Selection of Optimal Lag Length in Cointegrated VAR Models with Weak Form of Common Cyclical Features

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(1)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.

(2) ISSN 1518-3548 CGC 00.038.166/0001-05 Working Paper Series. Brasília. n. 139. Jun. 2007. P. 1-35.

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(4) 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 autoregressive (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 important 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) restriction of common cyclical features in the model in order to analyze the appropriate way to select the correct lag order. Two methodologies 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 MonteCarlo simulation is used in the analysis. 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 comments. The authors are responsible for any remaining errors in this paper. Carlos Enrique C. Gutiérrez acknowledges the support of CAPES-Brazil. † Corresponding author, cgutierrez@fgvmail.br, Graduate School of Economics, FGV-EPGE-RJ, Brazil. ‡ reinaldo@ele.puc-rio.br, DEE-PUC-RJ, Brazil § osmani.guillen@bcb.gov.br, Banco Central do Brasil and Ibmec-RJ, Brazil. 3.

(5) 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 determination 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 cointegration 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 restrictions, 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) as a measure of common cyclical feature. SCCF restrictions might be imposed in a covariance stationary 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) defined a weak version of SCCF restrictions which they denominated it as weak-form (WF) common cyclical restric4.

(6) 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 cointegration 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 standard 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 restrictions. 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), and b) 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.

(7) 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 econometric 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: yt =. p X. Ai yt−i + εt. (1). i=1. 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 6= τ , where Ω is non singular}. The model (1) could be written equivalently as; P Π (L) yt = εt where L represents the lag operator and Π (L) = In − pi=1 Ai Li that P when L = 1, Π (1) = In − pi=1 Ai . If cointegration is considered in (1) the (n × n). matrix Π (·) satisfies two conditions: a) Rank (Π (1)) = r, 0 < r < n, such that Π (1) can be expressed as Π (1) = −αβ 0 , where α and β are (n × r) matrices with full column rank, r. b) The characteristic equation |Π (L)| = 0 has n − r roots equal to 1 and all other are outside the unit circle. These assumptions imply that yt is cointegrated of order (1, 1). The elements of α are the adjustment coefficients and the columns of β span the space of cointegration vectors. We can represent a VAR 2. This is quite recent in the literature (see, Hecq et al., 2006). 6.

(8) 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: 0. ∆yt = αβ yt−1 +. p−1 X. (2). Γi ∆yt−i + εt. i=1. where: αβ 0 = −Π (1), Γj = −. Pp. k=j+1. Ak for j = 1, ...., p − 1 and Γ0 = In . The. VAR(p) model can include additional 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 (cointegration), there exists a (n × s) matrix β̃ of rank s, whose columns span the cofeature 0. 0. 0. space, such that β̃ (∆yt − αβýt−1 ) = β̃ εt , where β̃ εt 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 β̃ that satisfies the following assumption: 0. Assumption 1 : β̃ Γj = 0s×n for. j = 1, ...., p − 1.. Imposing WF restrictions is convenient because it allows the study of both cointegration and common cyclical feature without the constraint r + s ≤ n. We can rewrite the VECM with WF restrictions as a model of reduced-rank structure. In 0 0 , .....∆yt−p+1 ]0 and Φ = [Γ1 , ...., Γp−1 ], therefore we get: (2) let Xt−1 = [∆yt−1. (3). ∆yt = αβýt−1 + ΦXt−1 + εt. If assumption (1) holds matrices Γi , i = 1, ..., p are all of rank (n − s) then we can write Φ = β̃ ⊥ Ψ = β̃ ⊥ [Ψ1 , ...., Ψp−1 ], where, β̃ ⊥ 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 0. (n − s) × n. Hence, given assumption (1), there exists β̃ of n × s such that β̃ β̃ ⊥ = 0. 7.

(9) That is, β̃ ⊥ n × (n − s) is a full column rank orthogonal to the complement of β̃ with rank(β̃, β̃ ⊥ ) = n. Rewriting model (3) we have: ∆yt = αβýt−1 + β̃ ⊥ (Ψ1 , Ψ2 , ..., Ψp−1 ) Xt−1 + εt. (4) (5). = αβýt−1 + β̃ ⊥ ΨXt−1 + εt. 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 Anderson (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 MaximumLikelihood 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 analysis3 . 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 eigenvalues λ̂1 > λ̂2 > λ̂3 ....... > λ̂n . 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 procedure 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): ξ r = −T 3. n X. i=r+1. 2. Ln (1 − λ̂i ). i = 1, ..., n. This estimation is referred as Full Information Maximum Likelihood - FIML 8. (6).

(10) where the eigenvalues 0 < λ̂n < ... < λ̂1 are the solution of : |λm11 − m−1 10 m00 m01 | = P 0, where mij , i, j = 0.1, are the second moment matrices: m00 = T1 Tt=1 ũ0t ũ00t , P P P m10 = T1 Tt=1 ũ1t ũ00t , m01 = T1 Tt=1 ũ0t ũ01t , m11 = T1 Tt=1 ũ1t ũ01t of the residuals ũ0t. and ũ1t obtained in the multivariate least squares regressions ∆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 0. 0. 0. for long-run effects which are white noise (i.e., β̃ (∆yt − αβýt−1 ) = β̃ εt where β̃ εt 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. Using the lag length chosen in the previous step, 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.

(11) The above procedure is followed when there is evidence of cointegration restrictions. 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 cointegrated 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 performance of IC(p, s) superior to that of IC(p)? This is an important question we aim to answer in this work. The procedure of selecting the lag order and the rank of the structure of shortrun is carried out by minimizing the following modified information criteria (see Hecq, 2006). T X. AIC (p, s) =. i=n−s+1. HQ(p, s) =. T X. i=n−s+1. SC(p, s) =. T X. ln(1 − λ2i (p)) +. i=n−s+1. 2 ×N T. (7). 2 ln(ln T ) ×N T. (8). ln(1 − λ2i (p)) +. ln(1 − λ2i (p)) +. ln T ×N T. (9). 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, from the number 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.

(12) 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). As mentioned, Vahid and Issler (2002) suggested the use of IC(p, s) to simultaneously choose the order p and a number of reduced rank structure s 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 = αβ 0 yt−1 + β̃ ⊥ 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 (p̄, s̄) chosen by information criteria (7), (8) or (9), we estimate β (and so its rank, r = r̄) using Johansen cointegration test. Step2 : Estimation of β̃ ⊥ and Ψ. Taking β̂ estimated in step one, we proceed to estimate β̃ ⊥ and Ψ. Hence, 0. we run a regression of ∆yt and of Xt−1 on β̂ yt−1 . We labeled the residuals as u0 and u1 , respectively. Therefore, we obtain a reduced rank regression: u0 = β̃ ⊥ Ψu1 + εt 11. (11).

(13) ¡ ¢ where Ψ can be written as Ψ = C1 , ..., C(p̄−1) of (n − s̄) × n(p̄ − 1) and. β̃ ⊥ of n × (n − s̄). We estimate (11) by FIML. Thus, we can obtain β̃ ⊥ and Ψ̂. 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 cal2. culate the eigenvalues associated with Ψ̂, λ̂i i = 1, ..., s̄ and the matrix P of residuals max r̄, s=s̄ . Hence, we compute the ML function: # " ¯ max ¯ s̄ ¯ X ¯ X ´ ³ 2 T ¯ ¯ (12) L0max, r̄<n, s=s̄ = − ln 1 − λ̂i ln ¯ ¯− ¯r̄<n, s=s̄¯ 2 i=1. If r̄ = n, we use instead of (12) the derived log-likelihood: Lmax, r=n, s=s̄ = ¯ ¯P ¯ ¯ − T2 ln ¯ max r̄=n, s=s̄ ¯. The determinant of the covariance matrix for r̄ = n cointegration vector is calculated by ¯ max ¯ s̄ ¯ X ¯ ´ ³ ¯ ¯ X 2 ¯ ¯ −1 ¯ ¯ ln ¯ ln 1 − λ̂i ¯ = ln m00 − m01 m11 m10 − ¯r̄=n, s=s̄¯. (13). i=1. where mij refers to cross moment matrices obtained in multivariate least 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. h i In order to reestimate β we use the program CanCorr ∆yt , yt−1 | Ψ̂Xt−1 and thus using the new β̂ we can repeat step 2 to reestimate β̃ ⊥ and Ψ.. Then, we can calculate the new value of the ML function in the step 3. ¡ Henceforth, we obtain L1max, r=r̄, s=s̄ for calculating ∆L = L1max, r=r̄, s=s̄ - L0max, We repeat steps 1 to 4 to choose β̃ ⊥ and Ψ until convergence is reached ( i.e., ∆L < 10−7 ). In the end, optimal parameters p̄, r̄ and s̄ are obtained and it can be used for estimation and forecasting of a VECM with WF restrictions. 12. r=r̄, s=s̄. ¢. ..

(14) 4. Monte-Carlo Design. 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 = 1 and s = 2, respectively). β and β̃ satisfy:    1.0 β =  0.2  , β̃ =  −1.0      ε1t 0 1.0  ε2t  ∼ N  0  ,  0.6 0 0.6 ε3t.  1.0 0.1 0.0 1.0  0.5 −0.5 0.6 0.6 1.0 0.6 0.6 1.0.  . Consider the VAR(3) model: yt = A1 yt−1 + A2 yt−2 + A3 yt−3 + εt . The VECM. 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 cointegra0. tion restrictions: αβ 0 = (A1 + A2 + A3 − I3 ) ; b) WF restrictions: β̃ A3 = 0 (iii) 0. β̃ (A2 + A3 ) = 0 and c) covariance-stationary condition. Considering the cointegration restrictions we can rewrite (14) as the following VAR(1): ξ t = F ξ t−1 + vt. (15) .     −(A2 + A3 ) 4yt εt −A3 α      I3 4yt−1 , F = 0 0 0  and vt = ξt = 0 0 0 −β(A2 + A3 ) −β A3 β α + 1 β 0 εt β yt . 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 of the companion matrix all lie inside the unit circle. This may be carried out by 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.

(15) 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.. 5. Results. Values in Table I represent the percentage of time that the model selection criterion, 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.

(16) 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 can be summarized as follows: − All criteria (AIC, HQ and SC) choose the correct parameters more often when using IC(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 HannanQuinn 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.. 6. Conclusions. 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.

(17) 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 underparameterized model. In applied work, when the VAR model contains WF and cointegration restrictions, we suggest the use of AIC(p, s) criteria for simultaneously choosing the lagrank, 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 common 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 cointegration. The Review of Economics and Statistics 80: 675-679.. 16.

(18) [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 influence 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, estimation 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 Monetary Economics 35:83-113. [11] Guillén, Issler, Athanasopoulos. 2005. Forecasting Accuracy and Estimation Uncertainty using VAR Models with Short- and Long-Term Economic Restrictions: 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.

(19) [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 Economic Dynamics and Control 12: 231-54. [18] Johansen S. 1995. Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford Univ. Press. Z . [19] Kilian L. 2001. Impulse response analysis in vector autoregressions with unknown 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 International 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 American Statistical Association 83:97-107. [24] Tiao GC, Tsay RS. 1985. A canonical correlation approach to modeling multivariate time series., in Proceedings of the Business and Economic Statistics Section, American statistical Association, 112-120. 18.

(20) [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 identification. 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.

(21) A. Tables. Table I. Performance of information criterion, IC (p) in selecting the lag order p. 20.

(22) Table II. Performance of information criterion, IC(p, s) in selecting p and s simultaneously. 21.

(23) B. VAR Restrictions for the DGPs. Let’s consider the VAR(3) model :. yt = A1 yt−1 + A2 yt−2 + A3 yt−3 + εt (16)    1  a11 a112 a112 a211 a212 a212 with parameters: A1 =  a121 a122 a122 , A2 =  a221 a222 a222  and A3 = a131 a132 a132 a231 a232 a232    3 β 11 a11 a312 a312  a321 a322 a322  We consider the cointegration vectors β =  β 21 , the cofeaa331 a332 a332 β 31     β̃ 11 β̃ 12 α11 tures vectors β̃ =  β̃ 21 β̃ 22  and the adjustament matrix α =  α21  . The α31 β̃ 31 β̃ 32 0 long-run relation is defined by αβ = (A1 +A2 +A3 −I3 ). The VECM respresentation is: ∆yt = αβ 0 yt−1 − (A2 + A3 )∆yt−1 − A3 ∆yt−2 + εt. (17). Considering the cointegration restrictions we can rewrite (17) as the following VAR(1). . . . ξ t = F ξ t−1 + vt. −(A2 + A3 ) 4yt −A3 I3 0 where ξ t =  4yt−1  , F =  0 −β(A2 + A3 ) −β 0 A3 β yt. 1) Short-run restrictions (WF). (18)  εt α  and vt =  0  0 0 β 0 εt βα+1 . . Let us, G = −[R21 K + R31 ], K = [(R32 − R31 )/(R21 − R22 )], Rj1 = β̃ j1 /β̃ 11 , Rj2 = β̃ j2 /β̃ 12 (j = 2, 3) and S = β 11 G + β 21 K + β 31  −Ga331 −Ga332 −Ga333 0 (i) β̃ A3 = 0 ==> A3 =  −Ka331 −Ka332 −Ka333  −a331 −a332 −a333  −Ga231 −Ga232 −Ga233 0 0 (ii) β̃ (A2 + A3 ) = 0 ==> β̃ A2 = 0 ==> A2 =  −Ka231 −Ka232 −Ka233  −a231 −a232 −a233. 2) Long-run restrictions (cointegration) 22.

(24) (iv) β 0 (A2 + A3 ) = [−(a231 + a331 )S β 0 A3 = [−a331 S − a332 S − a333 S] (v) β 0 α + 1 = β =. £. β 11 β 21 β 31. ¤. − (a232 + a332 )S. − (a233 + a333 )S] and.  α11  α21  + 1 = β 11 α11 + β 21 α21 + β 31 α31 + 1 α31 . Therefore, considering short- and long-run restrictions, the companion matrix F is represented as:  −A3 α −(A2 + A3 ) = I3 0 0 F = 0 0 −β(A2 + A3 ) −β A3 β α + 1 .          . −G(a231 + a331 ) −K(a231 + a331 ) −(a231 + a331 ) 1 0 0 2 −(a31 + a331 )S. −G(a232 + a332 ) −G(a233 + a333 ) −Ga331 −Ga332 −Ga333 α11 −G(a232 + a332 ) −G(a233 + a333 ) −Ka331 −Ka332 −Ka333 α21 −G(a232 + a332 ) −(a233 + a333 ) −a331 −a332 −a333 α31 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 2 3 2 3 3 3 3 −(a32 + a32 )S −(a33 + a33 )S −a31 S −a32 S −a33 S b.          . with b = β 0 α + 1 = β 11 α11 + β 21 α21 + β 31 α31 + 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). λ7 + Ωλ6 + Θλ5 + Ψλ4 = 0. (20). The solution of (19) is:. where the parameters Ω, Θ, and Ψ are: Ω = G(a231 +a331 )+K(a232 +a332 )+a233 +a333 −b,. Θ = Ga331 + Ka332 − (a233 + a333 )b − Gb(a231 + a331 ) − Kb(a232 + a332 ) + α31 S(a233 + a333 ) + 23.

(25) Sα21 (a232 + a332 ) + Sα11 (a231 + a331 ) + a333 and Ψ = −a333 b − Ga331 b − Ka332 b + α31 a333 S +. a332 Sα21 + a331 Sα11 , and the first 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 for λ5 , we have: λ35 + Ωλ25 + Θλ5 + Ψ = 0. ..................................Eq1. for λ6 , we have: λ36 + Ωλ26 + Θλ6 + Ψ = 0. ..................................Eq2. for λ7 , we have: λ37 + Ωλ27 + Θλ7 + Ψ = 0. ..................................Eq3. (21). Solving Eq1, Eq2 and Eq3 we have: Ω = −λ7 − λ6 − λ5 , Θ = λ6 λ7 + λ6 λ5 + λ5 λ7 and Ψ = −λ5 λ6 λ7 . Equaling these parameters with relations above we have: a231 = −(−Ka232 − Ka232 b + α31 Sa233 − λ6 λ7 − λ6 − λ7 − a233 b − λ5 λ6 λ7 + b − λ5 λ7 −. λ5 λ6 − a233 + Sa232 α21 − λ5 )/(Sα11 − G − Gb). a332 = (−S 2 λ7 α11 α31 −b2 λ7 G−λ6 Gb2 +bλ7 Sα11 +λ6 Sα11 b−a331 Sα11 G+a331 S 2 α211 −. Ga331 bSα11 −λ5 Gb2 +λ5 Sα11 b−λ7 λ6 α31 SG−λ7 λ5 α31 SG−S 2 α11 λ5 α31 −S 2 α11 λ6 α31 +. Sλ5 Gbα31 +Sα31 λ6 Gb−λ5 λ7 λ6 G+λ6 λ7 Gb+λ5 λ7 Gb+λ5 λ6 Gb−SGb2 α31 +S 2 α11 bα31 − S 2 α11 α31 a233 + S 2 α231 a233 G + SG2 a331 α31 + Sα11 a233 b + Gb3 − Sα11 b2 − S 2 α11 Ka232 α31 − S 2 α11 α31 Ga331 +S 2 a232 α21 Gα31 −Sa232 α21 Gb+Sα31 G2 a331 b−Sα31 a233 Gb+Sα11 Ka232 b+. Sλ7 Gbα31 −λ5 λ6 α31 SG−λ5 λ7 λ6 α31 SG+λ5 λ7 λ6 Sα11 )/(Sα11 Kα31 −KGα31 +bGα21 −. Kα31 Gb − Sα11 α21 + Gα21 )/S a333 = −(Kb3 G − λ5 Gb2 K + Sα11 λ6 Kλ7 λ5 + Kbλ7 Sα11 − Kb2 λ7 G − S 2 α21 λ7 α11 +. λ6 GbSα21 +Sα21 λ7 Gb−λ6 Gb2 K +λ6 Sα11 Kb−λ6 S 2 α11 α21 +λ5 GbSα21 +λ5 Sα11 Kb− λ5 S 2 α11 α21 −λ7 λ6 Sα21 G+Kbλ7 λ6 G+Kbλ7 λ5 G+Kbλ5 λ6 G−λ7 λ6 KGλ5 −S 2 α11 α21 Ka232 + S 2 α11 α21 b−S 2 α11 α21 a233 +S 2 α221 a232 G−Sα11 Kb2 +Sα21 G2 a331 −Sα21 Gb2 +S 2 a331 Kα211 −. S 2 α11 α21 Ga331 +S 2 α21 a233 Gα31 +Sα11 Kˆ2ba232 +Sα11 Kba233 −Sα11 a331 KG−Sα11 KbGa331 −. SKba233 Gα31 +Sα21 G2 a331 b−Sα21 λ5 λ6 G−Sα21 λ5 λ7 λ6 G−Sα21 Ka232 Gb−Sα21 λ7 λ5 G)/(Sα11 Kα31 − KGα31 + bGα21 − Kα31 Gb − Sα11 α21 + Gα21 )/S 24.

(26) We can calculate a231 , a332 and a333 fixing the set λ1 = λ2 = λ3 = λ4 = 0 and sort independently from uniform distributions (−0.9; 0.9) the values of a331 , a232 , a233 , λ5 , 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.

(27) 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 Joel Bogdanski, Alexandre Antonio Tombini and Sérgio Ribeiro da Costa Werlang. Jul/2000. 2. Política Monetária e Supervisão do Sistema Financeiro Nacional no Banco Central do Brasil Eduardo Lundberg. Jul/2000. Monetary Policy and Banking Supervision Functions on the Central Bank Eduardo Lundberg. Jul/2000. 3. Private Sector Participation: a Theoretical Justification of the Brazilian Position Sérgio Ribeiro da Costa Werlang. Jul/2000. 4. An Information Theory Approach to the Aggregation of Log-Linear Models Pedro H. Albuquerque. Jul/2000. 5. The Pass-Through from Depreciation to Inflation: a Panel Study Ilan Goldfajn and Sérgio Ribeiro da Costa Werlang. Jul/2000. 6. Optimal Interest Rate Rules in Inflation Targeting Frameworks José Alvaro Rodrigues Neto, Fabio Araújo and Marta Baltar J. Moreira. Jul/2000. 7. Leading Indicators of Inflation for Brazil Marcelle Chauvet. Sep/2000. 8. The Correlation Matrix of the Brazilian Central Bank’s Standard Model for Interest Rate Market Risk José Alvaro Rodrigues Neto. Sep/2000. 9. Estimating Exchange Market Pressure and Intervention Activity Emanuel-Werner Kohlscheen. Nov/2000. 10. Análise do Financiamento Externo a uma Pequena Economia Aplicação da Teoria do Prêmio Monetário ao Caso Brasileiro: 1991–1998 Carlos Hamilton Vasconcelos Araújo e Renato Galvão Flôres Júnior. Mar/2001. 11. A Note on the Efficient Estimation of Inflation in Brazil Michael F. Bryan and Stephen G. Cecchetti. Mar/2001. 12. A Test of Competition in Brazilian Banking Márcio I. Nakane. Mar/2001. 26.

(28) 13. Modelos de Previsão de Insolvência Bancária no Brasil Marcio Magalhães Janot. Mar/2001. 14. Evaluating Core Inflation Measures for Brazil Francisco Marcos Rodrigues Figueiredo. Mar/2001. 15. Is It Worth Tracking Dollar/Real Implied Volatility? Sandro Canesso de Andrade and Benjamin Miranda Tabak. Mar/2001. 16. Avaliação das Projeções do Modelo Estrutural do Banco Central do Brasil para a Taxa de Variação do IPCA Sergio Afonso Lago Alves. Mar/2001. Evaluation of the Central Bank of Brazil Structural Model’s Inflation Forecasts in an Inflation Targeting Framework Sergio Afonso Lago Alves. Jul/2001. Estimando o Produto Potencial Brasileiro: uma Abordagem de Função de Produção Tito Nícias Teixeira da Silva Filho. Abr/2001. Estimating Brazilian Potential Output: a Production Function Approach Tito Nícias Teixeira da Silva Filho. Aug/2002. 18. A Simple Model for Inflation Targeting in Brazil Paulo Springer de Freitas and Marcelo Kfoury Muinhos. Apr/2001. 19. Uncovered Interest Parity with Fundamentals: a Brazilian Exchange Rate Forecast Model Marcelo Kfoury Muinhos, Paulo Springer de Freitas and Fabio Araújo. May/2001. 20. Credit Channel without the LM Curve Victorio Y. T. Chu and Márcio I. Nakane. May/2001. 21. Os Impactos Econômicos da CPMF: Teoria e Evidência Pedro H. Albuquerque. Jun/2001. 22. Decentralized Portfolio Management Paulo Coutinho and Benjamin Miranda Tabak. Jun/2001. 23. Os Efeitos da CPMF sobre a Intermediação Financeira Sérgio Mikio Koyama e Márcio I. Nakane. Jul/2001. 24. Inflation Targeting in Brazil: Shocks, Backward-Looking Prices, and IMF Conditionality Joel Bogdanski, Paulo Springer de Freitas, Ilan Goldfajn and Alexandre Antonio Tombini. Aug/2001. 25. Inflation Targeting in Brazil: Reviewing Two Years of Monetary Policy 1999/00 Pedro Fachada. Aug/2001. 26. Inflation Targeting in an Open Financially Integrated Emerging Economy: the Case of Brazil Marcelo Kfoury Muinhos. Aug/2001. 27. Complementaridade e Fungibilidade dos Fluxos de Capitais Internacionais Carlos Hamilton Vasconcelos Araújo e Renato Galvão Flôres Júnior. Set/2001. 17. 27.

(29) 28. Regras Monetárias e Dinâmica Macroeconômica no Brasil: uma Abordagem de Expectativas Racionais Marco Antonio Bonomo e Ricardo D. Brito. Nov/2001. 29. Using a Money Demand Model to Evaluate Monetary Policies in Brazil Pedro H. Albuquerque and Solange Gouvêa. Nov/2001. 30. Testing the Expectations Hypothesis in the Brazilian Term Structure of Interest Rates Benjamin Miranda Tabak and Sandro Canesso de Andrade. Nov/2001. 31. Algumas Considerações sobre a Sazonalidade no IPCA Francisco Marcos R. Figueiredo e Roberta Blass Staub. Nov/2001. 32. Crises Cambiais e Ataques Especulativos no Brasil Mauro Costa Miranda. Nov/2001. 33. Monetary Policy and Inflation in Brazil (1975-2000): a VAR Estimation André Minella. Nov/2001. 34. Constrained Discretion and Collective Action Problems: Reflections on the Resolution of International Financial Crises Arminio Fraga and Daniel Luiz Gleizer. Nov/2001. 35. Uma Definição Operacional de Estabilidade de Preços Tito Nícias Teixeira da Silva Filho. Dez/2001. 36. Can Emerging Markets Float? Should They Inflation Target? Barry Eichengreen. Feb/2002. 37. Monetary Policy in Brazil: Remarks on the Inflation Targeting Regime, Public Debt Management and Open Market Operations Luiz Fernando Figueiredo, Pedro Fachada and Sérgio Goldenstein. Mar/2002. 38. Volatilidade Implícita e Antecipação de Eventos de Stress: um Teste para o Mercado Brasileiro Frederico Pechir Gomes. Mar/2002. 39. Opções sobre Dólar Comercial e Expectativas a Respeito do Comportamento da Taxa de Câmbio Paulo Castor de Castro. Mar/2002. 40. Speculative Attacks on Debts, Dollarization and Optimum Currency Areas Aloisio Araujo and Márcia Leon. Apr/2002. 41. Mudanças de Regime no Câmbio Brasileiro Carlos Hamilton V. Araújo e Getúlio B. da Silveira Filho. Jun/2002. 42. Modelo Estrutural com Setor Externo: Endogenização do Prêmio de Risco e do Câmbio Marcelo Kfoury Muinhos, Sérgio Afonso Lago Alves e Gil Riella. Jun/2002. 43. The Effects of the Brazilian ADRs Program on Domestic Market Efficiency Benjamin Miranda Tabak and Eduardo José Araújo Lima. Jun/2002. 28.

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(31) 61. O Uso de Dados de Alta Freqüência na Estimação da Volatilidade e do Valor em Risco para o Ibovespa João Maurício de Souza Moreira e Eduardo Facó Lemgruber. Dez/2002. 62. Taxa de Juros e Concentração Bancária no Brasil Eduardo Kiyoshi Tonooka e Sérgio Mikio Koyama. Fev/2003. 63. Optimal Monetary Rules: the Case of Brazil Charles Lima de Almeida, Marco Aurélio Peres, Geraldo da Silva e Souza and Benjamin Miranda Tabak. Feb/2003. 64. Medium-Size Macroeconomic Model for the Brazilian Economy Marcelo Kfoury Muinhos and Sergio Afonso Lago Alves. Feb/2003. 65. On the Information Content of Oil Future Prices Benjamin Miranda Tabak. Feb/2003. 66. A Taxa de Juros de Equilíbrio: uma Abordagem Múltipla Pedro Calhman de Miranda e Marcelo Kfoury Muinhos. Fev/2003. 67. Avaliação de Métodos de Cálculo de Exigência de Capital para Risco de Mercado de Carteiras de Ações no Brasil Gustavo S. Araújo, João Maurício S. Moreira e Ricardo S. Maia Clemente. Fev/2003. 68. Real Balances in the Utility Function: Evidence for Brazil Leonardo Soriano de Alencar and Márcio I. Nakane. Feb/2003. 69. r-filters: a Hodrick-Prescott Filter Generalization Fabio Araújo, Marta Baltar Moreira Areosa and José Alvaro Rodrigues Neto. Feb/2003. 70. Monetary Policy Surprises and the Brazilian Term Structure of Interest Rates Benjamin Miranda Tabak. Feb/2003. 71. On Shadow-Prices of Banks in Real-Time Gross Settlement Systems Rodrigo Penaloza. Apr/2003. 72. O Prêmio pela Maturidade na Estrutura a Termo das Taxas de Juros Brasileiras Ricardo Dias de Oliveira Brito, Angelo J. Mont'Alverne Duarte e Osmani Teixeira de C. Guillen. Maio/2003. 73. Análise de Componentes Principais de Dados Funcionais – Uma Aplicação às Estruturas a Termo de Taxas de Juros Getúlio Borges da Silveira e Octavio Bessada. Maio/2003. 74. Aplicação do Modelo de Black, Derman & Toy à Precificação de Opções Sobre Títulos de Renda Fixa Octavio Manuel Bessada Lion, Carlos Alberto Nunes Cosenza e César das Neves. Maio/2003. 75. Brazil’s Financial System: Resilience to Shocks, no Currency Substitution, but Struggling to Promote Growth Ilan Goldfajn, Katherine Hennings and Helio Mori. 30. Jun/2003.

(32) 76. Inflation Targeting in Emerging Market Economies Arminio Fraga, Ilan Goldfajn and André Minella. Jun/2003. 77. Inflation Targeting in Brazil: Constructing Credibility under Exchange Rate Volatility André Minella, Paulo Springer de Freitas, Ilan Goldfajn and Marcelo Kfoury Muinhos. Jul/2003. 78. Contornando os Pressupostos de Black & Scholes: Aplicação do Modelo de Precificação de Opções de Duan no Mercado Brasileiro Gustavo Silva Araújo, Claudio Henrique da Silveira Barbedo, Antonio Carlos Figueiredo, Eduardo Facó Lemgruber. Out/2003. 79. Inclusão do Decaimento Temporal na Metodologia 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. Out/2003. 80. Diferenças e Semelhanças entre Países da América Latina: uma Análise de Markov Switching para os Ciclos Econômicos de Brasil e Argentina Arnildo da Silva Correa. Out/2003. 81. Bank Competition, Agency Costs and the Performance of the Monetary Policy Leonardo Soriano de Alencar and Márcio I. Nakane. Jan/2004. 82. Carteiras de Opções: Avaliação de Metodologias de Exigência de Capital no Mercado Brasileiro Cláudio Henrique da Silveira Barbedo e Gustavo Silva Araújo. Mar/2004. 83. Does Inflation Targeting Reduce Inflation? An Analysis for the OECD Industrial Countries Thomas Y. Wu. May/2004. 84. Speculative Attacks on Debts and Optimum Currency Area: a Welfare Analysis Aloisio Araujo and Marcia Leon. May/2004. 85. Risk Premia for Emerging Markets Bonds: Evidence from Brazilian Government Debt, 1996-2002 André Soares Loureiro and Fernando de Holanda Barbosa. May/2004. 86. Identificação do Fator Estocástico de Descontos e Algumas Implicações sobre Testes de Modelos de Consumo Fabio Araujo e João Victor Issler. Maio/2004. 87. Mercado de Crédito: uma Análise Econométrica dos Volumes de Crédito Total e Habitacional no Brasil Ana Carla Abrão Costa. Dez/2004. 88. Ciclos Internacionais de Negócios: uma Análise de Mudança de Regime Markoviano para Brasil, Argentina e Estados Unidos Arnildo da Silva Correa e Ronald Otto Hillbrecht. Dez/2004. 89. O Mercado de Hedge Cambial no Brasil: Reação das Instituições Financeiras a Intervenções do Banco Central Fernando N. de Oliveira. Dez/2004. 31.

(33) 90. Bank Privatization and Productivity: Evidence for Brazil Márcio I. Nakane and Daniela B. Weintraub. Dec/2004. 91. Credit Risk Measurement and the Regulation of Bank Capital and Provision Requirements in Brazil – A Corporate Analysis Ricardo Schechtman, Valéria Salomão Garcia, Sergio Mikio Koyama and Guilherme Cronemberger Parente. Dec/2004. 92. Steady-State Analysis of an Open Economy General Equilibrium Model for Brazil Mirta Noemi Sataka Bugarin, Roberto de Goes Ellery Jr., Victor Gomes Silva, Marcelo Kfoury Muinhos. Apr/2005. 93. Avaliação de Modelos de Cálculo de Exigência de Capital para Risco Cambial Claudio H. da S. Barbedo, Gustavo S. Araújo, João Maurício S. Moreira e Ricardo S. Maia Clemente. Abr/2005. 94. Simulação Histórica Filtrada: Incorporação da Volatilidade ao Modelo Histórico de Cálculo de Risco para Ativos Não-Lineares Claudio Henrique da Silveira Barbedo, Gustavo Silva Araújo e Eduardo Facó Lemgruber. Abr/2005. 95. Comment on Market Discipline and Monetary Policy by Carl Walsh Maurício S. Bugarin and Fábia A. de Carvalho. Apr/2005. 96. O que É Estratégia: uma Abordagem Multiparadigmática para a Disciplina Anthero de Moraes Meirelles. Ago/2005. 97. Finance and the Business Cycle: a Kalman Filter Approach with Markov Switching Ryan A. Compton and Jose Ricardo da Costa e Silva. Aug/2005. 98. Capital Flows Cycle: Stylized Facts and Empirical Evidences for Emerging Market Economies Helio Mori e Marcelo Kfoury Muinhos. Aug/2005. 99. Adequação das Medidas de Valor em Risco na Formulação da Exigência de Capital para Estratégias de Opções no Mercado Brasileiro Gustavo Silva Araújo, Claudio Henrique da Silveira Barbedo,e Eduardo Facó Lemgruber. Set/2005. 100 Targets and Inflation Dynamics Sergio A. L. Alves and Waldyr D. Areosa. Oct/2005. 101 Comparing Equilibrium Real Interest Rates: Different Approaches to Measure Brazilian Rates Marcelo Kfoury Muinhos and Márcio I. Nakane. Mar/2006. 102 Judicial Risk and Credit Market Performance: Micro Evidence from Brazilian Payroll Loans Ana Carla A. Costa and João M. P. de Mello. Apr/2006. 103 The Effect of Adverse Supply Shocks on Monetary Policy and Output Maria da Glória D. S. Araújo, Mirta Bugarin, Marcelo Kfoury Muinhos and Jose Ricardo C. Silva. Apr/2006. 32.

(34) 104 Extração de Informação de Opções Cambiais no Brasil Eui Jung Chang e Benjamin Miranda Tabak. Abr/2006. 105 Representing Roommate’s Preferences with Symmetric Utilities José Alvaro Rodrigues Neto. Apr/2006. 106 Testing Nonlinearities Between Brazilian Exchange Rates and Inflation Volatilities Cristiane R. Albuquerque and Marcelo Portugal. May/2006. 107 Demand for Bank Services and Market Power in Brazilian Banking Márcio I. Nakane, Leonardo S. Alencar and Fabio Kanczuk. Jun/2006. 108 O Efeito da Consignação em Folha nas Taxas de Juros dos Empréstimos Pessoais Eduardo A. S. Rodrigues, Victorio Chu, Leonardo S. Alencar e Tony Takeda. Jun/2006. 109 The Recent Brazilian Disinflation Process and Costs Alexandre A. Tombini and Sergio A. Lago Alves. Jun/2006. 110 Fatores de Risco e o Spread Bancário no Brasil Fernando G. Bignotto e Eduardo Augusto de Souza Rodrigues. Jul/2006. 111 Avaliação de Modelos de Exigência de Capital para Risco de Mercado do Cupom Cambial Alan Cosme Rodrigues da Silva, João Maurício de Souza Moreira e Myrian Beatriz Eiras das Neves. Jul/2006. 112 Interdependence and Contagion: an Analysis of Information Transmission in Latin America's Stock Markets Angelo Marsiglia Fasolo. Jul/2006. 113 Investigação da Memória de Longo Prazo da Taxa de Câmbio no Brasil Sergio Rubens Stancato de Souza, Benjamin Miranda Tabak e Daniel O. Cajueiro. Ago/2006. 114 The Inequality Channel of Monetary Transmission Marta Areosa and Waldyr Areosa. Aug/2006. 115 Myopic Loss Aversion and House-Money Effect Overseas: an Experimental Approach José L. B. Fernandes, Juan Ignacio Peña and Benjamin M. Tabak. Sep/2006. 116 Out-Of-The-Money Monte Carlo Simulation Option Pricing: the Join Use of Importance Sampling and Descriptive Sampling Jaqueline Terra Moura Marins, Eduardo Saliby and Joséte Florencio dos Santos. Sep/2006. 117 An Analysis of Off-Site Supervision of Banks’ Profitability, Risk and Capital Adequacy: a Portfolio Simulation Approach Applied to Brazilian Banks Theodore M. Barnhill, Marcos R. Souto and Benjamin M. Tabak. Sep/2006. 118 Contagion, Bankruptcy and Social Welfare Analysis in a Financial Economy with Risk Regulation Constraint Aloísio P. Araújo and José Valentim M. Vicente. Oct/2006. 33.

(35) 119 A Central de Risco de Crédito no Brasil: uma Análise de Utilidade de Informação Ricardo Schechtman. Out/2006. 120 Forecasting Interest Rates: an Application for Brazil Eduardo J. A. Lima, Felipe Luduvice and Benjamin M. Tabak. Oct/2006. 121 The Role of Consumer’s Risk Aversion on Price Rigidity Sergio A. Lago Alves and Mirta N. S. Bugarin. Nov/2006. 122 Nonlinear Mechanisms of the Exchange Rate Pass-Through: a Phillips Curve Model With Threshold for Brazil Arnildo da Silva Correa and André Minella. Nov/2006. 123 A Neoclassical Analysis of the Brazilian “Lost-Decades” Flávia Mourão Graminho. Nov/2006. 124 The Dynamic Relations between Stock Prices and Exchange Rates: Evidence for Brazil Benjamin M. Tabak. Nov/2006. 125 Herding Behavior by Equity Foreign Investors on Emerging Markets Barbara Alemanni and José Renato Haas Ornelas. Dec/2006. 126 Risk Premium: Insights over the Threshold José L. B. Fernandes, Augusto Hasman and Juan Ignacio Peña. Dec/2006. 127 Uma Investigação Baseada em Reamostragem sobre Requerimentos de Capital para Risco de Crédito no Brasil Ricardo Schechtman. Dec/2006. 128 Term Structure Movements Implicit in Option Prices Caio Ibsen R. Almeida and José Valentim M. Vicente. Dec/2006. 129 Brazil: Taming Inflation Expectations Afonso S. Bevilaqua, Mário Mesquita and André Minella. Jan/2007. 130 The Role of Banks in the Brazilian Interbank Market: Does Bank Type Matter? Daniel O. Cajueiro and Benjamin M. Tabak. Jan/2007. 131 Long-Range Dependence in Exchange Rates: the Case of the European Monetary System Sergio Rubens Stancato de Souza, Benjamin M. Tabak and Daniel O. Cajueiro. Mar/2007. 132 Credit Risk Monte Carlo Simulation Using Simplified Creditmetrics’ Model: the Joint Use of Importance Sampling and Descriptive Sampling Jaqueline Terra Moura Marins and Eduardo Saliby. Mar/2007. 133 A New Proposal for Collection and Generation of Information on Financial Institutions’ Risk: the Case of Derivatives Gilneu F. A. Vivan and Benjamin M. Tabak. Mar/2007. 134 Amostragem Descritiva no Apreçamento de Opções Européias através 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. Abr/2007. 34.

(36) 135 Evaluation of Default Risk for the Brazilian Banking Sector Marcelo Y. Takami and Benjamin M. Tabak. May/2007. 136 Identifying Volatility Risk Premium from Fixed Income Asian Options Caio Ibsen R. Almeida and José Valentim M. Vicente. May/2007. 137 Monetary Policy Design under Competing Models of Inflation Persistence Solange Gouvea e Abhijit Sen Gupta. May/2007. 138 Forecasting Exchange Rate Density Using Parametric Models: the Case of Brazil Marcos M. Abe, Eui J. Chang and Benjamin M. Tabak. May/2007. 35.

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