Size-dependent standard deviation for growth rates: Empirical results and theoretical modeling

We study annual logarithmic growth rates $R$ of various economic variables such as exports, imports, and foreign debt. For each of these variables we find that the distributions of $R$ can be approximated by double exponential (Laplace) distributions in the central parts and power-law distributions in the tails. For each of these variables we further find a power-law dependence of the standard deviation $\sigma \left(R\right)$ on the average size of the economic variable with a scaling exponent surprisingly close to that found for the gross domestic product (GDP) [Phys. Rev. Lett. 81, 3275 (1998)]. By analyzing annual logarithmic growth rates $R$ of wages of 161 different occupations, we find a power-law dependence of the standard deviation $\sigma \left(R\right)$ on the average value of the wages with a scaling exponent $\beta \approx 0.14$ close to those found for the growth of exports, imports, debt, and the growth of the GDP. In contrast to these findings, we observe for payroll data collected from 50 states of the USA that the standard deviation $\sigma \left(R\right)$ of the annual logarithmic growth rate $R$ increases monotonically with the average value of payroll. However, also in this case we observe a power-law dependence of $\sigma \left(R\right)$ on the average payroll with a scaling exponent $\beta \approx -0.08$. Based on these observations we propose a stochastic process for multiple cross-correlated variables where for each variable (i) the distribution of logarithmic growth rates decays exponentially in the central part, (ii) the distribution of the logarithmic growth rate decays algebraically in the far tails, and (iii) the standard deviation of the logarithmic growth rate depends algebraically on the average size of the stochastic variable.

Figure 1

Standard deviation $\sigma \left(R\right)$ of the logarithmic growth rate $R$ as a function of the average value $S$ of imports, exports, and debt. We find that, for each of the three economic variables, the standard deviation $\sigma \left(R\right)$ decays algebraically with $S$. Interestingly, all three scaling exponents $\beta$ are similar to each other and surprisingly similar to the scaling exponent $\beta \approx 0.15$ observed for the gross domestic product of countries (GDP) and firms.

Figure 2

Probability distribution of the country-specific scaling exponents $\beta$ for import data. For each country, we estimate $\beta$ by a linear regression of $\ln {(R-⟨R⟩)}^2$ versus $-2\beta \ln S$. We find that the scaling exponents $\beta$ vary strongly from country to country, ranging from strongly negative values smaller than $-2$ to strongly positive values greater than $+2$. Interestingly, the average scaling exponent $\beta \approx 0.16$ is positive and comparable to the value $\beta \sim 0.09$ obtained from the pooled data set of all countries.

Figure 3

Conditional probability distributions $P(R\mid S)$ of the logarithmic growth rate $R$ of exports for two values of $S$. We stratify the whole data set $(R_t^{\left(i\right)},S_t^{\left(i\right)})$ into three equally large subsets according to the values of $S_t^{\left(i\right)}$. The lower curve (filled circles) shows the conditional probability distribution $P(R\mid S)$ for the subset with the greatest values $S$ of exports, $S>8.8\times {10}^9$, and the upper curve (open circles) shows the conditional probability distribution $P(R\mid S)$ for the subset with the smallest values of exports, $S⩽9.2\times {10}^8$. Note that the distributions are shifted for the sake of clarity. We find that the central parts of both of the conditional distributions can be approximated by Laplace distributions centered at the median of $R$. We further find that the standard deviations $\sigma \left(R\right)$ of both distributions are significantly different. Specifically, we find that $\sigma \left(R\right)\sim 0.139$ for the group with the greatest values $S$ of exports is smaller than $\sigma \left(R\right)\sim 0.223$ for the group with the smallest values $S$ of exports, consistent with the observation from Fig. 1 that the standard deviation $\sigma \left(R\right)$ decreases with increasing $S$.

Figure 4

Probability distributions $P\left(R\right)$ of the logarithmic growth rates $R$ of (a) exports, (b) imports, and (c) debt. We find that, for each of the three economic variables, the central parts of $P\left(R\right)$ can be approximated by a Laplace distribution centered at the median of $R$, and $N$ denotes the number of data points. Consistent with Table the standard deviation $\sigma \left(R\right)$, the skewness of $R$, and the kurtosis of $R$ are greatest for debt and smallest for imports.

Figure 5

(a) Standard deviation $\sigma \left(R\right)$ of the logarithmic growth rate $R$ as a function of the average value $S$ of wages (expressed in U.S. dollars) of 161 different occupations and different countries. We divide the whole data set $(R_t^{\left(i\right)},S_t^{\left(i\right)})$ into five subsets of equal sizes according to the values of $S_t^{\left(i\right)}$. We find that the standard deviation $\sigma \left(R\right)$ decays algebraically with $S$. Interestingly, the scaling exponent $\beta$ is very similar to the scaling exponent $\beta \approx 0.15$ observed for the GDP of countries and firms, and to the scaling exponents $\beta$ found for exports, imports, and debt in Fig. 1. For the average, the median, and the standard deviation of $R$ we obtain $⟨R⟩=0.042$, $\text{median}=0.05$, and $\sigma =0.28$, respectively. (b) Probability distribution $P\left(R\right)$ of the logarithmic growth rates $R$ of wages. We find that the central parts of $P\left(R\right)$ can be approximated by a Laplace distribution centered at the median of $R$, where the far tails of $P\left(R\right)$ can be approximated by a power law $R^{-\alpha }$ with scaling exponent $\alpha \sim 4.2$.

Figure 6

Dependence of $\sigma \left(R\right)$ on $S$ for time series generated by the stochastic process of Eq. (9). We generate 300 times series $S_t$, each of the same length $n=200$, by the stochastic process of Eq. (9) with parameter values ${\mu }_0=0.01$, ${\sigma }_0=0.5$, $\gamma =0.15$, and $S_0=10000$. We compute the average size $⟨S⟩$ and the standard deviation $\sigma \left(R\right)$ from each of the 300 time series, and show a log-log scatter plot of $\sigma \left(R\right)$ versus $⟨S⟩$. We find that the dependence of $\sigma \left(R\right)$ on $⟨S⟩$ can be approximated by a power law, $\sigma \left(R\right)\propto {⟨S⟩}^{-\beta }$, with scaling exponent $\beta =\gamma$.

Figure 7

Probability distribution $P\left(R\right)$ for time series generated by the stochastic process of Eq. (9). For four different values of ${\sigma }_0$, we generate ${10}^6$ time series $S_t$, each of the same length $n=30$, by the stochastic process of Eq. (9) with parameter values ${\mu }_0=0.01$ and $\gamma =0.15$. Specifically, we choose the four values ${\sigma }_0=40$, ${\sigma }_0=20$, ${\sigma }_0=15$, and ${\sigma }_0=10$. For ${\sigma }_0=40$, we choose $S_0={10}^{10}$, and for ${\sigma }_0^{\prime }=30$, ${\sigma }_0^{\prime }=15$, and ${\sigma }_0^{\prime }=10$ we choose $S_0^{\prime }$ satisfying the relation ${\sigma }_0{S}_0^{\gamma }={\sigma }_0^{\prime }{S}_0^{\prime \gamma }$. We study the four probability distributions of the resulting $3\times {10}^7$ values of $R$ for these four different values of ${\sigma }_0$ and $S_0$. (a) We find that the four different probability distributions $P\left(R\right)$ collapse after rescaling of $R$, and that the central part of the collapsed probability distribution $P\left(R\right)$ can be approximated by a Laplace distribution. (b) We find that the tails of the four probability distributions $P\left(R\right)$ can be approximated by a power law $R^{-\alpha }$ with a positive scaling exponent $\alpha$ whose magnitude is monotonically decreasing with an increasing value of ${\sigma }_0$.

Figure 8

Standard deviation $\sigma \left(R\right)$ of the annual logarithmic growth rate $R$ as a function of the average size $⟨S⟩$ of total payroll expressed in U.S. dollars. In contrast to the observation found for countries (Fig. 1) and wages (Fig. 2), we find that for total payroll the standard deviation $\sigma \left(R\right)$ increases with $S$. However, in agreement with the observations of Figs. 1, 2, the dependence of $\sigma \left(R\right)$ on $S$ can be approximated by a power law, with a negative scaling exponent $(\beta =-0.08\pm 0.03)$. For the average and the median of $R$ we obtain $⟨R⟩=0.054$ and median of $R=0.057$, respectively. (b) Standard deviation $\sigma \left(R\right)$ as a function of $S$ for time series generated by the stochastic process of Eq. (9). We generate 300 time series $S_t$, each of the same length $n=200$, by the stochastic process of Eq. (9) with parameter values ${\mu }_0=0.02$, ${\sigma }_0=0.01$, $\gamma =-0.15$, and $S_0={10}^5$. We find qualitatively that for negative values of $\gamma$ the standard deviation $\sigma \left(R\right)$ grows monotonically with $S$ and quantitatively that the dependence of $\sigma \left(R\right)$ on $S$ can be approximated by a power law $S^{-\beta }$ with a negative scaling exponent $\beta \approx \gamma$.

Figure 9

Standard deviation $\sigma \left(R\right)$ of the growth rate $R$ as a function of the average size $⟨S⟩$ obtained for modeling business companies, denoted by $\alpha$, where $R=\ln (S_{\alpha ,t}∕S_{\alpha ,t-1})$ defines the growth rate of each company, and $S_{\alpha ,t}$ denotes the size of a company. Each company is comprised of $K_{\alpha }$ units where each company unit has size $s_{i,t}$, we assume $s_{i,t}$ are independent random variables, and it holds $S_{\alpha ,t}={\sum }_{i=1}^{K_{\alpha }}{s}_{i,t}$. The size of each company unit is governed by the stochastic process of Eq. (9) with parameter values ${\mu }_0=0.02$, ${\sigma }_0=0.7$, and $\gamma =0.15$. We draw initial sizes $s_{i,0}$ from a Gaussian distribution with a mean of ${10}^5$ and standard deviation of ${10}^4$. We consider 300 different companies, and for each company we generate a time series, each of the same length $n=200$. For each company we calculate the average size $⟨S⟩$ and the standard deviation $\sigma \left(R\right)$. We find that the standard deviation $\sigma \left(R\right)$ decreases with $S$ according to a power law with scaling exponent $\beta \approx \gamma$.

Figure 10

Time series of exports and imports of Germany expressed in U.S. dollars. We find that exports and imports are highly correlated.

Figure 11

Dependence of the magnitude $\mid R_{I_i,t}-{\mu }_0\mid$ of the logarithmic growth rate $R_{I_i,t}$ on the value of $I_i$ in the presence of cross correlations. We generate a bivariate time series $I_t^{\left(1\right)},I_t^{\left(2\right)}$ of length $n=10000$ by the stochastic process of Eqs. (11, 12) with parameter values ${\mu }_0^{\left(1\right)}=0.002$, ${\mu }_0^{\left(2\right)}=0.001$, ${\sigma }_0^{\left(1\right)}={\sigma }_0^{\left(2\right)}=0.1$, ${\gamma }^{\left(1\right)}={\gamma }^{\left(2\right)}=0.15$, $\varphi =0.9$, $I_{1,0}=2000$, and $I_{2,0}=3000$. We find that, for all $i=1,2,3$, $\mid R_{I_i,t}-{\mu }_0^{\left(i\right)}\mid$ is monotonically decreasing with $I_i,t$ and that the dependence of $\mid R_{I_i,t}-{\mu }_0\mid$ on $I_i,t$ can be approximated by a power law with a scaling exponent $\beta \approx \gamma$.