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Multifractal risk measures by Macroeconophysics perspective: The
case of Brazilian inflation dynamics
Leonardo H. S. Fernandesa,*, José W. L. Silvab, Fernando H. A. de Araujob
aDepartment of Economics and Informatics, Federal Rural University of Pernambuco, Serra Talhada, PE
56909-535 Brazil
bDepartment of Statistics and Informatics, Federal Rural University of Pernambuco, Recife, PE 52171-900
Brazil
Abstract
This paper examines the Brazilian inflation indexes dynamics using the multifractal
detrended fluctuations analysis (MF-DFA) and the multifractal detrended cross-correlation
analysis (MF-DCCA). We find that the Brazilian inflation indexes (𝛼0>0.5) and the pairs
of Brazilian Inflation indexes (Δ𝛼 > 0.5) display a persistent multifractal behaviour, high
complexity and skew symmetries. Also, we propose a novel multifractal risk measure (MR)
considering the multifractal cross-correlation measure (MRCC). The higher MR and MRCC
values indicate the more complex and persistent analysed phenomenon. In contrast, the
lowest MR value indicates less complexity and less persistence. From a Macroeconophysics
perspective, our findings clarify that the dynamics of Brazilian inflation indexes and the
pairs of Brazilian inflation indexes genuinely have a robust inertial component that makes
inflation last for a long time.
Keywords: Macroeconophysics, Brazilian inflation indexes; Multifractality;
Cross-correlation; Complexity; Risk measures.
*Corresponding author
Email addresses: leonardo.henrique@ufrpe.br (Leonardo H. S. Fernandes),
josewesley.silva@ufrpe.br (José W. L. Silva), fernando.araujo@ifpb.edu.br (Fernando H. A. de
Araujo)
Preprint submitted to Chaos, Solitons & Fractals July 26, 2022
Electronic copy available at: https://ssrn.com/abstract=4173225
1. Introduction
Based on a historical perspective, the Great Depression (1929) was a break-point with
classical economics paradigms, which forced the emergence of new economic thinking that
focused on economic aggregates (Labour market, Financial market and Goods and services
market) [1,2,3]. A central issue for the Keynes Macroeconomic doctrine is the need for state
intervention to guide economic development [4]. Thus, Keynes’s thesis is a clear opposition
to the self-regulation of the markets.
In this way, one of the mainstream macroeconomic issues is the adequate economic
system to maximize social welfare. We have three different economic systems, each one
with its characteristics. The socialist system was governed by high state intervention. This
economic system promoted overt planning of absolutely all economic production.
In contrast, the capitalist system defends private properties, meritocracy culture, and
the free market [5]. It is important to emphasize that Scandinavian countries have adopted
a hybrid economic system that simultaneously envisions economic growth and development
[6].
Currently, governments intervene in the economy only in extreme situations. Therefore,
they use the technical apparatus of monetary and fiscal policy to expand or retract the
economy.
In the ’90s [7], physicists realized that the financial market was formed by many elements
that present a non-linear dynamic characterized by a self-organized, collective, and emergent
behaviour [8]. Given this, the formalization of Econophysics promoted an evolution in the
investigation and modelling of economic and social systems [9].
Macroeconomic models with a microeconomic foundation and associated with an Econo-
physical perception were proposed to analyze the phenomenology inherent to the basic mech-
anisms related to the distribution of income [10], and wealth [11]. Also, we have investigated
the effects of saving propensity [12], money flow [13,14], and economic growth [15,16].
There was a classic prevailing economic problem related to a question asked by Adam
Smith "Why are there rich and poor nations?" To answer this question, we analyzed the
GDP of 26 countries and concluded that the economies have different economic growth
speeds (EGS) and instantaneous growth accelerations (IGA) [17].
Thus, the economies that present larger (EGS) and (IGA) are necessarily more efficient
in using the benefits arising from economic growth to generate a virtuous cycle of long-term
economic growth. In this same research, we formalize Macroeconophysics [17], which is a
new branch of Econophysics that aims to investigate macroeconomic phenomena.
We proposed Macroeconophysics [18] to provide a more reliable and precise perception
of the complex socio-economic challenges. Considering that Economics is segregated into
Microeconomics and Macroeconomics, it is natural to divide Econophysics into Microecono-
physics and Macroeconophysics [17].
Thus, we solve to examine the fundamentals of Macroeconophysics considering inflation
which is one of the most relevant Macroeconomics variables. Inflation is a permanent and
widespread increase in general prices in an economy [19]. It is a significant concern for
policymakers because of its ability to reduce welfare, discourage economic growth, reduce
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currency purchasing power, heighten economic uncertainty and instability, and discourage
investment [20,21].
Typically, governments set inflation targets to provide economic stability to maintain the
necessary balance between stimulating consumption and investment. If there is an imbalance
in this relationship, this implies complex economic scenarios. Rising inflation leads to an
increase in the interest rate that encourages investment and discourages consumption [22].
Another relevant macroeconomics issue is that the COVID-19 crisis negatively impacted
the dynamics of the worldwide economy [23]. This pandemic has led to abrupt changes
in people’s behavioural and consumption patterns. As a result of measures to contain the
spread of SARS-CoV-2 such as social distancing, staying at home, lockdown [24,25,26],
there was a real need for people to increase their spending on fast food and energy [27,28]
and decrease their spending on transport, travel [29] and oil demand [30].
This research aims to provide new insights into the complex dynamics of Brazilian in-
flation indexes considering a Macroeconophysics perspective. Therefore, we use two multi-
fractal approaches to explore the interplay between the significant Brazilian inflation index
(IPCA) and other Brazilian inflation indexes (IPA, INCC, IGP-DI, IPC, and INPC). This
paper contributed to the literature in several aspects:
(i) it provides the synergy between Macroeconomics and Macroeconophysics;
(ii) it draws new insights into the complex dynamics between the pairs of Brazilian infla-
tion;
(iii) it displays that the Brazilian inflation indexes and all pairs of Brazilian inflation indexes
are characterized by a persistent multifractal behaviour, high complexity and skew
symmetries;
(iv) it shows that the Brazilian inflation indexes and the pairs of Brazilian inflation indexes
are exhibit inertial dynamics;
(v) it proposes a novel multifractal risk measure and the multifractal risk cross-correlation
measure.
The rest of this paper is organized as follows. Section 2, presents the data and the
multifractal methods that we have used in this paper. Section 3, shows and discusses our
empirical results. Section 4, presents our concluding remarks.
2. Data and methodology
2.1. Data
This research examines the macroeconomic time series related to the six Brazilian infla-
tion price indexes (IPA, INCC, IGP-DI, IPC, IPCA, and INPC). These time series present
monthly granularity. The periods cover more than 41 years, from January 1980 until August
2021, with 500 observations for all Brazilian inflation price indexes. We collected the data at
https://www.ipea.gov.br/portal/. Table 1provides details of the Brazilian inflation indexes.
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Table 1: Descriptive list of Brazilian inflation indexes encompassing name and abbreviation
Item Brazilian inflation index Abbreviation
1 Wholesale Price Index IPA
2 National Civil Construction Index INCC
3 General Price Index - Internal Availability IGP-DI
4 Consumer Price Index IPC
5 Broad National Consumer Price Index IPCA
6 National Consumer Price Index INPC
Also, we show the descriptive statistics for the Brazilian inflation indexes and the volatil-
ity time series in Table 2. The skewness is positive for all Brazilian inflation indexes, and
for almost all, kurtosis is negative. The only exception is the IPA.
Table 2: Descriptive statistics of Brazilian inflation indexes.
Series N. Mean Standard
Deviation Median Minimun Maximun Skewness Kurtosis
IGP-DI 500 264.060 263.468 192.135 5.860E-10 1071.615 0.757 -0.290
INCC 500 271.561 268.342 194.387 6.607E-10 939.699 0.645 -0.815
IPA 500 285.524 295.861 192.471 6.233E-10 1305.535 0.925 0.319
IPC 500 203.099 180.356 186.775 1.711E-09 589.620 0.338 -1.108
INPC 500 2011.002 1860.350 1660.755 8.919E-09 6087.840 0.462 -1.018
IPCA 500 1962.417 1805.625 1670.930 8.122E-09 5876.050 0.450 -1.022
For each Brazilian inflation index, we calculate the volatility, which is a valuable measure
to estimate the economic risk [19] that affects the welfare cost [31] and reduce the economic
growth [32] by
𝑉𝑖(𝑡) = |𝑙𝑛𝑃𝑖(𝑡+ Δ𝑡)−𝑙𝑛𝑃𝑖(𝑡)|(1)
where Δ𝑡represents 1 month, 𝑃𝑖(𝑡)is the monthly price inflation index 𝑖at time 𝑡.
2.2. Methodology
We promote Macroeconophysics insights from the Brazilian inflation indexes using two
multifractal approaches. Specifically, we employ the Multifractal Detrended Fluctuation
Analysis (MF-DFA) [33] and the Multifractal Detrended Cross-Correlation Analysis (MF-
DCCA) [34].
2.2.1. Multifractal Detrended Fluctuation Analysis (MF-DFA)
The MF-DFA method aims to investigate the multifractal properties in non-stationary
time series. This approach is a generalization of the classical Detrended Fluctuations Analy-
sis (DFA) method [35]. It has been applied successfully in most diverse areas of Science such
as Macroeconomics [19], Econophysics [36], Finance [37,38], Hydrology [39,40], Energy [41]
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and others [42,43]. It follows a sequence of steps, considering that 𝑍𝑘is a time series of
length 𝑁. Then,
(i) Specific the corresponding profile of the original time series 𝑍𝑘
𝑊(𝑖)≡
𝑖
𝑗=1
[𝑍𝑗−¯
𝑍],(2)
where 𝑖= 1, ..., 𝑁 and ¯
𝑍is the mean value of the time series.
(ii) Segregate the integrated series 𝑊(𝑖)into 𝑁𝑙≡[𝑁/𝑙]non-overlapping segments of equal
length 𝑙.
(iii) For all segments 𝑁𝑙, its computes the local trend by fitting the least squares of the
series. So, the variance without trend is compute with the following expression
𝐹2=1
𝑙
𝑙
𝑖=1
{𝑊[(𝑣−1) 𝑙+ 1] 𝑦𝑣(𝑖)}2,(3)
where 𝑣= 1, ..., 𝑁𝑙and 𝑦𝑣(𝑖)is the polynomial regression fit.
(iv) The fluctuation function to the 𝑞order for a segment of size 𝑙is given by:
𝐹𝑞(𝑙) = ⎧
⎨
⎩
1
𝑁𝑙
𝑁𝑙
𝑣=1 𝐹2(𝑙, 𝑣)𝑞
2⎫
⎬
⎭
1
𝑞
,(4)
where, 𝑞can assume any real value except zero.
(v) Ascertain the scaling behavior concerning the fluctuation functions by examining log-
log plots 𝐹𝑞(𝑙)versus 𝑙for each value of 𝑞. If long-term correlations are present, 𝐹𝑞(𝑙)
will increase with 𝑙as a power law
𝐹𝑞(𝑙)∼𝑙ℎ(𝑞),(5)
where the scaling exponent ℎ(𝑞)is compute as the slope of the linear regression of log
𝐹𝑞(𝑙)versus log 𝑙.
The slope of the log 𝐹𝑞(𝑙)versus log 𝑙can furnish a family of scaling exponents ℎ(𝑞)
also called generalized Hurst exponent. Taking into account the values of ℎ(𝑞)is can be
feasible to uncover whether a process is fractal or multifractal [44]. In this way, we have two
possibilities: (i) ℎ(𝑞) = ℎ, i.e. independent of 𝑞implies in fractal processes and their scaling
behavior is uniquely determined by the constant ℎ[45]. (ii) ℎ(𝑞)not constant implies in
multifractal processes and each moment scales with a different exponent [19].
For multifractal series ℎ(𝑞)is a decreasing function of 𝑞, which in itself represents a sort
of "magnifying glass": negative 𝑞values enhance small fluctuations, while large positive 𝑞
values correspond to large fluctuations [33].
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Moreover, the ℎ(𝑞)obtain through the application of MF-DFA can be related to the
Rényi exponent by
𝜏(𝑞) = 𝑞ℎ(𝑞)−1,(6)
where 𝜏(𝑞)is the the Rényi expoent, 𝑞can admit positive and negative values.
Another measure to study the multifractal properties of a time series is calculated the
singularity spectrum, also called multifractal spectrum 𝑓(𝛼). The multifractal spectrum is
given by
𝑓[𝛼(𝑞, 𝑙)] = 𝑞𝛼(𝑞)−𝜏(𝑞),(7)
where 𝛼(𝑞) = 𝜕 𝜏 (𝑞)
𝜕𝑞
The singularity spectrum describes the multifractal measure in terms of interlaced sets
with singularity force 𝛼where 𝑓(𝛼)is the dimension of the contour subset characterized by
𝛼. For a monofractal structure, the uniqueness of the spectrum produces a single point.
However, for multifractal structures, the uniqueness of the spectrum is provided by a down-
ward concave function, whose degree of multifractality is assessed by 𝑓(𝛼). Given this, to
quantify the complexity inherent of the time series it should be fit the singularity spectrum
to a fourth-degree polynomial [46]
𝑓(𝛼) = 𝐴+𝐵(𝛼−𝛼𝑜) + 𝐶(𝛼−𝛼𝑜)2+𝐷(𝛼−𝛼𝑜)3+𝐸(𝛼−𝛼𝑜)4.(8)
The process related to the distinction of the multifractal spectrum 𝑓(𝛼)quantitatively,
it is recommendable to calculate the width of the spectrum 𝑊(𝛼𝑚𝑎𝑥 −𝛼𝑚𝑖𝑛)achieved from
equating the fitted curve to zero, and the skew parameter 𝑟= (𝛼𝑚𝑎𝑥 −𝛼0)/(𝛼𝑚𝑖𝑛 −𝛼0),
where 𝛼0represents the overall Hurst exponent, 𝑟= 1 for symmetric shapes, 𝑟 > 1for right-
skewed shapes, and 𝑟 < 1for left-skewed shapes. The skew parameter 𝑟displays that the
scaling behaviour of small fluctuations dominates the multifractal behaviour if the spectrum
is right-skewed, and the scaling behaviour of large fluctuations dominates if the spectrum is
left-skewed. These three parameters 𝛼0, 𝑊, 𝑟 can be employed to measure complexity where
a series with a high value of 𝛼0, a wide range 𝑊of scaling exponents, and a right-skewed
shape can be considered more complex than one with the opposite characteristics [47].
Furthermore, we propose a novel multifractal measure also called multifractal risk mea-
sure (MR), which is apply to understand the relation between the 𝑊(width of the spectrum)
and the 𝛼0(persistent or anti-persistent) of the phenomena analyzed. The MR is calculate
by
Λ = 𝑊
𝛼0
(9)
The higher MR value indicates that the more complex and persistent analysed phe-
nomenon. Otherwise, the lowest MR value indicates less complexity and less persistence.
The MR suggest a new way to understand the Brazilian inflation indexes and perform a dy-
namical analysis for these macroeconomic variables taking into account a Macroeconophysics
perspective.
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2.2.2. Multifractal Detrended Cross-Correlation Analysis (MF-DCCA)
The MF-DCCA [34] is a mix model between MF-DFA and DCCA [48]. It is a powerful
approach to explore the multifractal properties considering two non-stationary time series.
It has been useful for applications in Biophysics [49,50], Hydroclimatology [51], Meteorology
[52], Econophysics [53,54]. The MF-DCCA method is implemented through the following
sequence of steps:
(i) Considering two time series denotes by {𝑥𝑖, 𝑦𝑖, 𝑖 = 1,2, ..., 𝑁 },𝑁is the length of time
series. Determine the profile as:
𝑋𝑡=
𝑡
𝑘=1
(𝑥𝑘−¯𝑥), 𝑡 = 1,2, ..., 𝑁 (10)
𝑌𝑡=
𝑡
𝑘=1
(𝑦𝑘−¯𝑦), 𝑡 = 1,2, ..., 𝑁 (11)
where ¯𝑥and ¯𝑦denote the mean of the time series 𝑥𝑡and 𝑦𝑡.
(ii) Divide the time series 𝑋and 𝑌into 𝑁𝑠≡[𝑁/𝑠]non-overlapping segments of equal
length 𝑠, where 𝑠denotes the time scale. Thus, 2𝑁𝑠segments are obtained.
(iii) For each subsegment 𝑣, we apply least squares method to obtain the local trends with
an k-th order polynomial fit.
𝑥𝑣(𝑖) = 𝑎1𝑖𝑘+... +𝑎2𝑖𝑘−1+... +𝑎𝑘𝑖+𝑎𝑘−1, 𝑖 = 1,2, ..., 𝑆;𝑘= 1,2, ... (12)
𝑦𝑣(𝑖) = 𝑏1𝑖𝑘+... +𝑏2𝑖𝑘−1+... +𝑏𝑘𝑖+𝑏𝑘−1, 𝑖 = 1,2, ..., 𝑆;𝑘= 1,2, ... (13)
(iv) Compute the detrended covariance 𝐹2(𝑠, 𝑣). When 𝑣= 1,2, ..., 𝑁𝑠,
𝐹2(𝑠, 𝑣) = 1
𝑠
𝑠
𝑖=1
{|𝑋[(𝑣−1) 𝑠+𝑖]| |𝑌[(𝑣−1) 𝑠+𝑖]−𝑦𝑣(𝑖)|} (14)
When 𝑣=𝑁𝑠+ 1, 𝑁𝑠+ 2, ..., 2𝑁𝑠,
𝐹2(𝑠, 𝑣) = 1
𝑠
𝑠
𝑖=1
{|𝑋[𝑁−(𝑣−𝑁𝑠)𝑠+𝑖]−𝑥𝑣(𝑖)| |𝑌[𝑁−(𝑣−𝑁𝑠)𝑠+𝑖]−𝑦𝑣(𝑖))|}
(15)
(v) Averaging the detrended covariances to obtain the qth-order wave function as
𝐹𝑞(𝑠) = 1
2𝑁𝑠
2𝑁𝑠
𝑣=1 𝐹2(𝑠, 𝑣)𝑞
2
1
𝑞
(16)
(vi) When 𝑞= 0, according to Lopida’s law,
𝐹𝑞(𝑠) = exp 1
2𝑁𝑠
2𝑁𝑠
𝑣=1
ln 𝐹2(𝑠, 𝑣)(17)
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If the scale behavior is verified, the Power-Law correlations must satisfy 𝐹𝑞(𝑠)∝𝑠ℎ𝑥𝑦 (𝑞)·
ℎ𝑥𝑦 (𝑞)represent the Generalized Hurst exponent versus 𝑞the extent of multifractality can
be derived by calculating the range of ℎ𝑥𝑦 (𝑞), a larger Δ𝐻𝑥𝑦 =ℎ𝑥𝑦 (𝑞𝑚𝑖𝑛)−ℎ𝑥𝑦 (𝑞𝑚𝑎𝑥 ))
means stronger multifractal feature.
So if 𝑞= 2, the MF-DCCA becomes the DCCA, if ℎ𝑥𝑦 (2) = 0.5, it reveals that
the two time series have no cross-correlations with each other, when ℎ𝑥𝑦 (2) >0.5, the
cross-correlations are positive persistent, when ℎ𝑥𝑦 (2) <0.5, the cross-correlations are anti-
persistent.
The mass exponent spectrum is defined as,
𝜏𝑥𝑦 (𝑞) = 𝑞ℎ𝑥𝑦 (𝑞)−1(18)
where ℎ𝑥𝑦 (𝑞)is calculated from MF-DCCA. The singularity strength 𝛼𝑥𝑦, which depicts the
singular degree of each segment in a complex system; and the singularity spectrum 𝑓𝑥𝑦 (𝛼)
, which describes fractal dimension of 𝛼𝑥𝑦 are calculate by
𝛼=ℎ𝑥𝑦 (𝑞) + 𝑞ℎ′
𝑥𝑦 (𝑞)(19)
𝑓𝑥𝑦 (𝛼) = 𝑞[𝛼𝑥𝑦 −ℎ𝑥𝑦 (𝑞)] + 1 (20)
The range of the singularity strength Δ𝛼𝑥𝑦 =𝛼𝑥𝑦𝑚𝑎𝑥 −𝛼𝑥𝑦𝑚𝑖𝑛 specific the strength of
multifractality. So, the larger 𝛼𝑥𝑦 reveals a more intense fluctuation.
Moreover, we provide a novel multifractal risk cross-correlation (MRCC) measure, which
is use to examine the relation between the 𝑊(width of the spectrum) and the 𝛼0(persistent
or anti-persistent) of the phenomena analyzed. The MRCC is calculate by
Γ = 𝑊
Δ𝛼(21)
The higher MRCC value suggested that the relation between the two-time series inves-
tigated is more complex and persistent. In contrast, the lowest MRCC value indicates less
complexity and less persistence considering these two-time series. The MRCC presents a
new way to explore the cross-correlation of the Brazilian inflation indexes and perform a dy-
namical analysis for these macroeconomic variables taking into account a Macroeconophysics
perspective.
3. Empirical Results
An economy is an open system where endogenous and exogenous [55] forces generate
price shocks causing them to move up and down permanently. It occurs due to the market
adjustment itself, which Adam Smith defined as the "invisible hand" [56]. However, inflation
is the sudden and continuous increase in the general price level of an economy.
It is unlikely that the market will self-regulate in an economic situation characterized
by inflation: demand, cost, or inertial. That is, inflation is such an extreme event that it
directly undermines a fundamental characteristic of a complex system that is self-organized.
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Initially, we examine the temporal evolution of Brazilian inflation indexes from January 1980
until August 2021. Fig. 1displays the plots of the timeline for these indexes.
0
2000
4000
6000
0 100 200 300 400 500
t (Months)
IPCAi (t)
IPCA
(a)
0
200
400
600
0 100 200 300 400 500
t (Months)
IPCi (t)
IPC
(b)
0
250
500
750
0 100 200 300 400 500
t (Months)
INCCi (t)
INCC
(c)
0
300
600
900
0 100 200 300 400 500
t (Months)
IGPi (t)
IGP
(d)
0
500
1000
0 100 200 300 400 500
t (Months)
IPAi (t)
IPA
(e)
0
2000
4000
6000
0 100 200 300 400 500
t (Months)
INPCi (t)
INPC
(f)
Figure 1: The timeline of the six Brazilian inflation indexes’ monthly price variations time series covers the
period from January 1980 until August 2021.
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We apply the Box plot to examine the extreme situations considering the monthly price
variations for the six Brazilian inflation indexes from January 1980 until August 2021. For
each Brazilian inflation index, Fig. 2depicts the Box plot.
●
●
●
●
0
500
1000
IGP INCC IPA IPC
Indice
(a)
0
2000
4000
6000
INPC IPCA
Indice
(b)
Figure 2: For each time series for the monthly price variations, the Box plots identify the location referring
to 50% of the most probable values, the median and the extreme values for the investigated phenomenon.
It’s a proper nonparametric technique that allows studying and comparing the variable variation between
the different groups.
Then, we employ two multifractal approaches (MF-DFA and MF-DCCA) to analyze
the multifractal characteristics considering each Brazilian inflation index and the pairs of
Brazilian inflation indexes fixed the IPCA, which is the most relevant index. Fig. 3shows
the fluctuation values of the Brazilian indexes and the pairs of Brazilian inflation indexes.
For both multifractal approaches, the inflation increase with segment 𝑠means that the
Brazilian inflation indexes exhibit long-range correlations.
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●●●●
●
Hxy(2) = 1.29
Hxy(4) = 1.08
Hx(2) = 1.22
Hx(4) = 1.02
Hy(2) = 1.24
Hy(4) = 1.06
−8
−6
−4
−2
23456
Log(s)
Log[Fy MF−DFA, Fxy MF−DCCA, Fx MF−DFA](s)
●●●
Fx−IPCA Fxy: MF−DCCA Fy−IPC q−Order ●4 2
(a)
●
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●
●
●●●●
●●
●●
●●●●
●
●
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●●
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●
Hxy(2) = 1.23
Hxy(4) = 1.03
Hx(2) = 1.22
Hx(4) = 1.02
Hy(2) = 1.08
Hy(4) = 0.95
−8
−6
−4
−2
23456
Log(s)
Log[Fy MF−DFA, Fxy MF−DCCA, Fx MF−DFA](s)
●●●
Fx−IPCA Fxy−DCCA Fy−INCC q−Order ●4
(b)
●
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●●
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●
●
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●●
●●
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●
●
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●●
●●
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●
Hxy(2) = 1.27
Hxy(4) = 1.06
Hx(2) = 1.22
Hx(4) = 1.02
Hy(2) = 1.21
Hy(4) = 1.02
−8
−6
−4
−2
23456
Log(s)
Log[Fy MF−DFA, Fxy MF−DCCA, Fx MF−DFA](s)
q−Order ●4 2 ●●●
Fx−IPCA Fxy−DCCA Fy−IGP
(c)
●
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●
●
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●
●
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●
Hxy(2) = 1.27
Hxy(4) = 1.05
Hx(2) = 1.22
Hx(4) = 1.02
Hy(2) = 1.19
Hy(4) = 1.01
−8
−6
−4
−2
23456
Log(s)
Log[Fy MF−DFA, Fxy MF−DCCA, Fx MF−DFA](s)
●●●
Fx−IPCA Fxy−DCCA Fy−IPA q−Order ●4
(d)
●
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●
●
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●●
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●
●
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●●
●●
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●
Hxy(2) = 1.26
Hxy(4) = 1.05
Hx(2) = 1.22
Hx(4) = 1.02
Hy(2) = 1.22
Hy(4) = 1.02
−8
−6
−4
−2
23456
Log(s)
Log[Fy MF−DFA, Fxy MF−DCCA, Fx MF−DFA](s)
q−Order ●4 2 ●●●
Fx−IPCA Fxy−DCCA Fy−INPC
(e)
Figure 3: Power-law scaling in 𝐹𝑥𝑦 , 𝐹𝑥and 𝐹𝑦with respect to 𝑠for 𝑞= 2 and 𝑞= 4.
11
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Also, we calculate 𝛼with 𝑞from a range −10 to 10 to obtain the multifractality quanti-
tatively for each Brazilian inflation index, and the pairs of Brazilian inflation indexes fixed
the IPCA. Fig. 5displays the Generalized Hurst exponent for these phenomena.
12
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●●●●
●
●
●●●●
0.9
1.1
1.3
1.5
369
q−order
α
●
Fx MF−DFA:IPCA Fxy MF−DCCA Fy MF−DFA:IPC
(a)
●●●●
●
●
●●●●
1.0
1.5
3 6 9
q−order
α
●
Fx MF−DFA:IPCA Fxy MF−DCCA Fy MF−DFA:INCC
(b)
●●●
●
●
●
●●●●
0.75
1.00
1.25
1.50
3 6 9
q−order
α
●
Fx MF−DFA:IPCA Fxy MF−DCCA Fy MF−DFA:IGP
(c)
●●●●
●
●
●●●●
0.9
1.2
1.5
1.8
3 6 9
q−order
α
●
Fx MF−DFA:IPCA Fxy MF−DCCA Fy MF−DFA:IPA
(d)
●●●
●
●
●
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0.9
1.1
1.3
1.5
3 6 9
q−order
α
●
Fx MF−DFA:IPCA Fxy MF−DCCA Fy MF−DFA:INPC
(e)
Figure 4: The results of 𝛼as a function of 𝑞between IPCA and Brazilian inflation indices when polynomial
𝑚= 4. The red curves and blue curves are calculated by MF-DFA, black curves are obtained by MF-DCCA.
13
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We present the Generalized Hurst exponent denoted by ℎ(𝑞)for each volatility time
series considering the Brazilian inflation indexes in Table 3.
Table 3: Generalized Hurst Exponent to order 𝑞values in MF-DFA to the volatility time series of Brazilian
inflation indexes.
q IPCA IPC IGP INCC IPA INPC
-10 1.40150 1.36642 1.56358 1.71318 1.68570 1.28815
-8 1.38388 1.34720 1.53906 1.68666 1.66181 1.26730
-6 1.35657 1.31837 1.49859 1.64358 1.62337 1.23797
-4 1.30919 1.27062 1.42444 1.56281 1.55475 1.19648
-2 1.21631 1.17308 1.29489 1.36811 1.41314 1.14339
01.29190 1.22112 1.27578 1.07898 1.24132 1.30537
21.22408 1.23858 1.21157 1.08245 1.19440 1.22084
41.02196 1.06019 1.01980 0.95022 1.01211 1.01748
60.94309 0.98546 0.93896 0.87879 0.93062 0.93793
80.90209 0.94629 0.89642 0.83944 0.88752 0.89658
10 0.87694 0.92211 0.87027 0.81505 0.86105 0.87121
Note that the volatility times series of the Brazilian inflation indexes, the ℎ(𝑞)is de-
creasing a function of 𝑞. Given this, negative 𝑞values enhance small fluctuations, while
large positive 𝑞values correspond to large fluctuations. Moreover, we display the the cross-
correlation Generalized Hurst exponents ℎ𝑥𝑦 (𝑞)for the pairs of volatility times series of the
Brazilian inflation indexes in Table 4.
Table 4: Cross-correlation Generalized Hurst exponent to order 𝑞values in MF-DCCA to Brazilian inflation
indices.
q IPC x IPCA IGP x IPCA INCC x IPA IPA x IPCA INPC x IPCA
-10 1.18436 1.34384 1.28062 1.35253 1.23081
-8 1.16971 1.32124 1.25944 1.32978 1.21616
-6 1.15050 1.28720 1.22782 1.29662 1.19586
-4 1.12566 1.23576 1.17751 1.24793 1.16704
-2 1.10094 1.17525 1.10326 1.18436 1.13557
01.33201 1.34281 1.24554 1.31992 1.35555
21.28588 1.27308 1.23276 1.26700 1.26398
41.07642 1.05601 1.02537 1.05134 1.05079
60.99682 0.97017 0.94131 0.96512 0.96902
80.95601 0.92589 0.89834 0.92063 0.92697
10 0.93108 0.89903 0.87233 0.89367 0.90130
In addition, it can be noted that the pairs of the volatility times series for the Brazilian
inflation indexes ℎ𝑥𝑦 (𝑞)decline with the increase of 𝑞, which indicate that multifractal
14
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properties characterize these time series pairs. Also, we observe that for all the pairs of the
volatility times series for the Brazilian inflation indexes, the values inherent to 𝑞= 2, the
cross-correlation Generalized Hurst exponent are larger than 0.5. It means that the pairs of
the volatility times series for the Brazilian inflation indexes present persistence behaviour.
Table 5show the values of Δ𝐻𝑥𝑦 for the pairs of the volatility times series for the Brazilian
inflation indexes.
Table 5: Values of the Δ𝐻𝑥𝑦 get by MF-DCCA to Brazilian inflation indices.
Cross Correlation ΔHxy
IPC x IPCA 0.25329
IGP x IPCA 0.44481
INCC x IPCA 0.40829
IPA x IPCA 0.45886
INPC x IPCA 0.40829
The values of Δ𝐻𝑥𝑦 for the pairs of the volatility times series for the Brazilian inflation
indexes reveal the degree of multifractality. The pairs of IPA x IPCA display the high value
of Δ𝐻𝑥𝑦.
We apply a fourth-order polynomial regression on the singularity spectrum 𝑓(𝛼)to
obtain the position of 𝛼0and the zeros of the polynomial, 𝛼𝑚𝑎𝑥 and 𝛼𝑚𝑖𝑛, which are employ
to calculate the complexity parameters, more specifically the width of spectrum 𝑊and the
asymmetry parameter 𝑟. Fig. 5presents the multifractal spectrum for 𝐹𝑥, 𝐹𝑦and 𝐹𝑥𝑦
15
Electronic copy available at: https://ssrn.com/abstract=4173225
●
●
●
●
● ●
●
●
●
●
0.00
0.25
0.50
0.75
1.00
0.9 1.1 1.3 1.5
α
f(α)
●
Fx MF−DFA:IPCA Fxy MF−DCCA Fy MF−DFA:IPC
(a)
●
●
●
●
●●
●
●
●
●
0.00
0.25
0.50
0.75
1.00
0.9 1.2 1.5 1.8
α
f(α)
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Fx MF−DFA:IPCA Fxy MF−DCCA Fy MF−DFA:INCC
(b)
●
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●
●
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●
0.00
0.25
0.50
0.75
1.00
0.75 1.00 1.25 1.50
α
f(α)
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Fx MF−DFA:IPCA Fxy MF−DCCA Fy MF−DFA:IGP
(c)
●
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●
●
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●
●
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●
0.00
0.25
0.50
0.75
1.00
0.75 1.00 1.25 1.50 1.75
α
f(α)
●
Fx MF−DFA:IPCA Fxy MF−DCCA Fy MF−DFA:IPA
(d)
●
●
●
●
● ●
●
●
●
●
0.00
0.25
0.50
0.75
1.00
0.9 1.1 1.3 1.5
α
f(α)
●
Fx MF−DFA:IPCA Fxy MF−DCCA Fy MF−DFA:INPC
(e)
Figure 5: Multifractal spectrum 𝑓(𝛼)resulting 𝐹𝑥, 𝐹𝑦and 𝐹𝑥𝑦 , for the period January 1980 to August 2021.
The lines represent regression curves to the fourth order polynomial form.
16
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In this way, we perform a complexity analysis considering the multifractal parameters.
Table 6presents the multifractal parameters for all Brazilian inflation indexes volatility time
series (MF-DFA), more specifically, the 𝛼0,𝑊,𝑟and the novel risk multifractal measure.
Also, it shows the multifractal parameters for all the pairs of the volatility times series for
the Brazilian inflation indexes (MF-DCCA), 𝛼𝑥𝑦(0),𝑊𝑥𝑦,𝑟𝑥𝑦 and the novel risk multifractal
cross-correlation.
Table 6: Multifractal parameters for the volatility time series of Brazilian inflation indexes and the multi-
fractal parameters for the pairs of the volatility time series of Brazilian inflation indexes.
Series 𝛼0W r ΛCross
Correlation 𝛼xy(0)𝑊𝑥𝑦 𝑟𝑥𝑦 Γ
IPC 1.18137 0.61788 0.73572 0.52301 IPCA x IPC 1.04921 0.45452 1.08633 0.43320
IGP 1.36273 1.12096 0.87488 0.82259 IPCA x IGP 1.17477 0.69388 0.80900 0.59065
INCC 1.21970 1.09918 1.18863 0.90119 IPCA x INCC 1.12908 0.59368 0.64561 0.52581
IPA 1.22825 0.80952 0.69656 0.65908 IPCA x IPA 1.14901 0.55593 0.52262 0.48383
INPC 1.49363 1.26711 0.75046 0.84835 IPCA x INPC 1.20971 0.72198 0.75631 0.59682
IPCA 1.19186 0.69563 0.67400 0.58365 - - - - -
The volatility time series of Brazilian inflation indexes display persistence behaviour
𝛼0>0.5, a higher degree of multifractality, the dominance of higher fractal exponents,
and long-term correlations for small and large fluctuations. Based on the complexity risk
measure, we conclude that the INCC presents the higher complexity risk measure (higher
the width of the spectrum and higher 𝛼0).
The values of the 𝑟parameter display that for the almost volatility time series of Brazil-
ian inflation indexes, the multifractality is more influenced by large fluctuations’ scaling be-
haviour dominate the left-skewed spectrum. Given this, the volatility time series of Brazilian
inflation indexes are more persistent to negative shocks than to positive ones. The only ex-
ception is the INCC. Also, the INCC shows a higher value of MR. The INCC is the Brazilian
inflationary index that presents greater complexity and greater risk. These findings are in
line with information asymmetry effects [57,58,59].
Looking into the multifractal parameters of the volatility time series of the pairs of
Brazilian inflation indexes, we find that all the pairs present 𝛼𝑥𝑦(0) >0.5. It implies
that a standard multifractal cross-correlation behaviour is related to persistence dynamics.
Furthermore, we observed that the cross-correlation between the pairs of volatility series of
Brazilian inflation indices led to a lower 𝛼𝑥𝑦(0) than the 𝛼0of each volatility series measured
separately. The lower value ratifies it by the multifractal cross-correlation measure than the
multifractal measure.
Note that for all volatility time series of the pairs of Brazilian inflation indexes, the
values of 𝑊are significantly smaller than the values of this same multifractal parameter for
the volatility time series of Brazilian inflation indexes. Moreover, the values of 𝑟parameter
present that for the almost volatility time series of the pairs of Brazilian inflation indexes,
the multifractality is more influenced by large fluctuations’ scaling behaviour dominates the
left-skewed spectrum exclude the pair IPCA x IPC. Again, we find that the volatility time
series of the pairs of Brazilian inflation indexes are more persistent to negative shocks than
17
Electronic copy available at: https://ssrn.com/abstract=4173225
to positive ones. In addition, the values of MRCC shows that the pair IPCA x INPC exhibits
greater complexity and more significant risk than the other pairs.
4. Conclusion
Based on Macroeconophysics, we explore the complex dynamics of the Brazilian inflation
indexes. Therefore, we use two multifractal approaches (MF-DFA and the MF-DCCA)
considering the time series of these indexes. For each Brazilian inflation index, we calculate
the volatility time series, a proper proxy strategy to reflect the risk in an Economical.
For both multifractal approaches, the results related to our fitting procedure display
that the volatility time series of the Brazilian inflation indexes (𝛼0>0.5) and the volatility
times series of the pairs of Brazilian inflation indexes (Δ𝛼 > 0.5) present overall persistent
behaviour, a higher degree of multifractality, the dominance of higher fractal exponents,
and long-term correlations for small and large fluctuations. Brazilian inflation dynamics are
inertial, resulting from an automatic price feedback process.
Specifically, inertial inflation is formed by two economic components, the trend that
constitutes the autonomous component of self-reproduction and the shock that presents the
part responsible for the variation in the level of inflation.
Our findings indicate that the multifractal parameters for the Brazilian inflation indices
are significantly lower for the volatility time series of the pairs of these indexes. Thus, our
empirical results shed light on a novel way to understand the complex behaviour of one of
the most relevant Macroeconomics phenomena concerning Macroeconophysics.
It is undeniable that economic problems have evolved and encompassed a greater level
of complexity. In this sense, Macroeconophysics effectively seeks elegant and usual solu-
tions to Macroeconomic issues. In addition, we simultaneously propose a multifractal risk
measure and a multifractal cross-correlation measure. In this sense, these measures can
complement policymakers’ technological apparatus in developing more effective measures to
draw inflation targetting, ensuring a stable economy, maximising the welfare state.
Furthermore, in future research, we intend to apply the multifractal risk measure and
the multifractal cross-correlation measure to analyze financial assets and other relevant
macroeconomic variables such as income, wealth, poverty, gross domestic product (GDP).
The contributions of our propositions are not restricted to Macroeconophysical analysis.
They can necessarily be used to investigate the most diverse phenomena of nature.
Other articles by the authors, see [60,61,62,63,64,65,66,67,68,69]
5. Declaration of Competing Interest
The authors declare that this work has no conflicting personal or financial influences.
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