Estimation in functional linear quantile regression

This paper studies estimation in functional linear quantile regression in which the dependent variable is scalar while the covariate is a function, and the conditional quantile for each fixed quantile index is modeled as a linear functional of the covariate. Here we suppose that covariates are discretely observed and sampling points may differ across subjects, where the number of measurements per subject increases as the sample size. Also, we allow the quantile index to vary over a given subset of the open unit interval, so the slope function is a function of two variables: (typically) time and quantile index. Likewise, the conditional quantile function is a function of the quantile index and the covariate. We consider an estimator for the slope function based on the principal component basis. An estimator for the conditional quantile function is obtained by a plug-in method. Since the so-constructed plug-in estimator not necessarily satisfies the monotonicity constraint with respect to the quantile index, we also consider a class of monotonized estimators for the conditional quantile function. We establish rates of convergence for these estimators under suitable norms, showing that these rates are optimal in a minimax sense under some smoothness assumptions on the covariance kernel of the covariate and the slope function. Empirical choice of the cutoff level is studied by using simulations.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1066 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonparametric Inferences on Conditional Quantile Processes

This paper is concerned with tests of restrictions on the sample path of conditional quantile processes. These tests are tantamount to assessments of lack of fit for models of conditional quantile functions or more generally as tests of how certain covariates affect the distribution of an outcome variable of interest. This paper extends tests of the generalized likelihood ratio (GLR) type as introduced by Fan, Zhang and Zhang (2001) to nonparametric inference problems regarding conditional quantile processes. As such, the tests proposed here present viable alternatives to existing methods based on the Khmaladze (1981, 1988) martingale transformation. The range of inference problems that may be addressed by the methods proposed here is wide, and includes tests of nonparametric nulls against nonparametric alternatives as well as tests of parametric specifications against nonparametric alternatives. In particular, it is shown that a class of GLR statistics based on nonparametric additive quantile regressions have pivotal asymptotic distributions given by the suprema of squares of Bessel processes, as in Hawkins (1987) and Andrews (1993). The tests proposed here are also shown to be asymptotically rate-optimal for nonparametric hypothesis testing according to the formulations of Ingster (1993) and of Spokoiny (1996).quantile regression, nonparametric inference, minimax rate, additive models, local polynomials, generalized likelihood ratio

A Semiparametric Estimator for Dynamic Optimization Models

We develop a new estimation methodology for dynamic optimization models with unobserved state variables Our approach is semiparametric in the sense of not requiring explicit parametric assumptions to be made concerning the distribution of these unobserved state variables We propose a two-step pairwise-difference estimator which exploits two common features of dynamic optimization problems: (1) the weak monotonicity of the agent's decision (policy) function in the unobserved state variables conditional on the observed state variables; and (2) the state-contingent nature of optimal decision-making which implies that conditional on the observed state variables the variation in observed choices across agents must be due to randomness in the unobserved state variables across agents We apply our estimator to a model of dynamic competitive equilibrium in the market for milk production quota in Ontario Canada

Nonparametric Estimation of a Nonseparable Demand Function under the Slutsky Inequality Restriction

We present a method for consistent nonparametric estimation of a demand function with nonseparable unobserved taste heterogeneity subject to the shape restriction implied by the Slutsky inequality. We use the method to estimate gasoline demand in the United States. The results reveal differences in behavior between heavy and moderate gasoline users. They also reveal variation in the responsiveness of demand to plausible changes in prices across the income distribution. We extend our estimation method to permit endogeneity of prices. The empirical results illustrate the improvements in finite-sample performance of a nonparametric estimator from imposing shape restrictions based on economic theory