A Functional Linear Regression Model in the Space of Probability Density Functions

Author Name ARATA Yoshiyuki (Fellow, RIETI)
Creation Date/NO. March 2017 17-E-015
Research Project Sustainable Growth and Macroeconomic Policy
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In this paper, we propose a functional linear regression model in the space of probability density functions. We treat a cross-sectional distribution of individual earnings as an infinite dimensional random variable. By an isometric transformation of density functions, the constrained nature of density functions is explicitly taken into account. Then, we introduce a regression model where the income distribution is a dependent variable. Asymptotic results for the significance test statistics of the coefficients are obtained. Applying this method to Japanese data, we figure out a functional relationship of the income distribution with economic growth. It is found that the change in income distribution associated with economic growth is characterized by a disproportional increase in the lower income class, reduction of the middle income earners, and irresponsiveness of the higher income earners. Since the information that the income distribution offers is preserved as a density function, this method enables us to obtain implications ignored by the usual statistical ones.