ISSN 1518-3548

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

Carlos Enrique Carrasco Gutiérrez, Reinaldo Castro Souza andOsmani Teixeira de Carvalho Guillén

June, 2007

Working Paper Series

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: workingpaper@bcb.gov.br Editor: Benjamin Miranda Tabak – E-mail: benjamin.tabak@bcb.gov.br Editorial Assistent: Jane Sofia Moita – E-mail: jane.sofia@bcb.gov.br Head of Research Department: Carlos Hamilton Vasconcelos Araújo – E-mail: carlos.araujo@bcb.gov.br 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: editor@bcb.gov.br

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 inCointegrated VAR Models with WeakForm 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 viewsof the Banco Central do Brasil. The views expressed in the papers arethose of the author(s) and not necessarily reflect those of the BancoCentral 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 VARmodel depends on the correct model specification. Literature has shown im-portant studies of how to select the lag order of a nonstationary VAR modelsubject to cointegration restrictions. In this work, we consider an additionalweak form (WF) restriction of common cyclical features in the model in orderto analyze the appropriate way to select the correct lag order. Two method-ologies have been used: the traditional information criteria (AIC, HQ andSC) and an alternative criterion (IC(p, s)) which select simultaneously thelag order p and the rank structure s due to the WF restriction. A Monte-Carlo simulation is used in the analysis. The results indicate that the cost ofignoring additional WF restrictions in vector autoregressive modelling can behigh 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 BrazilianEconometric 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, 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

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) as a measure of

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) defined a weak version of SCCF

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), and b) the use of an

1When a VAR model has cointegration restriction it can be represented as a VECM. Thisrepresentation 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 VARmodel 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 =

pXi=1

Aiyt−i + εt (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;

Π (L) yt = εt where L represents the lag operator and Π (L) = In −Pp

i=1AiLi that

when L = 1, Π (1) = In −Pp

i=1Ai. 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) = −αβ0, where α and β are (n× r) matrices with

full column rank, r. b) The characteristic equation |Π (L)| = 0 has n−r roots equalto 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 VAR2This 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:

∆yt = αβ0yt−1 +p−1Xi=1

Γi∆yt−i + εt (2)

where: αβ0 = −Π (1), Γj = −Pp

k=j+1Ak 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 (cointe-

gration), there exists a (n×s) matrix β of rank s, whose columns span the cofeature

space, such that β0(∆yt − αβyt−1) = β

0εt , where β

0εt 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 : β0Γj = 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. We can

rewrite the VECM with WF restrictions as a model of reduced-rank structure. In

(2) let Xt−1 = [∆y0t−1, .....∆y0t−p+1]0 and Φ = [Γ1, ....,Γp−1], therefore we get:

∆yt = αβyt−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 = αβyt−1 + β⊥ (Ψ1,Ψ2, ...,Ψp−1)Xt−1 + εt (4)

= αβyt−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

analysis3. Therefore, we can use the expression CanCorrXt, Zt|Wt that denotesthe 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-

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 whereWt = [∆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):

ξr = −TnX

i=r+1

Ln (1− λ2

i ) i = 1, ..., n (6)

3This estimation is referred as Full Information Maximum Likelihood - FIML

8

where the eigenvalues 0 < λn < ... < λ1 are the solution of : |λm11−m−110m00m01| =

0, where mij, i, j = 0.1, are the second moment matrices: m00 =1T

PTt=1 u0tu

00t,

m10 =1T

PTt=1 u1tu

00t, m01 =

1T

PTt=1 u0tu

01t, m11 =

1T

PTt=1 u1tu

01t of the residuals u0t

and u1t 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

for long-run effects which are white noise (i.e., β0(∆yt − αβyt−1) = β

0εt where β

0εt

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.

4Cointegration 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)? 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 short-

run is carried out by minimizing the following modified information criteria (see

Hecq, 2006).

AIC (p, s) =TX

i=n−s+1ln(1− λ2i (p)) +

2

T×N (7)

HQ(p, s) =TX

i=n−s+1ln(1− λ2i (p)) +

2 ln(lnT )

T×N (8)

SC(p, s) =TX

i=n−s+1ln(1− λ2i (p)) +

lnT

T×N (9)

N = [n× (n× (p− 1)) + n× r]− [s× (n× (p− 1) + (n− s))]

The number of parametersN 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). Theeigenvalues 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). 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 = 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 β estimated in step one, we proceed to estimate β⊥ and Ψ. Hence,

we run a regression of ∆yt and of Xt−1on β0yt−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). We estimate (11) by FIML. Thus, we can obtain β⊥

and Ψ.

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-

culate the eigenvalues associated with Ψ, λ2

i i = 1, ..., s and the matrix

of residualsPmax

r, s=s. Hence, we compute the ML function:

L0max, r<n, s=s = −T

2

"ln

¯¯

maxXr<n, s=s

¯¯−

sXi=1

ln³1− λ

2

i

´#(12)

If r = n, we use instead of (12) the derived log-likelihood: Lmax, r=n, s=s =

−T2ln¯Pmax

r=n, s=s

¯. The determinant of the covariance matrix for r = n

cointegration vector is calculated by

ln

¯¯

maxXr=n, s=s

¯¯ = ln ¯m00 −m01m

−111m10

¯− sXi=1

ln³1− λ

2

i

´(13)

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.

In order to reestimate β we use the programCanCorrh∆yt, yt−1 | ΨXt−1

iand thus using the new β we can repeat step 2 to reestimate β⊥ and Ψ.

Then, we can calculate the new value of the ML function in the step 3.

Henceforth, we obtainL1max, r=r, s=s for calculating∆L =¡L1max, r=r, s=s - L0max, r=r, s=s

¢.

We repeat steps 1 to 4 to choose β⊥and Ψ until convergence is reached ( i.e.,

∆L < 10−7). 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

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 β satisfy:

β =

1.00.2−1.0

, β = 1.0 0.10.0 1.00.5 −0.5

ε1t

ε2tε3t

∼ N

000

, 1.0 0.6 0.60.6 1.0 0.60.6 0.6 1.0

Consider the VAR(3) model: yt = A1yt−1 + A2yt−2 + A3yt−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 cointegra-

tion restrictions: αβ0 = (A1 + A2 + A3 − I3) ; b) WF restrictions: β0A3 = 0 (iii)

β0(A2 +A3) = 0 and c) covariance-stationary condition. Considering the cointegra-

tion restrictions we can rewrite (14) as the following VAR(1):

ξt = F ξt−1 + vt (15)

ξt =

4yt4yt−1β0yt

, F = −(A2 +A3) −A3 α

I3 0 0−β(A2 +A3) −β0A3 β0α+ 1

and vt =

εt0

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

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

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 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 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 modelmore 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 frequentlyselected by all the traditional criteria.

− Table I shows that ignoring WF restrictions the standard SC has theworst 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.

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

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

A Tables

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)

with parameters: A1 =

a111 a112 a112a121 a122 a122a131 a132 a132

, A2 = a211 a212 a212

a221 a222 a222a231 a232 a232

and A3 = a311 a312 a312a321 a322 a322a331 a332 a332

We consider the cointegration vectors β = β11

β21β31

, the cofea-tures vectors β =

β11 β12β21 β22β31 β32

and the adjustament matrix α =

α11α21α31

. Thelong-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 = F ξt−1 + vt (18)

where ξt =

4yt4yt−1β0yt

, F = −(A2 +A3) −A3 α

I3 0 0−β(A2 +A3) −β0A3 β0α+ 1

and vt = εt

0β0εt

1) Short-run restrictions (WF)

Let us, G = −[R21K + R31], K = [(R32 − R31)/(R21 − R22)], Rj1 = βj1/β11,

Rj2 = βj2/β12 (j = 2, 3) and S = β11G+ β21K + β31

(i) β0A3 = 0 ==> A3 =

−Ga331 −Ga332 −Ga333−Ka331 −Ka332 −Ka333−a331 −a332 −a333

(ii) β

0(A2 +A3) = 0 ==> β

0A2 = 0 ==> A2 =

−Ga231 −Ga232 −Ga233−Ka231 −Ka232 −Ka233−a231 −a232 −a233

2) Long-run restrictions (cointegration)

22

(iv) β0(A2 + A3) = [−(a231 + a331)S − (a232 + a332)S − (a233 + a333)S] and

β0A3 = [−a331S − a332S − a333S]

(v) β0α+1 = β =£β11 β21 β31

¤ α11α21α31

+1 = β11α11+β21α21+β31α31+1

Therefore, considering short- and long-run restrictions, the companion matrix F

is represented as:

F =

−(A2 +A3) −A3 αI3 0 0

−β(A2 +A3) −β0A3 β0α+ 1

=

−G(a231 + a331) −G(a232 + a332) −G(a233 + a333) −Ga331 −Ga332 −Ga333 α11−K(a231 + a331) −G(a232 + a332) −G(a233 + a333) −Ka331 −Ka332 −Ka333 α21−(a231 + a331) −G(a232 + a332) −(a233 + a333) −a331 −a332 −a333 α31

1 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 0

−(a231 + a331)S −(a232 + a332)S −(a233 + a333)S −a331S −a332S −a333S b

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, and Ψ are: Ω = G(a231+a331)+K(a

232+a

332)+a

233+a

333−b,

Θ = Ga331+Ka332− (a233+ a333)b−Gb(a231+ a331)−Kb(a232+ a332) +α31S(a233+ a333) +

23

Sα21(a232+ a332)+Sα11(a

231+a331)+ a333 and Ψ = −a333b−Ga331b−Ka332b+α31a

333S+

a332Sα21+a331Sα11, 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 (21)

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

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:

a231 = −(−Ka232−Ka232b+α31Sa233−λ6λ7−λ6−λ7−a233b−λ5λ6λ7+ b−λ5λ7−

λ5λ6 − a233 + Sa232α21 − λ5)/(Sα11 −G−Gb)

a332 = (−S2λ7α11α31−b2λ7G−λ6Gb2+bλ7Sα11+λ6Sα11b−a331Sα11G+a331S2α211−Ga331bSα11−λ5Gb2+λ5Sα11b−λ7λ6α31SG−λ7λ5α31SG−S2α11λ5α31−S2α11λ6α31+Sλ5Gbα31+Sα31λ

6Gb−λ5λ7λ6G+λ6λ7Gb+λ5λ7Gb+λ5λ6Gb−SGb2α31+S2α11bα31−S2α11α31a

233+S2α231a

233G+SG2a331α31+Sα11a

233b+Gb3−Sα11b

2−S2α11Ka232α31−S2α11α31Ga

331+S

2a232α21Gα31−Sa232α21Gb+Sα31G2a331b−Sα31a233Gb+Sα11Ka232b+

Sλ7Gbα31−λ5λ6α31SG−λ5λ7λ6α31SG+λ5λ7λ6Sα11)/(Sα11Kα31−KGα31+bGα21−Kα31Gb− Sα11α21 +Gα21)/S

a333 = −(Kb3G−λ5Gb2K+Sα11λ6Kλ7λ5+Kbλ7Sα11−Kb2λ7G−S2α21λ7α11+

λ6GbSα21+Sα21λ7Gb−λ6Gb2K+λ6Sα11Kb−λ6S2α11α21+λ5GbSα21+λ5Sα11Kb−

λ5S2α11α21−λ7λ6Sα21G+Kbλ7λ6G+Kbλ7λ5G+Kbλ5λ6G−λ7λ6KGλ5−S2α11α21Ka232+

S2α11α21b−S2α11α21a233+S2α221a232G−Sα11Kb2+Sα21G2a331−Sα21Gb2+S2a331Kα211−

S2α11α21Ga331+S

2α21a233Gα31+Sα11Kˆ2ba

232+Sα11Kba233−Sα11a331KG−Sα11KbGa331−

SKba233Gα31+Sα21G2a331b−Sα21λ5λ6G−Sα21λ5λ7λ6G−Sα21Ka232Gb−Sα21λ7λ5G)/(Sα11Kα31−

KGα31 + bGα21 −Kα31Gb− Sα11α21 +Gα21)/S

24

We can calculate a231, a332and a333 fixing the set λ1 = λ2 = λ3 = λ4 = 0 and sort

independently from uniform distributions (−0.9; 0.9) the values of a331, a232, a

233, 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

26

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 Monetary Policy and Banking Supervision Functions on the Central Bank Eduardo Lundberg

Jul/2000

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

27

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 Evaluation of the Central Bank of Brazil Structural Model’s Inflation Forecasts in an Inflation Targeting Framework Sergio Afonso Lago Alves

Mar/2001

Jul/2001

17 Estimando o Produto Potencial Brasileiro: uma Abordagem de Função de Produção Tito Nícias Teixeira da Silva Filho Estimating Brazilian Potential Output: a Production Function Approach Tito Nícias Teixeira da Silva Filho

Abr/2001

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

28

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

29

44 Estrutura Competitiva, Produtividade Industrial e Liberação Comercial no Brasil Pedro Cavalcanti Ferreira e Osmani Teixeira de Carvalho Guillén

Jun/2002

45 Optimal Monetary Policy, Gains from Commitment, and Inflation Persistence André Minella

Aug/2002

46 The Determinants of Bank Interest Spread in Brazil Tarsila Segalla Afanasieff, Priscilla Maria Villa Lhacer and Márcio I. Nakane

Aug/2002

47 Indicadores Derivados de Agregados Monetários Fernando de Aquino Fonseca Neto e José Albuquerque Júnior

Set/2002

48 Should Government Smooth Exchange Rate Risk? Ilan Goldfajn and Marcos Antonio Silveira

Sep/2002

49 Desenvolvimento do Sistema Financeiro e Crescimento Econômico no Brasil: Evidências de Causalidade Orlando Carneiro de Matos

Set/2002

50 Macroeconomic Coordination and Inflation Targeting in a Two-Country Model Eui Jung Chang, Marcelo Kfoury Muinhos and Joanílio Rodolpho Teixeira

Sep/2002

51 Credit Channel with Sovereign Credit Risk: an Empirical Test Victorio Yi Tson Chu

Sep/2002

52 Generalized Hyperbolic Distributions and Brazilian Data José Fajardo and Aquiles Farias

Sep/2002

53 Inflation Targeting in Brazil: Lessons and Challenges André Minella, Paulo Springer de Freitas, Ilan Goldfajn and Marcelo Kfoury Muinhos

Nov/2002

54 Stock Returns and Volatility Benjamin Miranda Tabak and Solange Maria Guerra

Nov/2002

55 Componentes de Curto e Longo Prazo das Taxas de Juros no Brasil Carlos Hamilton Vasconcelos Araújo e Osmani Teixeira de Carvalho de Guillén

Nov/2002

56 Causality and Cointegration in Stock Markets: the Case of Latin America Benjamin Miranda Tabak and Eduardo José Araújo Lima

Dec/2002

57 As Leis de Falência: uma Abordagem Econômica Aloisio Araujo

Dez/2002

58 The Random Walk Hypothesis and the Behavior of Foreign Capital Portfolio Flows: the Brazilian Stock Market Case Benjamin Miranda Tabak

Dec/2002

59 Os Preços Administrados e a Inflação no Brasil Francisco Marcos R. Figueiredo e Thaís Porto Ferreira

Dez/2002

60 Delegated Portfolio Management Paulo Coutinho and Benjamin Miranda Tabak

Dec/2002

30

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

Jun/2003

31

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

32

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

33

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

34

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

35

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