Two Methodologies to Build Inflation Leading Indicators for
Brazil
Marcelle Chauvet
Solange Gouvea
Marta Baltar Moreira
José Ricardo da Costa e Silva
Research and Studies Department
Central Bank of Brazil
July 2000
Abstract
The goal of the paper is to describe and compare the performance of two
different methodologies to build composite leading inflation indicators: a
dynamic factor model, estimated through Kalman Filter, and a linear
regression framework. The models encompass four variables that present
predictive content in forecasting inflation from a total of 19 pre-selected
potential candidates. In the case of the dynamic factor, all possible
combinations of these 19 variables entail C19
4  3876 potential cases. In the
case of the linear regressions with k-step ahead (k=1,…12), there are
12  C19
4  46512 possible combinations. Both methodologies built effective
leading inflation indicators, according to several statistical tests.
The authors gratefully acknowledge helpful comments from Fabio Araújo and Pedro Henrique Esteves
Albuquerque. The views and opinions expressed in this article are solely those of the authors and should
not be attributed to the Central Bank of Brazil.
E-mails: [email protected], [email protected], [email protected] and
[email protected].

Department of Economics, University of California at Riverside.

These authors work for Research and Studies Department of Central Bank of Brazil.
1. INTRODUCTION
Over the last 12 months, monetary policy in Brazil has been guided by the inflation
targeting regime. Considering that the Real had already floated, there was no concern
about the necessary flexibility to pursue inflation as an overriding objective. Still, it was
essential to get rid of a possible fiscal dominance, to set an explicit target for inflation
and to make decisions transparent. In addition, the role of the monetary authority
remained to be established. The Central Bank was to be assigned operational
independence to fix monetary policy instruments and the responsibility to meet the
targets. Therefore, when implementing this new system the government put in place
some arrangements in both institutional and fiscal areas in order to better fulfill these
requirements.
Once those main decisions were taken, the Central Bank of Brazil focused on the
technical challenges implicit in the inflation targeting framework. It was necessary to
develop alternative methods to forecast inflation. The inflation targeting system
demands a preemptive stance from the Central Bank when setting the instruments of
monetary policy due to the existence of lags in its transmission mechanisms. In this
sense, it was essential to deal with econometric forecast techniques in order to properly
assess the repercussions of eventual shocks that may deviate the future path of inflation
from its previously fixed target.
Macroeconomic forecasts aim to provide the policy maker with the best possible set of
information to support the decision making process. There are several different
approaches to either forecast an event itself or its magnitude. In the first place, smallscale models of the monetary policy transmission mechanism were built using
theoretical relations among macroeconomic variables. Secondly, non-structural models
like Vector Autoregressive (VAR) have been used to provide short-term inflation
forecasts. A third method involves the construction of leading indicators for inflation
with the main purpose of detecting early-warning signals of turning points in the future
inflation path. In contrast to macroeconomic models, the leading indicator approach is
based on very little economic theory.
2
Leading indicators are a forecast tool to predict cyclical movements of an economic
time series1. Differently from other forecasting methods, the main goal of leading
indicators is not to predict future values of the dependent variable, but rather their
turning points. Predictions are based on the observation of the leading indicator once the
cycles and the lead are already identified. Actually, it is a qualitative forecast approach.
The first and main proposal of the leading indicators was to predict changes in
economic activity. It was based on the idea that its cyclical fluctuations may be related
to some economic indicators. The National Bureau of Economic Research started an
important research program on business cycle in the late 30s. As part of this program,
Burns and Mitchell (1946) classified economic time series as coincident, lagged or
leading depending whether they reproduce business cycles simultaneously, with a lag in
time or anticipating them.
Although the construction of leading indicators is based on atheoretical statistical study,
there are some economic reasons why indicators may anticipate the behavior of the
business cycles. Leeuw (1991) identifies five reasons out of which the first three have
been extensively discussed in the literature. They can be summarized as follows:

Production time: It refers to the fact that in the production process some activities
occur earlier than others in a concatenate sequence. An expansion of the economic
activity should begin with a previous increase in demand for consumer goods,
materials, and new contract order for plant and equipment. As an example, increases
in the production of cardboard should happen before growth in industrial sales.

Ease of adaptation: It considers the different flexibility of some variables to react in
earlier stages of an expansion or decline in the economic activity. This is due to the
fact that some decisions related to investment cost less than others. Obviously, it is
less risky to expand hours of work than the number of workers when an expansion of
the economy appears to be on the road.
1
See Chauvet (1998) and Klein (1982)
3

Market expectation: It accounts for the fact that some economic indicators are more
sensitive to future changes in economic activity than others. Stock price, some
material and commodities price are good examples of those.

Primer Movers: It indicates that some economic time series may record in advance
the behavior of economic fundamentals that eventually lead to a future short run
movement in economic activity. Fiscal and monetary variables should represent this
kind of leading indicators.

Changes versus level: It represents the idea that changes in the growth rate of
economic indicators may be signaling future direction changes. In other words, the
first difference of the economic indicators could be interpreted as a lead to a change
of their level.
A single economic time series, like unemployment, hours of work, and others, can be a
leading indicator candidate. However, the literature suggests that a composite index
would perform better than one variable alone. The main explanation for that relies on
the fact that a compounded index can neutralize false signs of its components. For
example, an increase in commodity price may not be anticipating a demand increase,
but reflecting a cartel action. Assuming that there are several reasons to justify the
ability of an indicator to anticipate future changes in economic activity, it is useful to
work with a set of different indicators.
Although leading economic indicators have been mostly used to predict movements in
economic activity, it can also be applied to anticipate inflation turning points. The
rationale is that both are characterized by the existence of cycles and can be forecast by
some economic indicators.
Recent literature proposes the usage of leading indicators to anticipate inflation turning
points. Moore (1986)2, Roth (1991), Webb (1995), Dasgupta (1991), and Chauvet
(1999a, 1999b) are good examples. According to Moore (1989), import price, domestic
credit and the percentage of working-age population employed are important sensitive
2
George Moore and the Center for International Business Cycles Research at Columbia University built
two composite leading indicators for inflation in early 1986, according to Roth (1989).
4
indicators for future inflation. Boughton and Branson (1989) conclude that Commodity
prices may lead consumer price when the data are denominated in currency other than
US dollar.
The objective of this paper is to compare two alternative methods to build leading
inflation indicators for Brazil.
Section 2 sets out the whole methodology involved in the construction of the leading
inflation indicator, including variable selection and both estimation procedures to
compose the indicators. Section 3 presents an in-sample analysis, including tuning point
analysis, the Quadratic Probability Score and the lead average and standard deviation
approach. Section 4 introduces and out-of-sample analysis of the selected indicators,
and section 5 concludes.
2. METHODOLOGY
I.
Variable Selection and Treatment 3
In order to build leading inflation indicator for Brazil, more than 200 economic time
series that could anticipate inflation movement were collected and analyzed. They were
selected from different fields like monetary and financial sectors, being monetary
aggregates, interest rates and stock indexes good examples of those. Analogously,
exchange rate, trade balance, public sector net external debt, commodity price indexes
and others were chosen to represent external pressures. In addition, industrial
production indexes, industrial sales, employment rates, wage indexes and others were
selected for being related to the concept of level of activity, while wholesales, aggregate
and consumer price indexes to prices.
A careful research about data quality and reliability was implemented to discard those
series with changes in methodology or collection procedure problems. In addition,
timeliness was also a criterion used for selection4. This investigation proved 115 time
series to be inadequate.
3
4
The methodology described here is similar to the one in Chauvet (1999a, 1999b).
Otherwise, composite index of those series would not be useful to support monetary policy decisions.
5
The series were tested for the presence of unit root5 and transformed to achieve
stationarity6 using first difference, if the tendency were found to be stochastic or
detrended in the deterministic case. The same scale for all these series was achieved
through normalization7.
The series used as the reference variable is the first difference of the National Consumer
Price Index (IPCA8), detrended using an exponential function9.
After this treatment, another data selection was performed using two econometric
procedures. One was to evaluate their ability to Granger-cause inflation. This test was
performed for each lag from two to twelve. Based on these results, 49 time series were
selected according to their predictive content in explaining inflation.
In the latter procedure, variables were categorized according to their lead-lag maximum
cross-correlation with inflation. A maximum correlation10 around lags 0, -6 or 6
corresponded to a coincident, lagging or leading indicator classification. A total of
twenty variables turned out to belong to the leading group11. However, one more
variable was discarded, due to multicolinearity12. These selection criteria yield 19
variables.
II.
Methods to compose the index
The literature presents several ways to combine variables to build a leading indicator.
Webb (1995) and Cifuents (2000) used arithmetic mean, Contador (1999) suggested the
5
It was used Augmented Dickey-Fuller unit root test and Phillipe-Perron test. In order to determine the
number of lag to be used in the test, Akaike information criterion and Schwert (1989) lag-selection rule
were adopted.
6
If any divergence were detected, both procedures were applied meaning that in some cases variables
were duplicated.
7
The series were normalized by subtracting their unconditional mean and dividing them by their standard
deviation.
8
This Index is used as a measure to target inflation
9
Thenceforth, whenever the expression inflation appears it refers to this series.
10
Throughout this paper, the expression maximum correlation refers to its maximum absolute value.
11
These variables presented maximum correlation in the range between 4 to 10 months.
12
It presented correlation above 0.95 with another series.
6
use of weighted mean13 and Chauvet (1999a, 1999b) used a dynamic factor model to
extract the common movement underlying a set of variables that lead inflation. The
resulting factor is a linear combination of the variables used weighted by the estimated
factor loadings.
This work uses two distinct approaches to construct the composite indexes. One is based
on Chauvet (1999a, 1999b). The other uses a linear regression to find the best set of
coefficients to be used as weights on the composition of the indexes.
II.1
The Dynamic Factor Exercise
Chauvet (1999a, 1999b) suggested the combination of four variables to build the
leading inflation indicator based on the idea that each of them would capture different
sources of inflation pressure14. The leading inflation indicators were estimated as an
unobserved variable using the Kalman Filter15. The idea is to extract a common factor
of all component economic time series. Following this rationale, all possible
combinations of four variables with those 19 pre-selected variables were considered16.
The model used was suggested by Chauvet (1999a, 1999b) and is built from a dynamic
factor as described in the following equation, and estimated through Kalman Filter
algorithm:
(1) Yt =  LIIt + t
(2) Lt = LIIt-1 + t
where t ~ i.i.d. N(0,) and t ~ i.i.d. N(0, ). Yt represents a 4x1 vector that contains
the four economic time series previously selected,  is 4x1 vector that measure the
sensibility of the composing series to the leading inflation indicator, t is the 4x1 vector
13
This method suggests the usage of the best correlation coefficient between each variable and lagged
inflation to work as weights.
14
These results were derived from the exam of the dynamic behavior of the components of the IPCA.
External and internal shocks, energy shocks and seasonal changes were identified as the main factors
driving increases in inflation for the Brazilian case.
15
Stock and Watson (1991, 1993) used a similar method to build coincident indicators.
16
Since leading indicators are essentially atheoretical method to anticipate cyclical movements of a
dependent variable, there is no reason for not studying all possible combinations.
7
of measurement errors, and t is the scalar transition shock. LIIt is the unobserved
dynamic factor, i.e. the leading inflation indicator.
The model estimated assumes that  is a diagonal matrix, that is, the errors associated
with each variable on the measurement equation (1) are uncorrelated. In addition, is
uncorrelated with .17
Originally, all possible combinations of variables generated 3876 potential leading
inflation indicators, depending on its predictive power. However, those indexes that
were highly18 correlated to one of the four component series were discarded. The
intention was to avoid obtaining indicators showing the same predictive properties as
the original series. This procedure reduced the number of potential indicators to 690.
In order to analyze the causality relation between each composite index and inflation,
Granger Causality Test was performed for each lag from 2 to 12. Those indicators that
were found not to cause inflation, using 5% as the level of significance for all lags were
discarded.
Afterwards, the correlation coefficients of each indicator with led inflation (from 0 to 12
months) were evaluated. The maximum correlation of each index was used to identify
the lead of the indicator and also to rank them. Those leading indicators presenting
maximum correlation below 0.45 were eliminated.
In the Table 1, the shadowed line represent the number of months that better anticipate
inflation, given the occurrence of maximum correlation.
17
The variance of the factor is set to one to assign a scale to it.
That is, those composite indexes presenting correlation coefficients equal or higher than 98% with any
of its component series.
18
8
Table 1: Correlation between Inflation and Dynamic Factor Indicators
Absolute Value of Correlation
LII_5
LII_6
0.1225
0.1786
0.1446
0.1719
0.2552
0.1934
0.0476
0.0366
0.1591
0.2261
0.3776
0.3735
0.5430
0.4959
0.4606
0.4078
0.1485
0.0795
0.0190
0.0028
0.0318
0.0839
0.0971
0.0511
0.0431
0.0578
Lead
0
1
2
3
4
5
6
7
8
9
10
11
12
LII_7
0.1068
0.1126
0.2429
0.0675
0.1287
0.3704
0.5321
0.4801
0.1588
0.0122
0.0296
0.1150
0.0598
In order to avoid negligible contribution or predominance of variables, indicators whose
correlation with the component variables was below 0.12 or above 0.85 were cast off.
Finally, it was implemented a recursive n-step ahead out of sample stability test
(ROSST) and a stochastic simulation test (SST) to select the most stable indicators. The
first test consists of successive re-estimations of the index through the Kalman Filter by
subtracting from the component variables one observation each time until reaching half
of the sample19. Comparison between each re-estimated indicator and the one generated
using the full sample should show similar behavior. Figure 1 shows the 32 re-estimated
indicators.
3
2
1
Out/99
Ago/99
Abr/99
Jun/99
Fev/99
Out/98
Dez/98
Jun/98
Ago/98
Abr/98
Fev/98
Out/97
Dez/97
Ago/97
Abr/97
Jun/97
Fev/97
Out/96
Dez/96
Ago/96
Abr/96
Jun/96
Fev/96
Dez/95
Out/95
Ago/95
Abr/95
Jun/95
Fev/95
Out/94
Dez/94
Ago/94
0
-1
-2
-3
-4
Figure 1: Stability Test of LII_7
19
Which means 32 observations, i.e. 32 re-estimation for each indicator.
9
The idea behind SST is to identify and discard those indicators that might present a
different behavior when they are re-estimated incorporating a new simulated
observation. This simulation was made by adding to the last observation of each
component positive and negative shocks based on their measurement errors. For each
indicator all possible combinations of shocks were implemented assuming just one level
of shock one-step ahead. All the procedures described above yielded three leading
inflation indicators.
The following figure presents two of these indicators and their relation with inflation
phases. The shadowed areas represent periods of growing inflation. The indicators were
lagged on 6 months in order to compare their movements with inflation.
6 Months Leaded Kalman Filter Indicators and Inflation
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
Mar-95
Apr-95
May-95
Jun-95
Jul-95
Aug-95
Sep-95
Oct-95
Nov-95
Dec-95
Jan-96
Feb-96
Mar-96
Apr-96
May-96
Jun-96
Jul-96
Aug-96
Sep-96
Oct-96
Nov-96
Dec-96
Jan-97
Feb-97
Mar-97
Apr-97
May-97
Jun-97
Jul-97
Aug-97
Sep-97
Oct-97
Nov-97
Dec-97
Jan-98
Feb-98
Mar-98
Apr-98
May-98
Jun-98
Jul-98
Aug-98
Sep-98
Oct-98
Nov-98
Dec-98
Jan-99
Feb-99
Mar-99
Apr-99
May-99
Jun-99
Jul-99
Aug-99
Sep-99
Oct-99
Nov-99
Dec-99
Jan-00
Feb-00
Mar-00
Apr-00
May-00
-3
LI 5
LI 7
IPCA
Figure 2: Comparison between inflation and the Dynamic Factor indicators
II.2
Regression Exercise20
As an alternative to the dynamic factor model, a weighted mean approach was
implemented to combine the same 19 variables and generate composite indexes.
20
Maher (1957) firstly used regression to estimate leading indicators.
10
The literature proposes many different ways to build leading indicators using weighted
mean. The rationale used to define the weights may involve economic discretionary or
statistical relations. In this work, it was adopted the second approach.
Using OLS, inflation was regressed against lagged leading series as a way to find the
best statistical weights. This procedure incorporates not only an alternative
methodology to compose the leading index, but also the intention to impose causality
between the indicator and inflation.
The 19 pre-selected variables were combined in groups of four, keeping the rationale
above mentioned. This also allows comparison between the indicators yielded from this
method with those from the dynamic factor model.
The weights were estimated by the following model:
(3) t = Yt-k + t
t ~ N(0, 2)
where Yt represents a 4x1 vector that contains the four economic time series previously
selected,  is 1x4 weight vector and t is a scalar error.
The indicator is built using the following equation: 21
(4) LSt = Yt
where LSt is the leading inflation indicator.
Similarly to the other exercise, all possible combinations of the variables were
estimated. For each combination of variables, 12 different regressions were run (one for
each lag k from 1 to 12). This means that model (3) was run 46512 times.
21
Recall that the series were normalized to have zero mean and unit standard deviation. Therefore, a
constant term was not included.
11
For each set of 12 estimations generated from a combination of variables, the best lead
relation (k) was chosen based on the maximum correlation between LSt-k and t. Then,
the indexes were grouped by their best lead relation (1 to 12) and ranked according to
the their correlation with t within each group. Those leading indicators presenting
correlation below 0.55 were eliminated.
In the Table 2, the shadowed line represent the number of months that better anticipate
inflation, given the occurrence of maximum correlation.
Table 2: Absolute Value of Correlations between the Regression Indicators and Inflation
Lead
0
1
2
3
4
5
6
7
8
9
10
11
12
Absolute Value of Correlations between Regression Indicators and Inflation
LS_2
LS_6
LS_1
LS_3
LS_4
LS_5
0.1227
0.1349
0.1177
0.1112
0.0278
0.0347
0.1300
0.1489
0.1900
0.1425
0.2458
0.2085
0.3439
0.3333
0.3312
0.3337
0.4663
0.3356
0.2591
0.2319
0.2116
0.2325
0.3436
0.2529
0.2314
0.2126
0.1868
0.1360
0.4476
0.1844
0.3534
0.3198
0.2849
0.3054
0.5134
0.2923
0.5550
0.4822
0.4541
0.4483
0.4982
0.4535
0.6233
0.5750
0.5671
0.5931
0.6695
0.5776
0.3618
0.3657
0.3557
0.3526
0.4891
0.3057
0.2126
0.1745
0.2137
0.1269
0.2066
0.3116
0.2943
0.2451
0.1806
0.2560
0.2353
0.3682
0.1782
0.1766
0.2058
0.1801
0.1353
0.1721
0.2707
0.2721
0.2678
0.2813
0.2150
0.3075
LS_7
0.0084
0.2505
0.4077
0.3752
0.5079
0.4549
0.4380
0.5893
0.2917
0.1909
0.3313
0.1437
0.1432
It is interesting to emphasize that the most correlated indexes with inflation belonged to
the groups presenting best lead relation within the range 4 to 10 months. This result is
consistent because the selected component variables also showed maximum correlation
with inflation within the same range.
Differently from the dynamic factor exercise, Granger Causality Test 22 was not useful to
eliminate any of the candidate indicators. It was quite expected given that this
construction method itself implicitly imposes causality relation.
Aiming to avoid negligible contribution or predominance of a variable, the indicators
that presented coefficients three times larger than any of the others were rejected.
22
The level of significance chosen was 5%.
12
In order to analyze the stability of the regressions coefficients, it was applied a modified
recursive least square test23. Its different approach relies on the fact that it compares the
coefficients of each re-estimation with those obtained using full sample. Based on these
results a measure of total dispersion was created to rank the candidate leading
indicators. Similarly to the Dynamic Factor exercise, ROSST was also applied. Finally,
seven leading inflation indicators came out of the whole procedures outlined above.
The following figure presents two of these indicators and their relation with inflation
phases. The shadowed areas represent periods of growing inflation. The indicators were
lagged on 7 months in order to compare their movements with inflation.
In f la t io n a n d 7 M o n t h s L a g g e d In d ic a t o r s
2
1 .5
1
0 .5
0
- 0 .5
-1
-2
M a r /9 5
A b r /9 5
M a i/9 5
J u n /9 5
J u l/9 5
A g o /9 5
S e t/9 5
O u t/9 5
N o v /9 5
D e z /9 5
J a n /9 6
F e v /9 6
M a r /9 6
A b r /9 6
M a i/9 6
J u n /9 6
J u l/9 6
A g o /9 6
S e t/9 6
O u t/9 6
N o v /9 6
D e z /9 6
J a n /9 7
F e v /9 7
M a r /9 7
A b r /9 7
M a i/9 7
J u n /9 7
J u l/9 7
A g o /9 7
S e t/9 7
O u t/9 7
N o v /9 7
D e z /9 7
J a n /9 8
F e v /9 8
M a r /9 8
A b r /9 8
M a i/9 8
J u n /9 8
J u l/9 8
A g o /9 8
S e t/9 8
O u t/9 8
N o v /9 8
D e z /9 8
J a n /9 9
F e v /9 9
M a r /9 9
A b r /9 9
M a i/9 9
J u n /9 9
J u l/9 9
A g o /9 9
S e t/9 9
O u t/9 9
N o v /9 9
D e z /9 9
J a n /0 0
F e v /0 0
M a r /0 0
A b r /0 0
M a i/0 0
- 1 .5
In f la t io n
LS_2
LS_6
Figure 3: Comparison between Regression Indicators and Inflation
3. LEADING INFLATION INDICATORS: IN-SAMPLE EVALUATION
As stressed in the introduction of this paper, the main purpose of a leading indicator is
to provide qualitative inference about the future state of a variable to be used as an early
alert. In this sense, turning point analysis of both the leading inflation indicator and the
inflation itself is an essential first step. The beginning of downturns and upturns periods
are marked by peaks and troughs based on which a binary variable is constructed. These
are used to calculate the QPS (Quadratic Probability Score) that is one approach used to
determine the lead relation between the composite indicator and inflation. In addition,
13
the procedure to anticipate inflation movements is based on the behavior of this binary
variable constructed for the composite indicator. In other words, the objective is to
evaluate whether the indicators signal future cyclical inflation phases, based on a
dichotomous version of the mean squared error.
I.
Turning Point Analysis24
After the indicators were obtained, a turning point analysis was implemented to
compare the performance of the two sets of indicators. The method to identify turning
points of the composite indicators and inflation takes into account both duration and
amplitude of cyclical changes in the series. Obviously, the implications of not properly
taking this into consideration are to miss or to wrongly identify a turning point. It is
essential to distinguish a turning point from minor changes in the series.
Techniques applied in this paper can be summarized by the use of two-state Markov
switching model as proposed by Hamilton (1989) and Hodrick-Prescott Filter. In
addition, , some binding rules were established as an attempt to minimize turning point
errors regarding future inflation states. The existence of an interval narrows the
flexibility of the decision about the existence of a turning point.
The following steps were implemented to determine turning points in the leading
indicators:
 Thresholds based on the frequency distribution of the indicators were used as upper
and lower bounds to identify peaks and troughs. This allows evaluation of the
significance of changes in the series based on their amplitude. An observation was
considered a turning point if it surpassed the upper or the lower limits of the
threshold. Assuming that x is the vector nx1 of the estimated leading index or
inflation25, the upper and lower limits are defined as:
23
This test is a standard stability test available in Eviews. The results are series of coefficients that were
generated on the re-estimation. Its implementation considered half of the sample.
24
The whole procedure described here was suggested by Chauvet (2000a). Complementary approaches
can be found on Neftci (1991), Pagan (1991), Stekler (1991), Webb (1991), Wecker (1979).
25
Some criteria use the growth of the series instead of its values.
14
(5) UL  x  0.5   x
(6) UL  x  0.5   x
where x is the unconditional mean of the series, x is the standard deviation, UL is
the upper limit, and LL is the lower bound.
Considering that the above construction of the thresholds incorporate x, it is
important to examine the dynamics of the volatility of the series in the sample. If
volatility changes over time, it is important to take this into consideration when
building the thresholds. In this way the thresholds can be flexible enough to capture
major changes in the amplitude of series due to potential changes in regimes..
 The behavior of the volatility of the indicators was investigated to determine whether
there exist in the sample different volatility phases that could affect evaluation of the
thresholds. This analysis is based on the evidence of pulse structural breaks in the
sample corresponding to the periods around recent currency crisis. To determine the
potential different volatility phases, Hamilton’s (1989) Markov switching model
was fitted to the leading indicators:
xt - st = (xt-1 - st-1) + st
st ~ N(0, st2 ), || < 1,
st =1,2
where xt are the leading indicators. The volatility of the indicator, st2 , may be in a
phase of high (st = 1) or low (st = 2)26, according to the transition probabilities ruling
changes in the Markov chain, st:
P(st = 1| st-1 = 1) = p11
P(st = 2| st-1 = 1) = p12
P(st = 1| st-1 = 2) = p21
p11 + p12 + p21 + p22 = 1
P(st = 2| st-1 = 2) = p22
Changes in the term st are not significant and are dominated by changes in the volatility phase. This is
partially due to the structural break in volatility during periods of currency crisis in Brazil such as in the
26
15
 Upon determination of the different volatility phases, the “switching thresholds”
were applied to identify turning points in the leading indicators.
 Finally, the leading indicators were smoothed using the H-P filter27 as an additional
tool to help the decision of ambiguous turning points. One of the rules adopted to
determine turning points is that changes in the direction of the leading indicators
should last at least two months.28
Figure 429 shows the turning point analysis and the whole set of instruments applied to
one of the leading indicators. The shadowed areas represent increasing phases of the
indicator and the dashed lines the thresholds for different periods of volatility.
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
Nov-99
Sep-99
May-99
Jul-99
Jan-99
Mar-99
Nov-98
Sep-98
Jul-98
May-98
Mar-98
Nov-97
H-P
Jan-98
Sep-97
May-97
Jul-97
Lower Limit
Mar-97
Nov-96
Jan-97
Sep-96
May-96
Jul-96
Mar-96
Nov-95
Jan-96
Sep-95
Jul-95
May-95
Jan-95
Mar-95
Nov-94
Sep-94
L5
Upper Limit
Figure 4: Turning Point Analysis
II.
Quadratic Probability Score (QPS)
The QPS measures the accuracy between turning points of the indicator and the ones
identified for inflation, given an existing lag between these two series. Diebold and
beginning of 1999. In addition, this result is due to the fact that the indicators were estimated using
normalized variables (zero mean, unit variance).
27
An advantage of the H-P filter is that it does not distort the timing of turning points, given that it is
zero-phase filter.
28
Since the objective is real time forecast of turning points due to both cyclical and seasonal changes, we
use this two-month rule-of-thumb for change in the indicators. This has been proving to be a reasonable
criterion for the Brazilian inflation process. In addition, a lower frequency rule-of-thumb would
compromise real time monthly forecasting of inflation phases.
29
The second upturn phase was identified despite of the fact that its peak is below the upper limit.
16
Rudebusch (1989) suggest this methodology as a way to evaluate leading indicator
turning point forecasts.
The QPS is a counterpart of the Mean Squared Errors and is defined as follows:
(7)
QPS 
2 T
 Ft  S t  
T t 1
where Ft and St are 0/1 variables for inflation and indicator, respectively30. The phases of
the leading composites and inflation are represented by variables taking the value of one
for upturn phases and the value of zero for downturn phases31.
The QPS range varies from 0 to 2, being 0 the perfect accuracy. This method allows
comparison of the different indicators.
Furthermore, the QPS can also be used to identify the best lead relation between an
indicator and inflation32. This result is obtained using lagged indicator and (from 0 to 12
months) inflation. The smallest QPS indicates the best lead relation.
The tables below present the QPS calculated for each of the leading indicators obtained
in the exercises. It is important to highlight that the lowest QPS for all indicators
generated by the dynamic factor exercise occurred at the lag 6, which means that these
indicators best anticipate inflation six months ahead. It is a reassuring result, given that
this is the same conclusion found using maximum correlation between the indicator and
lagged inflation.
The same approach was applied to the indicators obtained on the regression exercise.
For all indicators, the smallest QPS occurred at lag 7, except for two of them. This
result is similar to the one obtained previously. The maximum correlation occurred at
lag 7 for all indicators33, as shown on Table 2.
30
See Chauvet (1999b) and Webb (1991).
Peaks are included in the upturn phase while troughs are included in the downturn phase.
32
Although the best lead relation was already identified, the QPS would not be neglected as an instrument
for this purpose.
31
17
The following tables show the results of the QPS for both exercises. The shadowed cells
represent the minimal QPS for each indicator, identifying its best lead relation with
inflation.
Table 3: QPS of Regression Indicators
0
1
2
3
4
5
6
7
8
9
10
11
12
LS 1
1.1642
1.1212
1.1385
1.0000
0.9206
0.6129
0.4918
0.5333
0.9492
1.3103
1.4737
1.3214
1.0182
LS 2
1.1045
1.0000
1.0154
1.0625
1.1111
0.8065
0.4918
0.3333
0.7458
1.1724
1.4737
1.4643
1.2364
LS 3
1.1940
1.0909
1.0462
1.0313
1.0159
0.7742
0.5246
0.4333
0.7797
1.2069
1.4386
1.4286
1.2000
QPS
LS 4
0.9851
0.9394
0.9538
1.0938
1.1111
0.9355
0.6885
0.4667
0.7458
1.0345
1.1579
1.1786
1.0545
LS 5
1.0746
1.0909
1.1692
1.0938
1.0794
0.8387
0.7213
0.7000
0.9831
1.2759
1.3684
1.2857
1.0545
LS 6
1.1940
1.0909
1.1077
1.0313
1.0159
0.7097
0.4590
0.4333
0.8475
1.2759
1.4386
1.3571
1.1273
LS 7
1.0448
1.1212
1.2000
1.1563
0.9206
0.6452
0.4590
0.5333
0.9153
1.3103
1.4386
1.2857
0.9818
Table 4: QPS of the Dynamic Factor Indicators
0
1
2
3
4
5
6
7
8
9
10
11
12
LII 5
1.0448
1.0000
1.0462
0.9688
0.9206
0.6774
0.4918
0.6000
0.9831
1.3448
1.4035
1.2143
0.9455
QPS
LII 6
1.0149
1.0909
1.1385
1.0000
0.8254
0.5806
0.5246
0.7667
1.2203
1.4483
1.3684
1.1429
0.8364
LII 7
0.9851
0.9394
1.0462
1.0313
0.9841
0.7419
0.5574
0.6667
1.0169
1.3103
1.3333
1.1429
0.8727
Comparing the QPS presented in the tables 3 and 4, it is possible to rank the indicators.
Those generated by the regression exercise seem to be slightly superior to the ones
produced by the dynamic factor model.
33
Even though two of the indicators built showed different lead relation when distinct methodologies
were applied, the results were close (6 months instead of 7).
18
III.
Lead Average and Standard Deviation Approach
Analysis of turning point accuracy was also performed through examination of turning
point errors. It was performed the comparison between the turning points of the leading
indicators and those identified for inflation. This procedure intends to measure the
consistency of the leading indicator by avoiding the occurrence of three situations:
1. The indicator misses any inflation turning point;
2. The indicator overestimates the number of turning points;
3. The indicator lead relation with inflation varies intensively.
The following tables present the results obtained from this analysis. For each inflation
turning point it shows the number of lead months in which the leading indicators
anticipated this event. The blank cells represent that the indicator missed the inflation
turning point. Only one leading indicator overestimated the number of turning points.
The indicator LS_4, from the regression exercise, identified one false trough and one
false peak.
Table 5: Lead of each Inflation Turning Point
Dynamic factor Exercise
LI 5
LI 6
5
1995.5 (P)
5
5
1995.8 (T)
1995.12 (P)
6
7
1996.3 (T)
7
6
1996.6 (P)
6
4
1996.9 (T)
4
3
1997.1 (P)
1
5
1997.8 (T)
5
6
1998.1 (P)
6
7
1998.8 (T)
7
6
1999.3 (P)
6
7
1999.6 (T)
7
8
1999.10 (P)
3
5.8
Mean
5.3
6.0
Median
6.0
1.4
St Deviation
1.8
LI 7
5
5
7
6
4
1
5
6
7
6
7
8
5.6
6.0
1.8
19
Table 6: Lead of each Inflation Turning Point
1995.5 (P)
1995.8 (T)
1995.12 (P)
1996.3 (T)
1996.6 (P)
1996.9 (T)
1997.1 (P)
1997.8 (T)
1998.1 (P)
1998.8 (T)
1999.3 (P)
1999.6 (T)
1999.10 (P)
Mean
Median
St Deviation
LS 1
5
7
7
6
7
3
5
6
7
7
5
3
5.7
6.0
1.5
Regression Exercise
LS 2
LS 3
LS 4
7
8
6
7
7
7
7
7
7
7
7
6
6
6
7
7
7
3
3
7
7
4
7
6
6
6
7
7
7
6
6
3
7
5
2
3
3
3
6.1
5.8
5.8
7.0
6.0
7.0
1.5
1.6
1.8
LS 6
7
7
LS 7
6
7
5
3
7
6
7
3
7
6
7
6
5
3
8
6
4
3
5
6
7
6
5
7
5.8
6.0
1.6
5.9
6.5
1.6
5.8
6.0
1.4
LS 5
5
7
8
7
6
5
3
7
6
7
It was calculated the mean, median and standard deviation of the turning points
anticipation for each indicator. Those presenting the most similar mean and median to
the pre-identified lead relation and the lowest standard deviation were selected as the
most consistent ones. This analysis was used to select two indicators from each
exercise.
It is interesting to emphasize, that in both cases, the leading indicators chosen were
exactly those that presented the smallest QPS. This evidences consistency between the
two constructing methods developed here.
4. OUT-OF-SAMPLE ANALYSIS
The peculiarity concerning out-of-sample tests for leading indicators is the fact that
these indexes are not used to predict magnitude of the dependent variable, but to
anticipate its cyclical movements.
20
However, selected indicators are expected to perform well in a qualitative out-of-sample
test because previous analysis34 had already attested to their stable behavior and
accuracy to anticipate inflation turning points.
An out-of-sample test was performed as following:

The indicators were successively re-estimated adding each time one observation to
the component variables. This procedure started taking half of the sample (1994.8 to
1997.2).

It was built an out-of-sample index where the first half of the sample came from the
full-sample estimated indicator and the other half was composed from the last
observation of each re-estimated indicator35.

The same turning point analysis is applied to the out-of-sample index to verify its
ability to forecast inflation turning points. Its forecasting performance is then
compared to the indicator obtained using the full sample.
The out-of-sample analysis of the four selected leading indicators is consistent with
those of the in-sample tests, as it can be concluded from the Table 7. It indicates
robustness reinforcing reliability of their future performance.
Figure 5 shows the comparison between the out-of-sample and the full sample indexes
applied to LS_2. The shadowed areas identify the growing phases of the indicator
estimated with full sample. The out-of-sample index presents the same growing phases.
Table 7 shows the number of lead months in which the out of sample index anticipate
inflation turning points.
34
35
Out-of-sample stability tests.
It is a time series built from the recursively estimated indicators.
21
LS_2 Out-of-Sample Analysis
2
1.5
1
0.5
0
-0.5
-1
-1.5
LS_2
Out-of-Sample
Figure 5: Out-of-Sample Analysis
Table 7: Out-of-Sample Analysis for the Indicators
1997.8 (T)
1998.1 (P)
1998.8 (T)
1999.3 (P)
1999.6 (T)
1999.10 (P)
Out-of-Sample Analysis
LI 4
LI 7
LS 2
5
5
6
6
8
7
6
6
7
7
9
8
LS 6
7
6
7
6
7
3
7
6
7
6
5
3
4. CONCLUSIONS
The main purpose of this exercise was to build composite leading inflation indicators
and compare their efficiency using two different methodologies.
First, 19 economic time series were selected and combined in groups of four variables
to build the composite leading indicators. This exercise presented the analysis of all
possible combination of the selected variable, focusing on those that better forecasted
inflation turning points.
22
Oct-99
Jun-99
Aug-99
Apr-99
Feb-99
Oct-98
Dec-98
Jun-98
Aug-98
Apr-98
Feb-98
Oct-97
Dec-97
Jun-97
Aug-97
Apr-97
Feb-97
Oct-96
Dec-96
Jun-96
Aug-96
Apr-96
Feb-96
Dec-95
Oct-95
Jun-95
Aug-95
Apr-95
Feb-95
Oct-94
Dec-94
Aug-94
-2
The first methodology used to compose the leading index was the dynamic factor,
finding the dynamic correlation underlying four economic time series. In the second
exercise, the best weight combinations of the components were evaluated considering
the highest correlation with inflation. In this approach inflation was regressed against all
possible combinations of variables using OLS.
Both methods generated a large number of indicators that were selected through several
tests such as the recursive one-step ahead out-of-sample stability test (ROSST) and the
stochastic simulation test (SST).
Four leading indicators came out of these analyses. Two built from the dynamic factor
exercise and two from the regression exercise. All these leading indicators showed to be
very efficient and consistent.
It is important to stress that the sample size that could be used to implement the analysis
is still very small to be confident on the statistic tests. This is also true regarding
spectral analysis. This fact is intensified by structural changes in the Brazilian economy
such as the currency crisis in January of 1999. Additionally, it remains to be verified
how the selected leading indicators would perform in a real time forecasting exercise.
Nevertheless, there are some good conclusions from the analysis of the selected indexes
that should be highlighted:
1. The lead-lag relation was identified using maximum correlation, QPS and lead
average methodologies. All of them yielded similar results: 6 months to the dynamic
factor model indicators and 7 months to the regression ones;
2. The correlations found between inflation and the composite leading indexes are
much higher than those found between inflation and any other economic time series.
3. The QPS analysis shows that the chosen composite indexes performed well in
anticipating turning points in-sample.
23
4. The selected indicators were found to be stable through time in spite of the sample
size and structural break.
5. The out-of-sample analysis shows a similar result for the indicators built using both
methodologies. There was no occurrence of missing turning points or shifts on their
lead identification.
The comparison between the two methodologies does not allow concluding that one is
superior to the other regarding the fact that both yielded similar results.
24
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26