ISSN 1518-3548
Working Paper Series
Selection of Optimal Lag Length in Cointegrated VAR
Models with Weak Form of Common Cyclical Features
Carlos Enrique Carrasco Gutiérrez, Reinaldo Castro Souza and
Osmani Teixeira de Carvalho Guillén
June, 2007
ISSN 1518-3548
CGC 00.038.166/0001-05
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n. 139
Jun
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P. 1-35
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Selection of Optimal Lag Length in
Cointegrated VAR Models with Weak
Form of Common Cyclical Features∗
Carlos Enrique Carrasco Gutiérrez†
Reinaldo Castro Souza‡
Osmani Teixeira de Carvalho Guillén§
The Working Paper should not be reported as representing the views
of the Banco Central do Brasil. The views expressed in the papers are
those of the author(s) and not necessarily reflect those of the Banco
Central do Brasil.
Abstract
An important aspect of empirical research based on the vector autoregressive (VAR) model is the choice of the lag order, since all inference in the VAR
model depends on the correct model specification. Literature has shown important studies of how to select the lag order of a nonstationary VAR model
subject to cointegration restrictions. In this work, we consider an additional
weak form (WF) restriction of common cyclical features in the model in order
to analyze the appropriate way to select the correct lag order. Two methodologies have been used: the traditional information criteria (AIC, HQ and
SC) and an alternative criterion (IC(p, s)) which select simultaneously the
lag order p and the rank structure s due to the WF restriction. A MonteCarlo simulation is used in the analysis. The results indicate that the cost of
ignoring additional WF restrictions in vector autoregressive modelling can be
high specially when SC criterion is used.
Keywords: Cointegration; Common Cyclical Features; Reduced Rank Model;
Estimation; Information Criteria.
JEL Codes: C32, C53.
∗
Acknowledgments: We are grateful to comments and suggestions given by João Victor Issler,
Wagner Gaglianone, Ricardo Cavalcanti, Luiz Renato Lima and participants of the Brazilian
Econometric Meeting 2006. Special thanks are due to Alain Hecq for solving doubts and comments. The authors are responsible for any remaining errors in this paper. Carlos Enrique C.
Gutiérrez acknowledges the support of CAPES-Brazil.
†
Corresponding author, [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 determination of the lag order of the autoregressive lag polynomial, since all inference in the
VAR model depends on the correct model specification. In several contributions,
the effect of lag length selection has been demonstrated: Lütkepohl (1993) indicates
that selecting a higher order lag length than the true lag length causes an increase
in the mean square forecast errors of the VAR and that underfitting the lag length
often generates autocorrelated errors. Braun and Mittnik (1993) show that impulse
response functions and variance decompositions are inconsistently derived from the
estimated VAR when the lag length differs from the true lag length. When cointegration restrictions are considered in the model, the effect of lag length selection on
the cointegration tests has been demonstrated. For example, Johansen (1991) and
Gonzalo (1994) point out that VAR order selection may affect proper inference on
cointegrating vectors and rank.
Recently empirical works have considered another kind of restrictions on the
VAR model (e.g., Engle and Issler, 1995; Caporale, 1997; Mamingi and Sunday,
2003). Engle and Kozicki (1993) showed that VAR models can have another type of
restrictions, called common cyclical features, which are restrictions on the short-run
dynamics. These restrictions are defined in the same way as cointegration restrictions, while cointegration refers to relations among variables in the long-run, the
common cyclical restrictions refer to relations in the short-run. Vahid and Engle
(1993) proposed the Serial Correlation Common Feature (SCCF) as a measure of
common cyclical feature. SCCF restrictions might be imposed in a covariance stationary VAR model or in a cointegrated VAR model. When short-run restrictions
are imposed in cointegrated VAR models it is possible to define a weak version of
SCCF restrictions. Hecq, Palm and Urbain (2006) defined a weak version of SCCF
restrictions which they denominated it as weak-form (WF) common cyclical restric4
tions. A fundamental difference between SCCF and WF restrictions is in the form
which each one imposes restrictions on the Vector Error Correction Model (VECM)
representation1 . When SCCF are imposed, all matrices of a VECM have rank less
than the number of variables analyzed. On the other hand with WF restrictions
all matrices, except the long-run matrix, have rank less than a number of variables
in analysis. Hence, WF restrictions impose less restriction on VECM parameters.
Some advantages emerge when WF restrictions are considered. First, due to the
fact that WF restrictions does not impose restrictions on the cointegration space;
the rank of common cyclical features is not limited by the choice of cointegrating
rank. Another advantage is that WF restrictions is invariant over reparametrization
in VECM representation.
The literature has shown how to select an adequate lag order of a covariance
stationary VAR model and an adequate lag order of a VAR model subject to cointegration restrictions. Among the classical procedures, there are the information
criteria such as Akaike (AIC), Schwarz (SC) and Hannan-Quinn (HQ) (Lütkepohl,
1993). Kilian (2001) study the performance of traditional AIC, SC and HQ criterion
of a covariance stationary VAR model. Vahid and Issler (2002) analyzed the standard information criterion in a covariance stationary VAR model subject to SCCF
restriction and more recently Guillén, Issler and Athanasopoulos (2005) studied the
standard information criterion in VAR models with cointegration and SCCF restrictions. However, when cointegrated VAR models contain additional weak form
of common cyclical feature, there are no reported work on how to appropriately
determine the VAR model order.
The objective of this paper is to investigate the performance of information
criterion in selecting the lag order of a VAR model when the data are generated from
a true VAR with cointegration and WF restrictions that is referred as the correct
model. It will be carried out following two procedures: a) the use of standard criteria
as proposed by Vahid and Engle (1993), referred here as IC (p), and b) the use of an
1
When a VAR model has cointegration restriction it can be represented as a VECM. This
representation is also known as Granger Representation Theorem (Engle and Granger, 1987).
5
alternative procedure of model selection criterion (see, Vahid and Issler, 2002; Hecq
et al., 2006) consisting in selecting simultaneously the lag order p and the rank s do
to the weak form of common cyclical feature, which is referred to as IC(p, s)2 . The
most relevant results can be summarized as follows. The information criterion that
selects simultaneously the pair (p, s) has better performance than the model chosen
by conventional criteria. The cost of ignoring additional WF restrictions in vector
autoregressive modelling can be high specially when SC criterion is used.
The remaining of this work is organized as follows. Section 2 shows the econometric model. In section 3 the information criteria are mentioned. Monte Carlo
simulation is shown in section 4 and the results in section 5. Finally, the conclusions
are shown in section 6.
2
The Econometric Model
We show the VAR model with short-run and long-run restrictions. First, we consider
a Gaussian vector autoregression of finite order p, so-called VAR(p), such that:
yt =
p
X
Ai yt−i + εt
(1)
i=1
where, yt is a vector of n first order integrated series, I(1), Ai , i = 1, . . . , p are
matrices of dimension n × n, εt ∼ Normal (0, Ω) and {Ω, if t = τ and 0n×n , if
t 6= τ , where Ω is non singular}. The model (1) could be written equivalently as;
P
Π (L) yt = εt where L represents the lag operator and Π (L) = In − pi=1 Ai Li that
P
when L = 1, Π (1) = In − pi=1 Ai . If cointegration is considered in (1) the (n × n)
matrix Π (·) satisfies two conditions: a) Rank (Π (1)) = r, 0 < r < n, such that
Π (1) can be expressed as Π (1) = −αβ 0 , where α and β are (n × r) matrices with
full column rank, r. b) The characteristic equation |Π (L)| = 0 has n − r roots equal
to 1 and all other are outside the unit circle. These assumptions imply that yt is
cointegrated of order (1, 1). The elements of α are the adjustment coefficients and
the columns of β span the space of cointegration vectors. We can represent a VAR
2
This is quite recent in the literature (see, Hecq et al., 2006).
6
model as VECM. Decomposing the polynomial matrix Π (L) = Π (1) L + Π∗ (L) ∆,
where ∆ ≡ (1 − L) is the difference operator, a Vector Error Correction Model
(VECM) is obtained:
0
∆yt = αβ yt−1 +
p−1
X
(2)
Γi ∆yt−i + εt
i=1
where: αβ 0 = −Π (1), Γj = −
Pp
k=j+1
Ak for j = 1, ...., p − 1 and Γ0 = In . The
VAR(p) model can include additional short-horizon restrictions as shown by Vahid
and Engle (1993). We consider an interesting WF restriction (as defined by Hecq,
Palm and Urbain, 2006) that does not impose restrictions over long-run relations.
Definition 1 Weak Form-WF holds in (2) if, in addition to assumption 1 (cointegration), there exists a (n × s) matrix β̃ of rank s, whose columns span the cofeature
0
0
0
space, such that β̃ (∆yt − αβ´yt−1 ) = β̃ εt , where β̃ εt is a s-dimensional vector that
constitutes an innovation process with respect to information prior to period t, given
by {yt−1 , yt−2 , ..., y1 } .
Consequently we considerate WF restrictions in the VECM if there exists a
cofeature matrix β̃ that satisfies the following assumption:
0
Assumption 1 : β̃ Γj = 0s×n for
j = 1, ...., p − 1.
Imposing WF restrictions is convenient because it allows the study of both cointegration and common cyclical feature without the constraint r + s ≤ n. We can
rewrite the VECM with WF restrictions as a model of reduced-rank structure. In
0
0
, .....∆yt−p+1
]0 and Φ = [Γ1 , ...., Γp−1 ], therefore we get:
(2) let Xt−1 = [∆yt−1
(3)
∆yt = αβ´yt−1 + ΦXt−1 + εt
If assumption (1) holds matrices Γi , i = 1, ..., p are all of rank (n − s) then we can
write Φ = β̃ ⊥ Ψ = β̃ ⊥ [Ψ1 , ...., Ψp−1 ], where, β̃ ⊥ is n×(n−s) full column rank matrix,
Ψ is of dimension (n − s) × n(p − 1), the matrices Ψi , i = 1, ..., p − 1 all of rank
0
(n − s) × n. Hence, given assumption (1), there exists β̃ of n × s such that β̃ β̃ ⊥ = 0.
7
That is, β̃ ⊥ n × (n − s) is a full column rank orthogonal to the complement of β̃
with rank(β̃, β̃ ⊥ ) = n. Rewriting model (3) we have:
∆yt = αβ´yt−1 + β̃ ⊥ (Ψ1 , Ψ2 , ..., Ψp−1 ) Xt−1 + εt
(4)
(5)
= αβ´yt−1 + β̃ ⊥ ΨXt−1 + εt
Estimation of (5) is carried out via the switching algorithms (see, Hecq, 2006) that
use the procedure in estimating reduced-rank regression models suggested by Anderson (1951). There is a formal connection between a reduced-rank regression and the
canonical analysis as noted by Izenman (1975), Box and Tiao (1977), Tso (1980)
and Veleu et al. (1986). When the multivariate regression has all of its matrix
coefficients of full rank, it may be estimated by usual Least Square or MaximumLikelihood procedures. But when the matrix coefficients are of reduced-rank they
have to be estimated using the reduced-rank regression models of Anderson (1951).
The use of canonical analysis may be regarded as a special case of reduced-rank
regression. More specifically, the maximum-likelihood estimation of the parameters
of the reduced-rank regression model may result in solving a problem of canonical
analysis3 . Therefore, we can use the expression CanCorr{Xt , Zt |Wt } that denotes
the partial canonical correlations between Xt and Zt : both sets concentrate out the
effect of Wt that allows us to obtain canonical correlation, represented by the eigenvalues λ̂1 > λ̂2 > λ̂3 ....... > λ̂n . The Johansen test statistic is based on canonical
correlation. In model (2) we can use the expression CanCorr{∆yt , yt−1 |Wt } where
Wt = [∆yt−1 , ∆yt−2 , ....., ∆yt+p−1 ] that summarizes the reduced-rank regression procedure used in the Johansen approach. It means that one extracts the canonical
correlations between ∆yt and yt−1 : both sets concentrated out the effect of lags of
Wt . In order to test for the significance of the r largest eigenvalues, one can rely on
Johansen’s trace statistic (6):
ξ r = −T
3
n
X
i=r+1
2
Ln (1 − λ̂i )
i = 1, ..., n
This estimation is referred as Full Information Maximum Likelihood - FIML
8
(6)
where the eigenvalues 0 < λ̂n < ... < λ̂1 are the solution of : |λm11 − m−1
10 m00 m01 | =
P
0, where mij , i, j = 0.1, are the second moment matrices: m00 = T1 Tt=1 ũ0t ũ00t ,
P
P
P
m10 = T1 Tt=1 ũ1t ũ00t , m01 = T1 Tt=1 ũ0t ũ01t , m11 = T1 Tt=1 ũ1t ũ01t of the residuals ũ0t
and ũ1t obtained in the multivariate least squares regressions ∆yt = (∆yt−1 , ...∆yt−p+1 )+
u0t and yt−1 = (∆yt−1 , ...∆yt−p+1 )+u1t respectively (see, Hecq et al., 2006; Johansen,
1995). The result of Johansen test is a superconsistent estimated β. Moreover, we
could also use a canonical correlation approach to determine the rank of the common
features space due to WF restrictions. It is a test for the existence of cofeatures
in the form of linear combinations of the variables in the first differences, corrected
0
0
0
for long-run effects which are white noise (i.e., β̃ (∆yt − αβ´yt−1 ) = β̃ εt where β̃ εt
is a white noise). Canonical analysis is adopted in the present work in estimating,
testing and selecting lag-rank of VAR models as shown in next sections.
3
Model Selection Criteria
In model selection we use two procedures to identify the VAR model order. The
standard selection criteria, IC(p) and the modified informational criteria, IC(p, s),
novelty in the literature, which consists on identifying p and s simultaneously.
The model estimation following the standard selection criteria, IC(p), used by
Vahid and Engle (1993) entails the following steps:
1. Estimate p using standard informational criteria: Akaike (AIC),
Schwarz (SC) and Hanna-Quinn (HQ). We choose the lag length of the
VAR in levels that minimize the information criteria.
2. Using the lag length chosen in the previous step, find the number of
cointegration vector, r using Johansen cointegration test4 .
3. Conditional on the results of cointegration analysis, a final VECM is
estimated and then the multi-step ahead forecast is calculated.
4
Cointegration rank and vectors are estimated using the FIML as shown in Johansen (1991).
9
The above procedure is followed when there is evidence of cointegration restrictions. We check the performance of IC(p) when WF restrictions contain the
true model. Additionally we check the performance of alternative selection criteria
IC(p, s). Vahid and Issler (2002) analyzed a covariance-stationary VAR model with
SCCF restrictions. They showed that the use of IC(p, s) has better performance
than IC(p) in VAR model lag order selection. In the present work we analyze cointegrated VAR model with WF restrictions in order to analyze the performance of
IC(p) and IC(p, s) for model selection. The question investigated is: is the performance of IC(p, s) superior to that of IC(p)? This is an important question we aim
to answer in this work.
The procedure of selecting the lag order and the rank of the structure of shortrun is carried out by minimizing the following modified information criteria (see
Hecq, 2006).
T
X
AIC (p, s) =
i=n−s+1
HQ(p, s) =
T
X
i=n−s+1
SC(p, s) =
T
X
ln(1 − λ2i (p)) +
i=n−s+1
2
×N
T
(7)
2 ln(ln T )
×N
T
(8)
ln(1 − λ2i (p)) +
ln(1 − λ2i (p)) +
ln T
×N
T
(9)
N = [n × (n × (p − 1)) + n × r] − [s × (n × (p − 1) + (n − s))]
The number of parameters N is obtained by subtracting the total number of mean
parameters in the VECM (i.e., n2 × (p − 1) + nr), for given r and p, from the number
of restrictions the common dynamics imposes from s×(n×(p−1))−s×(n−s). The
eigenvalues λi are calculated for each p. To calculate the pair (p, s) we assume that
no restriction of cointegration exists, that is, r = n (see Hecq, 2006). We fix p in
model (3) and then find λi i = 1, 2...n using the program cancorr(∆yt , Xt−1 | yt−1 ).
This procedure is followed for every p and in the end we choose the p and s that
10
minimizes the IC(p, s). After selecting the pair (p, s) we can test the cointegration
relation using the procedure of Johansen. Finally we estimate the model using the
switching algorithms as shown in the next chapter. Notice that in this simultaneous
selection, testing the cointegration relation is the last procedure to follow, so we are
inverting the hierarquical procedure followed by Vahid and Engle (1993) where the
first step is the selection of the number of cointegration relations. It may be an
advantage specially when r is over-estimated. Few works have been dedicated to
analyze the order of the VAR models considering modified IC(p, s). As mentioned,
Vahid and Issler (2002) suggested the use of IC(p, s) to simultaneously choose the
order p and a number of reduced rank structure s on covariance stationary VAR
model subject to SCCF restrictions. However, no work has analyzed the order of
the VAR model with cointegration and WF restrictions using a modified criterion,
which is exactly the contribution of this paper.
To estimate the VAR model considering cointegration and WF restrictions we
use the switching algorithms model as considered by Hecq (2006). Consider the
VECM given by:
∆yt = αβ 0 yt−1 + β̃ ⊥ ΨXt−1 + εt
(10)
A full description of switching algorithms is presented below in four steps:
Step1 : Estimation of the cointegration vectors β.
Using the optimal pair (p̄, s̄) chosen by information criteria (7), (8) or
(9), we estimate β (and so its rank, r = r̄) using Johansen cointegration
test.
Step2 : Estimation of β̃ ⊥ and Ψ.
Taking β̂ estimated in step one, we proceed to estimate β̃ ⊥ and Ψ. Hence,
0
we run a regression of ∆yt and of Xt−1 on β̂ yt−1 . We labeled the residuals as u0 and u1 , respectively. Therefore, we obtain a reduced rank
regression:
u0 = β̃ ⊥ Ψu1 + εt
11
(11)
¡
¢
where Ψ can be written as Ψ = C1 , ..., C(p̄−1) of (n − s̄) × n(p̄ − 1) and
β̃ ⊥ of n × (n − s̄). We estimate (11) by FIML. Thus, we can obtain β̃ ⊥
and Ψ̂.
Step3 : Estimate of the Maximum Likelihood (ML) function.
Given the parameters estimated in steps 1 and 2 we use a recursive
algorithm to estimate the Maximum Likelihood (ML) function. We cal2
culate the eigenvalues associated with Ψ̂, λ̂i i = 1, ..., s̄ and the matrix
P
of residuals max
r̄, s=s̄ . Hence, we compute the ML function:
#
" ¯ max ¯
s̄
¯ X ¯ X
´
³
2
T
¯
¯
(12)
L0max, r̄<n, s=s̄ = −
ln 1 − λ̂i
ln ¯
¯−
¯r̄<n, s=s̄¯
2
i=1
If r̄ = n, we use instead of (12) the derived log-likelihood: Lmax, r=n, s=s̄ =
¯
¯P
¯
¯
− T2 ln ¯ max
r̄=n, s=s̄ ¯. The determinant of the covariance matrix for r̄ = n
cointegration vector is calculated by
¯ max ¯
s̄
¯ X ¯
´
³
¯
¯ X
2
¯
¯
−1
¯
¯
ln ¯
ln 1 − λ̂i
¯ = ln m00 − m01 m11 m10 −
¯r̄=n, s=s̄¯
(13)
i=1
where mij refers to cross moment matrices obtained in multivariate least
square regressions from ∆yt and Xt−1 on yt−1 . In this case, estimation
does not imply an iterative algorithm yet because the cointegrating space
spans Rn .
Step4 : Reestimation of β.
We reestimate β to obtain a more appropriated value for the parameters.
h
i
In order to reestimate β we use the program CanCorr ∆yt , yt−1 | Ψ̂Xt−1
and thus using the new β̂ we can repeat step 2 to reestimate β̃ ⊥ and Ψ.
Then, we can calculate the new value of the ML function in the step 3.
¡
Henceforth, we obtain L1max, r=r̄, s=s̄ for calculating ∆L = L1max, r=r̄, s=s̄ - L0max,
We repeat steps 1 to 4 to choose β̃ ⊥ and Ψ until convergence is reached ( i.e.,
∆L < 10−7 ). In the end, optimal parameters p̄, r̄ and s̄ are obtained and it can be
used for estimation and forecasting of a VECM with WF restrictions.
12
r=r̄, s=s̄
¢
.
4
Monte-Carlo Design
One of the critical issues regarding Monte-Carlo experiments is the data generating
processes. To build the data generating processes we consider a VAR model with
three variables, one cointegration vector, and two cofeatures vectors (i.e., n = 3,
r = 1 and s = 2, respectively). β and β̃ satisfy:



1.0
β =  0.2  , β̃ = 
−1.0



 
ε1t
0
1.0
 ε2t  ∼ N  0  ,  0.6
0
0.6
ε3t

1.0 0.1
0.0 1.0 
0.5 −0.5
0.6 0.6
1.0 0.6
0.6 1.0


Consider the VAR(3) model: yt = A1 yt−1 + A2 yt−2 + A3 yt−3 + εt . The VECM
respresentation as a function of the VAR level parameters can be written as:
∆yt = (A1 + A2 + A3 − I3 )yt−1 − (A2 + A3 )∆yt−1 − A3 ∆yt−2 + εt
(14)
The VAR coefficients must simultaneously obey the restrictions: a) The cointegra0
tion restrictions: αβ 0 = (A1 + A2 + A3 − I3 ) ; b) WF restrictions: β̃ A3 = 0 (iii)
0
β̃ (A2 + A3 ) = 0 and c) covariance-stationary condition. Considering the cointegration restrictions we can rewrite (14) as the following VAR(1):
ξ t = F ξ t−1 + vt
(15)





−(A2 + A3 )
4yt
εt
−A3
α





I3
4yt−1 , F =
0
0
0 
and vt =
ξt =
0
0
0
−β(A2 + A3 ) −β A3 β α + 1
β 0 εt
β yt

Thus, the equation (15) will be covariance-stationary if all eigenvalues of matrix
F lie inside the unit circle. An initial idea to design the Monte-Carlo experiment may
consist of constructing the companion matrix (F ) and verify whether the eigenvalues
of the companion matrix all lie inside the unit circle. This may be carried out by
selecting their values from a uniform distribution, and then verifying whether or
not the eigenvalues of the companion matrix all lie inside the unit circle. However,
this strategy could lead to a wide spectrum of search for adequate values for the
13
companion matrix. Hence, we follow an alternative procedure. We propose an
analytical solution to generate a covariance-stationary VAR, based on the choice
of the eigenvalues, and then on the generation of the respective companion matrix.
In the appendix we present a detailed discussion of the final choice of these free
parameters, including analytical solutions. In our simulation, we constructed 100
data generating processes and for each of these we generate 1000 samples containing
1000 observations. In order to reduce the impact of initial values, we consider
only the last 100 and 200 observations. All the experiments were conducted in the
MatLab environment.
5
Results
Values in Table I represent the percentage of time that the model selection criterion, IC(p), chooses that cell corresponding to the lag and number of cointegration
vectors in 100000 realizations. The true lag-cointegrating vectors are identified by
bold numbers and the selected lag-cointegration vectors chosen more times by the
criterion are underlined. The results show that, in general, the AIC criterion choose
more frequently the correct lag length for 100 and 200 observations. For example,
for 100 observations, the AIC, HQ and SC criteria chose the true lag, p, 54.08%,
35.62% and 17.49% of the times respectively. Note that all three criteria chose
more frequently the correct rank of cointegration (r = 1). When 200 observations
are considered, the correct lag length was chosen 74.72%, 57.75% and 35.28% of
the time for AIC, HQ and SC respectively. Again all three criteria selected the
true cointegrated rank r = 1. Tables II contains the percentage of time that the
simultaneous model selection criterion, IC(p, s), chooses that cell, corresponding to
the lag-rank and number of cointegrating vectors in 100,000 realizations. The true
lag-rank-cointegration vectors are identified by bold numbers and the best lag-rank
combination chosen more times by each criterion are underlined. The results show
that, in general, the AIC criterion chooses more frequently lag-rank for 100 and
200 observations. For instance, for 100 observations, the AIC, HQ and SC criteria
14
choose more frequently the true pair (p, s) = (3, 1), 56.34%, 40.85% and 25.20% of
the times respectively. For 200 observations, AIC, HQ and SC criteria choose more
frequently the true pair (p, s) = (3, 1), 77.07%, 62.58% and 45.03% of the times
respectively. Note that all three criteria choose more frequently the correct rank of
cointegration (r = 1) in both samples.
The most relevant results can be summarized as follows:
− All criteria (AIC, HQ and SC) choose the correct parameters more
often when using IC(p, s).
− The AIC criterion has better performance in selecting the true model
more frequently for both the IC(p, s) and the IC(p) criteria.
− When the size of the sample decreases the true value p is less frequently
selected by all the traditional criteria.
− Table I shows that ignoring WF restrictions the standard SC has the
worst performance in choosing the true value of p.
It is known that literature suggests the use of the traditional SC and HQ criteria
in VAR model selection. The results of this work indicate that if additional WF
restrictions are ignored, the standard SC and HQ criteria select few times the true
value of p. That is, there is a cost of ignoring additional WF restrictions in the model
specially when SC criterion is used. In general, the standard Schwarz or HannanQuinn selection criteria should not be used for this purpose in small samples due
to the tendency of identifying an underparameterized model. In general, the use of
these alternative criteria of selection, IC(p, s) has better performance than the usual
criteria, IC(p), when the cointegrated VAR model has additional WF restriction.
6
Conclusions
In this work, we considered an additional weak form restriction of common cyclical
features in a cointegrated VAR model in order to analyze the appropriate way for
15
selecting the correct lag order. These additional WF restrictions are defined in
the same way as cointegration restrictions, while cointegration refers to relations
among variables in the long-run, the common cyclical restrictions refer to relations
in the short-run. Two methodologies have been used for selecting lag length; the
traditional information criterion, IC(p), and an alternative criterion (IC(p, s)) that
selects simultaneously the lag order p and the rank structure s due to the WF
restriction.
The results indicate that information criterion that selects the lag length and
the rank order simultaneously has better performance than the model chosen by
conventional criteria. When the WF restrictions are ignored there is a non trivial
cost in selecting the true model with standard information criteria. In general, the
standard Schwarz or Hannan-Quinn criteria selection criteria should not be used
for this purpose in small samples due to the tendency of identifying an underparameterized model.
In applied work, when the VAR model contains WF and cointegration restrictions, we suggest the use of AIC(p, s) criteria for simultaneously choosing the lagrank, since it provides considerable gains in selecting the correct VAR model. Since
no work in the literature has been dedicated to analyze a VAR model with WF common cyclical restrictions, the results of this work provide new insights and incentives
to proceed with this kind of empirical work.
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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
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Journal of. multivariate Analysis 5: 248-264.
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[21] Mamingi N, Iyare SO. 2003. Convergence and Common Features in International Output: A Case Study of the Economic Community of West African
States, 1975-1997. Asian-African-Journal-of-Economics-and-Econometrics. 3:
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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 = A1 yt−1 + A2 yt−2 + A3 yt−3 + εt
(16)


 1

a11 a112 a112
a211 a212 a212
with parameters: A1 =  a121 a122 a122 , A2 =  a221 a222 a222  and A3 =
a131 a132 a132
a231 a232 a232 

 3
β 11
a11 a312 a312
 a321 a322 a322  We consider the cointegration vectors β =  β 21 , the cofeaa331 a332 a332
β 31




β̃ 11 β̃ 12
α11
tures vectors β̃ =  β̃ 21 β̃ 22  and the adjustament matrix α =  α21  . The
α31
β̃ 31 β̃ 32
0
long-run relation is defined by αβ = (A1 +A2 +A3 −I3 ). The VECM respresentation
is:
∆yt = αβ 0 yt−1 − (A2 + A3 )∆yt−1 − A3 ∆yt−2 + εt
(17)
Considering the cointegration restrictions we can rewrite (17) as the following
VAR(1)



ξ t = F ξ t−1 + vt
−(A2 + A3 )
4yt
−A3
I3
0
where ξ t =  4yt−1  , F = 
0
−β(A2 + A3 ) −β 0 A3
β yt
1) Short-run restrictions (WF)
(18)

εt
α
 and vt =  0 
0
0
β 0 εt
βα+1


Let us, G = −[R21 K + R31 ], K = [(R32 − R31 )/(R21 − R22 )], Rj1 = β̃ j1 /β̃ 11 ,
Rj2 = β̃ j2 /β̃ 12 (j = 2, 3) and S = β 11 G + β 21 K + β 31

−Ga331 −Ga332 −Ga333
0
(i) β̃ A3 = 0 ==> A3 =  −Ka331 −Ka332 −Ka333 
−a331
−a332
−a333

−Ga231 −Ga232 −Ga233
0
0
(ii) β̃ (A2 + A3 ) = 0 ==> β̃ A2 = 0 ==> A2 =  −Ka231 −Ka232 −Ka233 
−a231
−a232
−a233
2) Long-run restrictions (cointegration)
22
(iv) β 0 (A2 + A3 ) = [−(a231 + a331 )S
β 0 A3 = [−a331 S − a332 S − a333 S]
(v) β 0 α + 1 = β =
£
β 11 β 21 β 31
¤
− (a232 + a332 )S
− (a233 + a333 )S] and

α11
 α21  + 1 = β 11 α11 + β 21 α21 + β 31 α31 + 1
α31

Therefore, considering short- and long-run restrictions, the companion matrix F
is represented as:

−A3
α
−(A2 + A3 )
=
I3
0
0
F =
0
0
−β(A2 + A3 ) −β A3 β α + 1











−G(a231 + a331 )
−K(a231 + a331 )
−(a231 + a331 )
1
0
0
2
−(a31 + a331 )S
−G(a232 + a332 ) −G(a233 + a333 ) −Ga331 −Ga332 −Ga333 α11
−G(a232 + a332 ) −G(a233 + a333 ) −Ka331 −Ka332 −Ka333 α21
−G(a232 + a332 ) −(a233 + a333 )
−a331
−a332
−a333 α31
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
2
3
2
3
3
3
3
−(a32 + a32 )S −(a33 + a33 )S −a31 S −a32 S −a33 S b










with b = β 0 α + 1 = β 11 α11 + β 21 α21 + β 31 α31 + 1
3) Restrictions of covariance-stationary in equation (18)
The equation (18) will be covariance-stationary, all eigenvalues of matrix F lie
inside the unit circle. Therefore, the eigenvalues of the matrix F is a number λ such
that:
|F − λI7 | = 0
(19)
λ7 + Ωλ6 + Θλ5 + Ψλ4 = 0
(20)
The solution of (19) is:
where the parameters Ω, Θ, and Ψ are: Ω = G(a231 +a331 )+K(a232 +a332 )+a233 +a333 −b,
Θ = Ga331 + Ka332 − (a233 + a333 )b − Gb(a231 + a331 ) − Kb(a232 + a332 ) + α31 S(a233 + a333 ) +
23
Sα21 (a232 + a332 ) + Sα11 (a231 + a331 ) + a333 and Ψ = −a333 b − Ga331 b − Ka332 b + α31 a333 S +
a332 Sα21 + a331 Sα11 , and the first four roots are λ1 = λ2 = λ3 = λ4 = 0.We calculated
the parameters of matrices A1 , A2 and A3 as function of roots (λ5 , λ6 and λ7 ) and
free parameters. Hence we have three roots satisfying equation (20)
λ3 + Ωλ2 + Θλ + Ψ = 0
for λ5 , we have: λ35 + Ωλ25 + Θλ5 + Ψ = 0
..................................Eq1
for λ6 , we have: λ36 + Ωλ26 + Θλ6 + Ψ = 0
..................................Eq2
for λ7 , we have: λ37 + Ωλ27 + Θλ7 + Ψ = 0
..................................Eq3
(21)
Solving Eq1, Eq2 and Eq3 we have: Ω = −λ7 − λ6 − λ5 , Θ = λ6 λ7 + λ6 λ5 + λ5 λ7
and Ψ = −λ5 λ6 λ7 . Equaling these parameters with relations above we have:
a231 = −(−Ka232 − Ka232 b + α31 Sa233 − λ6 λ7 − λ6 − λ7 − a233 b − λ5 λ6 λ7 + b − λ5 λ7 −
λ5 λ6 − a233 + Sa232 α21 − λ5 )/(Sα11 − G − Gb)
a332 = (−S 2 λ7 α11 α31 −b2 λ7 G−λ6 Gb2 +bλ7 Sα11 +λ6 Sα11 b−a331 Sα11 G+a331 S 2 α211 −
Ga331 bSα11 −λ5 Gb2 +λ5 Sα11 b−λ7 λ6 α31 SG−λ7 λ5 α31 SG−S 2 α11 λ5 α31 −S 2 α11 λ6 α31 +
Sλ5 Gbα31 +Sα31 λ6 Gb−λ5 λ7 λ6 G+λ6 λ7 Gb+λ5 λ7 Gb+λ5 λ6 Gb−SGb2 α31 +S 2 α11 bα31 −
S 2 α11 α31 a233 + S 2 α231 a233 G + SG2 a331 α31 + Sα11 a233 b + Gb3 − Sα11 b2 − S 2 α11 Ka232 α31 −
S 2 α11 α31 Ga331 +S 2 a232 α21 Gα31 −Sa232 α21 Gb+Sα31 G2 a331 b−Sα31 a233 Gb+Sα11 Ka232 b+
Sλ7 Gbα31 −λ5 λ6 α31 SG−λ5 λ7 λ6 α31 SG+λ5 λ7 λ6 Sα11 )/(Sα11 Kα31 −KGα31 +bGα21 −
Kα31 Gb − Sα11 α21 + Gα21 )/S
a333 = −(Kb3 G − λ5 Gb2 K + Sα11 λ6 Kλ7 λ5 + Kbλ7 Sα11 − Kb2 λ7 G − S 2 α21 λ7 α11 +
λ6 GbSα21 +Sα21 λ7 Gb−λ6 Gb2 K +λ6 Sα11 Kb−λ6 S 2 α11 α21 +λ5 GbSα21 +λ5 Sα11 Kb−
λ5 S 2 α11 α21 −λ7 λ6 Sα21 G+Kbλ7 λ6 G+Kbλ7 λ5 G+Kbλ5 λ6 G−λ7 λ6 KGλ5 −S 2 α11 α21 Ka232 +
S 2 α11 α21 b−S 2 α11 α21 a233 +S 2 α221 a232 G−Sα11 Kb2 +Sα21 G2 a331 −Sα21 Gb2 +S 2 a331 Kα211 −
S 2 α11 α21 Ga331 +S 2 α21 a233 Gα31 +Sα11 Kˆ2ba232 +Sα11 Kba233 −Sα11 a331 KG−Sα11 KbGa331 −
SKba233 Gα31 +Sα21 G2 a331 b−Sα21 λ5 λ6 G−Sα21 λ5 λ7 λ6 G−Sα21 Ka232 Gb−Sα21 λ7 λ5 G)/(Sα11 Kα31 −
KGα31 + bGα21 − Kα31 Gb − Sα11 α21 + Gα21 )/S
24
We can calculate a231 , a332 and a333 fixing the set λ1 = λ2 = λ3 = λ4 = 0 and sort
independently from uniform distributions (−0.9; 0.9) the values of a331 , a232 , a233 , λ5 ,
λ6 and λ7 . Hencefore, each parameter of the matrices A1 , A2 and A3 are defined
and so we can generate the DGPs of VAR(3) model with cointegration and WF
restrictions.
25
Banco Central do Brasil
Trabalhos para Discussão
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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
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Política Monetária e Supervisão do Sistema Financeiro Nacional no
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Eduardo Lundberg
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Monetary Policy and Banking Supervision Functions on the Central
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Private Sector Participation: a Theoretical Justification of the Brazilian
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Sérgio Ribeiro da Costa Werlang
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Leading Indicators of Inflation for Brazil
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The Correlation Matrix of the Brazilian Central Bank’s Standard Model
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26
13
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14
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Francisco Marcos Rodrigues Figueiredo
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15
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Tito Nícias Teixeira da Silva Filho
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A Simple Model for Inflation Targeting in Brazil
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Uncovered Interest Parity with Fundamentals: a Brazilian Exchange
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Credit Channel without the LM Curve
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Inflation Targeting in Brazil: Shocks, Backward-Looking Prices, and
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Inflation Targeting in an Open Financially Integrated Emerging
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27
28
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Crises Cambiais e Ataques Especulativos no Brasil
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Monetary Policy in Brazil: Remarks on the Inflation Targeting Regime,
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Volatilidade Implícita e Antecipação de Eventos de Stress: um Teste para
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The Determinants of Bank Interest Spread in Brazil
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Medium-Size Macroeconomic Model for the Brazilian Economy
Marcelo Kfoury Muinhos and Sergio Afonso Lago Alves
Feb/2003
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On the Information Content of Oil Future Prices
Benjamin Miranda Tabak
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A Taxa de Juros de Equilíbrio: uma Abordagem Múltipla
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Fev/2003
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Avaliação de Métodos de Cálculo de Exigência de Capital para Risco de
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Real Balances in the Utility Function: Evidence for Brazil
Leonardo Soriano de Alencar and Márcio I. Nakane
Feb/2003
69
r-filters: a Hodrick-Prescott Filter Generalization
Fabio Araújo, Marta Baltar Moreira Areosa and José Alvaro Rodrigues Neto
Feb/2003
70
Monetary Policy Surprises and the Brazilian Term Structure of Interest
Rates
Benjamin Miranda Tabak
Feb/2003
71
On Shadow-Prices of Banks in Real-Time Gross Settlement Systems
Rodrigo Penaloza
Apr/2003
72
O Prêmio pela Maturidade na Estrutura a Termo das Taxas de Juros
Brasileiras
Ricardo Dias de Oliveira Brito, Angelo J. Mont'Alverne Duarte e Osmani
Teixeira de C. Guillen
Maio/2003
73
Análise de Componentes Principais de Dados Funcionais – Uma
Aplicação às Estruturas a Termo de Taxas de Juros
Getúlio Borges da Silveira e Octavio Bessada
Maio/2003
74
Aplicação do Modelo de Black, Derman & Toy à Precificação de Opções
Sobre Títulos de Renda Fixa
Octavio Manuel Bessada Lion, Carlos Alberto Nunes Cosenza e César das
Neves
Maio/2003
75
Brazil’s Financial System: Resilience to Shocks, no Currency
Substitution, but Struggling to Promote Growth
Ilan Goldfajn, Katherine Hennings and Helio Mori
30
Jun/2003
76
Inflation Targeting in Emerging Market Economies
Arminio Fraga, Ilan Goldfajn and André Minella
Jun/2003
77
Inflation Targeting in Brazil: Constructing Credibility under Exchange
Rate Volatility
André Minella, Paulo Springer de Freitas, Ilan Goldfajn and Marcelo Kfoury
Muinhos
Jul/2003
78
Contornando os Pressupostos de Black & Scholes: Aplicação do Modelo
de Precificação de Opções de Duan no Mercado Brasileiro
Gustavo Silva Araújo, Claudio Henrique da Silveira Barbedo, Antonio
Carlos Figueiredo, Eduardo Facó Lemgruber
Out/2003
79
Inclusão do Decaimento Temporal na Metodologia
Delta-Gama para o Cálculo do VaR de Carteiras
Compradas em Opções no Brasil
Claudio Henrique da Silveira Barbedo, Gustavo Silva Araújo,
Eduardo Facó Lemgruber
Out/2003
80
Diferenças e Semelhanças entre Países da América Latina:
uma Análise de Markov Switching para os Ciclos Econômicos
de Brasil e Argentina
Arnildo da Silva Correa
Out/2003
81
Bank Competition, Agency Costs and the Performance of the
Monetary Policy
Leonardo Soriano de Alencar and Márcio I. Nakane
Jan/2004
82
Carteiras de Opções: Avaliação de Metodologias de Exigência de Capital
no Mercado Brasileiro
Cláudio Henrique da Silveira Barbedo e Gustavo Silva Araújo
Mar/2004
83
Does Inflation Targeting Reduce Inflation? An Analysis for the OECD
Industrial Countries
Thomas Y. Wu
May/2004
84
Speculative Attacks on Debts and Optimum Currency Area: a Welfare
Analysis
Aloisio Araujo and Marcia Leon
May/2004
85
Risk Premia for Emerging Markets Bonds: Evidence from Brazilian
Government Debt, 1996-2002
André Soares Loureiro and Fernando de Holanda Barbosa
May/2004
86
Identificação do Fator Estocástico de Descontos e Algumas Implicações
sobre Testes de Modelos de Consumo
Fabio Araujo e João Victor Issler
Maio/2004
87
Mercado de Crédito: uma Análise Econométrica dos Volumes de Crédito
Total e Habitacional no Brasil
Ana Carla Abrão Costa
Dez/2004
88
Ciclos Internacionais de Negócios: uma Análise de Mudança de Regime
Markoviano para Brasil, Argentina e Estados Unidos
Arnildo da Silva Correa e Ronald Otto Hillbrecht
Dez/2004
89
O Mercado de Hedge Cambial no Brasil: Reação das Instituições
Financeiras a Intervenções do Banco Central
Fernando N. de Oliveira
Dez/2004
31
90
Bank Privatization and Productivity: Evidence for Brazil
Márcio I. Nakane and Daniela B. Weintraub
Dec/2004
91
Credit Risk Measurement and the Regulation of Bank Capital and
Provision Requirements in Brazil – A Corporate Analysis
Ricardo Schechtman, Valéria Salomão Garcia, Sergio Mikio Koyama and
Guilherme Cronemberger Parente
Dec/2004
92
Steady-State Analysis of an Open Economy General Equilibrium Model
for Brazil
Mirta Noemi Sataka Bugarin, Roberto de Goes Ellery Jr., Victor Gomes
Silva, Marcelo Kfoury Muinhos
Apr/2005
93
Avaliação de Modelos de Cálculo de Exigência de Capital para Risco
Cambial
Claudio H. da S. Barbedo, Gustavo S. Araújo, João Maurício S. Moreira e
Ricardo S. Maia Clemente
Abr/2005
94
Simulação Histórica Filtrada: Incorporação da Volatilidade ao Modelo
Histórico de Cálculo de Risco para Ativos Não-Lineares
Claudio Henrique da Silveira Barbedo, Gustavo Silva Araújo e Eduardo
Facó Lemgruber
Abr/2005
95
Comment on Market Discipline and Monetary Policy by Carl Walsh
Maurício S. Bugarin and Fábia A. de Carvalho
Apr/2005
96
O que É Estratégia: uma Abordagem Multiparadigmática para a
Disciplina
Anthero de Moraes Meirelles
Ago/2005
97
Finance and the Business Cycle: a Kalman Filter Approach with Markov
Switching
Ryan A. Compton and Jose Ricardo da Costa e Silva
Aug/2005
98
Capital Flows Cycle: Stylized Facts and Empirical Evidences for
Emerging Market Economies
Helio Mori e Marcelo Kfoury Muinhos
Aug/2005
99
Adequação das Medidas de Valor em Risco na Formulação da Exigência
de Capital para Estratégias de Opções no Mercado Brasileiro
Gustavo Silva Araújo, Claudio Henrique da Silveira Barbedo,e Eduardo
Facó Lemgruber
Set/2005
100 Targets and Inflation Dynamics
Sergio A. L. Alves and Waldyr D. Areosa
Oct/2005
101 Comparing Equilibrium Real Interest Rates: Different Approaches to
Measure Brazilian Rates
Marcelo Kfoury Muinhos and Márcio I. Nakane
Mar/2006
102 Judicial Risk and Credit Market Performance: Micro Evidence from
Brazilian Payroll Loans
Ana Carla A. Costa and João M. P. de Mello
Apr/2006
103 The Effect of Adverse Supply Shocks on Monetary Policy and Output
Maria da Glória D. S. Araújo, Mirta Bugarin, Marcelo Kfoury Muinhos and
Jose Ricardo C. Silva
Apr/2006
32
104 Extração de Informação de Opções Cambiais no Brasil
Eui Jung Chang e Benjamin Miranda Tabak
Abr/2006
105 Representing Roommate’s Preferences with Symmetric Utilities
José Alvaro Rodrigues Neto
Apr/2006
106 Testing Nonlinearities Between Brazilian Exchange Rates and Inflation
Volatilities
Cristiane R. Albuquerque and Marcelo Portugal
May/2006
107 Demand for Bank Services and Market Power in Brazilian Banking
Márcio I. Nakane, Leonardo S. Alencar and Fabio Kanczuk
Jun/2006
108 O Efeito da Consignação em Folha nas Taxas de Juros dos Empréstimos
Pessoais
Eduardo A. S. Rodrigues, Victorio Chu, Leonardo S. Alencar e Tony Takeda
Jun/2006
109 The Recent Brazilian Disinflation Process and Costs
Alexandre A. Tombini and Sergio A. Lago Alves
Jun/2006
110 Fatores de Risco e o Spread Bancário no Brasil
Fernando G. Bignotto e Eduardo Augusto de Souza Rodrigues
Jul/2006
111 Avaliação de Modelos de Exigência de Capital para Risco de Mercado do
Cupom Cambial
Alan Cosme Rodrigues da Silva, João Maurício de Souza Moreira e Myrian
Beatriz Eiras das Neves
Jul/2006
112 Interdependence and Contagion: an Analysis of Information
Transmission in Latin America's Stock Markets
Angelo Marsiglia Fasolo
Jul/2006
113 Investigação da Memória de Longo Prazo da Taxa de Câmbio no Brasil
Sergio Rubens Stancato de Souza, Benjamin Miranda Tabak e Daniel O.
Cajueiro
Ago/2006
114 The Inequality Channel of Monetary Transmission
Marta Areosa and Waldyr Areosa
Aug/2006
115 Myopic Loss Aversion and House-Money Effect Overseas: an
Experimental Approach
José L. B. Fernandes, Juan Ignacio Peña and Benjamin M. Tabak
Sep/2006
116 Out-Of-The-Money Monte Carlo Simulation Option Pricing: the Join
Use of Importance Sampling and Descriptive Sampling
Jaqueline Terra Moura Marins, Eduardo Saliby and Joséte Florencio dos
Santos
Sep/2006
117 An Analysis of Off-Site Supervision of Banks’ Profitability, Risk and
Capital Adequacy: a Portfolio Simulation Approach Applied to Brazilian
Banks
Theodore M. Barnhill, Marcos R. Souto and Benjamin M. Tabak
Sep/2006
118 Contagion, Bankruptcy and Social Welfare Analysis in a Financial
Economy with Risk Regulation Constraint
Aloísio P. Araújo and José Valentim M. Vicente
Oct/2006
33
119 A Central de Risco de Crédito no Brasil: uma Análise de Utilidade de
Informação
Ricardo Schechtman
Out/2006
120 Forecasting Interest Rates: an Application for Brazil
Eduardo J. A. Lima, Felipe Luduvice and Benjamin M. Tabak
Oct/2006
121 The Role of Consumer’s Risk Aversion on Price Rigidity
Sergio A. Lago Alves and Mirta N. S. Bugarin
Nov/2006
122 Nonlinear Mechanisms of the Exchange Rate Pass-Through: a Phillips
Curve Model With Threshold for Brazil
Arnildo da Silva Correa and André Minella
Nov/2006
123 A Neoclassical Analysis of the Brazilian “Lost-Decades”
Flávia Mourão Graminho
Nov/2006
124 The Dynamic Relations between Stock Prices and Exchange Rates:
Evidence for Brazil
Benjamin M. Tabak
Nov/2006
125 Herding Behavior by Equity Foreign Investors on Emerging Markets
Barbara Alemanni and José Renato Haas Ornelas
Dec/2006
126 Risk Premium: Insights over the Threshold
José L. B. Fernandes, Augusto Hasman and Juan Ignacio Peña
Dec/2006
127 Uma Investigação Baseada em Reamostragem sobre Requerimentos de
Capital para Risco de Crédito no Brasil
Ricardo Schechtman
Dec/2006
128 Term Structure Movements Implicit in Option Prices
Caio Ibsen R. Almeida and José Valentim M. Vicente
Dec/2006
129 Brazil: Taming Inflation Expectations
Afonso S. Bevilaqua, Mário Mesquita and André Minella
Jan/2007
130 The Role of Banks in the Brazilian Interbank Market: Does Bank Type
Matter?
Daniel O. Cajueiro and Benjamin M. Tabak
Jan/2007
131 Long-Range Dependence in Exchange Rates: the Case of the European
Monetary System
Sergio Rubens Stancato de Souza, Benjamin M. Tabak and Daniel O.
Cajueiro
Mar/2007
132 Credit Risk Monte Carlo Simulation Using Simplified Creditmetrics’
Model: the Joint Use of Importance Sampling and Descriptive Sampling
Jaqueline Terra Moura Marins and Eduardo Saliby
Mar/2007
133 A New Proposal for Collection and Generation of Information on
Financial Institutions’ Risk: the Case of Derivatives
Gilneu F. A. Vivan and Benjamin M. Tabak
Mar/2007
134 Amostragem Descritiva no Apreçamento de Opções Européias através
de Simulação Monte Carlo: o Efeito da Dimensionalidade e da
Probabilidade de Exercício no Ganho de Precisão
Eduardo Saliby, Sergio Luiz Medeiros Proença de Gouvêa e Jaqueline Terra
Moura Marins
Abr/2007
34
135 Evaluation of Default Risk for the Brazilian Banking Sector
Marcelo Y. Takami and Benjamin M. Tabak
May/2007
136 Identifying Volatility Risk Premium from Fixed Income Asian Options
Caio Ibsen R. Almeida and José Valentim M. Vicente
May/2007
137 Monetary Policy Design under Competing Models of Inflation
Persistence
Solange Gouvea e Abhijit Sen Gupta
May/2007
138 Forecasting Exchange Rate Density Using Parametric Models:
the Case of Brazil
Marcos M. Abe, Eui J. Chang and Benjamin M. Tabak
May/2007
35