Understanding the non-Gaussian distribution of revealed comparative advantage index and its alternatives

The Balassa Index (BI) is a widely used index for evaluating a country's trade (dis) advantage or specialization. Unsatisfied with its unstable distribution and poor ordinal ranking property, which arise from its unstable mean, asymmetric distributional shape, skewness and variable upper bound, many alternatives of BI, including Logarithm of RCA (LRCA), Revealed Symmetric Comparative Advantage (RSCA), Additive RCA (ARCA), Weighted RCA (WRCA), Normalized RCA (NRCA), B∗, and RCAi,k based on a micro-founded Ricardian model, have been proposed in the past several decades. One guiding principle in constructing new indices is that the distribution follows as much as possible a Gaussian. However, this goal has never been satisfactorily realized. To understand the cause, we have systematically carried out empirical analysis of exports within and across countries. We find that the exports of all the goods of a country, as well as a fixed good exported by all the countries in the world follow exponentially truncated Zipf-Mandelbrot's law, after ranked in descending order. The BI amounts to be the ratio of two such distributions, one in the naturally descending order of the exponentially truncated Zipf-Mandelbrot’ law, the other being a permutation of the Zipf-Mandelbrot's law with truncation (possibly with different parameters). Only in very rare situations can these ratios follow a Gaussian distribution. We thus shed light on why BI and its alternatives may have unstable mean for different goods or countries, asymmetric distributional shape, skewness and variable upper bound. In particular, the last feature is a natural consequence of the log-normal distribution of BI, which we find to likely occur in certain situations.