Coupled criticality analysis of inflation and unemployment

In this paper, we are interested to focus on the critical periods in the economy which are characterized by large fluctuations in macroeconomic indicators. To capture unusual and large fluctuations of inflation and unemployment, we concentrate on the non-Gaussianity of their distributions. To this aim, by using the coupled multifractal approach, we analyze US data for a period of 70 years from 1948 until 2018 and measure the non-Gausianity of the distributions. Then, we investigate how the non-Gaussianity of the variables affects the coupling structure of them. By applying the multifractal method, one can see that the non-Gaussianity depends on the scales. While the non-Gaussianity of unemployment is noticeable only for periods smaller than 1 year and for longer periods tends to Gaussian behavior, the non-Gaussianities of inflation persist for all time scales. Also, it is observed that the coupling structure of these variables tends to a Gaussian behavior after $2$ years.


I. INTRODUCTION
Unemployment and inflation are two important economic indices; their relation is meaningful for policy makers. Historically, there have been hot debates over the relation between unemployment and inflation, -starting with the observation of Phillips [1].
Later, the huge debate was ignited about the strength and causal effect of the relation [2][3][4]. The matter has been important because of its influence on government policies during the times of recession, but at other times as well [5,6]. Discussion over the matter is still a serious controversial issue in economics, see for example [7][8][9].
Yet, few works have considered the theoretical complexity of these variables, thus leading to the research questions of this paper [10].
Economy can be considered as a big network of heterogenous agents which interact with each other and with their environment [18][19][20][21][22][23][24][25][26][27]. It is reasonable to expect that inflation and unemployment as outcomes of these complex systems inherit complexity considerations.
This suggests a thorough analysis of these variables along advanced techniques available in complexity theory approach and applications [28][29][30].
Looking at unemployment and inflation indices as simple variables may ignore some rich knowledge about their complexity. It has been shown for example that economic indicators and their coupling have nice scaling features [10,31,32]. In fact, many economic variables present some multifractal nature [33][34][35]. Such an observation suggests that inflation and unemployment data sets might exhibit non-Gaussian probability density function (PDF).
This behavior may originate from the occurrence of large fluctuations in the system at extreme values. In order to model the non-Gaussianity of some signal, a multifractal random walk (MRW) model has already been implemented; MRW is composed of the product of two processes with normal and log-normal PDFs [36,37]. The variance of the log-normal part determines the strength of non-Gaussianity of the original signal [38][39][40].
In order to go further, i.e. to capture the non-Gaussianity in the coupling of variables, we apply the bivariate Multifractal Random Walk (Bi-MRW) technique [41,42]. Therefore, this approach leads us to obtain and to analyze the type of cross-correlations between large fluctuations in inflation and unemployment measures. Whence, the paper is organized as follows. In section 2, we explain the MRW and Bi-MRW methods; in section 3 and section 4, we present our findings and conclusions respectively.

A. Multifractal Random Walk (MRW)
The Multifractal Random Walk (MRW) for analysing time series stems from turbulent cascade models [38]. These processes are useful for presenting the non-Gaussian behavior of time series.
The temporal fluctuations increment of a process at scale s , δ s x(t) = x(t + s) − x(t), can be modeled by the product of a normal and a log-normal process: where ǫ s (t) and ω s (t) are normal processes with zero mean and standard deviations equal to σ(s) and λ(s) respectively.
Based on Eq.(1), we can write a probability density function for δ s x(t) as: where and since λ 2 (s) is the variance of the log-normal part of the process. This parameter is the representative measure of the non-Gaussianity of the process; if λ 2 (s) → 0, the PDF of δ s x(t) converges to a Gaussian distribution. An estimation of λ 2 (s) versus a scale s is our way for presenting the effect of large fluctuations over different scales. Furthermore, for showing the effects of the rare events (in the PDF tails), high order moments of fluctuations of order q, denoted by m q , can be calculated of q, the process is called monofractal; otherwise, it is called a multifractal [43].

B. Joint Multifractal Approach: the Bi-MRW method
Muzy et al. [41,42] have proposed the bivariate Multifractal Random Walk (Bi-MRW) method for analyzing two non-Gaussian stochastic processes ( taneously, when the increments of each time series are supposed to be generated by the product of normal and log-normal processes: in which (ǫ 2 ) have joint normal PDF with zero mean. Muzy et al., generalizing the MRW approach, consider the cross-correlation of stochastic variances of two processes [41]. Practically, (ǫ where Σ(s) = ρ ǫ (s)σ 1 (s)σ 2 (s) [41].
The covariance matrix of (ω is denoted by Λ (s) and called a multifractal matrix [41]; it is given by where Λ(s) = ρ ω (s)λ 1 (s)λ 2 (s) and ρ ω (s) is the multifractal correlation coefficient. The PDFs of (ǫ 2 ) have the following form: Therefore, the joint PDF of the fluctuations increment vector (δ s x 1 , δ s x 2 ) is given by P s (δ s x 1 , δ s x 2 ) = d(ln σ 1 (s)) d(ln σ 2 (s))G s (ln σ 1 (s), ln σ 2 (s)) It follows from the above definitions of G s (ln σ 1 (s), ln σ 2 (s)) and F s δsx 1 and It can be observed that P s (δ s x 1 , δ s x 2 ) becomes equal to P s (δ s x 1 )P s (δ s x 2 ) when Λ(s) and Σ(s) tend to zero.
The q-th order moment of fluctuations increments for such two processes at scale s can be written as III. RESULTS Thereafter, we can analyze both inflation and unemployment rates, provided by the U.S. where P data (δ s x) is the joint PDF computed from data, while P theory (δ s x; Λ(s), Σ(s)) is the theoretical joint PDF proposed in Eq. (11).
The best value of Λ(s) for the theoretical joint PDF is obtained from the fit of the joint PDF to the data: The parameter λ 2 (s) is depicted for inflation, λ 2 1 (s), and unemployment, λ 2 2 (s), in Fig. 1. It is seen that λ 2 1 (s) is large at all scales, whereas λ 2 2 (s) is large at scales smaller than one year. Large values of λ 2 1 (s) imply that rare events occurring in the inflation rate make its PDF non-Gaussian. For unemployment, λ 2 2 (s) tends to zero at scales larger than one year. This scaling dependency of λ 2 2 (s) implies that the occurrence of rare events in unemployment provides a non-Gaussian behavior at short time scales, but after one year it tends to a Gaussian state.
Concerning the joint multifractal coefficient Λ(s), see Fig. 1, we observe that it has its highest values for scales lower than one year. Thereafter, Λ(s) decreases relatively fast over the scales below two years. This is compatible with some beliefs about the effect of inflation on the joint relation in the short run and its ineffectiveness in the long-run [48].
To improve our understanding about the behavior of the rare events, higher moments of the variables' increments have been measured for various orders and various time scales.
Recall that high moments are more influenced by the rare events in the tails of the PDF.
In Fig. 2, color intensity plot of such high moments are depicted for different time scales.
As it can be seen, the value of the high moments of the unemployment rates have their largest values for scales below six months. Above six months the moments drop rapidly.
In contrast to the unemployment case, the value of high moments for the inflation rates is relatively noticeable for all scales but is higher in at scales below 2 years.
The right panel in Fig.2 illustrates that the behavior of the joint moment is more similar to the inflation case where a noticeable reduction can be observed for scales above 2 years.
This means that large fluctuations in inflation and unemployment are more strongly coupled above this time interval.
At the next step, the behavior of high moments of inflation and unemployment are  If the inflation rate likely dominates the coupled relation at the post WWII time, in contrast, the unemployment rate seems to dominate the joint relation in both other cases.

IV. CONCLUSION
Inflation and unemployment are dependent variables with non-Gaussian PDFs; however, the scale and intensity of this dependency and their coupling effects have been, and are still, much debated. The controversy finds its importance when the government aims to impose an expansionary monetary policy over the economic crises. Many researchers have discussed the relation between inflation and unemployment; recall one Friedman [49], among others, [50]. In this work, we were interested in large and rare fluctuations in the measures of these economic variables, in order to observe possible enhancements in themselves and in their coupling. We have focused on the non-Gaussianity of the signals' PDFs and their coupling. The Multifractal Random Walk (MRW) is known as a good approach that detects the non-Gaussianity of PDFs through the parameter λ 2 (s), the variance of the log-normal process. Moreover, the bivariate Multifractal Random Walk (Bi-MRW) method is useful for analyzing two non-Gaussian stochastic processes through the corresponding variance Λ 2 (s).
By analysing 70 years of US data via these techniques, the non-Gaussianity of the PDFs of unemployment and inflation and also their joint relation have been detected. The non-Gaussianity parameter λ 2 (s) of the unemployment rate data is smaller than that for the inflation and that for the coupled relation Λ 2 (s). It is observed that after one year, the behavior of this parameter for the unemployment case tends toward a normal condition, but on the contrary, for the inflation data, the non-Gaussianity parameter persists for all studied scales. The non-Gaussianity of the joint relation decreases to small values for scales above 2 years.
Also, this behavior is observable in the high-order moments of these (three) variables.
Moving the observation window over time, it is discovered that the non-Gaussianities of the parameters and of their coupling substantially grow over the critical periods of the economy.