Working Paper Series - Banco Central Do Brasil SCCF are imposed, all matrices of a VECM have rank...
Transcript of Working Paper Series - Banco Central Do Brasil SCCF are imposed, all matrices of a VECM have rank...
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
Working Paper Series Edited by Research Department (Depep) – E-mail: [email protected] Editor: Benjamin Miranda Tabak – E-mail: [email protected] Editorial Assistent: Jane Sofia Moita – E-mail: [email protected] Head of Research Department: Carlos Hamilton Vasconcelos Araújo – E-mail: [email protected] The Banco Central do Brasil Working Papers are all evaluated in double blind referee process. Reproduction is permitted only if source is stated as follows: Working Paper n. 139. Authorized by Mário Mesquita, Deputy Governor for Economic Policy. General Control of Publications Banco Central do Brasil
Secre/Surel/Dimep
SBS – Quadra 3 – Bloco B – Edifício-Sede – M1
Caixa Postal 8.670
70074-900 Brasília – DF – Brazil
Phones: (5561) 3414-3710 and 3414-3567
Fax: (5561) 3414-3626
E-mail: [email protected]
The views expressed in this work are those of the authors and do not necessarily reflect those of the Banco Central or its members. Although these Working Papers often represent preliminary work, citation of source is required when used or reproduced. As opiniões expressas neste trabalho são exclusivamente do(s) autor(es) e não refletem, necessariamente, a visão do Banco Central do Brasil. Ainda que este artigo represente trabalho preliminar, citação da fonte é requerida mesmo quando reproduzido parcialmente. Consumer Complaints and Public Enquiries Center Address: Secre/Surel/Diate
Edifício-Sede – 2º subsolo
SBS – Quadra 3 – Zona Central
70074-900 Brasília – DF – Brazil
Fax: (5561) 3414-2553
Internet: http://www.bcb.gov.br/?english
Selection of Optimal Lag Length 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, [email protected], Graduate School of Economics, FGV-EPGE-RJ,Brazil.
‡[email protected], DEE-PUC-RJ, Brazil§[email protected], Banco Central do Brasil and Ibmec-RJ, Brazil
3
1 Introduction
In the modelling of economic and financial time series, the vectorial autoregressive
(VAR)model became a standard linear model used in empirical works. An important
aspect of empirical research in the specification of the VAR models is the determi-
nation of the lag order of the autoregressive lag polynomial, since all inference in the
VAR model depends on the correct model specification. In several contributions,
the effect of lag length selection has been demonstrated: Lütkepohl (1993) indicates
that selecting a higher order lag length than the true lag length causes an increase
in the mean square forecast errors of the VAR and that underfitting the lag length
often generates autocorrelated errors. Braun and Mittnik (1993) show that impulse
response functions and variance decompositions are inconsistently derived from the
estimated VAR when the lag length differs from the true lag length. When cointe-
gration restrictions are considered in the model, the effect of lag length selection on
the cointegration tests has been demonstrated. For example, Johansen (1991) and
Gonzalo (1994) point out that VAR order selection may affect proper inference on
cointegrating vectors and rank.
Recently empirical works have considered another kind of restrictions on the
VAR model (e.g., Engle and Issler, 1995; Caporale, 1997; Mamingi and Sunday,
2003). Engle and Kozicki (1993) showed that VAR models can have another type of
restrictions, called common cyclical features, which are restrictions on the short-run
dynamics. These restrictions are defined in the same way as cointegration restric-
tions, while cointegration refers to relations among variables in the long-run, the
common cyclical restrictions refer to relations in the short-run. Vahid and Engle
(1993) proposed the Serial Correlation Common Feature (SCCF) 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
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