Abstract
In economics, rank-size regressions provide popular estimators of tail exponents of heavy-tailed distributions. We discuss the properties of this approach when the tail of the distribution is regularly varying rather than strictly Pareto. The estimator then over-estimates the true value in the leading parametric income models (so the upper income tail is less heavy than estimated), which leads to test size distortions and undermines inference. For practical work, we propose a sensitivity analysis based on regression diagnostics in order to assess the likely impact of the distortion. The methods are illustrated using data on top incomes in the UK.
Highlights
Income distributions exhibit, like many other size distributions in economics and the natural science, upper tails that decay like power functions
We obtain the distributional theory for our estimator γ, before returning to the distortions induced by deviations from the strict Pareto model
The ordinary least squares (OLS) estimator of the slope coefficient in the rank size regression can suffer significant higher order distortions that arise from the slow decay of the nuisance function l in
Summary
Like many other size distributions in economics and the natural science, upper tails that decay like power functions (see e.g., Schluter and Trede 2017). If the tail of the distribution varies regularly, the Pareto QQ-plot will become linear only eventually. As x → ∞, log U ( x ) ∼ γ log( x ) since log l( x ) → 0 Replacing these population quantities with their empirical counterparts gives the Pareto QQ-plot, and γ is its ultimate slope. This qualification (usually ignored by practitioners in economics) has important consequences for the behaviour of the estimator: Since the OLS estimator estimates the slope parameter of this QQ-plot, deviations from the strict Pareto model -captured by the nuisance function l- will induce distortions. An empirical illustration in the context of top incomes in the UK using data on tax returns is the subject of Section 4
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