ISSN 1518-3548
Working Paper Series
Selection of Optimal Lag Length in Cointegrated VAR Models with Weak Form of Common Cyclical Features Carlos Enrique Carrasco Gutiérrez, Reinaldo Castro Souza and Osmani Teixeira de Carvalho Guillén June, 2007
ISSN 1518-3548 CGC 00.038.166/0001-05
Working Paper Series Brasília n. 139 Jun 2007 P. 1-35
Working Paper Series
Edited by Research Department (Depep) – E-mail: [email protected]
Editor: Benjamin Miranda Tabak – E-mail: [email protected] Editorial Assistent: Jane Sofia Moita – E-mail: [email protected] Head of Research Department: Carlos Hamilton Vasconcelos Araújo – E-mail: [email protected]
The Banco Central do Brasil Working Papers are all evaluated in double blind referee process.
Reproduction is permitted only if source is stated as follows: Working Paper n. 139.
Authorized by Mário Mesquita, Deputy Governor for Economic Policy.
General Control of Publications
Banco Central do Brasil Secre/Surel/Dimep SBS – Quadra 3 – Bloco B – Edifício-Sede – M1 Caixa Postal 8.670 70074-900 Brasília – DF – Brazil Phones: (5561) 3414-3710 and 3414-3567 Fax: (5561) 3414-3626 E-mail: [email protected]
The views expressed in this work are those of the authors and do not necessarily reflect those of the Banco Central or its members.
Although these Working Papers often represent preliminary work, citation of source is required when used or reproduced.
As opiniões expressas neste trabalho são exclusivamente do(s) autor(es) e não refletem, necessariamente, a visão do Banco Central do Brasil.
Ainda que este artigo represente trabalho preliminar, citação da fonte é requerida mesmo quando reproduzido parcialmente.
Consumer Complaints and Public Enquiries Center
Address: Secre/Surel/Diate Edifício-Sede – 2º subsolo SBS – Quadra 3 – Zona Central 70074-900 Brasília – DF – Brazil Fax: (5561) 3414-2553 Internet: http://www.bcb.gov.br/?english
Selection of Optimal Lag Length in Cointegrated VAR Models with Weak Form of Common Cyclical Features∗
Carlos Enrique Carrasco Gutiérrez† Reinaldo Castro Souza‡ Osmani Teixeira de Carvalho Guillén§
The Working Paper should not be reported as representing the views of the Banco Central do Brasil. The views expressed in the papers are those of the author(s) and not necessarily reflect those of the Banco Central do Brasil.
Abstract An important aspect of empirical research based on the vector autoregres- sive (VAR) model is the choice of the lag order, since all inference in the VAR model depends on the correct model specification. Literature has shown im- portant studies of how to select the lag order of a nonstationary VAR model subject to cointegration restrictions. In this work, we consider an additional weak form (WF)restrictionofcommoncyclicalfeaturesinthemodelinorder to analyze the appropriate way to select the correct lag order. Two method- ologies have been used: the traditional information criteria (AIC, HQ and SC) and an alternative criterion (IC(p, s)) which select simultaneously the lag order p and the rank structure s due to the WF restriction. A Monte- Carlosimulationisusedintheanalysis. The results indicate that the cost of ignoring additional WF restrictions in vector autoregressive modelling can be high specially when SC criterion is used.
Keywords: Cointegration; Common Cyclical Features; Reduced Rank Model; Estimation; Information Criteria. JEL Codes: C32, C53.
∗Acknowledgments: We are grateful to comments and suggestions given by João Victor Issler, Wagner Gaglianone, Ricardo Cavalcanti, Luiz Renato Lima and participants of the Brazilian Econometric Meeting 2006. Special thanks are due to Alain Hecq for solving doubts and com- ments. The authors are responsible for any remaining errors in this paper. Carlos Enrique C. Gutiérrez acknowledges the support of CAPES-Brazil. †Corresponding author, [email protected], Graduate School of Economics, FGV-EPGE-RJ, Brazil. ‡[email protected], DEE-PUC-RJ, Brazil §[email protected], Banco Central do Brasil and Ibmec-RJ, Brazil
3 1 Introduction
In the modelling of economic and financial time series, the vectorial autoregressive (VAR) model became a standard linear model used in empirical works. An important aspect of empirical research in the specification of the VAR models is the determi- nation of the lag order of the autoregressive lag polynomial, since all inference in the VAR model depends on the correct model specification. In several contributions, the effect of lag length selection has been demonstrated: Lütkepohl (1993) indicates that selecting a higher order lag length than the true lag length causes an increase in the mean square forecast errors of the VAR and that underfitting the lag length often generates autocorrelated errors. Braun and Mittnik (1993) show that impulse response functions and variance decompositions are inconsistently derived from the estimated VAR when the lag length differs from the true lag length. When cointe- gration restrictions are considered in the model, the effect of lag length selection on the cointegration tests has been demonstrated. For example, Johansen (1991) and Gonzalo (1994) point out that VAR order selection may affect proper inference on cointegrating vectors and rank. Recently empirical works have considered another kind of restrictions on the VAR model (e.g., Engle and Issler, 1995; Caporale, 1997; Mamingi and Sunday, 2003). Engle and Kozicki (1993) showed that VAR models can have another type of restrictions, called common cyclical features, which are restrictions on the short-run dynamics. These restrictions are defined in the same way as cointegration restric- tions, while cointegration refers to relations among variables in the long-run, the common cyclical restrictions refer to relations in the short-run. Vahid and Engle (1993) proposed the Serial Correlation Common Feature (SCCF)asameasureof common cyclical feature. SCCF restrictions might be imposed in a covariance sta- tionary VAR model or in a cointegrated VAR model. When short-run restrictions are imposed in cointegrated VAR models it is possible to define a weak version of SCCF restrictions. Hecq, Palm and Urbain (2006) definedaweakversionofSCCF restrictions which they denominated it as weak-form (WF) common cyclical restric-
4 tions. A fundamental difference between SCCF and WF restrictions is in the form which each one imposes restrictions on the Vector Error Correction Model (VECM) representation1. When SCCF are imposed, all matrices of a VECM have rank less than the number of variables analyzed. On the other hand with WF restrictions all matrices, except the long-run matrix, have rank less than a number of variables in analysis. Hence, WF restrictions impose less restriction on VECM parameters. Some advantages emerge when WF restrictions are considered. First, due to the fact that WF restrictions does not impose restrictions on the cointegration space; the rank of common cyclical features is not limited by the choice of cointegrating rank. Another advantage is that WF restrictions is invariant over reparametrization in VECM representation. The literature has shown how to select an adequate lag order of a covariance stationary VAR model and an adequate lag order of a VAR model subject to coin- tegration restrictions. Among the classical procedures, there are the information criteria such as Akaike (AIC), Schwarz (SC) and Hannan-Quinn (HQ) (Lütkepohl, 1993). Kilian (2001) study the performance of traditional AIC, SC and HQ criterion of a covariance stationary VAR model. Vahid and Issler (2002) analyzed the stan- dard information criterion in a covariance stationary VAR model subject to SCCF restriction and more recently Guillén, Issler and Athanasopoulos (2005) studied the standard information criterion in VAR models with cointegration and SCCF re- strictions. However, when cointegrated VAR models contain additional weak form of common cyclical feature, there are no reported work on how to appropriately determine the VAR model order. The objective of this paper is to investigate the performance of information criterion in selecting the lag order of a VAR model when the data are generated from a true VAR with cointegration and WF restrictions that is referred as the correct model. It will be carried out following two procedures: a) the use of standard criteria as proposed by Vahid and Engle (1993), referred here as IC (p),andb) the use of an
1 When a VAR model has cointegration restriction it can be represented as a VECM. This representation is also known as Granger Representation Theorem (Engle and Granger, 1987).
5 alternative procedure of model selection criterion (see, Vahid and Issler, 2002; Hecq et al., 2006) consisting in selecting simultaneously the lag order p and the rank s do to the weak form of common cyclical feature, which is referred to as IC(p, s)2.The most relevant results can be summarized as follows. The information criterion that selects simultaneously the pair (p, s) has better performance than the model chosen by conventional criteria. The cost of ignoring additional WF restrictions in vector autoregressive modelling can be high specially when SC criterion is used. The remaining of this work is organized as follows. Section 2 shows the econo- metric model. In section 3 the information criteria are mentioned. Monte Carlo simulation is shown in section 4 and the results in section 5. Finally, the conclusions are shown in section 6.
2 The Econometric Model
We show the VAR model with short-run and long-run restrictions. First, we consider a Gaussian vector autoregression of finite order p, so-called VAR(p), such that:
p
yt = Aiyt i + εt (1) − i=1 X where, yt is a vector of n first order integrated series, I(1), Ai, i =1,...,p are matrices of dimension n n, εt Normal(0, Ω) and {Ω, if t = τ and 0n n, if × ∼ × t = τ,whereΩ is non singular}. The model (1) could be written equivalently as; 6 p i Π (L) yt = εt where L represents the lag operator and Π (L)=In AiL that − i=1 p when L =1, Π (1) = In Ai. If cointegration is considered inP (1) the (n n) − i=1 × matrix Π ( ) satisfies two conditions:P a) Rank (Π (1)) = r, 0 2 This is quite recent in the literature (see, Hecq et al., 2006). 6 model as VECM. Decomposing the polynomial matrix Π (L)=Π (1) L + Π∗ (L) ∆, where ∆ (1 L) is the difference operator, a Vector Error Correction Model ≡ − (VECM) is obtained: p 1 − ∆yt = αβ0yt 1 + Γi∆yt i + εt (2) − − i=1 X p where: αβ0 = Π (1), Γj = Ak for j =1, ...., p 1 and Γ0 = In.The − − k=j+1 − VAR(p) model can include additionalP short-horizon restrictions as shown by Vahid and Engle (1993). We consider an interesting WF restriction (as defined by Hecq, Palm and Urbain, 2006) that does not impose restrictions over long-run relations. Definition 1 Weak Form-WF holds in (2) if, in addition to assumption 1 (cointe- gration), there exists a (n s) matrix β˜ of rank s, whose columns span the cofeature × ˜ ˜ ˜ space, such that β0(∆yt αβ´yt 1)=β0εt , where β0εt is a s-dimensional vector that − − constitutes an innovation process with respect to information prior to period t,given by yt 1,yt 2, ..., y1 . { − − } Consequently we considerate WF restrictions in the VECM if there exists a cofeature matrix β˜ that satisfies the following assumption: ˜ Assumption 1 : β0 Γj =0s n for j =1, ...., p 1. × − Imposing WF restrictions is convenient because it allows the study of both coin- tegration and common cyclical feature without the constraint r + s n.Wecan ≤ rewrite the VECM with WF restrictions as a model of reduced-rank structure. In (2) let Xt 1 =[∆yt0 1, .....∆yt0 p+1]0 and Φ =[Γ1, ...., Γp 1], therefore we get: − − − − ∆yt = αβ´yt 1 + ΦXt 1 + εt (3) − − If assumption (1) holds matrices Γi,i =1, ..., p are all of rank (n s) then we can − ˜ ˜ ˜ write Φ = β Ψ = β [Ψ1, ...., Ψp 1],where, β is n (n s) full column rank matrix, ⊥ ⊥ − ⊥ × − Ψ is of dimension (n s) n(p 1), the matrices Ψi,i =1, ..., p 1 all of rank − × − − (n s) n. Hence, given assumption (1), there exists β˜ of n s such that β˜0β˜ =0. − × × ⊥ 7 That is, β˜ n (n s) is a full column rank orthogonal to the complement of β˜ ⊥ × − with rank(β,˜ β˜ )=n. Rewriting model (3) we have: ⊥ ˜ ∆yt = αβ´yt 1 + β (Ψ1, Ψ2, ..., Ψp 1) Xt 1 + εt (4) − ⊥ − − ˜ = αβ´yt 1 + β ΨXt 1 + εt (5) − ⊥ − Estimation of (5) is carried out via the switching algorithms (see, Hecq, 2006) that use the procedure in estimating reduced-rank regression models suggested by Ander- son (1951). There is a formal connection between a reduced-rank regression and the canonical analysis as noted by Izenman (1975), Box and Tiao (1977), Tso (1980) and Veleu et al. (1986). When the multivariate regression has all of its matrix coefficients of full rank, it may be estimated by usual Least Square or Maximum- Likelihood procedures. But when the matrix coefficients are of reduced-rank they have to be estimated using the reduced-rank regression models of Anderson (1951). The use of canonical analysis may be regarded as a special case of reduced-rank regression. More specifically, the maximum-likelihood estimation of the parameters of the reduced-rank regression model may result in solving a problem of canonical 3 analysis . Therefore, we can use the expression CanCorr Xt,Zt Wt that denotes { | } the partial canonical correlations between Xt and Zt: both sets concentrate out the effect of Wt that allows us to obtain canonical correlation, represented by the eigen- ˆ ˆ ˆ ˆ values λ1 > λ2 > λ3...... > λn. The Johansen test statistic is based on canonical correlation. In model (2) we can use the expression CanCorr ∆yt,yt 1 Wt where { − | } Wt =[∆yt 1, ∆yt 2, ....., ∆yt+p 1] that summarizes the reduced-rank regression pro- − − − cedure used in the Johansen approach. It means that one extracts the canonical correlations between ∆yt and yt 1: both sets concentrated out the effect of lags of − Wt. In order to test for the significance of the r largest eigenvalues, one can rely on Johansen’s trace statistic (6): n ˆ2 ξr = T Ln (1 λi ) i =1, ..., n (6) − i=r+1 − X 3 This estimation is referred as Full Information Maximum Likelihood -FIML 8 ˆ ˆ 1 where the eigenvalues 0 < λn < ... < λ1 are the solution of : λm11 m− m00m01 = | − 10 | 1 T 0,wheremij,i,j=0.1, are the second moment matrices: m00 = T t=1 u˜0tu˜0t, 1 T 1 T 1 T m10 = T t=1 u˜1tu˜0t, m01 = T t=1 u˜0tu˜10 t, m11 = T t=1 u˜1tu˜10 t of the residualsP u˜0t and u˜1t obtainedP in the multivariateP least squares regressionsP ∆yt =(∆yt 1,...∆yt p+1)+ − − u0t and yt 1 =(∆yt 1,...∆yt p+1)+u1t respectively (see, Hecq et al., 2006; Johansen, − − − 1995). The result of Johansen test is a superconsistent estimated β.Moreover,we could also use a canonical correlation approach to determine the rank of the common features space due to WF restrictions. It is a test for the existence of cofeatures in the form of linear combinations of the variables in the first differences, corrected ˜ ˜ ˜ for long-run effects which are white noise (i.e., β0(∆yt αβ´yt 1)=β0εt where β0εt − − is a white noise). Canonical analysis is adopted in the present work in estimating, testing and selecting lag-rank of VAR models as shown in next sections. 3 Model Selection Criteria In model selection we use two procedures to identify the VAR model order. The standard selection criteria, IC(p) and the modified informational criteria, IC(p, s), novelty in the literature, which consists on identifying p and s simultaneously. The model estimation following the standard selection criteria, IC(p), used by Vahid and Engle (1993) entails the following steps: 1. Estimate p using standard informational criteria: Akaike (AIC), Schwarz (SC) and Hanna-Quinn (HQ). We choose the lag length of the VAR in levels that minimize the information criteria. 2.Usingthelaglengthchoseninthepreviousstep,find the number of cointegration vector, r using Johansen cointegration test4. 3. Conditional on the results of cointegration analysis, a final VECM is estimated and then the multi-step ahead forecast is calculated. 4 Cointegration rank and vectors are estimated using the FIML as shown in Johansen (1991). 9 The above procedure is followed when there is evidence of cointegration re- strictions. We check the performance of IC(p) when WF restrictions contain the true model. Additionally we check the performance of alternative selection criteria IC(p, s). Vahid and Issler (2002) analyzed a covariance-stationary VAR model with SCCF restrictions. They showed that the use of IC(p, s) has better performance than IC(p) in VAR model lag order selection. In the present work we analyze coin- tegrated VAR model with WF restrictions in order to analyze the performance of IC(p) and IC(p, s) for model selection. The question investigated is: is the perfor- mance of IC(p, s) superior to that of IC(p)? Thisisanimportantquestionweaim to answer in this work. The procedure of selecting the lag order and the rank of the structure of short- run is carried out by minimizing the following modified information criteria (see Hecq, 2006). T 2 AIC (p, s)= ln(1 λ2(p)) + N (7) i T i=n s+1 − × X− T 2ln(lnT ) HQ(p, s)= ln(1 λ2(p)) + N (8) i T i=n s+1 − × X− T ln T SC(p, s)= ln(1 λ2(p)) + N (9) i T i=n s+1 − × X− N =[n (n (p 1)) + n r] [s (n (p 1) + (n s))] × × − × − × × − − The number of parameters N is obtained by subtracting the total number of mean parameters in the VECM (i.e., n2 (p 1)+nr), for given r and p,fromthenumber × − of restrictions the common dynamics imposes from s (n (p 1)) s (n s).The × × − − × − eigenvalues λi are calculated for each p. To calculate the pair (p, s) we assume that no restriction of cointegration exists, that is, r = n (see Hecq, 2006). We fix p in model (3) and then find λi i =1, 2...n using the program cancorr(∆yt,Xt 1 yt 1). − | − This procedure is followed for every p and in the end we choose the p and s that 10 minimizes the IC(p, s). After selecting the pair (p, s) we can test the cointegration relation using the procedure of Johansen. Finally we estimate the model using the switching algorithms as shown in the next chapter. Notice that in this simultaneous selection, testing the cointegration relation is the last procedure to follow, so we are inverting the hierarquical procedure followed by Vahid and Engle (1993) where the first step is the selection of the number of cointegration relations. It may be an advantage specially when r is over-estimated. Few works have been dedicated to analyze the order of the VAR models considering modified IC(p, s).Asmentioned, Vahid and Issler (2002) suggested the use of IC(p, s) to simultaneously choose the order p andanumberofreducedrankstructures on covariance stationary VAR model subject to SCCF restrictions. However, no work has analyzed the order of the VAR model with cointegration and WF restrictions using a modified criterion, which is exactly the contribution of this paper. To estimate the VAR model considering cointegration and WF restrictions we use the switching algorithms model as considered by Hecq (2006). Consider the VECM given by: ˜ ∆yt = αβ0yt 1 + β ΨXt 1 + εt (10) − ⊥ − A full description of switching algorithms is presented below in four steps: Step1:Estimation of the cointegration vectors β. Using the optimal pair (¯p, s¯) chosen by information criteria (7), (8) or (9), we estimate β (and so its rank, r =¯r) using Johansen cointegration test. Step2:Estimation of β˜ and Ψ. ⊥ Taking βˆ estimated in step one, we proceed to estimate β˜ and Ψ. Hence, ⊥ ˆ0 we run a regression of ∆yt and of Xt 1on β yt 1. We labeled the resid- − − uals as u0 and u1, respectively. Therefore, we obtain a reduced rank regression: ˜ u0 = β Ψu1 + εt (11) ⊥ 11 where Ψ can be written as Ψ = C1, ..., C(¯p 1) of (n s¯) n(¯p 1) and − − × − β˜ of n (n s¯).Weestimate(11)byFIML.Thus,wecanobtain¡ ¢ β˜ ⊥ × − ⊥ and Ψˆ . Step3:Estimate of the Maximum Likelihood (ML) function. Given the parameters estimated in steps 1 and 2 we use a recursive algorithm to estimate the Maximum Likelihood (ML) function. We cal- ˆ ˆ2 culate the eigenvalues associated with Ψ, λi i =1, ..., s¯ and the matrix max of residuals r,¯ s=¯s. Hence, we compute the ML function: P max s¯ 0 T ˆ2 Lmax, r square regressions from ∆yt and Xt 1 on yt 1. In this case, estimation − − does not imply an iterative algorithm yet because the cointegrating space spans Rn. Step4:Reestimation of β. We reestimate β to obtain a more appropriated value for the parameters. In order to reestimate β we use the program CanCorr ∆yt,yt 1 ΨˆXt 1 − | − and thus using the new βˆ we can repeat step 2 to reestimateh β˜ and Ψ.i ⊥ Then, we can calculate the new value of the ML function in the step 3. 1 1 0 Henceforth, we obtain Lmax,r=¯r, s=¯s for calculating ∆L = Lmax,r=¯r, s=¯s - Lmax,r=¯r, s=¯s . ¡ ¢ Werepeatsteps1to4tochooseβ˜ and Ψ until convergence is reached ( i.e., ⊥ 7 ∆L<10− ). In the end, optimal parameters p¯, r¯ and s¯ are obtained and it can be used for estimation and forecasting of a VECM with WF restrictions. 12 4Monte-CarloDesign One of the critical issues regarding Monte-Carlo experiments is the data generating processes. To build the data generating processes we consider a VAR model with three variables, one cointegration vector, and two cofeatures vectors (i.e., n =3, r =1and s =2, respectively). β and β˜ satisfy: 1.0 1.00.1 β = 0.2 , β˜ = 0.01.0 1.0 0.5 0.5 − − ε1t 0 1.00.60.6 ε2t ∼ N 0 , 0.61.00.6 ε3t 0 0.60.61.0 ConsidertheVAR(3)model: yt = A1yt 1 + A2yt 2 + A3yt 3 + εt.TheVECM − − − respresentation as a function of the VAR level parameters can be written as: ∆yt =(A1 + A2 + A3 I3)yt 1 (A2 + A3)∆yt 1 A3∆yt 2 + εt (14) − − − − − − The VAR coefficients must simultaneously obey the restrictions: a) The cointegra- ˜ tion restrictions: αβ0 =(A1 + A2 + A3 I3) ; b) WF restrictions: β0A3 =0(iii) − ˜ β0(A2 + A3)=0and c) covariance-stationary condition. Considering the cointegra- tion restrictions we can rewrite (14) as the following VAR(1): ξt = Fξt 1 + vt (15) − yt (A2 + A3) A3 α εt 4 − − ξt = yt 1 ,F= I3 00and vt = 0 4 − β0yt β(A2 + A3) β0A3 β0α +1 β0εt − − Thus, the equation (15) will be covariance-stationary if all eigenvalues of matrix F lie inside the unit circle. An initial idea to design the Monte-Carlo experiment may consist of constructing the companion matrix (F) and verify whether the eigenvalues ofthecompanionmatrixalllieinsidetheunitcircle.Thismaybecarriedoutby selecting their values from a uniform distribution, and then verifying whether or not the eigenvalues of the companion matrix all lie inside the unit circle. However, this strategy could lead to a wide spectrum of search for adequate values for the 13 companion matrix. Hence, we follow an alternative procedure. We propose an analytical solution to generate a covariance-stationary VAR, based on the choice of the eigenvalues, and then on the generation of the respective companion matrix. In the appendix we present a detailed discussion of the final choice of these free parameters, including analytical solutions. In our simulation, we constructed 100 data generating processes and for each of these we generate 1000 samples containing 1000 observations. In order to reduce the impact of initial values, we consider only the last 100 and 200 observations. All the experiments were conducted in the MatLab environment. 5Results Values in Table I represent the percentage of time that the model selection crite- rion, IC(p), chooses that cell corresponding to the lag and number of cointegration vectors in 100000 realizations. The true lag-cointegrating vectors are identified by bold numbers and the selected lag-cointegration vectors chosen more times by the criterion are underlined. The results show that, in general, the AIC criterion choose more frequently the correct lag length for 100 and 200 observations. For example, for 100 observations, the AIC, HQ and SC criteria chose the true lag, p,54.08%, 35.62% and 17.49% of the times respectively. Note that all three criteria chose more frequently the correct rank of cointegration (r =1). When 200 observations are considered, the correct lag length was chosen 74.72%, 57.75% and 35.28% of the time for AIC, HQ and SC respectively. Again all three criteria selected the true cointegrated rank r =1. Tables II contains the percentage of time that the simultaneous model selection criterion, IC(p, s), chooses that cell, corresponding to the lag-rank and number of cointegrating vectors in 100,000 realizations. The true lag-rank-cointegration vectors are identified by bold numbers and the best lag-rank combination chosen more times by each criterion are underlined. The results show that, in general, the AIC criterion chooses more frequently lag-rank for 100 and 200 observations. For instance, for 100 observations, the AIC, HQ and SC criteria 14 choose more frequently the true pair (p, s)=(3, 1), 56.34%, 40.85% and 25.20% of the times respectively. For 200 observations, AIC, HQ and SC criteria choose more frequently the true pair (p, s)=(3, 1), 77.07%, 62.58% and 45.03% of the times respectively. Note that all three criteria choose more frequently the correct rank of cointegration (r =1)in both samples. The most relevant results canbesummarizedasfollows: All criteria (AIC, HQ and SC) choose the correct parameters more − oftenwhenusingIC(p, s). The AIC criterion has better performance in selecting the true model − more frequently for both the IC(p, s) and the IC(p) criteria. When the size of the sample decreases the true value p is less frequently − selected by all the traditional criteria. Table I shows that ignoring WF restrictions the standard SC has the − worst performance in choosing the true value of p. It is known that literature suggests the use of the traditional SC and HQ criteria in VAR model selection. The results of this work indicate that if additional WF restrictions are ignored, the standard SC and HQ criteria select few times the true value of p. That is, there is a cost of ignoring additional WF restrictions in the model specially when SC criterion is used. In general, the standard Schwarz or Hannan- Quinn selection criteria should not be used for this purpose in small samples due to the tendency of identifying an underparameterized model. In general, the use of these alternative criteria of selection, IC(p, s) has better performance than the usual criteria, IC(p), when the cointegrated VAR model has additional WF restriction. 6Conclusions In this work, we considered an additional weak form restriction of common cyclical features in a cointegrated VAR model in order to analyze the appropriate way for 15 selecting the correct lag order. These additional WF restrictions are defined in the same way as cointegration restrictions, while cointegration refers to relations among variables in the long-run, the common cyclical restrictions refer to relations in the short-run. Two methodologies have been used for selecting lag length; the traditional information criterion, IC(p), and an alternative criterion (IC(p, s)) that selects simultaneously the lag order p and the rank structure s due to the WF restriction. The results indicate that information criterion that selects the lag length and the rank order simultaneously has better performance than the model chosen by conventional criteria. When the WF restrictions are ignored there is a non trivial cost in selecting the true model with standard information criteria. In general, the standard Schwarz or Hannan-Quinn criteria selection criteria should not be used for this purpose in small samples due to the tendency of identifying an under- parameterized model. In applied work, when the VAR model contains WF and cointegration restric- tions, we suggest the use of AIC(p, s) criteria for simultaneously choosing the lag- rank, since it provides considerable gains in selecting the correct VAR model. Since no work in the literature has been dedicated to analyze a VAR model with WF com- mon cyclical restrictions, the results of this work provide new insights and incentives to proceed with this kind of empirical work. References [1] Anderson TW. 1951. Estimating linear restrictions on regression coefficients for multivariate normal distributions. Ann. Math. Statist. 22: 327-351. [Correction (1980) Ann. Statist 8 1400.] [2] Bewley R, Yang M. 1998. On the size and power of system tests for cointegra- tion. The Review of Economics and Statistics 80:675-679. 16 [3] Box GE, Tiao GC. 1977. A canonical analysis of multiple time series. Biometrika 64:355-365. [4] Brandner P, Kunst RM. 1990. Forecasting vector autoregressions - The influ- ence of cointegration: A Monte Carlo Study. Research Memorandum N0 265, Institute for Advanced Studies, Vienna. [5] Braun PA, Mittnik S. 1993. Misspecifications in Vector Autoregressions and Their Effects on Impulse Responses and Variance Decompositions. Journal of Econometrics 59,319-41. [6] Caporale GM. 1997. Common Features and Output Fluctuations in the United Kingdom. Economic-Modelling 14:1-9. [7] Cubadda G. 1999. Common serial correlation and common business cycles: A cautious note. Empirical Economics 24:529-535. [8] Engle, Granger. 1987. Cointegration and error correction: representation, esti- mation and testing. Econometrica 55:251-76. [9] Engle, Kozicki. 1993. Testing for common features. Journal of Business and Economic Statistics 11: 369-395. [10] Engle, Issler JV. 1995. Estimating Common Sectoral Cycles. Journal of Mone- tary Economics 35:83-113. [11] Guillén, Issler, Athanasopoulos. 2005. Forecasting Accuracy and Estimation Uncertainty using VAR Models with Short- and Long-Term Economic Restric- tions: A Monte-Carlo Study. Ensaios Econômicos EPGE 589. [12] Hamilton JD. 1994. Time Series Analysis. Princeton University Press, New Jersey. [13] Hecq A, Palm FC, Urbain JP. 2006. Testing for Common Cyclical Features in VAR Models with Cointegration, Journal of Econometrics 132: 117-141. 17 [14] Hecq A. 2006. Cointegration and Common Cyclical Features in VAR Models: Comparing Small Sample Performances of the 2-Step and Iterative Approaches. Mimeo. [15] Hecq A. 2000. Common Cyclical Features in Multiple Time Series and Panel Data: Methodological Aspects and Applications. [16] Izenman AJ. 1975. Reduced rank regression for the multivariate linear model, Journal of. multivariate Analysis 5: 248-264. [17] Johansen S. 1988. Statistical analysis of cointegration vectors. Journal of Eco- nomic Dynamics and Control 12:231-54. [18] Johansen S. 1995. Likelihood-based Inference in Cointegrated Vector Autore- gressive Models. Oxford Univ. Press. Z . [19] Kilian L. 2001. Impulse response analysis in vector autoregressions with un- known lag order. Journal of Forecasting 20:161-179. [20] Lutkepohl H. 1993. Introduction to Multiple Time Series Analysis. Springer, New York. Z . [21] Mamingi N, Iyare SO. 2003. Convergence and Common Features in Interna- tional Output: A Case Study of the Economic Community of West African States, 1975-1997. Asian-African-Journal-of-Economics-and-Econometrics. 3: 1-16. [22] Reinsel GC, Velu RP. 1998. Multivariate Reduced-rank Regression. Springer, New York. Z. [23] Stock JH, Watson MW. 1988. Testing for common trends. Journal of the Amer- ican Statistical Association 83:97-107. [24] Tiao GC, Tsay RS. 1985. A canonical correlation approach to modeling mul- tivariate time series., in Proceedings of the Business and Economic Statistics Section, American statistical Association, 112-120. 18 [25] Tiao GC, Tsay RS. 1989. Model specification in multivariate time series (with discussion). J. Roy. Statist. Soc. 51: 157-213. [26] Tsay RS, Tiao GC. 1985. Use of canonical analysis in time series model identi- fication. Biometrika 72 299 315. Z . [27] Tso MK 1981. Reduced rank regression and Canonical analysis. Journal of the Royal Statistical Society 43: 183-189. [28] Vahid F, Issler JV. 2002. The Importance of Common Cyclical Features in VAR Analysis: A Monte Carlo Study. Journal of Econometrics 109: 341-363. [29] Vahid F, Engle.1993. Common trends and common cycles. Journal of Applied Econometrics 8: 341-360. [30] Velu RP, Reinsel GC, Wichern DW. 1986. Reduced rank models for multiple times series. Biometrika 73: 105-118. 19 ATables Table I. Performance of information criterion, IC (p) in selecting the lag order p 20 Table II. Performance of information criterion, IC(p, s) in selecting p and s simultaneously 21 B VAR Restrictions for the DGPs Let’s consider the VAR(3) model : yt = A1yt 1 + A2yt 2 + A3yt 3 + εt (16) − − − 1 1 1 2 2 2 a11 a12 a12 a11 a12 a12 1 1 1 2 2 2 with parameters: A1 = a21 a22 a22 , A2 = a21 a22 a22 and A3 = 1 1 1 2 2 2 a31 a32 a32 a31 a32 a32 3 3 3 a11 a12 a12 β11 3 3 3 a21 a22 a22 We consider the cointegration vectors β = β21 ,thecofea- 3 3 3 a31 a32 a32 β31 ˜ ˜ β11 β12 α11 ˜ ˜ ˜ tures vectors β = β21 β22 and the adjustament matrix α = α21 . The ˜ ˜ β31 β32 α31 long-run relation is defined by αβ0 =(A1 +A2 +A3 I3). The VECM respresentation − is: ∆yt = αβ0yt 1 (A2 + A3)∆yt 1 A3∆yt 2 + εt (17) − − − − − Considering the cointegration restrictions we can rewrite (17) as the following VAR(1) ξt = Fξt 1 + vt (18) − yt (A2 + A3) A3 α εt 4 − − where ξt = yt 1 ,F = I3 00and vt = 0 4 − β0yt β(A2 + A3) β0A3 β0α +1 β0εt − − 1) Short-run restrictions (WF) ˜ ˜ Let us, G = [R21K + R31], K =[(R32 R31)/(R21 R22)], Rj1 = β /β , − − − j1 11 ˜ ˜ Rj2 = βj2/β12 (j =2, 3) and S = β11G + β21K + β31 3 3 3 Ga31 Ga32 Ga33 ˜0 − 3 − 3 − 3 (i) β A3 =0==>A3 = Ka31 Ka32 Ka33 − 3 − 3 − 3 a31 a32 a33 − − − 2 2 2 Ga 31 Ga32 Ga33 ˜0 ˜0 − 2 − 2 − 2 (ii) β (A2 + A3)=0==> β A2 =0==>A2 = Ka31 Ka32 Ka33 − a2 − a2 − a2 − 31 − 32 − 33 2) Long-run restrictions (cointegration) 22 2 3 2 3 2 3 (iv) β0(A2 + A3)=[ (a + a )S (a + a )S (a + a )S] and − 31 31 − 32 32 − 33 33 3 3 3 β0A3 =[ a S a S a S] − 31 − 32 − 33 α11 (v) β0α+1 = β = β β β α21 +1 = β α11 +β α21 +β α31 +1 11 21 31 11 21 31 α31 £ ¤ Therefore, considering short- and long-run restrictions, the companion matrix F is represented as: (A2 + A3) A3 α − − F = I3 00= β(A2 + A3) β0A3 β0α +1 − − 2 3 2 3 2 3 3 3 3 G(a31 + a31) G(a32 + a32) G(a33 + a33) Ga31 Ga32 Ga33 α11 − 2 3 − 2 3 − 2 3 − 3 − 3 − 3 K(a31 + a31) G(a32 + a32) G(a33 + a33) Ka31 Ka32 Ka33 α21 − 2 3 − 2 3 − 2 3 − 3 − 3 − 3 (a + a ) G(a + a ) (a + a ) a a a α31 − 31 31 − 32 32 − 33 33 − 31 − 32 − 33 1000000 0100000 0010000 (a2 + a3 )S (a2 + a3 )S (a2 + a3 )S a3 S a3 S a3 Sb − 31 31 − 32 32 − 33 33 − 31 − 32 − 33 with b = β0α +1=β11α11 + β21α21 + β31α31 +1 3) Restrictions of covariance-stationary in equation (18) The equation (18) will be covariance-stationary, all eigenvalues of matrix F lie inside the unit circle. Therefore, the eigenvalues of the matrix F is a number λ such that: F λI7 =0 (19) | − | The solution of (19) is: λ7 + Ωλ6 + Θλ5 + Ψλ4 =0 (20) where the parameters Ω, Θ,andΨ are: Ω = G(a2 +a3 )+K(a2 +a3 )+a2 +a3 b, 31 31 32 32 33 33 − 3 3 2 3 2 3 2 3 2 3 Θ = Ga + Ka (a + a )b Gb(a + a ) Kb(a + a )+α31S(a + a )+ 31 32 − 33 33 − 31 31 − 32 32 33 33 23 2 3 2 3 3 3 3 3 3 Sα21(a + a )+Sα11(a + a )+a and Ψ = a b Ga b Ka b + α31a S + 32 32 31 31 33 − 33 − 31 − 32 33 3 3 a32Sα21 + a31Sα11,andthefirst four roots are λ1 = λ2 = λ3 = λ4 =0.We calculated the parameters of matrices A1, A2 and A3 as function of roots (λ5,λ6 and λ7)and free parameters. Hence we have three roots satisfying equation (20) λ3 + Ωλ2 + Θλ + Ψ =0 (21) 3 2 for λ5,wehave: λ5 + Ωλ5 + Θλ5 + Ψ =0 ...... Eq1 3 2 for λ6,wehave: λ6 + Ωλ6 + Θλ6 + Ψ =0 ...... Eq2 3 2 for λ7,wehave: λ7 + Ωλ7 + Θλ7 + Ψ =0 ...... Eq3 Solving Eq1,Eq2 and Eq3 we have: Ω = λ7 λ6 λ5, Θ = λ6λ7 + λ6λ5 + λ5λ7 − − − and Ψ = λ5λ6λ7. Equaling these parameters with relations above we have: − 2 2 2 2 6 7 6 7 2 5 6 7 5 7 a = ( Ka Ka b + α31Sa λ λ λ λ a b λ λ λ + b λ λ 31 − − 32 − 32 33 − − − − 33 − − − 5 6 2 2 5 λ λ a + Sa α21 λ )/(Sα11 G Gb) − 33 32 − − − 3 2 7 2 7 6 2 7 6 3 3 2 2 a =( S λ α11α31 b λ G λ Gb +bλ Sα11+λ Sα11b a Sα11G+a S α 32 − − − − 31 31 11− 3 5 2 5 7 6 7 5 2 5 2 6 Ga bSα11 λ Gb +λ Sα11b λ λ α31SG λ λ α31SG S α11λ α31 S α11λ α31+ 31 − − − − − 5 6 5 7 6 6 7 5 7 5 6 2 2 Sλ Gbα31+Sα31λ Gb λ λ λ G+λ λ Gb+λ λ Gb+λ λ Gb SGb α31+S α11bα31 − − − 2 2 2 2 2 2 3 2 3 2 2 2 S α11α31a + S α a G + SG a α31 + Sα11a b + Gb Sα11b S α11Ka α31 33 31 33 31 33 − − 32 − 2 3 2 2 2 2 3 2 2 S α11α31Ga +S a α21Gα31 Sa α21Gb+Sα31G a b Sα31a Gb+Sα11Ka b+ 31 32 − 32 31 − 33 32 7 5 6 5 7 6 5 7 6 Sλ Gbα31 λ λ α31SG λ λ λ α31SG+λ λ λ Sα11)/(Sα11Kα31 KGα31+bGα21 − − − − Kα31Gb Sα11α21 + Gα21)/S − 3 3 5 2 6 7 5 7 2 7 2 7 a = (Kb G λ Gb K +Sα11λ Kλ λ +Kbλ Sα11 Kb λ G S α21λ α11 + 33 − − − − 6 7 6 2 6 6 2 5 5 λ GbSα21 +Sα21λ Gb λ Gb K+λ Sα11Kb λ S α11α21 +λ GbSα21 +λ Sα11Kb − − − 5 2 7 6 7 6 7 5 5 6 7 6 5 2 2 λ S α11α21 λ λ Sα21G+Kbλ λ G+Kbλ λ G+Kbλ λ G λ λ KGλ S α11α21Ka + − − − 32 2 2 2 2 2 2 2 2 3 2 2 3 2 S α11α21b S α11α21a +S α a G Sα11Kb +Sα21G a Sα21Gb +S a Kα − 33 21 32 − 31− 31 11− 2 3 2 2 2 2 3 3 S α11α21Ga +S α21a Gα31+Sα11Kˆ2ba +Sα11Kba Sα11a KG Sα11KbGa 31 33 32 33− 31 − 31− 2 2 3 5 6 5 7 6 2 7 5 SKba Gα31+Sα21G a b Sα21λ λ G Sα21λ λ λ G Sα21Ka Gb Sα21λ λ G)/(Sα11Kα31 33 31 − − − 32 − − KGα31 + bGα21 Kα31Gb Sα11α21 + Gα21)/S − − 24 2 3 3 We can calculate a31, a32and a33 fixing the set λ1 = λ2 = λ3 = λ4 =0and sort 3 2 2 independently from uniform distributions ( 0.9; 0.9) thevaluesof a ,a ,a ,λ5, − 31 32 33 λ6 and λ7. Hencefore, each parameter of the matrices A1, A2 and A3 are defined and so we can generate the DGPs of VAR(3) model with cointegration and WF restrictions. 25 Banco Central do Brasil Trabalhos para Discussão Os Trabalhos para Discussão podem ser acessados na internet, no formato PDF, no endereço: http://www.bc.gov.br Working Paper Series Working Papers in PDF format can be downloaded from: http://www.bc.gov.br 1 Implementing Inflation Targeting in Brazil Jul/2000 Joel Bogdanski, Alexandre Antonio Tombini and Sérgio Ribeiro da Costa Werlang 2 Política Monetária e Supervisão do Sistema Financeiro Nacional no Jul/2000 Banco Central do Brasil Eduardo Lundberg Monetary Policy and Banking Supervision Functions on the Central Jul/2000 Bank Eduardo Lundberg 3 Private Sector Participation: a Theoretical Justification of the Brazilian Jul/2000 Position Sérgio Ribeiro da Costa Werlang 4 An Information Theory Approach to the Aggregation of Log-Linear Jul/2000 Models Pedro H. Albuquerque 5 The Pass-Through from Depreciation to Inflation: a Panel Study Jul/2000 Ilan Goldfajn and Sérgio Ribeiro da Costa Werlang 6 Optimal Interest Rate Rules in Inflation Targeting Frameworks Jul/2000 José Alvaro Rodrigues Neto, Fabio Araújo and Marta Baltar J. Moreira 7 Leading Indicators of Inflation for Brazil Sep/2000 Marcelle Chauvet 8 The Correlation Matrix of the Brazilian Central Bank’s Standard Model Sep/2000 for Interest Rate Market Risk José Alvaro Rodrigues Neto 9 Estimating Exchange Market Pressure and Intervention Activity Nov/2000 Emanuel-Werner Kohlscheen 10 Análise do Financiamento Externo a uma Pequena Economia Mar/2001 Aplicação da Teoria do Prêmio Monetário ao Caso Brasileiro: 1991–1998 Carlos Hamilton Vasconcelos Araújo e Renato Galvão Flôres Júnior 11 A Note on the Efficient Estimation of Inflation in Brazil Mar/2001 Michael F. Bryan and Stephen G. Cecchetti 12 A Test of Competition in Brazilian Banking Mar/2001 Márcio I. Nakane 26 13 Modelos de Previsão de Insolvência Bancária no Brasil Mar/2001 Marcio Magalhães Janot 14 Evaluating Core Inflation Measures for Brazil Mar/2001 Francisco Marcos Rodrigues Figueiredo 15 Is It Worth Tracking Dollar/Real Implied Volatility? Mar/2001 Sandro Canesso de Andrade and Benjamin Miranda Tabak 16 Avaliação das Projeções do Modelo Estrutural do Banco Central do Mar/2001 Brasil para a Taxa de Variação do IPCA Sergio Afonso Lago Alves Evaluation of the Central Bank of Brazil Structural Model’s Inflation Jul/2001 Forecasts in an Inflation Targeting Framework Sergio Afonso Lago Alves 17 Estimando o Produto Potencial Brasileiro: uma Abordagem de Função Abr/2001 de Produção Tito Nícias Teixeira da Silva Filho Estimating Brazilian Potential Output: a Production Function Approach Aug/2002 Tito Nícias Teixeira da Silva Filho 18 A Simple Model for Inflation Targeting in Brazil Apr/2001 Paulo Springer de Freitas and Marcelo Kfoury Muinhos 19 Uncovered Interest Parity with Fundamentals: a Brazilian Exchange May/2001 Rate Forecast Model Marcelo Kfoury Muinhos, Paulo Springer de Freitas and Fabio Araújo 20 Credit Channel without the LM Curve May/2001 Victorio Y. T. Chu and Márcio I. Nakane 21 Os Impactos Econômicos da CPMF: Teoria e Evidência Jun/2001 Pedro H. Albuquerque 22 Decentralized Portfolio Management Jun/2001 Paulo Coutinho and Benjamin Miranda Tabak 23 Os Efeitos da CPMF sobre a Intermediação Financeira Jul/2001 Sérgio Mikio Koyama e Márcio I. Nakane 24 Inflation Targeting in Brazil: Shocks, Backward-Looking Prices, and Aug/2001 IMF Conditionality Joel Bogdanski, Paulo Springer de Freitas, Ilan Goldfajn and Alexandre Antonio Tombini 25 Inflation Targeting in Brazil: Reviewing Two Years of Monetary Policy Aug/2001 1999/00 Pedro Fachada 26 Inflation Targeting in an Open Financially Integrated Emerging Aug/2001 Economy: the Case of Brazil Marcelo Kfoury Muinhos 27 Complementaridade e Fungibilidade dos Fluxos de Capitais Set/2001 Internacionais Carlos Hamilton Vasconcelos Araújo e Renato Galvão Flôres Júnior 27 28 Regras Monetárias e Dinâmica Macroeconômica no Brasil: uma Nov/2001 Abordagem de Expectativas Racionais Marco Antonio Bonomo e Ricardo D. Brito 29 Using a Money Demand Model to Evaluate Monetary Policies in Brazil Nov/2001 Pedro H. Albuquerque and Solange Gouvêa 30 Testing the Expectations Hypothesis in the Brazilian Term Structure of Nov/2001 Interest Rates Benjamin Miranda Tabak and Sandro Canesso de Andrade 31 Algumas Considerações sobre a Sazonalidade no IPCA Nov/2001 Francisco Marcos R. Figueiredo e Roberta Blass Staub 32 Crises Cambiais e Ataques Especulativos no Brasil Nov/2001 Mauro Costa Miranda 33 Monetary Policy and Inflation in Brazil (1975-2000): a VAR Estimation Nov/2001 André Minella 34 Constrained Discretion and Collective Action Problems: Reflections on Nov/2001 the Resolution of International Financial Crises Arminio Fraga and Daniel Luiz Gleizer 35 Uma Definição Operacional de Estabilidade de Preços Dez/2001 Tito Nícias Teixeira da Silva Filho 36 Can Emerging Markets Float? Should They Inflation Target? Feb/2002 Barry Eichengreen 37 Monetary Policy in Brazil: Remarks on the Inflation Targeting Regime, Mar/2002 Public Debt Management and Open Market Operations Luiz Fernando Figueiredo, Pedro Fachada and Sérgio Goldenstein 38 Volatilidade Implícita e Antecipação de Eventos de Stress: um Teste para Mar/2002 o Mercado Brasileiro Frederico Pechir Gomes 39 Opções sobre Dólar Comercial e Expectativas a Respeito do Mar/2002 Comportamento da Taxa de Câmbio Paulo Castor de Castro 40 Speculative Attacks on Debts, Dollarization and Optimum Currency Apr/2002 Areas Aloisio Araujo and Márcia Leon 41 Mudanças de Regime no Câmbio Brasileiro Jun/2002 Carlos Hamilton V. Araújo e Getúlio B. da Silveira Filho 42 Modelo Estrutural com Setor Externo: Endogenização do Prêmio de Jun/2002 Risco e do Câmbio Marcelo Kfoury Muinhos, Sérgio Afonso Lago Alves e Gil Riella 43 The Effects of the Brazilian ADRs Program on Domestic Market Jun/2002 Efficiency Benjamin Miranda Tabak and Eduardo José Araújo Lima 28 44 Estrutura Competitiva, Produtividade Industrial e Liberação Comercial Jun/2002 no Brasil Pedro Cavalcanti Ferreira e Osmani Teixeira de Carvalho Guillén 45 Optimal Monetary Policy, Gains from Commitment, and Inflation Aug/2002 Persistence André Minella 46 The Determinants of Bank Interest Spread in Brazil Aug/2002 Tarsila Segalla Afanasieff, Priscilla Maria Villa Lhacer and Márcio I. Nakane 47 Indicadores Derivados de Agregados Monetários Set/2002 Fernando de Aquino Fonseca Neto e José Albuquerque Júnior 48 Should Government Smooth Exchange Rate Risk? Sep/2002 Ilan Goldfajn and Marcos Antonio Silveira 49 Desenvolvimento do Sistema Financeiro e Crescimento Econômico no Set/2002 Brasil: Evidências de Causalidade Orlando Carneiro de Matos 50 Macroeconomic Coordination and Inflation Targeting in a Two-Country Sep/2002 Model Eui Jung Chang, Marcelo Kfoury Muinhos and Joanílio Rodolpho Teixeira 51 Credit Channel with Sovereign Credit Risk: an Empirical Test Sep/2002 Victorio Yi Tson Chu 52 Generalized Hyperbolic Distributions and Brazilian Data Sep/2002 José Fajardo and Aquiles Farias 53 Inflation Targeting in Brazil: Lessons and Challenges Nov/2002 André Minella, Paulo Springer de Freitas, Ilan Goldfajn and Marcelo Kfoury Muinhos 54 Stock Returns and Volatility Nov/2002 Benjamin Miranda Tabak and Solange Maria Guerra 55 Componentes de Curto e Longo Prazo das Taxas de Juros no Brasil Nov/2002 Carlos Hamilton Vasconcelos Araújo e Osmani Teixeira de Carvalho de Guillén 56 Causality and Cointegration in Stock Markets: Dec/2002 the Case of Latin America Benjamin Miranda Tabak and Eduardo José Araújo Lima 57 As Leis de Falência: uma Abordagem Econômica Dez/2002 Aloisio Araujo 58 The Random Walk Hypothesis and the Behavior of Foreign Capital Dec/2002 Portfolio Flows: the Brazilian Stock Market Case Benjamin Miranda Tabak 59 Os Preços Administrados e a Inflação no Brasil Dez/2002 Francisco Marcos R. Figueiredo e Thaís Porto Ferreira 60 Delegated Portfolio Management Dec/2002 Paulo Coutinho and Benjamin Miranda Tabak 29 61 O Uso de Dados de Alta Freqüência na Estimação da Volatilidade e Dez/2002 do Valor em Risco para o Ibovespa João Maurício de Souza Moreira e Eduardo Facó Lemgruber 62 Taxa de Juros e Concentração Bancária no Brasil Fev/2003 Eduardo Kiyoshi Tonooka e Sérgio Mikio Koyama 63 Optimal Monetary Rules: the Case of Brazil Feb/2003 Charles Lima de Almeida, Marco Aurélio Peres, Geraldo da Silva e Souza and Benjamin Miranda Tabak 64 Medium-Size Macroeconomic Model for the Brazilian Economy Feb/2003 Marcelo Kfoury Muinhos and Sergio Afonso Lago Alves 65 On the Information Content of Oil Future Prices Feb/2003 Benjamin Miranda Tabak 66 A Taxa de Juros de Equilíbrio: uma Abordagem Múltipla Fev/2003 Pedro Calhman de Miranda e Marcelo Kfoury Muinhos 67 Avaliação de Métodos de Cálculo de Exigência de Capital para Risco de Fev/2003 Mercado de Carteiras de Ações no Brasil Gustavo S. Araújo, João Maurício S. Moreira e Ricardo S. Maia Clemente 68 Real Balances in the Utility Function: Evidence for Brazil Feb/2003 Leonardo Soriano de Alencar and Márcio I. Nakane 69 r-filters: a Hodrick-Prescott Filter Generalization Feb/2003 Fabio Araújo, Marta Baltar Moreira Areosa and José Alvaro Rodrigues Neto 70 Monetary Policy Surprises and the Brazilian Term Structure of Interest Feb/2003 Rates Benjamin Miranda Tabak 71 On Shadow-Prices of Banks in Real-Time Gross Settlement Systems Apr/2003 Rodrigo Penaloza 72 O Prêmio pela Maturidade na Estrutura a Termo das Taxas de Juros Maio/2003 Brasileiras Ricardo Dias de Oliveira Brito, Angelo J. Mont'Alverne Duarte e Osmani Teixeira de C. Guillen 73 Análise de Componentes Principais de Dados Funcionais – Uma Maio/2003 Aplicação às Estruturas a Termo de Taxas de Juros Getúlio Borges da Silveira e Octavio Bessada 74 Aplicação do Modelo de Black, Derman & Toy à Precificação de Opções Maio/2003 Sobre Títulos de Renda Fixa Octavio Manuel Bessada Lion, Carlos Alberto Nunes Cosenza e César das Neves 75 Brazil’s Financial System: Resilience to Shocks, no Currency Jun/2003 Substitution, but Struggling to Promote Growth Ilan Goldfajn, Katherine Hennings and Helio Mori 30 76 Inflation Targeting in Emerging Market Economies Jun/2003 Arminio Fraga, Ilan Goldfajn and André Minella 77 Inflation Targeting in Brazil: Constructing Credibility under Exchange Jul/2003 Rate Volatility André Minella, Paulo Springer de Freitas, Ilan Goldfajn and Marcelo Kfoury Muinhos 78 Contornando os Pressupostos de Black & Scholes: Aplicação do Modelo Out/2003 de Precificação de Opções de Duan no Mercado Brasileiro Gustavo Silva Araújo, Claudio Henrique da Silveira Barbedo, Antonio Carlos Figueiredo, Eduardo Facó Lemgruber 79 Inclusão do Decaimento Temporal na Metodologia Out/2003 Delta-Gama para o Cálculo do VaR de Carteiras Compradas em Opções no Brasil Claudio Henrique da Silveira Barbedo, Gustavo Silva Araújo, Eduardo Facó Lemgruber 80 Diferenças e Semelhanças entre Países da América Latina: Out/2003 uma Análise de Markov Switching para os Ciclos Econômicos de Brasil e Argentina Arnildo da Silva Correa 81 Bank Competition, Agency Costs and the Performance of the Jan/2004 Monetary Policy Leonardo Soriano de Alencar and Márcio I. Nakane 82 Carteiras de Opções: Avaliação de Metodologias de Exigência de Capital Mar/2004 no Mercado Brasileiro Cláudio Henrique da Silveira Barbedo e Gustavo Silva Araújo 83 Does Inflation Targeting Reduce Inflation? An Analysis for the OECD May/2004 Industrial Countries Thomas Y. Wu 84 Speculative Attacks on Debts and Optimum Currency Area: a Welfare May/2004 Analysis Aloisio Araujo and Marcia Leon 85 Risk Premia for Emerging Markets Bonds: Evidence from Brazilian May/2004 Government Debt, 1996-2002 André Soares Loureiro and Fernando de Holanda Barbosa 86 Identificação do Fator Estocástico de Descontos e Algumas Implicações Maio/2004 sobre Testes de Modelos de Consumo Fabio Araujo e João Victor Issler 87 Mercado de Crédito: uma Análise Econométrica dos Volumes de Crédito Dez/2004 Total e Habitacional no Brasil Ana Carla Abrão Costa 88 Ciclos Internacionais de Negócios: uma Análise de Mudança de Regime Dez/2004 Markoviano para Brasil, Argentina e Estados Unidos Arnildo da Silva Correa e Ronald Otto Hillbrecht 89 O Mercado de Hedge Cambial no Brasil: Reação das Instituições Dez/2004 Financeiras a Intervenções do Banco Central Fernando N. de Oliveira 31 90 Bank Privatization and Productivity: Evidence for Brazil Dec/2004 Márcio I. Nakane and Daniela B. Weintraub 91 Credit Risk Measurement and the Regulation of Bank Capital and Dec/2004 Provision Requirements in Brazil – A Corporate Analysis Ricardo Schechtman, Valéria Salomão Garcia, Sergio Mikio Koyama and Guilherme Cronemberger Parente 92 Steady-State Analysis of an Open Economy General Equilibrium Model Apr/2005 for Brazil Mirta Noemi Sataka Bugarin, Roberto de Goes Ellery Jr., Victor Gomes Silva, Marcelo Kfoury Muinhos 93 Avaliação de Modelos de Cálculo de Exigência de Capital para Risco Abr/2005 Cambial Claudio H. da S. Barbedo, Gustavo S. Araújo, João Maurício S. Moreira e Ricardo S. Maia Clemente 94 Simulação Histórica Filtrada: Incorporação da Volatilidade ao Modelo Abr/2005 Histórico de Cálculo de Risco para Ativos Não-Lineares Claudio Henrique da Silveira Barbedo, Gustavo Silva Araújo e Eduardo Facó Lemgruber 95 Comment on Market Discipline and Monetary Policy by Carl Walsh Apr/2005 Maurício S. Bugarin and Fábia A. de Carvalho 96 O que É Estratégia: uma Abordagem Multiparadigmática para a Ago/2005 Disciplina Anthero de Moraes Meirelles 97 Finance and the Business Cycle: a Kalman Filter Approach with Markov Aug/2005 Switching Ryan A. Compton and Jose Ricardo da Costa e Silva 98 Capital Flows Cycle: Stylized Facts and Empirical Evidences for Aug/2005 Emerging Market Economies Helio Mori e Marcelo Kfoury Muinhos 99 Adequação das Medidas de Valor em Risco na Formulação da Exigência Set/2005 de Capital para Estratégias de Opções no Mercado Brasileiro Gustavo Silva Araújo, Claudio Henrique da Silveira Barbedo,e Eduardo Facó Lemgruber 100 Targets and Inflation Dynamics Oct/2005 Sergio A. L. Alves and Waldyr D. Areosa 101 Comparing Equilibrium Real Interest Rates: Different Approaches to Mar/2006 Measure Brazilian Rates Marcelo Kfoury Muinhos and Márcio I. Nakane 102 Judicial Risk and Credit Market Performance: Micro Evidence from Apr/2006 Brazilian Payroll Loans Ana Carla A. Costa and João M. P. de Mello 103 The Effect of Adverse Supply Shocks on Monetary Policy and Output Apr/2006 Maria da Glória D. S. Araújo, Mirta Bugarin, Marcelo Kfoury Muinhos and Jose Ricardo C. Silva 32 104 Extração de Informação de Opções Cambiais no Brasil Abr/2006 Eui Jung Chang e Benjamin Miranda Tabak 105 Representing Roommate’s Preferences with Symmetric Utilities Apr/2006 José Alvaro Rodrigues Neto 106 Testing Nonlinearities Between Brazilian Exchange Rates and Inflation May/2006 Volatilities Cristiane R. Albuquerque and Marcelo Portugal 107 Demand for Bank Services and Market Power in Brazilian Banking Jun/2006 Márcio I. Nakane, Leonardo S. Alencar and Fabio Kanczuk 108 O Efeito da Consignação em Folha nas Taxas de Juros dos Empréstimos Jun/2006 Pessoais Eduardo A. S. Rodrigues, Victorio Chu, Leonardo S. Alencar e Tony Takeda 109 The Recent Brazilian Disinflation Process and Costs Jun/2006 Alexandre A. Tombini and Sergio A. Lago Alves 110 Fatores de Risco e o Spread Bancário no Brasil Jul/2006 Fernando G. Bignotto e Eduardo Augusto de Souza Rodrigues 111 Avaliação de Modelos de Exigência de Capital para Risco de Mercado do Jul/2006 Cupom Cambial Alan Cosme Rodrigues da Silva, João Maurício de Souza Moreira e Myrian Beatriz Eiras das Neves 112 Interdependence and Contagion: an Analysis of Information Jul/2006 Transmission in Latin America's Stock Markets Angelo Marsiglia Fasolo 113 Investigação da Memória de Longo Prazo da Taxa de Câmbio no Brasil Ago/2006 Sergio Rubens Stancato de Souza, Benjamin Miranda Tabak e Daniel O. Cajueiro 114 The Inequality Channel of Monetary Transmission Aug/2006 Marta Areosa and Waldyr Areosa 115 Myopic Loss Aversion and House-Money Effect Overseas: an Sep/2006 Experimental Approach José L. B. Fernandes, Juan Ignacio Peña and Benjamin M. Tabak 116 Out-Of-The-Money Monte Carlo Simulation Option Pricing: the Join Sep/2006 Use of Importance Sampling and Descriptive Sampling Jaqueline Terra Moura Marins, Eduardo Saliby and Joséte Florencio dos Santos 117 An Analysis of Off-Site Supervision of Banks’ Profitability, Risk and Sep/2006 Capital Adequacy: a Portfolio Simulation Approach Applied to Brazilian Banks Theodore M. Barnhill, Marcos R. Souto and Benjamin M. Tabak 118 Contagion, Bankruptcy and Social Welfare Analysis in a Financial Oct/2006 Economy with Risk Regulation Constraint Aloísio P. Araújo and José Valentim M. Vicente 33 119 A Central de Risco de Crédito no Brasil: uma Análise de Utilidade de Out/2006 Informação Ricardo Schechtman 120 Forecasting Interest Rates: an Application for Brazil Oct/2006 Eduardo J. A. Lima, Felipe Luduvice and Benjamin M. Tabak 121 The Role of Consumer’s Risk Aversion on Price Rigidity Nov/2006 Sergio A. Lago Alves and Mirta N. S. Bugarin 122 Nonlinear Mechanisms of the Exchange Rate Pass-Through: a Phillips Nov/2006 Curve Model With Threshold for Brazil Arnildo da Silva Correa and André Minella 123 A Neoclassical Analysis of the Brazilian “Lost-Decades” Nov/2006 Flávia Mourão Graminho 124 The Dynamic Relations between Stock Prices and Exchange Rates: Nov/2006 Evidence for Brazil Benjamin M. Tabak 125 Herding Behavior by Equity Foreign Investors on Emerging Markets Dec/2006 Barbara Alemanni and José Renato Haas Ornelas 126 Risk Premium: Insights over the Threshold Dec/2006 José L. B. Fernandes, Augusto Hasman and Juan Ignacio Peña 127 Uma Investigação Baseada em Reamostragem sobre Requerimentos de Dec/2006 Capital para Risco de Crédito no Brasil Ricardo Schechtman 128 Term Structure Movements Implicit in Option Prices Dec/2006 Caio Ibsen R. Almeida and José Valentim M. Vicente 129 Brazil: Taming Inflation Expectations Jan/2007 Afonso S. Bevilaqua, Mário Mesquita and André Minella 130 The Role of Banks in the Brazilian Interbank Market: Does Bank Type Jan/2007 Matter? Daniel O. Cajueiro and Benjamin M. Tabak 131 Long-Range Dependence in Exchange Rates: the Case of the European Mar/2007 Monetary System Sergio Rubens Stancato de Souza, Benjamin M. Tabak and Daniel O. Cajueiro 132 Credit Risk Monte Carlo Simulation Using Simplified Creditmetrics’ Mar/2007 Model: the Joint Use of Importance Sampling and Descriptive Sampling Jaqueline Terra Moura Marins and Eduardo Saliby 133 A New Proposal for Collection and Generation of Information on Mar/2007 Financial Institutions’ Risk: the Case of Derivatives Gilneu F. A. Vivan and Benjamin M. Tabak 134 Amostragem Descritiva no Apreçamento de Opções Européias através Abr/2007 de Simulação Monte Carlo: o Efeito da Dimensionalidade e da Probabilidade de Exercício no Ganho de Precisão Eduardo Saliby, Sergio Luiz Medeiros Proença de Gouvêa e Jaqueline Terra Moura Marins 34 135 Evaluation of Default Risk for the Brazilian Banking Sector May/2007 Marcelo Y. Takami and Benjamin M. Tabak 136 Identifying Volatility Risk Premium from Fixed Income Asian Options May/2007 Caio Ibsen R. Almeida and José Valentim M. Vicente 137 Monetary Policy Design under Competing Models of Inflation May/2007 Persistence Solange Gouvea e Abhijit Sen Gupta 138 Forecasting Exchange Rate Density Using Parametric Models: May/2007 the Case of Brazil Marcos M. Abe, Eui J. Chang and Benjamin M. Tabak 35