Top incomes, heavy tails, and rank-size regressions

Schluter, Christian Article Top incomes, heavy tails, and rank-size regressions Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Schluter, Christian (2018) : Top incomes, heavy tails, and rank-size regressions, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 6, Iss. 1, pp. 1-16, https://doi.org/10.3390/econometrics6010010 This Version is available at: https://hdl.handle.net/10419/195447 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ econometrics Article Top Incomes, Heavy Tails, and Rank-Size Regressions Christian Schluter 1,2 1Aix-Marseille School of Economics, 5 Boulevard Maurice Bourdet CS 50498, 13205 Marseille CEDEX 01, France; [email protected]; Tel.: +33-413-55-25-72 2Department of Economics, University of Southampton, Highfield, Southampton SO17 1BJ, UK Received: 19 November 2017; Accepted: 20 February 2018; Published: 2 March 2018 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. Keywords: top incomes; heavy tails; rank size regression; extreme value index; regular variation JEL Classification: D31; C13; C14 1. Introduction Income distributions exhibit, 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). The recent and rapidly growing literature on top incomes focuses on this upper tail, and its presence has important consequences for the measurement of inequality. 1 However, estimating the heaviness of the upper tail is challenging, since real world size distributions usually are Pareto-like (i.e., tails are regularly varying) rather than strictly Pareto. To be precise, let X1 , . . . , Xn be a sequence of positive independent and identically distributed random variables (e.g., incomes) with distribution function Fthat is regularly varying, so for large x 1−F(x) = x−1 γl(x),γ∈(0, ∞), (1) where l is slowly varying at infinity, i.e., l(tx)/l(x) = 1 as x→∞ . The parameter γ , usually referred to as extreme value index (and 1 /γ as the tail exponent), is unknown and needs to be estimated. Many estimators have been proposed in the statistical literature (see e.g., the textbook treatments in Embrechts et al. 1997 or Beirlant et al. 2004). An estimator popular among economists is based on a simple ordinary least squares (OLS) regression of log sizes on log ranks (e.g., Jenkins 2017 and Atkinson 2017, and references therein, in the income distribution and top incomes literature, this regression is ubiquitous in the city size literature). The enduring popularity of the OLS estimator is partly due to its simplicity, and partly due 1 See e.g., Schluter and Trede (2002) in the contexts of Lorenz curves, Davidson and Flachaire (2007) who propose a semi-parametric bootstrap, Cowell and Flachaire (2007) who advocate semi-parametric methods, or Burkhauser et al. (2012) who seek to reconcile survey and tax return data. Also observe that the pth moment of the income distribution is finite only if p< 1 /γ , so very heavy tails can directly affect the validity of some standard inequality measurement tools. For instance, statistical inference for the Generalised Entropy index with parameter 2 requires the existence of the fourth moment (Cowell 1989). Econometrics 2018,6, 10; doi:10.3390/econometrics6010010 www.mdpi.com/journal/econometrics Econometrics 2018,6, 10 2 of 16 to a powerful intuition based on a Pareto quantile-quantile (QQ)-plot, the regression estimating its slope coefficient. However, if the tail of the distribution varies regularly, the Pareto QQ-plot will become linear only eventually. In particular, (1) can be expressed equivalently, using the tail quantile function U(x) = inf{t: Pr(X>t) = 1 /x} where x> 1, as U(x) = xγ˜ l(x) where ˜ l(x) is a slowly varying function. Hence, as x→∞ , log U(x)∼γlog(x) since then 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 lwill induce distortions. The empirical importance of this is illustrated in Figure 1, which depicts the Pareto QQ-plot for our administrative income data for the UK (the subject of our empirical application developed in Section 4 below), using the 1000 largest incomes. The plot exhibits a pronounced kink, and approximate linearity of the QQ-plot only holds for the very highest upper order statistics. Panel (b) shows the consequences for the OLS estimates: As we move in the QQ-plot from the right to the left, the departures from linearity become progressively more severe, and the OLS estimates progressively fall. Based on this first diagnostic QQ-plot, once the lower upper order statistics have been discarded as a source of downward bias, the subsequent analysis can then more clearly focus on the approximate linear part, the remaining distortions, and the choice of the number of order statistics. Figure 2provides a further illustration for three Burr (Singh-Maddala) distributions (examined in detail in Section 3below, being the leading parametric income distribution model) possessing the same γ . Here, the speed of decay of the nuisance function l is parametrised by the absolute value of the parameter ρ . The smaller the magnitude of ρ , the greater the initial curvature and steepness of the Pareto QQ-plot, and the larger the induced positive distortions of the OLS estimator of the slope coefficient. 01234567 14 15 16 17 18 -log(j/(n+1)) log(Xn−j+1) (a) 0200 400 600 800 1000 0.2 0.6 1.0 1.4 k γ (b) Figure 1. Pareto quantile-quantile (QQ)-Plot: Top incomes in the UK. Based on administrative income tax return for the UK in 2009/10. The Survey of Personal Incomes (SPI) is described in Section 4. Panel ( a ): The Pareto QQ-plot (see Section 1.1) is based on the largest 1000 incomes. Panel ( b ): Estimates of γ for the k upper order statistics using the OLS regression (solid lines), and pointwise 95% symmetric confidence intervals (dashed lines). The distributional theory is stated in Equation (8). Econometrics 2018,6, 10 3 of 16 01234567 -10 -5 05 -log(j/(n+1)) log(Xn−j+1) ρ=−2 ρ=−0.5 ρ=−0.25 (a) 050 100 150 200 250 300 0.4 0.6 0.8 1.0 1.2 k γ ^ ρ=−2 ρ=−0.5 ρ=−0.25 (b) Figure 2. Pareto QQ-Plots: The Burr distribution. Based on the Burr distribution given by F(γ,ρ)(x) = 1−(1+x−ρ/γ)1/ρ with γ= 2 / 3, and ρ∈ {− 2, − 0.5, − 0.25 } . Panel ( a ): Pareto QQ-plots for 3 random samples drawn from the Burr distribution. Sample size is 1000. To aid comparison across cases, the points of each QQ-plot have been connected and rendered as lines. Panel ( b ): Mean of estimates ˆ γ across 1000 Monte Carlo simulations for given ρ , drawing samples of size 1000 in each iteration. The faint horizontal line is the population value γ=2/3. In this paper, we examine the asymptotic distortions of the OLS estimator that arise in these circumstances, caused by the slow decay of the nuisance function l and modeled here as higher order regular variation. The theory is presented in Section 2(proofs are collected in Appendix A), and numerical illustrations and quantifications of the distortions are provided in Section 3, as well as of the stark consequence for inference. More specifically, we show formally that the OLS estimator over-estimates the true value in the leading heavy-tailed model (i.e., the Hall class, which includes the Burr (Singh-Maddala) distribution, as well as the student, Fréchet, and Cauchy distributions). An empirical illustration in the context of top incomes in the UK using data on tax returns is the subject of Section 4. 1.1. The Log-Log Rank-Size Regression We briefly review the rank size regression. Let X1,n≤ ··· ≤ Xn,n denote the order statistics of X1 , ··· , Xn , and consider the k upper order statistics. Let ranks be shifted by a constant η< 1. The regression of sizes on ranks leads to the minimisation of the least squares criterion k ∑ j=1log Xn−j+1,n Xn−k,n−glog k+1 j−η2 (2) with respect to g , where η< 1 and 1 ≤j≤k<n . The classic case is η= 0. However, since the OLS estimator of the slope coefficient is not invariant to shifts in the data, it is conceivable that a purposefully chosen shift could yield an asymptotic refinement (Gabaix and Ibragimov 2011 consider this in the strict Pareto model 1 −F(x) = cx−1/γ). The analysis below allows for this possibility. The justification of considering regression (2) is based on a Pareto QQ-plot (Beirlant et al. 1996): For a sufficiently high threshold Xn−k,n where k<n , the Pareto quantile plot in model (1) with coordinates (−log(j/(n+ 1 )) , log Xn−j+1,n)j=1,···,k becomes ultimately linear. The line through point −log((k+ 1 )/(n+ 1 )) , log Xn−k,n) with slope g is thus given by y=log Xn−k,n+g[x+log((k+ Econometrics 2018,6, 10 4 of 16 1 )/(n+ 1 ))] and the data points are (x , y) = (−log(j/(n+ 1 )) , log Xn−j+1,n)j=1,···,k . The regression estimator estimates this slope parameter. In particular, the OLS estimator of the slope coefficient gis ˆ γ= 1 k∑k j=1log k+1 j−ηlog Xn−j+1,n−log Xn−k,n 1 k∑k j=1hlog k+1 j−ηi2≡Nn,k Dk ,η<1. (3) Note that the denominator Dk is a Riemann approximation to R1 0log2x d x= 2. An asymptotic expansion of the denominator reveals that Dk=2+O log2k k!(4) From Kratz and Resnick (1996, proof of their Equation 2.4, p. 704) we know that the numerator Nn,k converges in probability to 2 γ , hence the estimator is weakly consistent: ˆ γ→Pγ as k→∞ and k/n→ 0. We proceed in the next Section to refine this result by obtaining higher order expansions of the estimator in (3). The literature contains several variants of regression (2). Rather regressing log sizes on log ranks, one could regress log ranks on log sizes, thus obtaining the ‘dual’ regression. In view of (3), our asymptotic analysis of the numerator carries immediately over to this dual regression. Another variant of (2) includes the additional estimation of a regression constant: log Xn−j+1,n is regressed on a constant and log j .Kratz and Resnick (1996) obtain the distributional theory for this alternative estimator and show that its asymptotic variance is 2 γ2/k , which exceeds, as will be shown below, the asymptotic variance of ˆ γ given by (3). Hence this regression variant is less efficient. Schultze and Steinebach (1996) also prove weak consistency of the estimator in this setting. 2. Asymptotic Expansions and Distributional Theory 2.1. Preliminaries: Higher Order Regular Variation In order to obtain our asymptotic expansions, we use an equivalent representation of model (1) based on regular variation and extreme value theory. First we recall the definition of first-order regular variation, and then proceed to model the slowly varying nuisance function lin (1) by a refinement to second-order regular variation. We then show that most heavy-tailed distributions of interest (in the income, finance and urban literature) satisfy this condition. It is well known that model (1) has the equivalent (first-order regular variation) representation (e.g., Dekkers et al. 1989) lim t→∞ log U(tx)−log U(t) a(t)/U(t)=log x, (5) for all x> 0 where a is a positive norming function with the property a(t)/U(t)→γ . The problem for estimating the extreme value index γ is the behaviour of the slowly varying function l in (1). It is, therefore, common practice in the extreme value literature to model such second-order behaviour, thus strengthening model (1), by strengthening the first-order regular representation (5) to second-order regular variation. Following De Haan and Stadtmüller (1996), we assume that the following refinement of (5) holds lim t→∞ log U(tx)−log U(t) a(t)/U(t)−log x A(t)=Hγ,ρ(x)(6) for all x> 0, where Hγ>0,ρ<0(x) = 1 ρ(xρ−1 ρ−log x) with ρ< 0. This parameter ρ is the so-called second-order parameter of regular variation, and A(t) is a rate function that is regularly varying with index ρ , with A(t)→ 0 as t→∞ . As ρ falls in magnitude, the nuisance part of l in (1) decays more slowly. Our numerical illustrations will thus consider small magnitudes for ρ. Econometrics 2018,6, 10 5 of 16 Examples. Most heavy-tailed distributions of interest satisfy representation (6). Consider the Hall class of distributions (Hall 1982), given by, for large x, F(x) = 1−ax−1/γ[1+bxβ+o(xβ)] with γ , a> 0, b∈R , β< 0. In this class, the nuisance function l in model (1) converges to a constant at a polynomial rate. The Hall class nests, for instance, the Burr (Singh-Maddala), Student, Fréchet, and Cauchy distributions. 2 The tail quantile function is U(x) = cxγ[ 1 +dxρ+o(xρ)] where c=aγ , d=bγaγβ . This Hall class satisfies the second order representation (6) with ρ=γβ < 0, and rate function A(t) = ρ2 γdtρ. Figure 2illustrates the role of ρ for the Burr distribution (examined in greater detail in Section 3) in terms of the Pareto QQ-plot, and the implications for the estimator ˆ γ of its slope parameter. For ρ=− 2 the plot is close to linear, and the estimates close to the population value. However, as ρ falls in magnitude, the initial curvature increases, and the slope estimates consequently becomes more positively distorted as the number of upper order statistics kentering the estimator increases. 2.2. The Main Results We first state the higher order asymptotic expansion of the numerator Nn,k . We then obtain the distributional theory for our estimator ˆ γ , before returning to the distortions induced by deviations from the strict Pareto model (captured by second order regular variation). Asymptotic expansion. In theAppendix Awe prove the following higher order expansion of the numerator Nn,kunder the assumption of second-order regular variation (6). Throughout, we will consider an intermediate sequence k=kn of positive integers such that kn→∞ and kn/n→ 0 as n→∞. It is then true that, for γ>0 and ρ<0, Nn,k/γ=2−1 2−ηlog(k−η) k−1 2−ηlog2k 2k +Op1 k1/2 +O1 k+Oplog k k1/2 (7) +An k1 ρ2−ρ (1−ρ)2+Oplog k k+op(A(n/k)) A few comments are in order. The first two lines of this expression characterise the first-order behaviour of the numerator. It can be seen that setting the regression shift factor η to 1/2 eliminates the second and third term. However, the term Oplog k/k1/2 is still present. The asymptotic refinement due to second-order regular variation is given by the terms of line 3. Although A(t)→ 0 as t→∞ , this decay might be slow: A(t) is regularly varying with index ρ , and as ρ falls in magnitude the nuisance part of l in (1) decays more slowly. A slow decay then introduces a noticeable distortion in finite samples. We examine these distortions after stating the distributional theory for the estimator. 2 The Burr distribution F(γ,ρ)(x) = 1 −( 1 +x−ρ/γ)1/ρ is a member of the Hall class with parameters γ and ρ< 0, c= 1 and d=γ/ρ , as is the Student tδ distribution with δ degrees of freedom where γ= 1 /δ , ρ=− 2 /δ , d=γBC−2γ , B=− 0.5 δ2(δ+ 1 )/(δ+ 2 ) , and C=Γ((δ+ 1 )/ 2 )δ(δ−1)/2/(δπ)1/2Γ(δ/ 2 ) (valid for δ> 2); so is the Fréchet distribution Fγ(x) = exp(−x−1/γ) with ρ=− 1, c= 1, and d=− .5 γ , and the Cauchy distribution with γ= 1, ρ=− 2, c= 1 /π , and d=−0.5π2. Econometrics 2018,6, 10 6 of 16 Distributional theory. Beirlant et al. (1996) observe that our slope estimator ˆ γ , given by (3), is (to first order) a member of the class of kernel estimators discussed in Csorgo et al. (1985) with kernel K(t) = 1 −log t . Since R1 0K(t) d t= 2 and not unity, a scale correction is required. Since R1 0K2(t) d t= 5, the following result obtains as k→∞and k/n→0, and if √kA(n/k)→0 √k(ˆ γ−γ)→dN0, 5 4γ2(8) Higher order distortions. Asymptotically, the estimator is thus unbiased if √kA(n/k)→ 0. If this decay is slow, however, the estimator will suffer from a higher order distortion in finite samples. By (7), this distortion equals, for γ>0 and ρ<0, bk,n≡1 2 γ ρ 2−ρ (1−ρ)2A(n/k)(9) In particular, in the Hall model, A(t)=(ρ2/γ)dtρ . The sign of the higher order distortion of Nn,k and hence ˆ γ is, since ρ< 0, then given by -sgn (d) . For the Burr (Singh-Maddala), student, Fréchet, and Cauchy distributions it can be shown that d< 0, leading to a positive higher order distortion. We conclude that the higher order distortion induced by higher order regular variation is positive for many popular distribution -i.e., for which the nuisance function l in model (1) converges to a constant at a polynomial rateleading to an overestimation of γ.3 Simulation evidence for these theoretical results is presented next. We also quantify the higher order distortions and the consequences for statistical inference about γ. 3. Numerical Illustrations We illustrate numerically several of our results in a Monte Carlo study. First, we verify the distributional theory, then show that most of the empirical distortion is captured by the bias function bk,n . At the same time, we show that the distortions can be sizeable, leading to substantial test size distortions, while a bias correction using bk,nwould reconcile nominal and actual test sizes. Our Monte Carlo study is based on the Burr distribution, a member of the Hall class, parametrised here as F(γ,ρ)(x) = 1 −( 1 +x−ρ/γ)1/ρ with parameters γ and ρ< 0. In the income distribution and inequality literature, this distribution is also know as the Singh-Maddala distribution, and used frequently in parametric income models. Specifically, we set γ= 2 / 3, and ρ=− 1 / 2 to begin with. Qualitatively similar results are obtained for the student, Fréchet, and Cauchy distributions, all of which are members of the Hall class, and therefore not reported here. Since 1 < 1 /γ< 2 we consider a situation of fairly heavy tails (as second moments of the distribution do not exist). However, the qualitative insights depend little on the actual choice of γ . We have chosen ρ=− 1 / 2 as our leading example since we are interested in the consequences of deviating from a strict Pareto model. As ρ falls in magnitude the nuisance part of l in (1) decays more slowly. This is illustrated in Figure 2, where we depict three Pareto QQ-plots for different ρ . For ρ=− 2, the plot is almost linear throughout. The deviations from the strict Pareto model become increasingly more pronounced in the left part of the plot as ρfalls in magnitude. For the simulation study, we draw R= 1000 samples of size n= 10, 000 at first (then n= 1000), and consider the upper k order statistics. In order to choose a particular k , we follow standard practice and 3 De Haan and Ferreira (2006) consider the merit of shifting the tail for tail quantile functions U(t) = c0+c1tγ+c2tγ+τ+ o(tγ+τ) where c0 and c2 are not zero, c1> 0, and γ> 0 and τ< 0. It can then be shown that if τ<−γ , the second order parameter satisfies ρ=−γ . A data shift that eliminates c0 then results in ρ=τ , so the post shift second order parameter has increased in magnitude, leading to a decrease in the induced distortion. However, the reverse reasoning also applies. In particular, the Hall model is U(x) = cxγ[ 1 +dxρ+o(xρ)] . A data shift by c0 yields U(x) + c0=cxγ[ 1 + (c0/c)x−γ+ dxρ+o(xρ)], and increases the distortion if γ≤ |ρ|. Econometrics 2018,6, 10 7 of 16 minimise the theoretical asymptotic Mean Squared Error (AMSE) (e.g., Hall 1982,orBeirlant et al. 1996), given by b2 k,n+ ( 1 /k)( 5 / 4 )γ2 , trading off distortion and dispersion. The theoretical higher order bias in ˆ γinduced by higher order regular variation in this Burr case is bk,n=1 2γ2−ρ (1−ρ)2n kρ which is, of course, increasing in k . The theoretical AMSE is minimised around k∗= 200, which also corresponds to the minimiser of the empirical AMSE based on the R samples. The mean of ˆ γ at this k∗ is 0.739, and exceeds, as predicted by the theory, the population value γ=2/3. Figure 3depicts the results. In panel (a) we illustrate the distributional theory, given by (8), for k∗ , by plotting a kernel density estimate of √k∗ˆ γ (solid line), as well as a normal density with variance ( 5 / 4 )γ2 , centered on the empirical mean of the simulated data. The two are in close agreement. The figure also implies that any inferential problems are due to location shifts. In panel (b) we contrast the empirical distortions (solid line) with bk,n (dashed line). ˆ γ overestimates γ , and the distortion increases in k . It is evident that most of the distortion is captured by bk,n . In panel (c) we illustrate the consequences of the distortions for statistical inference, by plotting the empirical coverage error rates of the usual 95% symmetric confidence intervals. The higher order distortions lead to undermining inference because of the considerable size distortions. For instance, at k∗ , the empirical coverage error rate is 30% for a nominal 5% rate. Shifting the estimate by bk∗,n reduces the coverage error rate to 7%. -1 01234 0.0 0.2 0.4 Density (a) 050 100 150 200 250 300 0.00 0.05 0.10 0.15 bias k bias (b) 050 100 150 200 250 300 0.0 0.2 0.4 0.6 0.8 1.0 k coverage error rate Coverage errors (c) Figure 3. Bias and Inference: Burr. Monte Carlo study for the Burr distribution with parameters γ= 2 / 3 and ρ=− 0.5. Based on samples of size n=10,000 and R=1000 repetitions. k∗= 200 minimises the asymptotic Mean Squared Error (AMSE), and is depicted by the vertical lines in panels b and c. Panel ( a ): Density plot of √k∗ˆ γ (solid line) and shifted normal density with variance ( 5 / 4 )γ2 (dashed line). Panel ( b ): empirical bias (solid line) and higher order bias function bk,n (dashed line). Panel ( c ): Coverage error rate of the usual 95% symmetric confidence intervals for nominal rate of 5%, with no bias correction (solid line) and correction by the theoretical bk,n(dashed line). Next, we consider the role of the sample size n . Reducing the sample sizes in the Monte Carlo to n= 1000 yields results that are in line with the above theory, and therefore not depicted. The bias of ˆ γ increases by a factor predicted by the theory, namely bk,1000/bk,10,000 = 10 1/2 = 3.16. The optimal k∗ shrinks by a factor of 4, as now k∗= 50. The density of √k∗ˆ γ is in good agreement with the theory, and empirical coverage error rates at this k∗ are 32% for the uncorrected and 11% for the corrected estimator. The empirical coverage error rate for the uncorrected estimator rises steeply after k∗ , reaching 64% at k= 100. Reducing the sample sizes further to 100 results in k∗= 20, and an empirical coverage error rate for the uncorrected estimator of 46% at this k∗ . Biases are increased by a factor bk,100/bk,10,000 = 10. Finally, we illustrate the importance of the speed of decay in the nuisance function l of model (1). As ρ falls in magnitude, the nuisance function l decays more slowly. For the Burr case with γ= 2 / 3, Econometrics 2018,6, 10 8 of 16 we depict in Figure 4 bk,n as ρ falls in magnitude for n= 1, 000 and selected k . While for ρ=− 2 the distortions are negligible (in line with Figure 2, it is evident that for small magnitudes of ρ the higher order distortions cannot be ignored). -2.0 -1.5 -1.0 -0.5 0.0 020 40 60 80 100 ρ relative distortion [%] k=100 k=50 Figure 4. Relative distortions in the Burr model. Burr model with γ= 2 / 3 and n= 1000. Depicted is 100 ∗bk,n/γas ρvaries. As the purpose of our simulation study is the provision of numerical evidence for our theory, we have used the theoretical bias function bk,n in the Burr case. When no such external knowledge is available, estimating the bias function requires non-parametric estimates of the second order parameter ρ and the function A(·) . However, existing methods perform poorly, yielding excessively volatile estimates. The theory then informs a sensitivity analysis which is described in Section 4.1 in the context of our empirical application. 4. Empirical Illustration: Top incomes in the UK Our empirical application uses administrative income tax return data are from the public-release files of the Survey of Personal Incomes (SPI) for the year 2009/10 (see e.g., Jenkins 2017 for a detailed description, and an analysis that includes rank size regressions). The SPI data underlie the UK top income share estimates in the World Top Incomes Database (WTID), and is a stratified sample of the universe of tax returns. The unit of taxation is the individual, and we use total taxable income as the income variable. The file contains 674,715 individuals, and we consider the nlargest incomes. In Figure 1panel (a), we have depicted the Pareto QQ-plot for the 1000 largest incomes. It is evident that the data clearly reject a strict Pareto model: The plot exhibits a pronounced kink, and approximate linearity of the QQ plot only holds for the very highest upper order statistics. The function l in (1) captures this significant departure from the strict Pareto model. The Pareto QQ-plot thus conveys crucial information that is usually ignored by practitioners in economics, making it a key diagnostic device. For instance, a common mechanical approach is to set k by choosing ‘blindly’ (i.e., without reference to the Pareto QQ-plot) e.g., the top 1% or the top 1000 observations. Since the approximate linearity only obtains for about the 70 largest observations, the estimate of the slope parameter of the Pareto QQ-plot, i.e., the OLS estimator (3), will be severely biased if k is set to 1000 or higher. This is illustrated in panel (b) of the figure: The estimates fall for higher values of k , since the estimation procedure then attributes increasing weights to the left of the kink in the Pareto QQ-plot. In the light of these observations, we restrict our subsequent analysis to the range of k in which the Pareto QQ-plot is approximately linear. We confirm this in Figure 5panel (a), having restricted the plot to the n= 70 highest incomes. The plot now appears fairly linear. In panel (b), we depict the regression estimates ˆ γ and the 95% symmetric pointwise confidence intervals. One first visual way of Econometrics 2018,6, 10 15 of 16 Their Theorem 2 (or Theorem 1.1 in Beirlant et al. 1996) states the following. Under general conditions on kernel K and distribution function F so that there exists a nonrandom sequence Cn such that Cn(ˆ a−γ) converges weakly to a limiting N( 0, 1 ) distribution for some sequence k=kn→∞ with kn/n→0, it is necessary and sufficient that lim n→∞√kZ1 0b(kw/n)K(w)dw=0 where b is a function such that b( 1 /x)→ 0 as x→∞ , and that for the tail quantile function U(x) = xγ˜ l(x)with ˜ l(x) = c(x)exp Z1 1/x b(u) udu where c(x)→cas x→∞. If this condition is satisfied, then as k=kn→∞and kn/n→0 √kZ1 0K2(v)dv−1/2 (ˆ γkernel −γ)→dN(0, γ2). Beirlant et al. (1996) observe that our slope estimator ˆ γ , given by (3), is (to first order) a member of the class of kernel estimators ˆ γkernel with kernel K(t) = 1 −log t , and that the above condition holds under the regular variation hypothesis. Turning to the specific kernel K(t) = 1 −log t , since R1 0K(t) d t= 2 and not unity, a scale correction is required. As R1 0K2(t)dt=5, the stated result (8) follows. Proof of (10). Consider the mean weighted theoretical squared deviation 1 k k ∑ j=1 wj,kElog Xn−j+1,n Xn−k,n−γlog k+1 j2 for some weights wj,n. Using (A5) this equals, to first order, γ2 k k ∑ j=1 E k ∑ i=j Ei i−log k+1 j!+A(Yn−k,n)Hγ,ρ(Yk−i+1,k)!2 Then, recalling that Yk−j+1,k=d(Vj,k)−1 and proceeding as in Beirlant et al. (1996, Section 4), which involves approximating expectations E(f(Vj,k)) by the leading term f(j/(k+ 1 )) when applying the delta method yields, to first order, γ2 k˜ ck+dk(ρ)b2 k,n with ˜ ck= k ∑ j=1 wj,k  k−j+1 ∑ l=11 k−l+12 + k−j+1 ∑ l=1 1 k−l+1−log k+1 j!2  and dk(ρ) = (1 2 2−ρ (1−ρ)2)−2˜ dk(ρ)with ˜ dk(ρ) = 1 k k ∑ j=1 wj,k(j/(k+1))−ρ−1 ρ2 Finally, we set ck= (4/5)˜ ckand wj,k≡1 to arrive at (10). Remark : In order to obtain an estimate of the AMSE, Beirlant et al. 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