ISSN 1518-3548
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
Working Paper Series
ISSN 1518-3548 CGC 00.038.166/0001-05
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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 autoregres-sive (VAR) model is the choice of the lag order, since all inference in the VAR model depends on the correct model specification. Literature has shown im-portant studies of how to select the lag order of a nonstationary VAR model subject to cointegration restrictions. In this work, we consider an additional weak form (WF) restriction of common cyclical features in the model in order to analyze the appropriate way to select the correct lag order. Two method-ologies have been used: the traditional information criteria (AIC, HQ and SC) and an alternative criterion (IC(p, s)) which select simultaneously the lag order p and the rank structure s due to the WF restriction. A Monte-Carlo 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 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
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-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. This
alternative procedure of model selection criterion (see, Vahid and Issler, 2002; Hecq et al., 2006) consisting in selecting simultaneously the lag order p and the rank s do to the weak form of common cyclical feature, which is referred to as IC(p, s)2. The
most relevant results can be summarized as follows. The information criterion that selects simultaneously the pair (p, s) has better performance than the model chosen by conventional criteria. The cost of ignoring additional WF restrictions in vector autoregressive modelling can be high specially when SC criterion is used.
The remaining of this work is organized as follows. Section 2 shows the econo-metric model. In section 3 the information criteria are mentioned. Monte Carlo simulation is shown in section 4 and the results in section 5. Finally, the conclusions are shown in section 6.
2
The Econometric Model
We show the VAR model with short-run and long-run restrictions. First, we consider a Gaussian vector autoregression of finite order p, so-called VAR(p), such that:
yt= p
X
i=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−Ppi=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 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
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−1 X i=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− αβ´yt−1) = ˜β
0
εt , where ˜β 0
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 ˜β that satisfies the following assumption:
Assumption 1 : ˜β0 Γj = 0s×n for j = 1, ...., p− 1.
Imposing WF restrictions is convenient because it allows the study of both coin-tegration and common cyclical feature without the constraint r + s ≤ n. We can rewrite the VECM with WF restrictions as a model of reduced-rank structure. In (2) let Xt−1 = [∆y0
t−1, ...∆y0t−p+1]0 and Φ = [Γ1, ...., Γp−1], therefore we get:
∆yt= αβ´yt−1+ ΦXt−1+ εt (3)
If assumption (1) holds matrices Γi, i = 1, ..., p are all of rank (n − s) then we can
write Φ = ˜β⊥Ψ = ˜β⊥[Ψ1, ...., Ψp−1], where, ˜β⊥is n×(n−s) full column rank matrix,
Ψ is of dimension (n − s) × n(p − 1), the matrices Ψi, i = 1, ..., p− 1 all of rank
That is, ˜β⊥ n× (n − s) is a full column rank orthogonal to the complement of ˜β with rank(˜β, ˜β⊥) = n. Rewriting model (3) we have:
∆yt = αβ´yt−1+ ˜β⊥(Ψ1, Ψ2, ..., Ψp−1) Xt−1+ εt (4)
= αβ´yt−1+ ˜β⊥ΨXt−1+ εt (5)
Estimation of (5) is carried out via the switching algorithms (see, Hecq, 2006) that use the procedure in estimating reduced-rank regression models suggested by Ander-son (1951). There is a formal connection between a reduced-rank regression and the canonical analysis as noted by Izenman (1975), Box and Tiao (1977), Tso (1980) and Veleu et al. (1986). When the multivariate regression has all of its matrix coefficients of full rank, it may be estimated by usual Least Square or Maximum-Likelihood procedures. But when the matrix coefficients are of reduced-rank they have to be estimated using the reduced-rank regression models of Anderson (1951). The use of canonical analysis may be regarded as a special case of reduced-rank regression. More specifically, the maximum-likelihood estimation of the parameters of the reduced-rank regression model may result in solving a problem of canonical 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 Wtthat allows us to obtain canonical correlation, represented by the
eigen-values ˆλ1 > ˆλ2 > ˆλ3... > ˆλn. The Johansen test statistic is based on canonical
correlation. In model (2) we can use the expression CanCorr{∆yt, yt−1|Wt} where
Wt= [∆yt−1, ∆yt−2, ..., ∆yt+p−1]that summarizes the reduced-rank regression
pro-cedure used in the Johansen approach. It means that one extracts the canonical correlations between ∆yt and yt−1: both sets concentrated out the effect of lags of
Wt. In order to test for the significance of the r largest eigenvalues, one can rely on
Johansen’s trace statistic (6):
ξr =−T n
X
i=r+1
Ln (1− ˆλ2i) i = 1, ..., n (6)
where the eigenvalues 0 < ˆλn < ... < ˆλ1 are the solution of : |λm11− m−110m00m01| =
0, where mij, i, j = 0.1, are the second moment matrices: m00 = T1 PTt=1u˜0tu˜00t,
m10 = T1 PT t=1u˜1tu˜00t, m01 = T1 PT t=1u˜0tu˜01t, m11 = T1 PT t=1u˜1tu˜01t of the residuals ˜u0t
and ˜u1tobtained in the multivariate least squares regressions ∆yt= (∆yt−1, ...∆yt−p+1)+
u0tand yt−1 = (∆yt−1, ...∆yt−p+1)+u1trespectively (see, Hecq et al., 2006; Johansen,
1995). The result of Johansen test is a superconsistent estimated β. Moreover, we could also use a canonical correlation approach to determine the rank of the common features space due to WF restrictions. It is a test for the existence of cofeatures in the form of linear combinations of the variables in the first differences, corrected for long-run effects which are white noise (i.e., ˜β0(∆yt− αβ´yt−1) = ˜β
0
εt where ˜β 0
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.
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) = T X i=n−s+1 ln(1− λ2i(p)) + 2 T × N (7) HQ(p, s) = T X i=n−s+1 ln(1− λ2i(p)) + 2 ln(ln T ) T × N (8) SC(p, s) = T X i=n−s+1 ln(1− λ2i(p)) + ln T T × N (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).
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 ˆβestimated in step one, we proceed to estimate ˜β⊥and Ψ. Hence, we run a regression of ∆yt and of Xt−1on ˆβ
0
yt−1. We labeled the resid-uals as u0 and u1, respectively. Therefore, we obtain a reduced rank
regression:
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 ˆΨ.
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 ˆΨ, ˆλ2i i = 1, ..., ¯s and the matrix
of residualsPmaxr, s=¯¯ s. Hence, we compute the ML function: L0max, ¯r<n, s=¯s=−T 2 " ln ¯ ¯ ¯ ¯ ¯ max X ¯ r<n, s=¯s ¯ ¯ ¯ ¯ ¯− ¯ s X i=1 ln³1− ˆλ2i´ # (12)
If ¯r = n, we use instead of (12) the derived log-likelihood: Lmax, r=n, s=¯s=
−T2 ln ¯ ¯ ¯Pmaxr=n, s=¯¯ s ¯ ¯
¯. The determinant of the covariance matrix for ¯r = n cointegration vector is calculated by
ln ¯ ¯ ¯ ¯ ¯ max X ¯ r=n, s=¯s ¯ ¯ ¯ ¯ ¯= ln ¯ ¯m00− m01m−111m10 ¯ ¯ − ¯ s X i=1 ln ³ 1− ˆλ2i ´ (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 program CanCorrh∆yt, yt−1 | ˆΨXt−1
i and thus using the new ˆβ 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 obtain L1max, r=¯r, s=¯sfor 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
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 = 1.00.0 0.11.0 0.5 −0.5 εε1t2t ε3t ∼ N 00 0 , 1.00.6 0.61.0 0.60.6 0.6 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 restriccointegra-tions: αβ0 = (A1 + A2 + A3 − I3) ; b) WF restrictions: ˜β
0
A3 = 0 (iii)
˜
β0(A2+ A3) = 0 and c) covariance-stationary condition. Considering the
cointegra-tion restriccointegra-tions we can rewrite (14) as the following VAR(1):
ξt= F ξt−1+ vt (15) ξt = 4y4yt−1t β0yt , F = −(A2I+ A3 3) −A03 α0 −β(A2+ A3) −β0A3 β0α + 1 and vt= ε0t 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
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
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 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
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.
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A
Tables
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 = a 1 11 a112 a112 a1 21 a122 a122 a1 31 a132 a132 , A2 = a 2 11 a212 a212 a2 21 a222 a222 a2 31 a232 a232 and A3 = a 3 11 a312 a312 a3 21 a322 a322 a3 31 a332 a332
We consider the cointegration vectors β = ββ1121 β31 , the cofea-tures vectors ˜β = ˜ β11 β˜12 ˜ β21 β˜22 ˜ β31 β˜32
and the adjustament matrix α = αα1121
α31
. The long-run relation is defined by αβ0 = (A1+A2+A3−I3).The VECM respresentation
is:
∆yt= αβ0yt−1− (A2+ A3)∆yt−1− A3∆yt−2+ εt (17)
Considering the cointegration restrictions we can rewrite (17) as the following VAR(1) ξt= F ξt−1+ vt (18) where ξt = 4y4yt−1t β0yt , F = −(A2I+ A3 3) −A03 α0 −β(A2+ A3) −β0A3 β0α + 1 and vt= ε0t β0εt
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 = −Ga 3 31 −Ga332 −Ga333 −Ka3 31 −Ka332 −Ka333 −a3 31 −a332 −a333 (ii) ˜β0(A2+ A3) = 0 ==> ˜β 0 A2 = 0 ==> A2 = −Ga 2 31 −Ga232 −Ga233 −Ka2 31 −Ka232 −Ka233 −a2 31 −a232 −a233 (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 ¤ αα1121 α31 +1 = β11α11+ β21α21+ β31α31+ 1
Therefore, considering short- and long-run restrictions, the companion matrix F is represented as: F = −(A2I+ A3 3) −A0 3 α0 −β(A2+ A3) −β0A3 β0α + 1 = −G(a2
31+ a331) −G(a232+ a332) −G(a233+ a333) −Ga331 −Ga332 −Ga333 α11
−K(a231+ a331) −G(a232+ a332) −G(a332 + a333) −Ka331 −Ka332 −Ka333 α21
−(a2
31+ a331) −G(a232+ a332) −(a233+ a333) −a331 −a332 −a333 α31
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
−(a2
31+ 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(a2
31+a331)+K(a232+a332)+a233+a333−b,
Sα21(a232+ a332) + Sα11(a231+ a331) + a333 and Ψ = −a333b− Ga331b− Ka332b + α31a333S +
a3
32Sα21+ 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+ Ωλ 2 5 + Θλ5+ Ψ = 0 ...Eq1 for λ6, we have: λ36+ Ωλ 2 6 + Θλ6+ Ψ = 0 ...Eq2 for λ7, we have: λ37+ Ωλ 2 7 + 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− λ 6 λ7− λ6− λ7− a233b− λ 5 λ6λ7+ b− λ5λ7− λ5λ6− a2 33+ Sa232α21− λ5)/(Sα11− G − Gb) a332= (−S2λ 7 α11α31−b2λ7G−λ6Gb2+bλ7Sα11+λ6Sα11b−a331Sα11G+a331S2α211− Ga3 31bSα11−λ5Gb2+λ5Sα11b−λ7λ6α31SG−λ7λ5α31SG−S2α11λ5α31−S2α11λ6α31+ Sλ5Gbα31+Sα31λ6Gb−λ5λ7λ6G+λ6λ7Gb+λ5λ7Gb+λ5λ6Gb−SGb2α31+S2α11bα31− S2α 11α31a233+ S2α312 a233G + SG2a331α31+ Sα11a233b + Gb3− Sα11b2− S2α11Ka232α31− S2α11α31Ga331+S2a232α21Gα31−Sa232α21Gb+Sα31G2a331b−Sα31a233Gb+Sα11Ka232b+ Sλ7Gbα31−λ5λ6α31SG−λ5λ7λ6α31SG+λ5λ7λ6Sα11)/(Sα11Kα31−KGα31+bGα21− Kα31Gb− Sα11α21+ Gα21)/S a3 33=−(Kb3G− λ 5Gb2K + Sα 11λ6Kλ7λ5+ Kbλ7Sα11− Kb2λ7G− S2α21λ7α11+ λ6GbSα21+Sα21λ7Gb−λ6Gb2K +λ6Sα11Kb−λ6S2α11α21+λ5GbSα21+λ5Sα11Kb− λ5S2α 11α21−λ7λ6Sα21G+Kbλ7λ6G+Kbλ7λ5G+Kbλ5λ6G−λ7λ6KGλ5−S2α11α21Ka232+ S2α11α21b−S2α11α21a233+S2α221a232G−Sα11Kb2+Sα21G2a331−Sα21Gb2+S2a331Kα211− S2α
11α21Ga331+S2α21a233Gα31+Sα11Kˆ2ba232+Sα11Kba233−Sα11a331KG−Sα11KbGa331−
SKba233Gα31+Sα21G2a313 b−Sα21λ5λ6G−Sα21λ5λ7λ6G−Sα21Ka232Gb−Sα21λ7λ5G)/(Sα11Kα31−
We can calculate a2
31, a332and a333 fixing the set λ1 = λ2 = λ3 = λ4 = 0 and sort
independently from uniform distributions (−0.9; 0.9) the values of a3
31, 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.
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
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
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
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
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
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
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
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
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
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
