Quantile and Probability Curves Without Crossing

This paper introduces a natural monotonization of the empirical curves by sampling from the estimated non-monotone model, and then taking the resulting conditional quantile curves which by construction are monotone in the probability index. This construction of the monotone curve may be seen as a bootstrap and as a sorting or monotone rearrangement of the original non-monotone curve (see Hardy et al., 1952, and references given below). We show that the rearranged curve is closer to the true quantile curve in finite samples than the original curve is, and derive functional limit distribution theory for the rearranged curve to perform simultaneous inference on the entire quantile function. Our theory applies to both dependent and independent data, and to a wide variety of original estimators, with only the requirement that they satisfy a functional central limit theorem. Our results also apply to many other econometric problems with monotonicity restrictions, such as distribution and structural distribution functions, as well as demand and production functions, option pricing functions, and yield curves. 2 As an example, we provide an empirical application to estimation of structural distribution and quantile functions based on Abadie (2002) and Chernozhukov andHansen (2005, 2006).

There exist other methods to obtain monotonic fits for conditional quantile functions. He (1997), for example, proposed to impose a location-scale regression model, which naturally satisfies monotonicity. This approach is fruitful for location-scale situations, but in numerous cases the data do not satisfy the location-scale paradigm, as discussed in Lehmann (1974), Doksum (1974), and Koenker (2005). Koenker and Ng (2005) developed a computational method for quantile regression that imposes the non-crossing constraints in simultaneous fitting of a finite number of quantile curves. The statistical properties of this method have yet to be studied,

This includes all principal approaches to estimation of conditional quantile functions, such as the canonical quantile regression of Koenker and Bassett (1978) and censored quantile regression of Powell (1986). This also includes principal approaches to estimation of structural quantile functions, such as the instrumental quantile regression methods via control functions of Imbens and Newey (2001), Blundell and Powell (2003), Chesher (2003), and Koenker and Ma (2006), and instrumental quantile regression estimators of Chernozhukov andHansen (2005, 2006).

See Matzkin (1994) for more examples and additional references, and Chernozhukov et. al. (2009) for further theoretical results that cover the latter set of applications. and the method does not immediately apply to other quantile estimation methods. Mammen (1991) proposed two-step estimators, with mean estimation in the first step followed by isotonization in the second. 3 Similarly to Mammen (1991), we can employ quantile estimation in the first step followed by isotonization in the second, obtaining an interesting method whose properties have yet to be studied. In contrast, our method uses rearrangement rather than isotonization, and is better suited for quantile applications. The reason is that isotonization is best suited for applications with (near) flat target functions, while rearrangement is best suited for applications with steep target functions, as in typical quantile applications. Indeed, in a numerical example closely matching our empirical application, we find that rearrangement significantly outperforms isotonization. Finally, in an independent and contemporaneous work, Dette and Volgushev (2008) propose to obtain monotonic quantile curves by applying an integral transform to a local polynomial estimate of the conditional distribution function, and derive pointwise limit theory for this estimator. In contrast, we directly monotonize any generic estimate of a conditional quantile function and then derive generic functional limit theory for the entire monotonized curve. 4 In addition to resolving the problem of estimating quantile curves that avoid crossing, this paper develops a number of original theoretical results on rearranged estimators. It therefore makes both practical and theoretical contributions to econometrics and statistics. In order to discuss these contrib

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