Scaling of Growth Rate Volatility for Six Macroeconomic Variables

We study the annual growth rates of six macroeconomic variables: public debt, public health expenditures, exports of goods, government consumption expenditures, total exports of goods and services, and total imports of goods and services. For each variable, we find (i) that the distribution of the growth rate residuals approximately follows a double exponential (Laplace) distribution and (ii) that the standard deviation of growth rate residuals scales according to the size of the variable as a power law, with a scaling exponent similar to the scaling exponent found for GDP [Economics Letters 60, 335 (1998)]. We hypothesise that the volatility scaling we find for these GDP constituents causes the volatility scaling found in GDP data.


Introduction and Data Analysis
Volatility scaling is an important factor in describing the relationship between the "micro'' and "macro'' levels.
In particular, the way volatility changes under different measurement scales tiesthe microstructure of a given system to its macroscopic observables through scaling laws.
Therefore, empirical studies of volatility scaling may provide better insight into the fundamental processes governing systems at the "micro'' level, which then produce the observed patterns of scaling at the "macro'' level.
The study of volatility scaling has been applied to different levels of aggregation in macroeconomics, ranging from the "micro'' level of company products (see, e.g., Growiec et al., 2008) to the "macro'' level of countries (see, e.g., Canning et al., 1998, Podobnik et al., 2008. For countries, Barro (1991) assumed the existence of heteroscedasticity in growth rates of per capita real gross domestic product (GDP). Head (1995) argued that the higher GDP variances of smaller countries can be explained by their open economies.
However, the exact functional dependence between the volatility of GDP growth rates and country size was not understood until Canning et al. (1998) and Lee et al. (1998) Fu et al. (2005) found that (iii) power laws exist in the tails of ) (R P . These results, obtained from macroeconomic data, are consistent with results obtained from microeconomic data, such as the number of employees and company sales (see, e.g., Stanley et al. 1996). Regarding (iii), Podobnik et al (2011) reported that an asymmetric Levy distribution, which has power-law tails and is characterised by infinite variance, is a good model for several multiplecredit ratios that are used in financial accounting to quantify a firm's financial health, such as the Altman Z score (1968). The asymmetric Levy distribution also models changes in individual financial ratios.
The expenditure method is the most common way to calculate a country's GDP. In this method, GDP is calculated as the sum of five macroeconomic variables, including exports, imports, and government consumption expenditures. We identify patterns of volatility scaling for these three GDP constituents and hypothesise that volatility scaling in these factors contributes to volatility scaling observed in a country's GDP. We also show that similar volatility scaling exists for three other macroeconomic variables.
We analyse the scaling of annual growth rates, ) / ( l n , In Section 2, we find that the residuals, t a r , , are not normally distributed but are exponentially distributed.
This result may have useful implications for our study.
For example, the Schwarz Information Criterion (see Schwarz, 1978), often proposed as a statistical criterion for model selection, requires that data follow an exponential distribution. In Section 3, we use a rigorous statistical approach. For each of the six macroeconomic variables and total labor force, we find that the standard deviation of the growth rate residuals, ) ( a r , follow a power law with the size of variable a S . Thus, the two findings (i) and (ii) from above imply that for the six macroeconomic variables analysed, a r are neither normally distributed nor homoscedastic.

Graphical approach
Next, we investigate whether ) ( a r depends on the size of macroeconomic variable a S . First, we qualitatively explain the growth rate for each macroeconomic variable, a. We sort the data set for each a into three subsets of equal size (small, medium, and large a S ).
We plot the empirical pdf of the residuals for the smallest and largest subsets for public debt [

Least Squares Regression
Next, we quantitatively investigate how the volatility of growth rates changes with the size of a S . For each a S , we partition the entire sample into ten equal subintervals of log a S . Then, in Fig. 2(a), we plot the standard deviation, ) ( a r , of the growth rate residuals, a r , versus the size of a S in the corresponding interval for each macroeconomic variable a S . From Fig. 2(a) Table 1. Skewness, kurtosis, and power-law exponent. The three macroeconomic variables denoted by (*) are GDP constituents. (2)

Scaling of Growth Rate Volatility for Six Macroeconomic Variables
Using the method of least squares regression for each a S , we estimate the parameters a N and a for the regression a a a a S N r l n l n = ) ( l n + . The results are shown in Table 1. Surprisingly, the scaling exponents, a , are within the confidence interval of the scaling exponent, 0.03 0.15 = ± − , reported for GDP (Canning et al., 1998).
The classical regression model, along with many other models in economics, assumes a normal distribution of residuals and assumes that variance remains constant as the size of the variable increases. When the residuals obey the latter assumption, they are said to be homoscedastic. We show that for each of the macroeconomic variables studied, the residuals are neither normally distributed nor homoscedastic.
Labor force is another key macroeconomic variable. Hence, we also plot ) ( a r for labor force [ Fig.   2(b)]. Interestingly, we find this macroeconomic variable has residuals that are normally distributed. However, ) (r decreases with the size of the labor force, S , as a power law with an exponent, similar to the exponential values describing the six macroeconomic variables analysed above.  we reject the hypothesis that 0 = . We conclude that for each variable analysed, the power-law dependence between the standard deviation, ) ( a r , and the size of the variable, a S , are statistically significant.

Conclusions
By analysing many macroeconomic variables, we reject the microeconomic-level hypothesis that a country's economy is composed of entities with identically distributed Gaussian residuals. We hypothesise that the volatility scaling we find in GDP constituents results in the volatility scaling found in GDP data. Our finding that residuals for a broad range of macroeconomic variables are neither normal nor homoscedastic restricts the set of microeconomic variables that can be used to generate observed patterns of macroeconomic scaling (see Wu et al. 2001). In the maximum likelihood approach, when the number of model parameters is increased, researchers commonly employ a statistical criterion for model selection. The BIC or Schwarz Information Crite-

Scaling of Growth Rate Volatility for Six Macroeconomic Variables
rion is an asymptotic result that is derived under the assumption that data follows an exponential distribution.
We show that this assumption holds for many different macroeconomic variables.