Fabio Kanczuk

Paulo Picchetti

English

Research Papers in Economics

 
 
 
   
Título: An Application of Quah and Vahey's SVAR Methodology for Estimating Core 
Inflation in Brazil 
   
Autores:   Paulo Picchetti (pichetti@usp.br) 
      Fabio Kanczuk (kanczuk@usp.br) 
 
Universidade de São Paulo – Departamento de Economia 
Av. Luciano Gualberto, 908, Cidade Universitária 
São Paulo, SP, 05508-900, Brasil 
Fone: (011) 3818 6012 
Fax:  (011) 3818 6013 
 
Resumo:  A questão da mensuração do núcleo inflacionário e sua relevância para a 
realização de política econômica foi largamente discutida na literatura internacional, e na 
literatura aplicada ao Brasil. A despeito disso, não existe consenso acerca da melhor 
medida de núcleo inflacionário. Várias metodologias diferentes já foram empregadas para 
dados brasileiros, com resultados distintos. Aqui, nós e mpregamos a metodologia 
proposta por Quah e Vahey (1995) para dados do Brasil. Essa metodologia é atrativa dos 
pontos de vista de requerimento de dados e computação, e é a única baseada em 
argumentos de teoria econômica. Também discutimos como os resultados do modelo 
estimado podem ser utilizados para fornecer uma medida de produto potencial, também 
de grande relevância para a realização de política econômica. 
 
Palavras-Chave: Núcleo Inflacionário, Estimação de Vetores Auto-Regressivos 
Estruturais, estimação de produto potencial.  
 
Abstract:  The issue of core inflation measurement and its relevance for policy making 
has been thoroughly discussed both in the international, and on the Brazilian applied 
literature. Despite this relevance, there is no consensus on how best to measure core 
inflation. Several different methodologies have been employed for Brazilian data, with 
different results. Here we apply Quah and Vahey's (1995) methodology to Brazilian data. 
This methodology is attractive in computational and data requirement aspects, and is the 
only one based on economic theory. We also discuss how the results from the estimated 
model can be used to provide a measure of potential product, which is also a very 
relevant issue for policy making. 
 
Key-Words: Core Inflation, Structural VAR estimation, Potential Product estimation 
 
Área de Classificação da ANPEC: 02 (Macroeconomia) 
 
Código JEL: C22, E31 
 
 1 Introduction
The issue of core in‡ation measurement and its relevance for policy making
has been thoroughly discussed both in the international, and on the Brazilian
applied literature. Despite this relevance, there is no consensus on how best to
measure core in‡ation. For two very good references concerning applications of
di¤erent techniques to Brazilian data, the reader is referred to Figueiredo (2000)
and Bryan and Cecchetti (2001).
Several di¤erent methodologies have been employed for Brazilian data, in-
cluding trimmed means estimators, common stochastic components (a reference
for these two methods in the Brazilian case is Picchetti and Toledo (2000)), and
zero weighting of seasonal components. One of the methodologies implemented
in the international literature is the one proposed by Quah and Vahey (1995).
This methodology has three main attractive features: it uses relatively much
less data than its alternatives, it is computationally simple, and it is the only
one based on economic theory justi…cations.
Here we apply Quah and Vahey’s methodology to Brazilian data, and com-
pare the results with the shocks identi…ed by Bogdanski et alli (2000). We
also discuss how the results from the estimated model can be used to provide
a measure of potential product, which is also a very relevant issue for policy
making.
2 Quah and Vahey’s Model
Essentially, we estimate a VAR system where Y – the log of output – and ¼ –
the log of price-level – are the endogenous variables. Taking …rst di¤erences to
guarantee stationarity, we have:
Xt =
·
¢Yt
¢¼t
¸
; " =
·
"1
"2
¸
(vector of “shocks”), where
E["1] = E["2] = 0
Cov["1;"2] = 0
From the VMA representation, we know that:
Xt = C(0)"t + C(1)"t¡1 + C(2)"t¡2 + :::
=
1 X
j=0
C(j)"t¡j
1where C(j) =
·
c11(j) c12(j)
c21(j) c22(j)
¸
We can estimate the vector of disturbances e =
·
e1
. . .e2
¸0
from the estimation
of the VAR in reduced form, but we cannot directly estimate the vector of
shocks, since the parameters in C(j) are the ones from the standard form, not
identi…able unless we pose some restrictions. The problem is, as always happens
in identi…cation issues, to be in a situation where we have more unknows than
equations. The relationship between the estimated residuals and the original
shocks is given by:
e1t = c11(0)"1t + c12(0)"2t
e2t = c21(0)"2t + c22(0)"2t
so that if we knew c11(0);::: ;c22(0), we could directly recover the original
shocks from the estimated residuals, which are composites of them. But we
know that c11(0);::: ;c22(0) involve coe¢cients from the standard VAR, and
these are not identi…able by the estimated coe¢cients of the reduced form. The
main idea behind Blanchard and Quah’s decomposition method is to impose
four additional restrictions which create four equations that can be solved for
c11(0);::: ;c22(0). Three of these restrictions come from the covariance matrix of
the estimated residuals, while the fourth and …nal one come from the restriction
of one of the original shocks not having any long-run impact on one of the
VAR variables. This restriction is motivated on economic grounds. In our
application it translates into aggregate demand shocks not having any long-term
e¤ect on product. The two shocks resulting from the above restriction can then
be interpreted as an aggregate demand shock, and an aggregate supply shock.
The underlying movement in core in‡ation is associated with the disturbance
having no long-run e¤ect on output.
This …rst restriction comes from the variance of the …rst VAR residual:
V ar[e1] = V ar[c11(0)"1t + c12(0)"2t]
= c11(0)2 + c12(0)2
Similarly, we obtain the second restriction from the variance for the second
estimated residual, in the same manner:
V ar[e2] = c21(0)2 + c22(0)2
The third restriction comes from the covariance between the estimated residuals.
Note that, as said before, these are both composites of the original shocks, so
that their expected value is zero, while their covariance is not. These two facts
imply that their covariance is simply the expected value of their product:
Cov[e1;e2] = E[(e1 ¡ E[e1])(e2 ¡ E[e2])]
= c11(0)c21(0) + c12(0)c22(0)
2again using the facts that the variance of each shock is normalized to unit, and
their covariance is zero. The fourth restriction says that the long-run e¤ect of
one of the shocks in one of tha variables is zero:
1 X
k=0
c11(k)"1;t¡k = 0
after some algebra (since we don’t know "1), we get
"
1 ¡
1 X
k=0
a22(k)
#
c11(0) +
1 X
k=0
a12(k)c21(0) = 0
So, now we have four equations and four unknowns:
V ar[e1] = c11(0)2 + c12(0)2
V ar[e2] = c21(0)2 + c22(0)2
Cov[e1;e2] = c11(0)c21(0) + c12(0)c22(0)
0 =
"
1 ¡
1 X
k=0
a22(k)
#
c11(0) +
1 X
k=0
a12(k)c21(0)
Solving these equations for c11(0);::: ;c22(0), we can obtain the original shocks,
and proceed with the impulse-response and variance-decomposition analysis of
the VAR.
We will take IBGE’s industrial production monthly index as our measure of
output, and will examine, in turn, the results for the VAR with the IBGE-IPCA.
The estimated VAR includes nominal interest rates (SELIC) as an exogenous
variable, as well as seasonal dummies. The inclusion of interest rates as a
variable outside the system in this case is not without problems, since for some
part of the sample the monetary regime in Brazil was mainly characterized
by in‡ation targeting setting of nominal interest rates. However, the inclusion
of interest rates as endogenous would make the imposition and intepretation
of additional restrictions more cumbersome, and so we decided to include this
issue in our future research agenda along with the inclusion of other potentially
relevant variables, such as the exchange rate, and concentrate on the results we
have with the simpler model. The chosen speci…cation turned out to include a
single lag of each of the endogenous variables.
3 Results
Once the VAR is estimated, and the strutuctural shocks are obtained by the
above explained identi…cation restrictions, the impulse response functions are:
3-.03
-.02
-.01
.00
.01
.02
.03
.04
.05
2 4 6 8 10 12 14 16
Non-Core Core
Accumulated Response of DL_IND to Structural
One S.D. Innovations
.000
.002
.004
.006
.008
.010
.012
2 4 6 8 10 12 14 16
Non-Core Core
Accumulated Response of DL_IPCA to Structural
One S.D. Innovations
The proposed mesure for Core In‡ation can then be obtained by the un-
derlying movement in headline in‡ation associated only with the disturbances
representing “demand shocks”. The di¤erence between measured IPCA and our
4Core component can be seen in the graph below:
-.010
-.005
.000
.005
.010
.015
.020
.025
.030
1995 1996 1997 1998 1999 2000
CORE DL_IPCA
It is interesting to investigate these results in comparison to the shocks
analysed by Bogdansky et alli, for the period after the implementation of the
in‡ation-targeting program in Brazil:
5-.004
.000
.004
.008
.012
.016
.020
1999:01 1999:07 2000:01 2000:07 2001:01
CORE DL_IPCA
Expectations
Adjustment
Adverse Food
and Oil price
shocks
Trade Deficit
Robust
Fundamentals
Oil and
Food Price
shocks
The periods characterized by negative supply shocks in Bogdansky et alli
consistently depict a value for headline in‡ation above our measure of Core
in‡ation. This is an encouraging fact for the use of this measure of Core in‡ation.
Compared to some alternative measures, it presents some signi…cant advantages
in termos of data requirement and computational complexity. Whereas our
measure depends only on the availability of data for the general price index, the
industrial production index, and nominal interest rates, trimmed-means based
measures require availability of data on individual components of the index.
Also, the structural VAR estimation methods used here are already implemented
in some of the standard econometrics softwares, whereas measures such as the
ones proposed by Bryan and Cecchetti (1994), applied to Brazilian data by
Picchetti and Toledo, require some substantil programming to implement the
estimation procedures.
4 Implications for Potential Product Estimation
We now use this methodology to propose a measure of potential GDP. Following
the literature on in‡ation targeting (Taylor (1999)), the convenient de…nition
of potential GDP, or “natural rate of output”, is one that can be grasped from
the relation
6¼t = a(Yt ¡ ^ Yt) + b¼t+1 + "t
where ¼ is in‡ation, Y is GDP, ^ Y denotes the potential GDP, " is a white noise,
and the coe¢cients a and b are positive numbers.
The above speci…cation is known as the New Keynesian Phillips Curve, but
this equation can more generally take other forms that include lagged in‡ation,
wages and government-controled prices (see Bogdanski, Tombini and Goldfajn
(2001)). The important point to be taken here, which is common to all speci…-
cations, is that the “output gap”, or the deviation from current GDP to current
potential GDP has a positive impact on in‡ation. This means that if GDP is
higher that potential GDP then in‡ation is perpetually increasing; and GDP
lower that potential GDP implies that in‡ation is decreasing. The concept of
potential GDP is therefore the level of output for which in‡ation is stable.
The measure of potential GDP that is commonly used both by the academic
literature and central banks (see, however, Woodford (2000)) is the one obtained
by Hodrick-Prescott …ltering the actual output series, or by linear, quadratic or
spline …tting through minimum square errors minimization. In all these cases,
the previous concept of potential GDP is lost, since these measures of potential
GDP are simply di¤erent assessments of the low frequency components of the
actual GDP.
To better understand the problem with these measures, consider that the
economy was hit by a high frequency positive supply shock. Because all these
techniques exclude high frequency components from the actual series, the cal-
culated measure of potential GDP is not a¤ected by this suppy shock. On the
other hand, as in any positive supply shock, there will be a positive e¤ect on
output, and a negative e¤ect on prices. Therefore, the output gap will be posi-
tive, and the e¤ect on in‡ation will be negative, contrary to any Phillips Curve
speci…cation.
The theoretical explanation for this contradiction (which was already put
forward by Wicksell!) resides on the presence of technological shocks, that
a¤ect the marginal costs of production, and the pricing of goods. In the in‡ation
targetting language, it means that potential output should move together with
the aggregate supply. The empirical relevance of this point, however, is an open
question, since it depends on the relevance of high frequency supply shocks.
That is exactly the question that we try to answer in what follows.
We advocate that a more appropriate measure of potential GDP is one that
captures all supply shocks, including the high frequency ones. Because our
methodology to identify the VAR separates the existent supply and demand
shocks, such measure of potential GDP is a natural extention. It simply consti-
tutes the GDP that would have been caused exclusively by supply shocks.
In the graph below we depict the “actual GDP series” (in terms of its rates
of change) and our proposed measure of potential GDP in the 1999:1 to 2001:2
horizon (We have actually used the Industrial Production Index, as before). It
is clear that the series are very similar, and there seems to be no observation
7for which they are statistically di¤erent.
The conclusion that stems from this …gure is that roughly all GDP ‡uctu-
ations are caused by supply shocks, or, equivalently, that demand shocks have
no important impact on output. As a consequence, output gap is virtually zero
in all observations, and the Phillips Curve looses its relevance.
Striking as they may be, similar conclusions were already reached by other
empirical studies, and by the Real Business Cycle literature.
-.2
-.1
.0
.1
.2
1995 1996 1997 1998 1999 2000
DL_IND P_POTENC
These assertions seem to be con…rmed by examining the variance decompo-
sition results for the two kinds of shocks:
820
30
40
50
60
70
80
2 4 6 8 10 12 14 16
Non-Core Core
Variance Decomposition of DL_IND
5 Conclusions
The results of the estimated model show that Non-Core shocks have long-run
impacts on output, but negligible impact on measured in‡ation. Also, Core
shocks have long-run impacts on measured in‡ation, and impact on output is
slightly positive in the beggining, but negligible in the long-run. These are
expected from the restrictions imposed on the model, but seem to …t the data
well. Moreover, the Core measure seems a good indicator of turning points of
measured in‡ation, and is able to …lter out the main short-run supply shocks
identi…ed by the Brazilian Central Bank.
The estimation of potential product from the Structural VAR model dif-
fer substantially from the ones obtained by simpler …ltering of long-run trends
usually produced.
Comparing the Core in‡ation estimates produced here with the ones based
on trimmed means and stochastic common components (Picchetti and Toledo
(2000)), the results are considerably di¤erent, with less smooth paths for the
Core in‡ation measure. Ideally, all these methodologies are attempts to estimate
the same values, representing the long run components if changes in the price
level which respond to monetary policy, …ltering out what could be considered
as short run supply shocks to the price level. However, the comparisons of the
results from di¤erent methodologies, and di¤erent levels of aggregation for price
level data in the Brazilian case, show that further research is indeed necessary
to point out which methodology is preferable.
9The methodology implemented in this paper would bene…t, in this context,
from further research which would extended in two di¤erent dimensions: the
inclusion of more variables in the VAR system, and accounting for probable
regime-switching events.
6 References
Blanchard, O. and Quah, D. (1989) “The dynamic e¤ects of aggregate demand
and supply disturbances”, American Economic Review, Vol. 79.
Bogdanski, J., Freitas, P., Goldfajn, I. and Tombini, A. (2001) “In‡ation
Targeting in Brazil: Shcoks, Backward Looking Prices, and IMF Conditional-
ity”, mimeo – Programa de Seminários Acadêmicos do IPE/USP, Seminário n.
07/2001.
Bryan, M. and Cecchetti, S.G. (1994) “Measuring Core In‡ation” In: Mankiw,
G. (ed.), Monetary Policy: Chicago: The University of Chicago Press.
Bryan, M. and Cecchetti, S.G. (2001) “A Note on the E¢cient Estimation
of In‡ation in Brazil”, Working Paper Series 11, Banco Central do Brasil.
Figueiredo, F. (2000) “Evaluation of a measure of core in‡ation” , Manuscript
– Studies and Research Department, Central Bank of Brazil.
Picchetti, P. and Toledo, C. (2000) “How Much to Trim ? A Methodology for
Calculating Core In‡ation, Wich an Application for Brazil”, Economia Aplicada,
Vol.4 – N. 4
Quah, D. and Vahey, S. (1995) “Measuring Core In‡ation”, Economic Jour-
nal, 105.
Taylor, J. B. (1999), Monetary Policy Rules, The Univ. of Chicago Press
Woodford, M. (2000) “A Neo-Wicksellian Framework for the Analysis of
Monetary Policy”, Princeton Working Paper
10

A Note on the E¢cient Estimation of In‡ation in Brazil”, Working Paper Series 11, Banco Central do Brasil.

In‡ation Targeting in Brazil: Shcoks, Backward Looking Prices,

Measuring Core In‡ation”

The dynamic e¤ects of aggregate demand and supply disturbances”,

An Application of Quah And Vaheys Svar Methodology for Estimating Core Inflation in Brazil

http://www.anpec.org.br/encontro2001/artigos/200102165.pdf

An Application of Quah And Vaheys Svar Methodology for Estimating Core Inflation in Brazil

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