm response o his variable was significan and increasing beween quaniles 0.5 and 0.9. Conversely, he shor-erm response of he Selic rae o he oupu gap increased from quanile 0.2 o quanile 0.7 and was no saisically differen from zero a he exreme quaniles of he condiional ineres rae disribuion. We also perceived ha he shor-erm response of he Selic rae o expeced inflaion was significan from quanile 0.4, exhibiing an uprend. Regarding he long-erm response, resuls sugges he Selic rae responded srongly o curren and expeced inflaion when he ineres rae was above he median. On he oher hand, he long-erm response o he oupu gap was significan only a some quaniles on he [0.05, 0.7] inerval. This suggess ha he CBB does no reac o demand pressures when he ineres rae is oo high. When we included he real exchange rae as an ineres rae rule regressor, we noiced he CBB responded posiively o he real exchange rae boh in he condiional mean and across he ineres rae disribuion. Moreover, resuls show ha he reacion o he real exchange rae was, in general, sronger in he upper ail of he condiional Selic rae disribuion. Aside from his inroducion, his paper is organized ino four secions. Secion 2 inroduces he heoreical model used o derive he ineres rae rule adoped by he moneary auhoriy. Secion 3 describes he empirical specificaions of he CBB s reacion funcion and is esimaion mehod across differen quaniles of he condiional ineres rae disribuion. Secion 4 inerpres he resuls. Secion 5 concludes. 2 Theoreical Model 3 Chevaparakul e al. (2009) assess moneary policy conduc in he Unied Saes and in Japan, whereas Chevaparakul & Paez-Farrell (2014) focus heir analysis on Ausralia, Canada, and New Zealand. In urn, Wolers (2012) esimaes he Federal Reserve s reacion funcion. 4 For furher deails on he 2SQR mehod, see Amemiya (1982), Powell (1983) and Kim & Muller (2004, 2008). 4
The heoreical model used in his paper o analyze moneary policy opimal decisions is based on Clarida e al. (1999). The model employs he new Keynesian framework inroduced by hese auhors and consiss of hree componens. The firs componen is a sysem of equaions ha resric he moneary auhoriy s dynamic conrol problem. This sysem of equaions comprises: i) an IS curve, which governs oupu dynamics; and ii) a Phillips curve, which describes inflaion dynamics. The second componen concerns he cenral bank s quadraic loss funcion, which describes moneary policy goals. Finally, he hird componen is he moneary policy opimal rule, which shows how he cenral bank races he opimal pah for he nominal ineres rae. 2.1 Moneary auhoriy s opimizaion problem To assess moneary policy conduc, suppose ha he moneary auhoriy s decisions are made before demand shocks,, and before cos shocks,. Therefore, condiional on he informaion available a he end of he previous period, he cenral bank ries o choose he curren nominal ineres rae, i, and a sequence of fuure ineres raes so as o minimize: E j 1 L j j0 (1) where β (0,1) is he fixed discoun facor and he loss funcion a is denoed by: 1 L y y i i i i i i 2 * 2 * 1 2 2 2 where π is he inflaion rae, π * is he inflaion arge, y is he oupu gap (i.e., he difference beween acual oupu and poenial oupu), y is he relaive weigh of he deviaion of oupu from poenial oupu, and i and Δi are he relaive weighs of ineres rae sabilizaion around an implici arge, i *, and he ineres rae a -1, i -1. 5 The moneary auhoriy presumably sabilizes inflaion around he inflaion arge, keeps he oupu gap closed a zero, and sabilizes he nominal ineres rae around arge i * and he nominal ineres rae a -1. The moneary auhoriy s goal is o minimize (1) condiional on he following sysem of equaions describing he economic srucure: y E y ( i E ) u (3) 1 d 1 1 s 1 (2) E y u (4) where E y +1 and E π +1 are he expeced values for oupu gap and inflaion rae given he informaion se available a, and demand shocks ( u ) and cos shocks ( u ). These d s 5 The lieraure liss several reasons for ineres rae smoohing. Among hem, we may cie he following: i) presence of uncerainies abou he values of economic daa and of he coefficiens of he macroeconomic model; ii) remarkable changes in ineres raes could desabilize he exchange rae and financial markes; and iii) consan flucuaions in he shor-erm ineres rae, albei small, would srongly impac he aggregae demand and inflaion rae. For deails abou he smoohing of he moneary policy insrumen, see Clarida e al. (1998), Sack (2000), Woodford (1999, 2003) and Sack & Wieland (2000). 5
shocks follow firs-order auoregressive processes. Parameers σ and κ are posiive consans. 6 The IS curve, given by equaion (3), is a log-linearized version of Euler equaion for consumpion derived from household opimal decision on consumpion and savings afer imposiion of he marke clearing condiion. The value expeced for he oupu gap shows ha, as households prefer o smooh consumpion over ime, he expecaion for a higher consumpion level evenually increases curren consumpion, also boosing he curren demand for oupu. The Phillips curve, given by equaion (4), describes he characerisics of overlapping nominal prices, where companies show a consan probabiliy of mainaining he oupu price fixed in any ime period (Calvo, 1983). The discree naure of price adjusmen encourages every company o se a higher opimal price he higher he expecaion of fuure inflaion. In addiion, he presence of oupu gap in Phillips curve capures he movemens in marginal coss associaed wih excess demand. 2.2 Opimal moneary rule The cenral bank s opimizaion problem (1) is solved discreionally. 7 This implies ha he cenral bank akes he expecaions of fuure variables as given and chooses he curren ineres rae in each ime period. Since here is no endogenous persisence in inflaion and in oupu gap, he ineremporal opimizaion problem can be reduced o a sequence of saic opimizaion problems. Combining he firs-order condiions and solving i, we ge: * i (1 1) 0 1E 1 2E 1( y ) 1i 1 (5) where * 1 i ; ; 1 ;. This equaion shows 0 1 i 2 y i 1 i i i ha he opimal nominal ineres rae a responds linearly o deviaions of he expeced inflaion rae from he inflaion arge, and o he oupu gap expeced for ime. 3 Empirical specificaions In his secion, we iniially inroduce he CBB s reacion funcion o be esimaed in he condiional mean of he ineres rae. This linear funcion is based on he heoreical model described in he previous secion. Thereafer, we describe he moneary policy rule o be esimaed by quanile regression and he esimaion mehod for his funcion. Finally, we ake ino consideraion an alernaive specificaion of he CBB s reacion funcion. 3.1 The moneary policy rule in he condiional mean For esimaion purposes, we made four amendmens in he reduced form of reacion funcion (5). Firsly, we included an exogenous random shock for he ineres rae, m, in his equaion. This shock is assumed o be i.i.d and can be inerpreed as he moneary policy s purely random componen. Secondly, we consider a variable inflaion arge (π * ). This change is necessary because he inflaion arges esablished by he Brazilian 6 Equaions (1) and (2) are obained explicily from he opimizing behavior of firms and households in an economy wih currency and nominal price sickiness. For furher deails, see Clarida e al. (1999). 7 Palma & Porugal (2011) provide evidence in favor of a discreionary moneary policy in Brazil for he period 2000-2010. 6
Naional Moneary Council varied annually in he period 1999-2004. Thirdly, he nominal ineres rae a -2 is insered in he policy rule o avoid possible serial auocorrelaion problems. 8 Fourhly, he expeced values for inflaion and oupu gap in (5) are replaced wih heir observed values. By making hese amendmens, he specificaion of he policy rule o be esimaed is given by: i ( ) y i i (6) * 0 1 2 1 1 2 2 where β i = (1 θ 1 θ 2 ) β i, i=0,1,2, and 1, ( E 1( )) 2, ( y E 1( y )) m. The coefficiens β 1 and β 2 (β 1 and β 2 ) measure he shor-erm (long-erm) response of he ineres rae o inflaion and o oupu gap. Once inflaion and oupu gap forecas errors are an inegral par of erm ε, π and y are correlaed wih his error erm. In view of ha, (6) in he condiional mean of he moneary policy s ineres rae will be esimaed by IV and by he generalized mehod of momens (GMM). 3.2 The moneary policy rule across differen condiional quaniles Quanile regression models manage o deermine he heerogeneous impacs of variables a differen poins along a disribuion. Quanile regression was firs proposed by Koenker & Basse (1978) and has raher aracive feaures, namely: i) i can be used o assess he response of he dependen variable o explanaory variables a differen poins of he dependen variable disribuion; ii) quanile regression esimaors are more efficien han OLS esimaors when he error erm is non-gaussian; and iii) quanile regression esimaors are less sensiive o he presence of ouliers in he dependen variable (Koenker, 2005). Quariles spli observaions ino four segmens wih equal proporions of benchmark observaions in each segmen. Quiniles and deciles, similarly o quariles, spli observaions ino 5 and 10 segmens, respecively. Quaniles or perceniles refer o he general case (Koenker & Hallock, 2001). For our moneary policy problem, he τh condiional quanile is defined as q τ (i i -1, i -2, π π *, y ) such ha he likelihood of he nominal ineres rae being smaller han q τ (i i -1, i -2, π π *, y ) is equal o τ, i.e.: * q i, y, i 1, i 2 * f * i,,,,, 1, 2, (0,1) y i i di (7) i y i i 1 2 where f i i-1, i-2, π π*, y (i i -1, i -2, π π *, y ) is he condiional densiy of i given i -1, i -2, π π * and y. This is a nonparameric specificaion in which τ can vary coninually beween zero and one; hence, here are an infinie number of possible parameer vecors. 9 For τ = ½, equaion (7) shows he condiional median funcion of i given i -1, i -2, π π * and y. Taking (7), he CBB s reacion funcion a quanile τ can be expressed as: *,,, * q i y i 1 i 2 0 1 2 y 1 i 1 2 i (8) 2 According o equaion (8), he parameers of he CBB s reacion funcion can be esimaed a differen quaniles, hereby allowing for a complee descripion of he condiional disribuion of he moneary policy ineres rae. Unforunaely, by virue of he presence of endogenous regressors π and y, he esimaion of reacion funcion (8) by he quanile regression mehod proposed by 8 This procedure was also adoped by Aragón & Porugal (2010) and Minella & Souza-Sobrinho (2013). 9 This requires fewer deails abou he specificaion of he disribuion of y x (Greene, 2012). 7
Koenker & Basse (1978) yields biased esimaes (Kim & Muller, 2012). To circumven his problem, an alernaive would be o use wo-sage quanile regression (2SQR). This mehod is based on he wo-sage leas absolue deviaion esimaor developed by Amemiya (1982) and Powell (1983), and exended o quanile regression by Chen & Pornoy (1996) and Kim & Muller (2004, 2012). For our problem, he wo sages of he 2SQR mehod consis in: i) esimaing regressions on endogenous regressors π and y as a funcion of a se of seleced insrumens and calculaing he adjused values of hese regressors; ii) esimaing moneary rule (8) by quanile regression replacing π and y wih heir adjused (or prediced) values obained in sep (i). Alhough he 2SQR mehod yields consisen esimaors for slope parameers, he inercep esimaor is biased (Kim & Muller, 2012). Because of ha, we uilize he inverse quanile regression (IVQR) mehod, proposed by Chernozhukov and Hansen (2005, 2006). 10 The advanage of his procedure is ha i yields unbiased esimaes even when changes in endogenous regressors aler he condiional disribuion of he dependen variable. As poined ou by Wolers (2012), his appears o be he case of he esimaion of he moneary auhoriy s reacion funcion in which he nominal ineres rae exhibis a zero bound. Given such consrain, i is reasonable o assume ha a decrease in inflaion followed by a reducion in nominal ineres raes alers he condiional disribuion of his policy insrumen. In wha follows, we briefly describe he IVQR mehod. 3.2.1 Inverse quanile regression The IVQR mehod derives from he following momen condiion regarded as he major idenificaion consrain: P Y q D, X X, Z (9) where P(..) sands for he condiional probabiliy, Y is he dependen variable, D is a vecor of endogenous variables, X is a vecor of exogenous variables including he consan, and Z is a vecor of addiional insrumenal variables. In he case of ineres rae rule (8), Y is he policy insrumen i, D is made of inflaion oupu (π π * ) and oupu gap (y ), X is he vecor ha includes he inercep, i -1 and i -2, and Z is he vecor of addiional insrumens ha may include lagged values of inflaion gap and oupu gap. In IVQR, he momen condiion is equivalen o saing ha 0 is he τh quanile of he random variable Y q τ (D, X) condiional on (X, Z). Thus, equaion (9) is he ransform wihin an analogous sample. For ha reason, we have o find he parameers for funcion q τ (D, X) such ha zero is he soluion o he quanile regression problem, in which he error erm regressor is Y q τ (D, X) in any funcion of (X, Z). Le δ D = [β π-π* β y ] be he vecor of parameers of endogenous variables, δ X = [β 0 θ 1 θ 2 ] he vecor of parameers of exogenous variables and Λ a se of possible values for δ D. Therefore, he condiional quanile as a linear funcion is q τ (Y D, X) = D δ D (τ) + X δ X (τ). According o Wolers (2012), he algorihm ha implemens he IVQR esimaor can be summarized in hree seps. The firs sep consiss in esimaing regressions by leas squares, relaing endogenous regressors (D) o he vecors of exogenous variables (X) and insrumens (Z), and obaining he vecor of prediced values ( ˆD ). In he 10 This mehod is also known as insrumenal variable quanile regression (Chernozhukov & Hansen, 2006). 8
second sep, for all δ D Λ, we obain he esimaes for vecors δ X and δ Z as he soluion o he following minimizaion problem: T 1 ˆ Y D X D, arg min (10) X D Z D D X Z X D T 1 where φ τ (u) = (τ 1(u < 0))u is he asymmeric loss funcion of he leas absolue deviaion from he sandard quanile regression and δ Z is he vecor of parameers relaed o addiional insrumens in he regressions shown in he previous sep. In he hird sep, he esimae of δ D is obained as he soluion o he problem: arg min (11) D Z D Z D D This minimizaion ensures ha Y q τ (D, X) no longer depends on ˆD, i.e., on (X, Z). As noed in (10) and (11), he esimaes of he parameers of he model are obained by he esimaion of an array of sandard quanile regressions (in which convex opimizaion problems are solved in order o esimae δ X and δ Z, in combinaion wih a grid search only for he values of he vecor of parameers δ D. 11 3.2.2 Moving blocks boosrap To obain he sandard errors of he coefficiens of he reacion funcion esimaed by IVQR, we used moving blocks boosrap (MBB), proposed by Fizenberger (1997). This auhor demonsraes ha MBB yields sandard errors ha are robus o unknown forms of heeroskedasiciy and auocorrelaion, boh in linear regressions esimaed by OLS and in quanile regressions. As in Clarida e al. (1998) and Wolers (2012), we resriced he auocorrelaion o he ime horizon of 1 year, which is reasonable for monhly daa. Noe ha in MBB each boosrap block of he variables (including he dependen variable, he endogenous variables, he exogenous variables, and he insrumens) is obained randomly from he whole sample. Afer ha, he esimaes of he parameers by IVQR are obained for each of he 1000 boosraps, and he sandard errors are calculaed as he sandard deviaion of he 100 esimaes obained for each parameer. 12 3.3 An alernaive specificaion for he CBB s reacion funcion Consonan wih Minella e al. (2003), Aragón & Porugal (2010) and Minella & Souza- Sobrinho (2013), we also esimae a specificaion of he reacion funcion ha includes he deviaion of inflaion expecaions from he inflaion arge (or from he expeced inflaion gap). In his case, he reacion funcion wih consan parameers is given by: i Dj y i i (12) 0 1 2 1 1 2 2 Whereas he reacion funcion a quanile τ can be expressed as,,, q i Dj y i 1 i 2 0 1 Dj 2 y 1 i 1 2 i 2 (13) Wih variable Dj denoed as 11 For furher deails, see Koenker (2005) and Chernozhukov & Hansen (2006). 12 For more deails abou MBB, see Fizenberger (1997). 9
Dj 12 j * j * EjT T EjT 1 T 1 (14) 12 12 where j is he monhly index, E j T is he inflaion expecaion in monh j for year T, E j T+1 is he inflaion expecaion in monh j for year T+1, * T is he inflaion arge for year T and * T+1 is he inflaion arge for year T+1. As inflaion expecaions and oupu gap are poenially endogenous variables, he IVQR mehod will be used o esimae he coefficiens of moneary rule (13). 13 4 Resuls 4.1 Daa and uni roo ess To esimae he CBB s reacion funcions, we uilized monhly series for he period beween January 2000 and December 2013. The series were obained from he websies of he Applied Economics Research Insiue (IPEA) and CBB. The dependen variable, i, is he annualized Selic rae accumulaed on a monhly basis. This variable has been used as he major moneary policy insrumen in he inflaion-argeing regime. The inflaion rae,, is he inflaion accumulaed over he pas 12 monhs, measured by he broad consumer price index (IPCA). 14 Since inflaion arges are considered o be ime-varying, we inerpolaed he annual raes o obain he monhly series of he arge for he inflaion accumulaed over he nex 12 monhs. The variable Dj is buil from he inflaion arges se for years T and T+1, and from he inflaion expecaions series obained from he survey conduced by he CBB wih financing and consuling firms. In his survey, firms indicae he inflaion rae hey expec for years T (E j T ) and T+1 (E j T+1 ). The oupu gap (y ) is measured by he percenage difference beween he seasonally adjused indusrial producion index and poenial oupu. Poenial oupu is an unobservable variable and, for ha reason, i should be esimaed. We obained he proxy for poenial oupu using he Hodrick-Presco (HP) filer. The hisogram for he Selic rae and he behavior of his variable and of he deviaion of inflaion from is arge are depiced in Figure 1. By comparing he behavior of inflaion gap wih ha of he Selic rae, we noe ha he CBB has increased (decreased) he use of his policy insrumen in response o rises (reducions) in inflaion rae. The correlaion coefficien beween i and π π * was 0.72, suggesing a close relaionship beween hese series. The hisogram for he Selic rae indicaes ha he disribuion of his series is asymmeric and skewed o he righ and playkuric. 15 So, he Jarque-Bera saisic (6.66) indicaes he null hypohesis of normaliy of he Selic rae is rejeced a 5%. Addiionally, i should be noed ha he Selic rae is way above zero a he lower quaniles. This suggess ha he fear of a lower bound wih value zero canno explain possible asymmeric reacions of he CBB in he lower ail of he condiional disribuion of i. 13 For he deerminans of inflaion expecaions in Brazil, see Bevilaqua e al. (2008) and Carvalho & Minella (2012). 14 IPCA is calculaed by he Brazilian Insiue of Geography and Saisics (IBGE) and is he price index used by he Naional Moneary Council as benchmark for he inflaion-argeing regime. 15 The coefficien of asymmery was 0.45 and he coefficien of kurosis was 2.62. 10
(a) (b) 30 sdasds 25 14 12 20 10 15 8 10 6 5 4 0 2-5 00 01 02 03 04 05 06 07 08 09 10 11 12 13 0 8 10 12 14 16 18 20 22 24 26 Selic ineres rae Deviaion of inflaion from arge Figure 1 Selic rae and deviaion of inflaion from is arge (panel a) and hisogram for he Selic rae (panel b) Before resuming he esimaions, we checked wheher he variables used in his sudy are saionary. Iniially, we invesigaed he order of inegraion of he variables by he applicaion of hree ess: ADF (Augmened Dickey-Fuller), and MZ α GLS and MZ GLS ess, suggesed by Perron & Ng (1996) and Ng & Perron (2001). 16 As poined ou by Ng & Perron (2001), he selecion of he number of lags (k) was based on he modified Akaike informaion crierion (MAIC) regarded as he maximum number of lags of k max = in(12(t/100) 1/4 ) = 13. Consan (c) and a linear rend () were included as deerminisic componens for he cases in which hese componens were saisically significan. Table 1 Uni roo ess Exogenous Variable ADF(k) MZ GLS regressors α (k) MZ GLS (k) i c, -3.309 * (4) -11.471 (9) -2.386 (9) C -1.909(13) -13.77 ** (1) -2.599 *** (1) C -3.225 ** (0) -6.142 * (0) -1.698 * (0) * Dj C -2.088 (10) -11.75 ** (10) -2.410 ** (10) y C -3.508 *** (0) -18.95 *** (0) -3.053 *** (0) Noe: *** Significan a 1%. ** Significan a 5%. * Significan a 10%. The resuls in Table 1 show ha, in general, i is possible o rejec he uni roo hypohesis in inflaion, inflaion arge, oupu gap, and Dj series. For he Selic rae, he MZ α GLS and MZ GLS es resuls indicae his variable is nonsaionary in he level. Since he failure o rejec he uni roo null hypohesis in he Selic rae may be relaed o he exisence of a srucural break in he rend funcion, wo procedures were performed. 17 Firs, we used he Exp-W FS saisic, proposed by Perron & Yabu (2009), o es he null hypohesis of no srucural break in he rend funcion of he Selic rae agains he alernaive hypohesis of a break in inercep and slope of he rend funcion 16 The null hypohesis of he ess is ha he series is nonsaionary (or uni roo). 17 See, for insance, Perron (1989). 11
a an unknown dae. 18 The value of his saisic (9.42) implies rejecion of he hypohesis of no srucural break a a 1% significance level. Therefore, wo uni roo ess wih srucural breaks were run. Following Carrion-i-Silvesre e al. (2009), he MZ GLS α (λ 0 ) and MZ GLS (λ 0 ) saisics were used o es he uni roo null hypohesis, allowing for hree breaks in he rend funcion a an unknown dae under he null and alernaive hypoheses. The values obained for MZ GLS GLS α (-113.4) and MZ (-7.52) allow rejecing he uni roo hypohesis in he Selic rae a 1%. 4.2 The CBB s reacion funcion in he condiional mean Firs, we esimaed reacion funcions (6) and (12) in he condiional mean using IV and GMM wih he opimal weighing marix, aking ino accoun possible heeroskedasiciy and serial auocorrelaion in residuals. Specifically, we applied he mehod proposed by Newey & Wes (1987) wih he Barle kernel and fixed bandwidh o esimae he covariance marix. The following insrumens were used: a consan erm, lags 1-2 of he Selic rae and deviaion of (curren or expeced) inflaion from he arge, lags 2-3 of he oupu gap, and nominal exchange rae movemen a -1 (ΔE -1 ). 19 The se of insrumens implies hree overidenificaion consrains. We esed he validiy of hese consrains wih Hansen s (1982) J es. Addiionally, anoher wo ess were employed: i) Durbin-Wu-Hausman es o verify he null hypohesis of exogeneiy of regressors π π * and y in equaion (6), and Dj and y in equaion (12); and ii) Cragg- Donald s F es, proposed by Sock & Yogo (2005), o es he null hypohesis ha he insrumens are weak. 20,21 The resuls of hese ess, shown in Table 2, indicae we may rejec he hypoheses ha (curren or expeced) inflaion gap and oupu gap are exogenous and ha he insrumens used in he regressions are weak. Also, he J es shows we canno rejec he hypohesis ha he overidenificaion consrains are me. The esimaes of he CBB s reacion funcion parameers obained by IV and GMM are quie similar. For specificaion (6), he values of he coefficiens ha measure shor-erm (β 1 ) and long-erm (β 1 ) responses of he Selic rae o inflaion were no saisically differen from zero in he condiional ineres rae mean. This suggess ha he CBB has no adoped a sabilizaion policy for he curren inflaion around he inflaion arge, as he increase in inflaion has no been followed by a significan increase in he Selic rae. On he oher hand, he Selic rae responded o he changes in oupu gap. The long-erm coefficiens of his variable were equal o 2.2 and 2.4 for rule (6) esimaed by IV and GMM, respecively, and were significan a 1%. Finally, he Selic rae smoohing (θ 1 +θ 2 ) yielded approximaely 0.98. This resul is consisen wih 18 Perron & Yabu (2009) presen some ess for he srucural break in he rend funcion ha do no require knowing a priori wheher he noise componen of he series is saionary or has a uni roo. These auhors also demonsrae ha, in he case in which he srucural break is unknown, he Exp-W FS funcional of Wald s es provides a es wih almos idenical limi values for a noise componen I(0) or I(1). Therefore, es procedures wih similar sizes can be performed for hose wo cases. 19 Exchange rae movemen is he percenage variaion of he Real/Dollar nominal exchange rae (mean for he period). 20 As underscored by Sock & Yogo (2005), he presence of weak insrumens may yield biased IV esimaors. Thus, following hese auhors, we considered insrumens o be weak when he bias of he IV or GMM esimaor relaive o he bias of he OLS esimaor was greaer han any value b (for example, b = 5%). 21 The criical values of his es are described in Sock &Yogo (2005). 12
he lieraure on shor-erm ineres rae smoohing and indicaes he adjusmen of his policy insrumen a discree inervals and in discree amouns. 22 Table 2 Esimaes of he CBB s reacion funcions Parameers Eqn. (6) Eqn. (12) IV GMM IV GMM β 0 0.179 *** (0.067) 0.171 *** (0.061) 0.120 (0.088) 0.134 * (0.076) β 1 0.025 (0.017) 0.016 (0.014) 0.115 *** (0.029) 0.111 *** (0.030) β 2 0.037 *** (0.009) 0.039 *** (0.008) 0.042 *** (0.010) 0.043 *** (0.010) θ 1 1.753 *** (0.062) 1.716 *** (0.058) 1.627 *** (0.068) 1.627 *** (0.063) θ 2-0.770 *** (0.062) -0.732 *** (0.057) -0.644 *** (0.070) -0.645 *** (0.063) β 1 1.452 (1.005) 0.995 (0.798) 7.067 ** (3.104) 6.377 *** (2.190) β 2 2.196 *** (0.771) 2.407 *** (0.865) 2.602 ** (1.125) 2.469 *** (0.931) J-saisic (p-value) 0.213 0.486 0.803 0.664 Hausman es (p-value) 0.008 0.036 0.000 0.017 Cragg-Donald F-sa 26.61 26.61 24.00 24.00 R 2 -adjused 0.996 0.996 0.996 0.996 Noe: *** Significan a 1%. ** Significan a 5%. * Significan a 10%. Sandard deviaion (in brackes). Indicaes ha he relaive bias of he IV (or GMM) in relaion o he OLS esimaor corresponds o a mos 5%. Wih respec o moneary rule (12), he esimaes of coefficien β 1 indicae ha, in he condiional Selic rae mean, he CBB has reaced srongly o he deviaion of expeced inflaion from he inflaion arge. Specifically, he values obained for his parameer show he moneary policy rule fulfills he Taylor principle (1993), i.e., he CBB has increased he Selic rae jus enough o rise he real ineres rae in response o an increase in expeced inflaion. This resul is in line wih hose encounered by Minella e al. (2003), Moura & Carvalho (2010), Sanches-Fung (2011), Aragón & Medeiros (2013) and Minella & Souza-Sobrinho (2013). Compared o he esimaes of β 1 for reacion funcion (6), he CBB has responded more srongly o expeced inflaion han o curren inflaion. This procedure is consisen wih a forward-looking policy rule and indicaes he CBB has been concerned mainly wih anchoring inflaion expecaions o he inflaion arge se by he Naional Moneary Council. In regard o coefficien β 2, he resuls were analogous o hose obained for moneary rule (6) and show he Brazilian moneary auhoriy has also reaced o he demand pressure. 4.3 Quanile regression resuls Now, we presen he resuls for he CBB s reacion funcion esimaed by IVQR. Table 3 conains he coefficiens esimaed by quanile regressions and heir respecive sandard errors (in brackes) for specificaion (8). The esimaes for each quanile τ ϵ {0.05,0.1, 0.2,...,0.9, 0.95} are shown. Unlike IV and GMM resuls, he shor-erm Selic ineres rae response o inflaion gap, β 1 (τ), is saisically differen from zero from quanile 0.5 o quanile 0.9. In conras, he response o inflaion is no significan for he 22 For shor-erm ineres rae smoohing, see Goodfriend (1991) and Rudebusch (1995). 13
lower quaniles of he condiional Selic rae disribuion. Hence, resuls reveal ha he CBB s response o inflaion gap is sronger when he Selic rae is adjused o a higher level han is condiional median. In addiion, he response o inflaion is more inense beween quaniles 0.5 and 0.9. This suggess he CBB has reaced more aggressively o inflaion for higher levels of he Selic rae (and of he inflaion gap). This resul is also observed by Chevaparakul e al. (2009) and Wolers (2012) for he Federal Reserve, and by Chevaparakul & Paez-Farrell (2014) for he Cenral Bank of Ausralia. Table 3 also shows ha he shor-erm response of he Selic rae o oupu gap (β 2 ) is significan from quanile 0.1 o quanile 0.9 and is no saisically differen from zero a he exreme quaniles of he condiional ineres rae disribuion. In comparison wih he IV resuls, he response o he oupu gap in he condiional mean is, in general, sronger han he esimaes obained for he quaniles. However, his difference is suble as he confidence inerval for he IV esimae includes hose esimaes obained by IVQR. Table 3 IVQR esimaes for reacion funcion (8) Quanile β 1 β 2 θ 1 θ 2 0.05-0.020 (0.042) 0.035 (0.022) 1.796 *** (0.107) -0.838 *** (0.108) 0.1-0.043 (0.032) 0.032 ** (0.015) 1.784 *** (0.110) -0.813 *** (0.110) 0.2 0.002 (0.025) 0.027 ** (0.011) 1.764 *** (0.125) -0.784 *** (0.124) 0.3-0.001 (0.018) 0.027 *** (0.010) 1.758 *** (0.101) -0.777 *** (0.100) 0.4 0.013 (0.018) 0.029 *** (0.011) 1.734 *** (0.090) -0.749 *** (0.088) 0.5 0.030 * (0.017) 0.032 *** (0.011) 1.683 *** (0.082) -0.696 *** (0.080) 0.6 0.048 *** (0.018) 0.034 *** (0.011) 1.678 *** (0.075) -0.691 *** (0.074) 0.7 0.062 *** (0.023) 0.041 *** (0.013) 1.656 *** (0.081) -0.667 *** (0.080) 0.8 0.083 *** (0.029) 0.034 ** (0.017) 1.626 *** (0.087) -0.639 *** (0.089) 0.9 0.087 ** (0.037) 0.023 (0.021) 1.635 *** (0.114) -0.646 *** (0.113) 0.95 0.091 (0.060) 0.034 (0.027) 1.655 *** (0.189) -0.674 *** (0.183) Noe: *** Significan a 1%. ** Significan a 5%. * Significan a 10%. The resuls regarding he ineres rae smoohing coefficiens are significanly differen from zero. Beween quaniles 0.05 and 0.8, here was a reducion in coefficien θ 1 (τ), whereas θ 2 (τ) increased. By adding up θ 1 (τ) + θ 2 (τ), we verify ha he Selic rae smoohing wen up from 0.959 a quanile 0.05 o 0.981 a quanile 0.95. This demonsraes ha he CBB s moneary policy is characerized by large smoohing of he Selic rae and ha his smoohing increases a he higher quaniles along he disribuion. Figure 2 depics he long-erm responses of he Selic rae o deviaions of inflaion from he arge and o oupu gap for specificaion (8). The solid line shows he coefficiens obained by IVQR and he horizonal lines show he IV esimaes wih a 90%CI (dashed lines). Consonan wih Wolers (2012), we do no provide he confidence inerval for he coefficiens a he quaniles because, in general, we had high 14
sandard errors which, consequenly, implied raher broad confidence inervals. 23 A possible explanaion for ha is ha he sum of he smoohing parameers is very close o 1, yielding very high esimaes for he sandard errors obained by he Dela mehod. 24 Tha being said, we may noe ha, when he Selic rae is in he lower ail of he condiional disribuion, he reacion o inflaion and o oupu gap is more passive and becomes more acive as we move owards he righ side of he disribuion. In addiion, we verified ha, in he upper ail of he disribuion, he reacion of he ineres rae o inflaion was sronger han ha obained by IV. Noe ha he esimaes of β 1 (τ) were significan a quaniles 0.6 (3.81 wih a sandard error of 1.89) and 0.7 (5.59 wih a sandard error of 3.31), whereas he IV esimae was no saisically differen from zero. This suggess ha he response of he Selic rae o inflaion is sronger when his ineres rae is above is condiional median. The upper ail of he disribuion exhibis a weaker response o inflaion han in he IV esimaion, alhough he coefficiens are insignifican in boh cases. Compared wih he coefficien of inflaion, he long-erm response o oupu gap is more sable along he whole disribuion, as he poin esimaes obained by quanile regression usually fall wihin he confidence inerval of he IV esimae. Figure 2 Long-erm response of he Selic rae o inflaion (β 1 ) and o oupu gap (β 2 ) for reacion funcion (8). Noe: Dashed lines denoe a 90% CI for he coefficiens esimaed by IV. Table 4 shows he shor-erm coefficiens of moneary rule (13) esimaed for he quaniles and heir respecive sandard errors (in brackes). The shor-erm response of he Selic rae o he expeced inflaion gap is saisically differen from quanile 0.4 onwards. Resuls also demonsrae ha his response has an uprend as we move owards he righ side of he condiional Selic rae disribuion. Moreover, noe ha from quanile 0.6, he esimae of β 1 (τ) is higher han he esimae obained by IV. Noneheless, his difference is no significan, as he confidence inervals of he esimaes a he quaniles include he poin IV esimae. Finally, when we compare hese resuls wih hose shown in Table 3, we verify ha he shor-erm response of he Selic rae o he expeced inflaion gap is sronger han ha o he curren inflaion gap beween quaniles 0.4 and 0.95. This indicaes ha he CBB s forward-looking behavior 23 The sandard errors of he long-erm responses of he Selic rae may be provided by he auhors upon reques. 24 Chevaparakul e al. (2009) solve his problem by esimaing he original Taylor rule, i.e., wihou he smoohing parameer. However, as he shor-erm ineres rae smoohing is observed in CBB s moneary policy, we oped no o follow Chevaparakul e al. (2009), as we would have misspecificaion of he reacion funcion o be esimaed. 15
is observed no only in he condiional mean, bu also in mos of he Selic rae disribuion. Table 4 IVQR esimaes for reacion funcion (13) Quanile β 1 β 2 θ 1 θ 2 0.05 0.027 (0.099) 0.065 *** (0.024) 1.787 *** (0.135) -0.829 *** (0.138) 0.1-0.042 (0.088) 0.045 ** (0.023) 1.765 *** (0.136) -0.797 *** (0.142) 0.2 0.052 (0.064) 0.040 ** (0.016) 1.640 *** (0.138) -0.660 *** (0.141) 0.3 0.060 (0.046) 0.044 *** (0.013) 1.687 *** (0.108) -0.705 *** (0.109) 0.4 0.092 ** (0.041) 0.035 *** (0.012) 1.652 *** (0.098) -0.664 *** (0.097) 0.5 0.114 *** (0.034) 0.034 *** (0.010) 1.584 *** (0.087) -0.593 *** (0.086) 0.6 0.142 *** (0.028) 0.035 *** (0.010) 1.563 *** (0.076) -0.573 *** (0.075) 0.7 0.133 *** (0.032) 0.032 *** (0.011) 1.572 *** (0.076) -0.578 *** (0.075) 0.8 0.134 *** (0.047) 0.030 ** (0.015) 1.530 *** (0.089) -0.534 *** (0.088) 0.9 0.154 ** (0.073) 0.037 * (0.022) 1.505 *** (0.138) -0.506 *** (0.133) 0.95 0.201 * (0.109) 0.036 (0.026) 1.663 *** (0.167) -0.668 *** (0.159) Noe: *** Significan a 1%. ** Significan a 5%. * Significan a 10%. The response of oupu gap is significan beween quaniles 0.05 and 0.9 and shows a downrend along he condiional ineres rae disribuion. Wih respec o ineres rae smoohing, i should be noed ha he coefficien θ 1 (τ) has a downrend whereas he coefficien θ 2 (τ) exhibis he opposie behavior. As wih moneary rule (8), he sum θ 1 (τ) + θ 2 (τ) indicaes larger smoohing a he upper quaniles of he Selic rae disribuion. Figure 3 Long-erm responses of he Selic rae o Dj (β 1 ) and oupu gap (β 2 ) for reacion funcion (13). Noe: Dashed lines denoe a 90% CI for he coefficiens esimaed by IV. 16
Figure 3 displays he long-erm responses of he Selic rae o Dj and o he oupu gap for specificaion (13). Noe ha he response of he ineres rae o hese variables is increasing along he condiional disribuion. However, he sandard errors allow us o say ha he esimae of β 1 (τ) is significan only a quanile 0.6 (15.41 wih a sandard error of 8.87). On he oher hand, he esimaes of he coefficien of oupu gap (β 2 ) were significan a quaniles 0.05 (1.56 wih a sandard error of 0.80) and 0.3 (2.33 wih a sandard error of 1.10), bu insignifican a he oher quaniles of he condiional disribuion. 4.4 Robusness of he resuls In his secion, we check he robusness of he resuls by performing wo exercises: i) we use differen oupu gap measures; ii) we include he exchange rae in he CBB s reacion funcion. 4.4.1 Differen oupu gap measures Table 5 shows he resuls esimaed by IV, GMM, and IVQR for reacion funcion (8) for wo differen oupu gap measures. In he firs half of he able, we consider he oupu gap (y TL ) obained from a linear rend model, whereas in he second one, we use he oupu gap (y TQ ) calculaed from a quadraic rend model for he naural log of oupu. For hese specificaions, we idenify similariies o he resuls ha consider he oupu gap obained wih he HP filer. For boh specificaions, he shor-erm response of he ineres rae o inflaion is increasing along he disribuion. In addiion, in he upper ail of he condiional disribuion, his response has been sronger han he resuls esimaed by IV and GMM and saisically differen from zero beween quaniles 0.5 and 0.9. Regarding he shor-erm response o he oupu gap, i is significan from quanile 0.05 o quanile 0.7 and shows an uprend. As far as long-erm responses of he Selic rae are concerned, wo resuls should be highlighed. Firs, he response of inflaion gap in he condiional mean and beween quaniles 0.5 and 0.7 is saisically differen from zero and saisfies he Taylor (1993) principle. Second, he response of he Selic rae o oupu gap is saisically differen from zero up o quanile 0.7. Neverheless, all he significan par of he IVQR is wihin he confidence inerval esimaed by IV for he condiional mean. Thus, we may infer ha he long-erm response of he Selic rae o oupu gap is more sable han ha of he inflaion gap along he disribuion of his policy insrumen. Table 6 shows he resuls obained by IV, GMM, and IVQR for reacion funcion (13) aking ino accoun differen oupu gap measures (linear rend and quadraic rend). As demonsraed above, here are nonlineariies in he shor-erm response of he Selic rae o expeced inflaion. Paricularly, we noe ha he CBB s shor-erm response o expeced inflaion is significan beween quaniles 0, 3 and 0.95, bu no in he exreme ail of he disribuion. In urn, he response of he ineres rae o oupu gap is no saisically differen from zero a he quaniles below 0.2 and above 0.8. 17
Table 5 IVQR esimaes for reacion funcion (8) β 1 β 2 θ 1 θ 2 β 1 β 2 Quanile Specificaion wih y TL IV 0.036 * (0.019) 0.022 *** (0.007) 1.768 *** (0.063) -0.792 *** (0.064) 1.560 * (0.797) 0.945 *** (0.237) GMM 0.038 * (0.021) 0.022 *** (0.007) 1.706 *** (0.060) -0.728 *** (0.059) 1.714 * (0.873) 0.997 *** (0.280) 0.05-0.023 (0.045) 0.030 ** (0.012) 1.784 *** (0.107) -0.819 *** (0.105) -0.662 (1.595) 0.870 * (0.492) 0.1-0.029 (0.034) 0.030 * (0.009) 1.791 *** (0.109) -0.822 *** (0.108) -0.905 (1.220) 0.955 ** (0.439) 0.2 0.009 (0.032) 0.019 ** (0.009) 1.828 *** (0.137) -0.853 *** (0.134) 0.360 (1.226) 0.769 * (0.400) 0.3 0.026 (0.025) 0.016 * (0.008) 1.755 *** (0.117) -0.780 *** (0.114) 1.060 (0.942) 0.652 * (0.344) 0.4 0.026 (0.023) 0.015 * (0.009) 1.758 *** (0.091) -0.777 *** (0.089) 1.394 (1.087) 0.783 * (0.402) 0.5 0.048 ** (0.024) 0.016 * (0.009) 1.694 *** (0.084) -0.716 *** (0.082) 2.160 ** (0.942) 0.722 ** (0.313) 0.6 0.067 * (0.026) 0.019 ** (0.009) 1.702 *** (0.081) -0.720 *** (0.079) 3.664 *** (1.351) 1.042 *** (0.364) 0.7 0.083 * (0.030) 0.028 *** (0.010) 1.666 *** (0.083) -0.683 *** (0.082) 4.884 ** (2.125) 1.645 *** (0.600) 0.8 0.093 * (0.033) 0.019 (0.014) 1.630 *** (0.090) -0.645 *** (0.089) 6.583 (4.206) 1.342 (0.887) 0.9 0.081 * (0.042) 0.008 (0.021) 1.631 *** (0.121) -0.641 *** (0.116) 8.315 (13.41) 0.864 (1.744) 0.95 0.099 (0.067) 0.011 (0.026) 1.707 *** (0.186) -0.716 *** (0.178) 10.62 (26.38) 1.164 (2.884) Quanile Specificaion wih y TQ IV 0.032 * (0.019) 0.023 *** (0.008) 1.771 *** (0.064) -0.791 *** (0.064) 1.691 * (0.968) 1.190 *** (0.363) GMM 0.034 * (0.020) 0.023 *** (0.007) 1.705 *** (0.060) -0.724 *** (0.059) 1.887 * (1.075) 1.288 *** (0.427) 0.05-0.034 (0.044) 0.028 ** (0.014) 1.785 *** (0.108) -0.811 *** (0.106) -1.263 (2.317) 1.057 (0.742) 0.1-0.033 (0.034) 0.032 *** (0.010) 1.793 *** (0.109) -0.820 *** (0.109) -1.218 (1.465) 1.202 * (0.662) 0.2 0.009 (0.030) 0.021 ** (0.009) 1.786 *** (0.133) -0.807 *** (0.131) 0.406 (1.378) 0.999 * (0.578) 0.3 0.018 (0.024) 0.023 *** (0.008) 1.756 *** (0.115) -0.777 *** (0.113) 0.885 (1.084) 1.124 ** (0.520) 0.4 0.027 (0.021) 0.018 ** (0.009) 1.745 *** (0.090) -0.761 *** (0.088) 1.618 (1.188) 1.078 * (0.544) 0.5 0.045 * (0.023) 0.017 * (0.009) 1.701 *** (0.084) -0.720 *** (0.082) 2.379 ** (1.124) 0.895 ** (0437) 0.6 0.065 *** (0.025) 0.020 ** (0.010) 1.703 *** (0.080) -0.718 *** (0.079) 4.238 ** (1.794) 1.286 ** (0.584) 0.7 0.075 *** (0.028) 0.023 ** (0.010) 1.689 *** (0.082) -0.701 *** (0.082) 6.191 * (3.386) 1.941 * (1.037) 0.8 0.084 *** (0.032) 0.016 (0.013) 1.649 *** (0.091) -0.656 *** (0.091) 11.33 (13.13) 2.182 (2.727) 0.9 0.080 ** (0.040) 0.009 (0.021) 1.631 *** (0.120) -0.638 *** (0.117) 10.34 (19.39) 1.186 (2.780) 0.95 0.100 (0.066) 0.010 (0.027) 1.713 *** (0.181) -0.714 *** (0.174) 139.8 (4690.0) 14.35 (472.1) Noe: *** Significan a 1%. ** Significan a 5%. * Significan a 10%. 18
Table 6 IVQR esimaes for reacion funcion (13) β 1 β 2 θ 1 θ 2 β 1 β 2 Quanile Specificaion wih y TL IV 0.149 *** (0.029) 0.031 *** (0.007) 1.613 *** (0.070) -0.634 *** (0.071) 7.165 *** (2.268) 1.465 *** (0.407) GMM 0.165 *** (0.026) 0.033 *** (0.006) 1.582 *** (0.061) -0.605 *** (0.063) 7.380 *** (2.051) 1.472 *** (0.392) 0.05 0.052 (0.105) 0.037 ** (0.017) 1.754 *** (0.168) -0.804 *** (0.170) 1.049 (2.089) 0.752 * (0.391) 0.1 0.021 (0.102) 0.020 (0.014) 1.750 *** (0.158) -0.800 *** (0.163) 0.425 (2.052) 0.409 (0.311) 0.2 0.135 (0.073) 0.030 *** (0.011) 1.586 *** (0.156) -0.612 *** (0.157) 5.087 (3.371) 1.104 * (0.591) 0.3 0.119 *** (0.046) 0.028 *** (0.008) 1.650 *** (0.117) -0.671 *** (0.116) 5.858 ** (2.767) 1.359 ** (0.606) 0.4 0.127 *** (0.039) 0.021 *** (0.008) 1.655 *** (0.101) -0.671 *** (0.099) 7.878 ** (3.383) 1.316 ** (0.597) 0.5 0.151 *** (0.033) 0.023 *** (0.006) 1.608 *** (0.092) -0.624 *** (0.090) 9.168 *** (2.707) 1.386 *** (0.503) 0.6 0.155 *** (0.029) 0.022 *** (0.006) 1.556 *** (0.078) -0.569 *** (0.077) 11.71 *** (4.095) 1.654 ** (0.686) 0.7 0.160 *** (0.031) 0.023 *** (0.007) 1.574 *** (0.070) -0.584 *** (0.069) 17.24 * (9.359) 2.488 * (1.424) 0.8 0.172 *** (0.046) 0.023 ** (0.010) 1.511 *** (0.092) -0.513 *** (0.090) 102.2 (534.8) 13.52 (70.15) 0.9 0.150 * (0.090) 0.014 (0.020) 1.530 *** (0.165) -0.539 *** (0.156) 15.80 (28.12) 1.440 (2.510) 0.95 0.214 * (0.120) 0.027 (0.025) 1.665 *** (0.193) -0.658 *** (0.182) -30.37 (93.41) -3.821 (12.94) Quanile Specificaion wih y TQ IV 0.141 *** (0.029) 0.032 *** (0.007) 1.622 *** (0.071) -0.638 *** (0.072) 8.714 ** (3.398) 1.966 *** (0.693) GMM 0.156 *** (0.026) 0.034 *** (0.007) 1.585 *** (0.063) -0.602 *** (0.064) 9.127 *** (3.239) 1.994 *** (0.690) 0.05 0.025 (0.105) 0.027 (0.018) 1.770 *** (0.161) -0.815 *** (0.163) 0.537 (2.290) 0.591 (0.448) 0.1-0.016 (0.101) 0.031 (0.016) 1.782 *** (0.153) -0.826 *** (0.159) -0.363 (2.313) 0.711 (0.454) 0.2 0.095 (0.072) 0.028 ** (0.011) 1.680 *** (0.152) -0.708 *** (0.155) 3.365 (3.022) 0.997 * (0.594) 0.3 0.106 ** (0.046) 0.028 *** (0.008) 1.660 *** (0.116) -0.675 *** (0.116) 6.908 (4.326) 1.829 * (1.038) 0.4 0.102 *** (0.039) 0.022 *** (0.009) 1.685 *** (0.098) -0.700 *** (0.097) 7.142 * (3.773) 1.523 * (0.791) 0.5 0.151 *** (0.033) 0.025 *** (0.007) 1.607 *** (0.093) -0.620 *** (0.091) 12.21 ** (4.869) 2.036 ** (0.946) 0.6 0.149 *** (0.029) 0.024 *** (0.007) 1.573 *** (0.078) -0.581 *** (0.078) 18.89 (11.55) 3.068 (2.041) 0.7 0.159 *** (0.031) 0.025 *** (0.008) 1.559 *** (0.072) -0.565 *** (0.072) 27.31 (23.72) 4.303 (3.922) 0.8 0.166 *** (0.047) 0.023 ** (0.010) 1.527 *** (0.095) -0.525 *** (0.093) -77.32 (325.8) -10.789 (45.58) 0.9 0.149 * (0.085) 0.014 (0.021) 1.531 *** (0.163) -0.537 *** (0.155) 23.66 (60.67) 2.175 (5.598) 0.95 0.197 * (0.118) 0.026 (0.025) 1.662 *** (0.193) -0.646 *** (0.184) -12.74 (19.02) -1.651 (2.969) Noe: *** Significan a 1%. ** Significan a 5%. * Significan a 10%. 19