TÍTULO: TESTING THE RATIONALITY
OF
MARKET EXPECTATIONS
ON
BRAZILIAN
INFLATION: EVIDENCE FROM AN UNBALANCED PANEL
Autor: Fabia A. de Carvalho
Afiliação: Banco Central do Brasil
Endereço para Correspondência:
SQS 202 Bloco D Apt 106
Asa Sul – Brasilia – DF
70232-040
61-414-3404 e 61-414-3507 (Fax)
fabia.carvalho@bcb.gov.br
Key words: Rationality Tests, Rational Expectations, GMM, Unbalanced Panel Data,
Serial Correlated and Moving Average Errors
JEL: C12, C23, E52
1
TESTING THE RATIONALITY OF MARKET EXPECTATIONS ON
BRAZILIAN INFLATION: EVIDENCE FROM AN UNBALANCED PANEL
Fabia A. de Carvalho
ABSTRACT
This paper tests the rationality of inflation forecasts surveyed by the
Central Bank of Brazil. It employs panel data techniques to avoid the
aggregation bias that usually leads to rejection of the rationality hypothesis.
Additional techniques are employed to account for the specificities of the
Brazilian panel. Most distinguishably, one procedure for treating
unbalanced panel data is extended to the case in which it is necessary to
estimate a consistent covariance matrix that accounts for distortions caused
by errors that present serial correlation and are also correlated across
individuals. For the entire sample available, joint hypothesis tests refute
rationality for all forecasting horizons but the very short-term one, in spite
of the sharp reduction in test statistics of unbiasedness after the proper
covariance matrix is employed. However, after reducing the sample to
include only those periods for which information is more reliable, the
estimated parameters approach those suggested by unbiasedness. The failure
in efficiency tests for all sampling periods and forecasting horizons but the
very short-term one indicate that forecasters are not making proper use of
important information available. This result may be highly influenced by the
fact that under an inflation targeting regime which is still building up
reputation, the power of the past to predict the future is very uncertain.
Key words: Rationality Tests, Rational Expectations, GMM, Unbalanced
Panel Data, Serial Correlated and Moving Average Errors
JEL Classification: C12, C23, E52

Research Department. Banco Central do Brasil. E-mail: fabia.carvalho@bcb.gov.br
2
1. INTRODUCTION
In an inflation-targeting regime, private forecasts for inflation play a number of
important roles. Their behavior is usually seen as a proxy of the dynamics of the
expectations transmission channel to inflation. Operationally, private forecasts function
as a thermometer that measures the success of the regime in keeping inflation within a
desired path. In this sense, central banks use them as an instrument to assess likely
sources of friction in the inflation dynamics that may not be encompassed by their own
inflation forecasting models.
A way to measure the importance of private forecasts to monetary policy
decisions is to estimate central banks’ reaction functions. For the Brazilian case, Minella
et al. (2003) showed that the coefficient of market forecasts in the reaction function of
the Central Bank of Brazil is above two1. The strong reaction to market forecasts proves
that “the central bank conducts monetary policy on a forward-looking basis, and
responds to inflationary pressures.”2
In Brazil, the most comprehensive survey of private forecasts for inflation has
been Central Bank’s Investor Relations Office survey of private forecasts. This survey
has been going on daily since June 1999. Figure 1 depicts a sample of daily evolutions
of the median forecast in the survey for the Brazilian CPI. After the median response
started to strongly decouple from inflation target bands, the survey began to catch more
attention of the media, market analysts and the Central Bank of Brazil itself. The
median forecasts proved to be unable to anticipate the extent to which inflation would
be influenced by the severe sequence of shocks that hit the Brazilian economy, most
markedly in 2002. The predictive power of forecasts in the survey was then highly
criticized, as they seemed to no longer reflect the real pricing dynamics of the economy.
A number of people claimed that market forecasts had become simple representations of
Vector Autoregressive models, meaning that they had become strictly backward
looking, when inflation itself seemed to still have a significant forward-looking
component.
1
Punctual coefficient estimation.
For more comprehensive studies on the inflation targeting regime in Brazil, refer to Bogdanski,et al
(2000, 2001), Alves (2001), Muinhos (2001), Bonomo & Brito (2001), Minella (2001), Minella et al.
(2002), and Fraga et al. (2003)
2
3
Figure 1: Market Forecasts vs. Brazilian Consumer Price Inflation (IPCA)
August 2002
September 2002
0.80
0.80
0.70
0.60
%
0.60
Sep-02
Sep-02
Jul-02
Aug-02
Jul-02
Jun-02
Apr-02
Apr-02
May-02
Actual
November 2002
1.31
3.5
1.20
3.0
1.00
2.5
0.80
%
%
Mar-02
Median
Actual
October 2002
1.40
Feb-02
Jan-02
Feb-02
Nov-01
Sep-02
Jul-02
Aug-02
Jul-02
Jun-02
May-02
Apr-02
Median
May-02
Mar-02
Mar-02
Jan-02
0.20
Feb-02
0.30
0.20
Jan-02
0.30
Dec-01
0.40
Nov-01
0.40
Dec-01
0.50
Nov-01
0.50
Nov-01
%
0.72
0.70
0.65
3.02
2.0
1.5
0.60
1.0
0.40
0.5
Oct-02
Nov-02
Dec-02
Jun-03
Sep-02
Aug-02
Jul-02
Aug-02
Jun-02
May-02
Apr-02
Apr-02
Mar-02
Feb-02
Median
May-03
Actual
Jan-02
Dec-01
Nov-01
Nov-01
Oct-02
Sep-02
Aug-02
Jul-02
0.0
May-03
Median
Jun-02
May-02
Apr-02
Mar-02
Feb-02
Jan-02
Dec-01
Nov-01
0.20
Actual
June 2003
December 2002
1.0
2.5
0.8
2.10
2.0
0.6
%
%
1.5
0.4
0.2
1.0
0.0
-0.15
0.5
-0.2
Median
Actual
Median
Apr-03
Mar-03
Jan-03
Feb-03
Dec-02
Nov-02
Oct-02
Sep-02
Aug-02
Jul-02
Jul-02
Jun-02
-0.4
May-02
Jan-03
Dec-02
Nov-02
Oct-02
Sep-02
Aug-02
Jul-02
Jun-02
May-02
Apr-02
Mar-02
Feb-02
Jan-02
Dec-01
Nov-01
0.0
Actual
To restore credibility in the targets, the Central Bank of Brazil engaged in a
strong effort to coordinate these expectations. Interest rate decisions were made taking
expectations strongly into account, as indicated in the minutes of the Monetary Policy
Committee meetings throughout the beginning of 2003. However, interest rate decisions
were not the sole instrument used by the central bank to get expectations into track
again. There was a strong effort to communicate central bank’s forecasts, in addition to
the board’s understanding of the inflation dynamics. In particular, a number of verbal
and written communications emphasized central bank’s assessment that some unusual
4
phenomena that had been observed in inflation, such as a higher inertia coefficient and
asymmetric passthrough of the exchange rate to prices, were temporary. As these
phenomena actually proved to be short-lived, expectations quickly returned to the path
compatible with inflation targets.
Credibility in these forecasts was undermined, though. The question that
naturally arises in this context is: are private forecasts to be so much relied upon? Are
they really good proxies of future price setting behavior? One way to answer this
question is to test the rationality of private forecasts. Muth (1961) argued that
expectations are rational when they are formed based on an optimizing behavior of
agents within an economic system. Rationality then implies that forecasts are not
systematically biased in any direction, or, in other words, that they perfectly predict
inflation, and that forecasts are efficient, meaning that forecasts fully employ all
available information up to the moment they are made. 3
If the surveyed forecasts are unbiased, central banks have the proper
instruments to observe the effects of their decisions on the “expectations transmission
channel”. If they are inflation targeting, then it is highly commendable that they react
strongly and fast to private inflation forecasts, as they are really good indicators of the
future price setting behavior. If forecasts are systematically biased, shouldn’t the central
bank attempt to measure the real bias before responding to them? If surveyed forecasts
are not unbiased and central banks still react strongly to them, some other way to infer
their influence in future price setting behavior should be investigated. If forecasts are
unbiased, but are not efficient, are they based on optimizing, and thus, predictable
behaviors? At the Central Bank of Brazil, inflation targeting models assume rational
expectations upon estimations and employ private forecasts as proxy of rational
expectations upon simulations. This paper then attempts to address the issue of whether
private forecasts in Brazil are really rational.
There is a vast literature on testing the rationality of market forecasts for
inflation or price indices. Most of them employ unbiasedness and efficiency tests. The
great majority has focused on surveys particular to the USA, and the results obtained
most frequently refute the rationality assumption. These papers have, nonetheless, relied
upon aggregate measures of forecasts and used simplifications that led the errors
3
Other concepts of rationality are also used in the literature, although to a lesser extent. One example
would be tests of the convergence dynamics of forecasts to actual inflation and vice versa This line of
rationality test is well exploited by Grant & Thomas (1999).
5
covariance to be wrongly specified. In a seminal paper, Keane and Runkle (1990)
managed to prove rationality of expectations reported to the ASA_NBER inflation
survey of economic forecasters by employing more suitable econometric techniques. In
their test, they used individual forecasts in a panel data and estimated an errors
covariance matrix that accounted for aggregate shocks that hit all forecasters alike and
an autoregressive behavior of the errors that is inherent to most expectations models.
In the literature concerning Brazil, the author is unaware of rationality tests of
inflation forecasts. Alves (2001) went close to this by assessing the predictive power of
Central Bank of Brazil’s inflation model comparatively to benchmark models and
median market forecasts for inflation collected by the Central Bank of Brazil. To
compare the three types of forecasts, he calculated the average deviation of forecasts to
actual inflation outcomes.
This paper tests the hypothesis of rationality of Central Bank of Brazil’s
inflation forecast survey avoiding the shortfalls identified in Keane & Runkle (1990).
The most complex task in this journey was to construct a covariance matrix that
accounted for both serial correlation and errors correlated across individuals, with a
sample of unbalanced panel data.
4
The standard prescription for treating unbalanced
panels —using a subset of data that is balanced —could not be applied here as such
subset would be statistically insignificant and thus bias all results. In addition, some
forecasters were more active in participating in the survey in different periods and these
periods were qualitatively different in terms of economic shocks that hit the economy.
The use of the entire available information was then also commendable to assure
representativeness of the sample.
With the use of appropriate econometric techniques, this paper shows that
private forecasts for inflation surveyed since June 1999 by the Central Bank of Brazil
are rational only for very short-term forecasting horizons. This result gets weaker,
however, when the sample is reduced to include only those periods for which the survey
is more reliable. Sampling after July 2001, when quality procedures are adopted to
ensure timeliness in survey responses, or after November 2001, when the survey starts
to be carried out through the Internet, results in estimated coefficients that converge to
those expected in unbiased forecasts. For most forecasting horizons, individual t-tests
do not allow for refuting the assumption that forecasts are unbiased. However, if the
4
In unbalanced panel data, there is an unequal number of observations for different individuals.
6
restrictions are jointly tested, using  statistics that are more adequate when errors are
non-spherical, unbiasedness should be refuted for all forecasting horizons. The sample
size may be a drawback here, though, as any of these statistics are more reliable as the
sample size increases5.
Efficiency tests, in turn, strongly suggest that private forecasters in Brazil are not
making proper use of information that is available for any forecasting horizon. This
inefficiency may be a natural consequence of an inflation-targeting regime that is still
building up credibility. In this circumstance, forecasts will usually be a combination of
the past behavior of key observable variables, such as macroeconomic data, and of
future unobservable ones, such as the future conduct of monetary policy. Moreover,
forecasters, as well as the central bank, are still learning about the workings of each
transmission channel, which adds a lot of uncertainty to the forecasts.
The organization of the paper is as follows. Section 2 discusses rationality tests
in the economic literature. Section 3 describes in details the model used to test the
rationality of private forecasts for the Brazilian inflation. Section 4 brings about some
important data issues. Section 5 shows the results of the rationality tests and finally
Section 6 concludes the paper.
2. RATIONALITY TESTS
Rationality tests in Muth (1961)’s sense encompass two criteria: that of
unbiasedness and that of efficiency. Suppose that  t  k is the actual outcome for
inflation in period t+k, for k  0 , and Et  t  k is some sort of aggregate survey response
as to the future inflation rate (at period t+k) using information available up to period t.
 0
Most of literature then claims that if H o :     holds in the following regression,
1
forecasts are unbiased:

1

    t  k
 t  k  
 Et  t  k 
(1)
The efficiency criterion tests whether forecasters make efficient use of all
information available at period t. Let  t be the vector of information available to the
5
For a clear discussion of hypothesis testing when errors are non-spherical, refer to Greene (2001-pp 279281)
7
forecaster up to period t. For forecasts to be efficient, H o :   0 1 0  0'
should hold in the following regression:
 t k
 1 


  Et  t  k     t  k
  
t


(2)
As pointed out by Keane and Runkle (1990, 1998) almost all of the tests
conducted in the literature have refuted rationality as a result of one or more of the
problems listed below:
a)
use of some sort of aggregate survey response, such as average or
median. Rationality fails in this case because “average forecasts that
are conditional on different information sets are not rational forecasts
conditional on any particular information set”. In addition, aggregate
forecasts neutralize the biases that some analysts may have and that
may provide valuable information for the econometrician;
b)
use of revised data when indeed the forecaster was attempting to
predict the unrevised number or was using unrevised available data to
predict the future;
c)
use of predictions made by forecasters who are not professional and
thus have no costs in providing imprecise information; and
d)
failure to account for the proper covariance structure of the errors.
The procedure adopted in this paper overcomes all of these sources of improper
rationality rejection. The data used are a panel of consumer price inflation (IPCA)
forecasts surveyed by Central Bank of Brazil’s Investor Relations Group since 1999. As
there is no revision in the figures released for the IPCA, the second source of problems
identified by Keane and Runkle (1990,1998) is not an issue here.
The issue regarding the professionalism of forecasters should be carefully
addressed. In a technical paper released by the Brazilian Institute of Applied Economics
(IPEA), the professionalism of Central Bank of Brazil’s survey was questioned on the
grounds of a game theory argument: if economic agents perceive their power to affect
the monetary policy decision, they will have incentives not to disclose their correct
forecast. However, in Central Bank of Brazil’s survey, at least three strong incentives
are in place for analysts to report their “true” forecasts.
8
First, more than 90% of the participants in the survey are chief economists of
either financial institutions or economic consultancies, who thus receive monetary
rewards to produce accurate forecasts. Second, the participants in the survey either sell
their projections on the market or disclose them publicly as a means to add value to the
products and services sold by the institutions to which they are affiliated. Therefore, the
accuracy of their projections can be easily verified by the Central Bank of Brazil. Third,
the Central Bank of Brazil has been frequently publishing the rank of the best
forecasting institutions during a time range. Some financial institutions are beginning to
reward their forecasting teams whenever they are ranked as top 5.
The last source of improper rationality rejection pointed out by Keane and
Runkle (1990) is the misspecification of the covariance structure of the forecast errors
across forecasters. Some forms of misspecification may be outlined. First, employing
OLS errors estimates to test the rationality hypothesis in the presence of non-spherical
errors yield inconsistent error covariances and thus may render the tests invalid. In a
panel of inflation forecasters, aggregate shocks are likely to affect all cross sections, and
thus errors will not be independent across forecasters (Zarnowitz (1984, 1985)).
Shocks may also be a source of auto correlation of the errors. As Grant &
Thomas (1999) argue “because of overlapping forecast intervals, agents are frequently
unaware of the stochastic nature of the inflation-generating process. If shocks to
inflation are assumed to be temporary when in fact they are permanent, agents may
consistently under- or overpredict inflation for several consecutive forecast periods”.
Under overlapping forecast intervals and uncertainty in regard to the persistence of a
certain shock, “the structure of autocorrelated errors may yield valuable information that
can be used to confirm agent’s rationality” rather than refute it. 6
The sample period used to perform rationality tests for the Brazilian case
includes a number of shocks that most likely affected all cross sections. Most of them
were originally supply or cost shocks whose intensity could be hardly ever anticipated,
and thus could not be in the information set of the forecaster, such as the energy
rationing, the Argentine crisis and the September 11th attacks in 2001, a confidence
crisis in 2002 regarding the future conduct of economic policy in Brazil, and the
accounting problems in large American companies also in 2002.
6
Grant & Thomas (1999)
9
In order to treat the auto correlation that arises from overlapping forecast
intervals, one procedure has been more advocated in the literature. If one samples a
variable xt at every kth integer, the regression equation will present a serially
uncorrelated disturbance term. The drawback of this procedure is to use only a part of
the sample at the cost of missing relevant information. The sample may thus cease to be
representative.
In fact, on comparing both suggestions for treating auto correlation, Hansen &
Hodrick (1980) show that, under a sample size T, hypothesis testing on the
 parameters using the entire sample dominate the sampling procedure.
Auto correlation can also arise due to the fact that when the forecaster is
predicting inflation for period t+k with information available up to period t, the specific
information on inflation at period t may not be available. This presence of auto
correlation should not be considered a failure in the rationality test as the forecast error
 ti,k   t k  Et tik should not be orthogonal to  ti .
Estimating and employing the correct covariance matrix will not necessarily
yield valid test statistics. The use of GLS, which is the “standard econometric technique
for estimation in the presence of serially correlated errors”7, requires the regressors to be


exogenous, meaning that E  ti,k / xt , xt 1 , xt 1 ,...  0 when xt  Et  t  k or any variable in
the information set of forecaster i. As Hansen & Hodrick (1980) argue, “the strict
exogeneity assumption is a claim that knowledge of future xt ’s would be useless in
determining the optimal forecast for ‘  t  k ’8. That is clearly not the case here. If
forecasters knew with certainty the future stance of the output gap, for instance, it would
be much likelier that their inflation projections would be more accurate. When the
exogeneity assumption fails, an alternative estimation method is through GMM.
Estimation of a covariance matrix in the case of auto correlation of the errors
and aggregate shocks is a straightforward extension of the techniques proposed in
Keane and Runkle (1990) and Hansen & Hodrick (1980). However, with an unbalanced
panel, this estimation becomes very cumbersome. The procedures for estimating in this
environment are described in the following section.
7
Hansen & Hodrick (1980)
10
3. THE MODEL
The model for testing the hypothesis of rationality of the Brazilian inflation
survey is an extension of the techniques proposed by Keane and Runkle (1990) and
Hansen & Hodrick (1980) to the unbalanced panel data case.
Let Yraw be a tx1 vector of actual inflation, in which t represents the total time
period considered for sampling expectations. Let X raw be the matrix built from the
inflation forecasts of each of the n cross sections formed throughout the t months. The
columns of this matrix refer to observations in each cross section. In each column there
is a number of missing information that varies throughout cross sections. For
computational purposes, the cells where there is missing information should contain a
non-numeric entry, which will be later considered “invalid observation”. The total
number of valid observations in X raw will be denoted by S. Note that if the panel was
balanced, S would equal nt. Let Ti be a vector of length S in which cell (i,1) contains the
number of valid information for cross section i, denoted by t i .
By pooling all the information, valid and invalid, in X raw , replicating Yraw , and
grouping the two resulting matrices into one matrix of size nt x 2, one can exclude the
lines that contain missing information for expectations and proceed with the
estimations. (The resulting matrix, hereafter referred to as DATA, will be of size S x 2).
For OLS estimations, the procedure is straightforward. Letting  t be the first column of
DATA, and xt be an S x 2 matrix of ones in the first column and the second column built
with the second column of DATA, the system is ready for the regression in (1).
The construction of the covariance matrix that is consistent in the case of serial
correlation across individuals and moving average errors is an extension to the
unbalanced case of the technique proposed by Keane & Runkle (1990). In order not to
exhaust the number of degrees of freedom in the data, some assumptions need to be
made:

i, t , l , m, s.t. l  m  k
E  til ,k  ti m,k   l m ,
otherwise
 0,


(3)
and
8
In their paper,
 t  k is replaced by y t  k as they are testing the zero rate of return hypothesis in forward
exchange markets.
11

i, h, t , l , m, s.t. i  h and l  m  k
E  til ,k  th m,k   l m ,
0,
otherwise



(4)
The first restriction is only a translation to panel data sets of the result of the
application of the technique proposed by Hansen & Hodrick (1980) to consistently
estimate a covariance matrix in time series models in the presence of MA errors. The
second restriction assumes that the data are not conditionally heteroskedastic. This
assumption may be unrealistic as an important source of heteroskedasticity in
forecasters panels is the fact that different cross sections may present asymmetric loss
functions, in the sense that one analyst may be more penalized for underprediction of
inflation than others, for instance, or contrarily forecasters could have the relationship
they built with policymakers jeopardized by being labeled “too pessimistic”. However,
considering these biases in the covariance matrix would not be consistent with the
rationality assumption.
Computationally, to derive the covariance matrix of equation (1) for the
unbalance panel data case, a number of steps should precede9. A matrix of dates should
be constructed by indexing each line of X raw . After X raw is pooled, the matrix of dates
should be replicated until the size equals that of the pooled X raw . Afterwards, the lines
in the matrix of dates corresponding to those in the pooled X raw in which information is
missing should be deleted. A second matrix should be constructed to index the
information belonging to each cross section in DATA.
The system will thus be:
1 E1i 1 k 
 x1 


 
. 
.
. 
.  assuming, for ease of exposition, but not
X S   .  , in which x i   .


 
. 
.
. 
1 E ii  i 
x n 
 
t
t k 

necessarily in the calculations, that the observation in period 1 is always available for
each cross section.
Letting the estimate of  lm be
1
S
i i


t j
t ,k
 t  j ,k , in which  is the number of
observations available for the cases in which l  m  0 , and  l m be
1
S
i h


t j
t ,k
 t  j ,k ,
12
i  h,
the
R A
 B
R
ˆ
form   
 
 B
 R
estimated
covariance
matrix
of
size
SxS
will
have
the
RB  RB 

RA  RB 
, in which R A will have the elements ̂ lm allocated
B
  R

R B  R A 
according to the time period to which they belong and R B will have the elements
ˆ l  m allocated also according to the time period to which they belong.
The GMM estimation will be based on this estimate of  . In fact,
ˆ
GMM
1
 X ' Z  Z '
ˆ Z  Z ' X 




 S  S 
S 




1
1
 X ' Z  Z '
ˆ Z  Z ' Y 



where the Z’s are instruments
 S  S  S 




that are expected to be correlated with X but not with the errors. For the estimations in
this paper, the Z’s were chosen from the information set available to the forecaster at
time t.
Modified t- and  2 statistics can then be calculated for  to test the hypothesis
in (1) and (2). t and F-tests are not usually recommended when the errors are not
normally distributed, in which case they are only asymptotically valid. However, if the
regression coefficients are looked at individually rather than jointly, the t-statistics do
not allow for rejection of H 0 in (1) in reasonable levels of significance for most cases
analyzed here. By looking only at the  2 -statistics, rationality can be refuted in all
cases, in spite of the fact that the results are much closer to accept rationality had the
covariance matrix not been properly estimated. 10
9
The covariance matrix for equation (2) is a straightforward extension.
The computation of the robust covariance matrix alongside with all estimations was carried out in the
Matlab environment, as Beck & Katz (1995) advise. As they argue “many standard packages, such as
RATS or SHAZAM compute robust standard errors based on White’s method; none of these take into
account the panel structure of the errors, even within a panel estimation module.” Their Monte Carlo
simulations for testing their proposal of a covariance matrix “show that researchers should not use robust
standard errors calculated by typical computer packages to approximate ‘panel corrected standard
errors’”.
10
13
4. THE DATA
At this point, a few remarks on the data sampled for this paper are necessary.
Central Bank of Brazil’s Investor Relations Office has been surveying private forecasts
for inflation on a daily basis since June 199911.
The professionalism of forecasts informed to the Central Bank of Brazil at the
beginning of the survey is not immune to criticism. In the period prior to July 2001,
although participants belonged to a selected group of chief economists, very few
qualitative procedures were adopted to assure the timeliness of the information input as
these forecasters’ expectations. During this period, private forecasts were obtained
through sporadic client reports, phone calls or spreadsheets sent to the Central Bank of
Brazil through fax or e-mails. There was no established routine to update these
expectations on a frequent basis. From July 2001 to November 2001, the quality of the
survey enhanced through significant efforts by the Central Bank of Brazil to contact all
participants and update the series as timely as possible.
The participants also played an important role in improving quality of the
survey. With more importance being attributed to the Top 5 ranking of forecasters,
published monthly by the Central Bank of Brazil, the participants became more active in
making sure that their forecasts had been input correctly in the survey. However, as the
database was in Excel and participants’ access to the entries that corresponded to their
forecasts was not instantaneous, besides being available only upon request, input
problems might have occurred. After November 2001, these problems were highly
reduced as the database started to be fed through the Internet directly by the survey’s
participants. Monitoring the timeliness of the information was also enforced with the
procedure of excluding a participant’s forecast from the current sample in case it had
not been updated —which does not mean ‘altered’— within the past 30 days.
Additionally, some upper and lower limits were programmed to help avoid input
mistakes, such as switched signs.
In regard to the data used in this paper, in order to simplify operational
procedures, one representative day in every month was selected for sampling. To avoid
unnecessary sources of heteroscedasticity, the day chosen would have to be the one in
which the information set of forecasters were similar. Some institutions have their own
price surveys and therefore have more timely information regarding the evolution of
14
prices in the month, prior to the publication of the index. In order to eliminate this bias
of direct available information on price behavior for only a part of the cross sections, the
day chosen for sampling forecasts for inflation in month t was the last day of the month
t-1. As an illustration, for the IPCA of January, which is usually published on the first
half of February, and refers to the evolution of prices throughout January, the sampling
day chosen was December 31st.1213
Outliers or forecasters that scarcely participated in the survey were not removed
from the sample used in this paper so as to preserve representativeness of the sample.
5. TEST RESULTS
Table 1 shows the results of the tests of unbiasedness of inflation forecasts for
the current month (  t ), the following month (  t 1 ), accumulated in the following three
months (  3 m ), six months (  6 m ), and twelve months (  12m ). Figures in parenthesis right
below the estimated parameters are t-tests for   0 . The t-tests for  2  1 are
presented below the tests for  2  0 .
Sampling from June 1999 on14, t-ratios do not validate the rejection of
unbiasedness for  t . For all other forecast horizons, unbiasedness should be rejected. If
the sample is reduced to begin in July 2001, when quality procedures to assure the
timeliness of the information provided to the survey were enforced, unbiasedness can
not be rejected with 95% of significance in forecasts for  t ,  3 m , and  6 m .
For the sake of comparison, estimations were carried out for  3 m and  6 m
sampling from November 2001 on, the month in which the database was transferred to
the Internet and monitoring was enhanced. In all these cases, t-ratios could not refute
unbiasedness.
11
Marques et al. (2003) provide more details on the operational aspects of this survey as well as an
analysis of the reasons for the recent behavior of the aggregate responses.
12
Some files in the database were corrupted. Information for the IPCA projections on the last business
days of March, April, May and June 2000 was not available. In March, the latest information available
referred to the 24th of the month. In April, it was the 18th of the month, which was too far away from the
last business day of the month. The files for the first business days of May were also corrupted, the first
reliable one being that of the May 4th, which was what was used here. In May, the information available
was for the 25th of the month. In June, the information used was that of the 28th of the month.
13
Some banks that participated in the survey were either acquired by other banks or merged. For
simplification, they were considered different forecasters as long as they participated in the survey
separately, as the teams projecting were different. After the acquisition or merging, the forecaster was the
institution that continued to participate in the survey and “missing information” was used for the other.
14
Information on expected inflation in the next twelve months was available only from November 2001
on and the number of cross sections in the survey was smaller.
15
The use of standard t or F-tests is, nonetheless, questioned in the literature when
the errors are non-spherical. In this case, the use Wald tests based on  2 distributions is
advisable, but the test resuts are still more reliable as the sample enlarges. The first
important conclusion of inference with  2 statistics is that for all forecasting horizons,
the use of the correct model specification, namely, an estimated covariance matrix that
accounts for errors that are correlated across individuals and serially correlated, renders
much lower  2 ratios.
The  2 ratios suggest that private forecasts surveyed by the Central Bank of
Brazil are unbiased only for forecasting horizons no longer than 30 days and when the
sample covers June 1999 through March 2003. For longer forecasting horizons, except
that of 12-month ahead inflation, as the sample reduces to begin in July 2001 or
November 2001, the estimated punctual parameters approached those suggested by
unbiasedness, but asymptotic variance of the parameters do not allow for accepting the
unbiasedness hypothesis.
These results suggest caution when using the entire series available from the
survey to infer any future behavior of price setters. Strictly speaking, private forecasts
are systematically over- or underestimating future inflation, even after the average effect
of aggregate shocks to forecasts is controlled for. For any analysis of this series, it is
more advisable to use information available after July 2001 or November 2001. For this
sample, unbiasedness could not be strictly accepted, but as the time series enlarges with
time going by, it seems likely that unbiasedness will not be rejected.
For the past two years, the central bank has been attempting to shift the focus of
market analysts’ and society’s attention to twelve-month ahead inflation instead of
inflation accumulated at the end of a certain year. Therefore, private forecasts for
twelve-month ahead inflation have also been in the spotlight. In simulations, this
forecast horizon is also the one that is mostly used. However, for 12-month ahead
inflation, the tests run here suggest that forecasts are strongly biased. The sample size is
very small, which does not allow for any extreme advice. The least that the Central
Bank of Brazil is advised to do is then to add high doses of caution when dealing in any
sense with these forecasts.
16
Table 1 – Tests for Unbiasedness of Expectations for the Brazilian CPI inflation
(IPCA)
Dependent variable:  t
Sample: June 1999 to March 2003
N. of cross sections: 47
N. of observations: 513

OLS
2SLS
GMM


0.03
1.05
27.66
(1.40)
(41.47)
(1.82)
0.03
(1.40)
1.05
(41.47)
(1.83)
25.86
0.03
1.05
3.77
(0.55)
(15.65)
(0.68)
Dependent variable:  t
Sample: July 2001 to March 2003
N. of cross sections: 31
N. of observations: 283

OLS


0.12
1.01
(4.25)
(38.85)
(0.37)
60.49
2SLS
0.12
(4.25)
1.01
(38.87)
(0.38)
60.20
GMM
0.14
0.99
9.31
(1.95)
(15.21)
(0.15)
Dependent variable:  t 1
Sample: June 1999 to March 2003
N. of cross sections: 46
N. of observations: 491

OLS
2SLS
GMM


0.13
1.22
(2.99)
(18.27)
(3.27)
0.13
1.22
(2.99)
(18.27)
(3.27)
(0.21)
1.82
(1.81)
(10.60)
(4.77)
145.84
164.05
37.41
17
Dependent variable:  t 1
Sample: July 2001 to March 2003
N. of cross sections: 30
N. of observations: 266

2SLS
GMM


0.10
1.48
(1.48)
(4.66)
(1.50)
0.11
179.96
1.48
25.96
(7.61)
(2.45)
Dependent variable:  3 m
Sample: June 1999 to February 2003
N. of cross sections: 45
N. of observations: 439


OLS
2SLS
0.32
1.18
2.81
22.09
(3.35)
0.91
0.87
(4.83)
(9.08)

153.81
133.14
(1.40)
GMM
0.91
(2.38)
0.87
(4.87)
(0.75)
Dependent variable:  3 m
Sample: July 2001 to February 2003
N. of cross sections: 29
N. of observations: 237


2SLS
GMM
0.68
1.20
(3.65)
(14.34)
(2.39)
0.68
(1.62)
1.20
(6.97)
(1.16)
12.07

206.00
22.60
18
Dependent variable:  3 m
Sample: November 2001 to February 2003
N. of cross sections: 20
N. of observations: 205



2SLS
GMM
0.77
1.18
(3.65)
(13.04)
(2.00)
0.77
(1.59)
1.18
(6.34)
(0.97)
185.42
19.32
Dependent variable:  6 m
Sample: June 1999 to November 2002
N. of cross sections: 38
N. of observations: 304

OLS


1.85
1.01
135.48
(3.94)
(7.38)
(0.10)
2SLS
7.24
(6.74)
(0.67)
(2.03)
5.07
112.27
GMM
7.24
(3.97)
(0.67)
(1.28)
(3.20)
17.14
Dependent variable:  6 m
Sample: July 2001 to November 2002
N. of cross sections: 29
N. of observations: 179

2SLS
GMM


1.26
1.62
(2.08)
(8.64)
(3.30)
1.26
(0.90)
1.62
(3.96)
233.61
26.86
(1.51)
19
Dependent variable:  6 m
Sample: November 2001 to November 2002
N. of cross sections: 19
N. of observations: 150



2SLS
GMM
1.70
1.57
(2.58)
(8.05)
(2.92)
1.70
1.57
(1.07)
(3.57)
(1.30)
224.89
25.02
Dependent variable:  12m
Sample: November 2001 to July 2002
N. of cross sections: 16
N. of observations: 83

OLS
2SLS


17.38
(0.65)
(6.76)
(1.29)
(3.27)
46.24
(6.37)
(3.53)
(2.46)
650.53
227.55
(2.84)
GMM
46.24
(6.37)
(11.34)
(8.31)
(9.62)
174.52
Table 2 shows the results for the efficiency tests made for the previous
prediction error, the lagged exchange rate and the lagged real interest rate. As
unbiasedness was strongly refuted for inflation accumulated in the next 12 months,
efficiency tests for this sample are not shown here.
Through t-tests, efficiency can not be refuted only for  t . Joint tests also do not
allow for refuting efficiency in this forecast horizon, except for when the regressors
include lagged exchange rate values, which may bear unit root problems. As the
forecast horizon enlarges,  2 statistics worsen. Refuting efficiency implies that
forecasters are not making proper use of information available to them at the time they
make their forecasts. This issue should be carefully addressed, however, when monetary
policy is guided through an inflation-targeting regime. In a regime that is still building
20
up reputation, the failure in efficiency tests may be due to the high uncertainty
prevailing among forecasters in regard to the extent that the past influences future
inflation. These uncertainties stem not only from shocks, which were controlled for
through the use of the estimated covariance matrix, but also from how credible the
inflation targeting regime is. If it is fully credible, inflation forecasts will always
converge to inflation targets and the dynamics of inflation will be known with certainty.
If credibility is still being built up, the regime will be pushing forecasters to look ahead
instead of backwards, but the very inflation dynamics will still have a significant degree
of inertia. The degree of inertia in inflation will be changing throughout the disinflation
process, but estimating exactly what it is adds a lot of uncertainty to forecasts, which
may then end up being inefficient.
21
Table 2 – Tests for Efficiency of Expectations for the Brazilian CPI inflation
(IPCA)
Regressors
Const, Expectation,
Prediction Error (-1)
Dependent
Variable

t
0.02
(0.28)
Const, Expectation,
Exchange Rate (-1),
Exchange Rate (-2)
t
(0.29)
(1.98)
Const, Expectation,
Real Interest Rate (-1),
Real Interest Rate (-2)
t
0.21
(2.32)
Const, Expectation,
Prediction Error (-1)
 t+1
0.27
(2.35)
Const, Expectation,
Exchange Rate (-1),
Exchange Rate (-2)
 t+1
(0.65)
(2.22)
Const, Expectation,
Real Interest Rate (-1),
Real Interest Rate (-2)
 t+1
1.06
(6.68)
Const, Expectation,
Prediction Error (-1)
 3m
0.79
(3.84)
Const, Expectation,
Exchange Rate (-1),
Exchange Rate (-2)
 3m
(2.02)
(3.94)
Const, Expectation,
Real Interest Rate (-1),
Real Interest Rate (-2)
 3m
3.71
(7.54)
Const, Expectation,
Prediction Error (-1)
 6m
1.96
(3.92)
Const, Expectation,
Exchange Rate (-1),
Exchange Rate (-2)
 6m
(1.49)
(0.81)
Const, Expectation,
Real Interest Rate (-1),
Real Interest Rate (-2)
 6m
12.68
(8.22)

1.07
(15.59)
(0.96)
0.97
(12.76)
(0.38)
0.84
(8.44)
(1.61)
0.74
(4.43)
(1.53)
0.27
(1.38)
(3.74)
(0.30)
(1.34)
(5.75)
0.63
(6.16)
(3.63)
0.65
(4.99)
(2.65)
(0.54)
(2.33)
(6.62)
0.46
(3.18)
(3.72)
0.01
(0.04)
(4.18)
(2.08)
(4.85)
(7.17)


(0.03)
(0.30)

4.28
0.53
(1.98)
(0.38)
(1.37)
13.94
(0.07)
(1.15)
(0.04)
(0.85)
7.71
(0.54)
(4.18)
32.13
1.33
(2.43)
(0.80)
(1.40)
39.64
(0.37)
(3.31)
(0.09)
(0.84)
48.98
(0.94)
(9.75)
131.14
4.08
(4.42)
(2.74)
(2.89)
72.99
(1.35)
(4.72)
(0.24)
(1.02)
63.10
(0.98)
(14.67)
325.85
13.25
(4.92)
(10.74)
(3.44)
105.27
(4.18)
(5.04)
0.71
(0.86)
76.12
6. CONCLUSION
This paper tested the rationality assumption in Muth (1961)’s sense for a panel
of forecasts made by professional analysts for the Brazilian consumer price inflation
since June 1999. Due to the presence of severe aggregate shocks that impacted all cross-
22
sections in the period sampled and to evidence of serial correlation in the errors,
rationality tests based on OLS render invalid results. To obtain consistency in
estimations, the technique proposed by Keane & Runkle (1990) of employing a
covariance matrix that accounts for the non-spherical structure of the errors was
extended to an unbalanced panel data case.
The results showed that with t-tests, inflation forecasts for the current month and
accumulated in the next three and six months are unbiased. However, t-statistics do not
perform well in the presence of non-spherical errors. Through the analysis of
 2 statistics, which are more suitable for the case analyzed in this paper, unbiasedness
should be refuted for all cases but the very short-term forecasting horizon. Nonetheless,
these results should be looked at carefully. By sampling the forecasts after July 2001 or
November 2001, dates after which the database became more reliable, the punctual
estimated parameters approach unbiased ones, except for the longer forecasting horizon
(12 months). In all cases, the tests may perform better as the sample size increases,
which indicates that one should wait longer until the more reliable database contains
more information for testing the unbiasedness assumption with a higher degree of
confidence in the sample. Until then, a monetary policy that reacts strongly to private
forecasts risks biasing its decisions.
The tests in the sample of the longer forecasting horizon, in turn, may be
jeopardized by the reduced number of cross sections available. However, as this is the
forecast horizon that is mostly analyzed and used as a benchmark at the Central Bank of
Brazil, the least that one could advise is caution when dealing with them.
Efficiency was also refuted for all samples and forecast horizons but the shorter
one. Although this indicates that forecasters are not making proper use of the
information available, in an inflation targeting regime, these results are not surprising.
One of the goals of this regime is to switch forecasters’ attention to the future, rather
than the past. If, on the other hand, the credibility in the regime is still being built,
uncertainty embedded in the forecasts regarding the forward and backward looking
component of future inflation is higher.
23
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