Average Derivative Estimation Research Articles

AVERAGE DERIVATIVE ESTIMATION UNDER MEASUREMENT ERROR

In this paper, we derive the asymptotic properties of the density-weighted average derivative estimator when a regressor is contaminated with classical measurement error and the density of this error must be estimated. Average derivatives of conditional mean functions are used extensively in economics and statistics, most notably in semiparametric index models. As well as ordinary smooth measurement error, we provide results for supersmooth error distributions. This is a particularly important class of error distribution as it includes the Gaussian density. We show that under either type of measurement error, despite using nonparametric deconvolution techniques and an estimated error characteristic function, we are able to achieve a $\sqrt {n}$ -rate of convergence for the average derivative estimator. Interestingly, if the measurement error density is symmetric, the asymptotic variance of the average derivative estimator is the same irrespective of whether the error density is estimated or not. The promising finite sample performance of the estimator is shown through a Monte Carlo simulation.

Semiparametric estimation of dynamic discrete choice models

We consider the estimation of dynamic binary choice models in a semiparametric setting, in which the per-period utility functions are known up to a finite number of parameters, but the distribution of utility shocks is left unspecified. This semiparametric setup differs from most of the existing identification and estimation literature for dynamic discrete choice models. To show identification we derive and exploit a new recursive representation for the unknown quantile function of the utility shocks. Our estimators are straightforward to compute, and resemble classic closed-form estimators from the literature on semiparametric regression and average derivative estimation. Monte Carlo simulations demonstrate that our estimator performs well in small samples.

Semiparametric Estimation of Dynamic Discrete Choice Models

We consider the estimation of dynamic discrete choice models in a semiparametric setting, in which the per-period utility functions are known up to a finite number of parameters, but the distribution of utility shocks is left unspecified. This semiparametric setup differs from most of the existing identification and estimation literature for dynamic discrete choice mod- els. To show identification we derive and exploit a new Bellman-like recursive representation for the unknown quantile function of the utility shocks. Our estimators are straightforward to compute; some are simple and require no iteration, and resemble classic estimators from the literature on semiparametric regression and average derivative estimation. Monte Carlo simulations demonstrate that our estimator performs well in small samples. To highlight features of this estimator, we estimate a structural model of dynamic labor supply for New York City taxicab drivers.

Semiparametric Estimation of Dynamic Discrete Choice Models

We consider the estimation of dynamic discrete choice models in a semiparametric setting, in which the per-period utility functions are known up to a finite number of parameters, but the distribution of utility shocks is left unspecified. This semiparametric setup differs from most of the existing identification and estimation literature for dynamic discrete choice mod- els. To show identification we derive and exploit a new Bellman-like recursive representation for the unknown quantile function of the utility shocks. Our estimators are straightforward to compute; some are simple and require no iteration, and resemble classic estimators from the literature on semiparametric regression and average derivative estimation. Monte Carlo simulations demonstrate that our estimator performs well in small samples. To highlight features of this estimator, we estimate a structural model of dynamic labor supply for New York City taxicab drivers.

Estimating coefficients of single-index regression models by minimizing variation

ABSTRACTThis paper proposes a novel estimation of coefficients in single-index regression models. Unlike the traditional average derivative estimation [Powell JL, Stock JH, Stoker TM. Semiparametric estimation of index coefficients. Econometrica. 1989;57(6):1403–1430; Hardle W, Thomas M. Investigating smooth multiple regression by the method of average derivatives. J Amer Statist Assoc. 1989;84(408):986–995] and semiparametric least squares estimation [Ichimura H. Semiparametric least squares (sls) and weighted sls estimation of single-index models. J Econometrics. 1993;58(1):71–120; Hardle W, Hall P, Ichimura H. Optimal smoothing in single-index models. Ann Statist. 1993;21(1):157–178], the procedure developed in this paper is to estimate the coefficients directly by minimizing the mean variation function and does not involve estimating the link function nonparametrically. As a result, it avoids the selection of the bandwidth or the number of knots, and its implementation is more robust and easier. The resultant estimator is shown to be consistent. Numerical results and real data analysis also show that the proposed procedure is more applicable against model free assumptions.

A Comparative Study of Semiparametric Estimation in Partially Linear Single-index Models

We consider a semiparametric method based on partial splines for estimating the unknown function and partially linear regression parameters in partially linear single-index models. Three methods—project pursuit regression (PPR), average derivative estimation (ADE), and a boosting method—are considered for estimating the single-index parameters. Simulations revealed that PPR with partial splines was superior in estimating single-index parameters, while the boosting method with partial splines performed no better than PPR and ADE. All three methods performed similarly in estimating the partially linear regression parameters. The relative performances of the methods are also illustrated using a real-world data example.

Nonparametric estimation of varying-coefficient single-index models

The varying-coefficient single-index model has two distinguishing features: partially linear varying-coefficient functions and a single-index structure. This paper proposes a nonparametric method based on smoothing splines for estimating varying-coefficient functions and an unknown link function. Moreover, the average derivative estimation method is applied to obtain the single-index parameter estimates. For interval inference, Bayesian confidence intervals were obtained based on Bayes models for varying-coefficient functions and the link function. The performance of the proposed method is examined both through simulations and by applying it to Boston housing data.

Average Derivative Estimation from Biased Data

We investigate the estimation of the density-weighted average derivative from biased data. An estimator integrating a plug-in approach and wavelet projections is constructed. We prove that it attains the parametric rate of convergence 1/n under the mean squared error.

Generalized Jackknife Estimators of Weighted Average Derivatives

With the aim of improving the quality of asymptotic distributional approximations for nonlinear functionals of nonparametric estimators, this article revisits the large-sample properties of an important member of that class, namely a kernel-based weighted average derivative estimator. Asymptotic linearity of the estimator is established under weak conditions. Indeed, we show that the bandwidth conditions employed are necessary in some cases. A bias-corrected version of the estimator is proposed and shown to be asymptotically linear under yet weaker bandwidth conditions. Implementational details of the estimators are discussed, including bandwidth selection procedures. Consistency of an analog estimator of the asymptotic variance is also established. Numerical results from a simulation study and an empirical illustration are reported. To establish the results, a novel result on uniform convergence rates for kernel estimators is obtained. The online supplemental material to this article includes details on the theoretical proofs and other analytic derivations, and further results from the simulation study.

SMALL BANDWIDTH ASYMPTOTICS FOR DENSITY-WEIGHTED AVERAGE DERIVATIVES

This paper proposes (apparently) novel standard error formulas for the density-weighted average derivative estimator of Powell, Stock, and Stoker (Econometrica 57, 1989). Asymptotic validity of the standard errors developed in this paper does not require the use of higher-order kernels, and the standard errors are “robust” in the sense that they accommodate (but do not require) bandwidths that are smaller than those for which conventional standard errors are valid. Moreover, the results of a Monte Carlo experiment suggest that the finite sample coverage rates of confidence intervals constructed using the standard errors developed in this papercoincide (approximately) with the nominal coverage rates across a nontrivial range of bandwidths.

Cleaning up the kitchen sink: Specification tests and average derivative estimators for growth econometrics

Theory and case-study evidence suggest that non-linearities are pervasive in the growth process. Growth empirics have attempted to characterize these non-linearities with regression trees, additively separable non-parametric estimates, or simple interaction terms. Each method requires specific assumptions about functional form which we demonstrate may not be defensible. We provide two alternate mechanisms for making inference about the growth effects of production–function shifters that do not make a priori assumptions about functional form: monotonicity tests and average derivative estimation. Our results suggest that the growth effects of policies are country-specific while the effects of institutions are more robustly monotonic.

ESTIMATION OF BINARY CHOICE MODELS WITH LINEAR INDEX AND DUMMY ENDOGENOUS VARIABLES

This paper presents computationally simple estimators for the index coefficients in a binary choice model with a binary endogenous regressor without relying on distributional assumptions or on large support conditions and yields root-n consistent and asymptotically normal estimators. We develop a multistep method for estimating the parameters in a triangular, linear index, threshold-crossing model with two equations. Such an econometric model might be used in testing for moral hazard while allowing for asymmetric information in insurance markets. In outlining this new estimation method two contributions are made. The first one is proposing a novel “matching” estimator for the coefficient on the binary endogenous variable in the outcome equation. Second, in order to establish the asymptotic properties of the proposed estimators for the coefficients of the exogenous regressors in the outcome equation, the results of Powell, Stock, and Stoker (1989, Econometrica 75, 1403–1430) are extended to cover the case where the average derivative estimation requires a first-step semiparametric procedure.

GLOBAL BAHADUR REPRESENTATION FOR NONPARAMETRIC CENSORED REGRESSION QUANTILES AND ITS APPLICATIONS

This paper is concerned with the nonparametric estimation of regression quantiles of a response variable that is randomly censored. Using results on the strong uniform convergence rate of U-processes, we derive a global Bahadur representation for a class of locally weighted polynomial estimators, which is sufficiently accurate for many further theoretical analyses including inference. Implications of our results are demonstrated through the study of the asymptotic properties of the average derivative estimator of the average gradient vector and the estimator of the component functions in censored additive quantile regression models.

Identification and Estimation of Nonseparable Single-Index Models in Panel Data with Correlated Random Effects

Abstract: The identification of parameters in a nonseparable single-index models with correlated random effects is considered in the context of panel data with a fixed number of time periods. The identification assumption is based on the correlated random-effect structure: the distribution of individual effects depends on the explanatory variables only by means of their time-averages. Under this assumption, the parameters of interest are identified up to scale and could be estimated by an average derivative estimator based on the local polynomial smoothing. The rate of convergence and asymptotic distribution of the proposed estimator are derived along with a test whether pooled estimation using all available time periods is possible. Finally, a Monte Carlo study indicates that our estimator performs quite well in finite samples.

Identification and Estimation of Nonseparable Single-Index Models in Panel Data with Correlated Random Effects

The identification of parameters in a nonseparable single-index models with correlated random effects is considered in the context of panel data with a fixed number of time periods. The identification assumption is based on the correlated random-effect structure: the distribution of individual effects depends on the explanatory variables only by means of their time-averages. Under this assumption, the parameters of interest are identified up to scale and could be estimated by an average derivative estimator based on the local polynomial smoothing. The rate of convergence and asymptotic distribution of the proposed estimator are derived along with a test whether pooled estimation using all available time periods is possible. Finally, a Monte Carlo study indicates that our estimator performs quite well in finite samples.

Smoothing Spline Estimation for Partially Linear Single-index Models

In this article, the partially linear single-index models are discussed based on smoothing spline and average derivative estimation method. This proposed technique consists of two stages: one is to estimate the vector parameter in the linear part using the smoothing cubic spline method, simultaneously, obtaining the estimator of unknown single-index function; the other is to estimate the single-index coefficients in the single-index part by the using average derivative estimator procedure. Some simulated and real examples are presented to illustrate the performance of this method.

Smoothness adaptive average derivative estimation

Many important models utilize estimation of average derivatives of the conditional mean function. Asymptotic results in the literature on density weighted average derivative estimators (ADE) focus on convergence at parametric rates; this requires making stringent assumptions on smoothness of the underlying density; here we derive asymptotic properties under relaxed smoothness assumptions. We adapt to the unknown smoothness in the model by consistently estimating the optimal bandwidth rate and using linear combinations of ADE estimators for different kernels and bandwidths. Linear combinations of estimators (i) can have smaller asymptotic mean squared error (AMSE) than an estimator with an optimal bandwidth and (ii) when based on estimated optimal rate bandwidth can adapt to unknown smoothness and achieve rate optimality. Our combined estimator minimizes the trace of estimated MSE of linear combinations. Monte Carlo results for ADE confirm good performance of the combined estimator.

Testing the martingale hypothesis for futures prices: Implications for hedgers

AbstractThe martingale hypothesis for futures prices is investigated using a nonparametric approach where it is assumed that the expected futures returns depend (nonparametrically) on a linear combination of predictors. We first collapse the predictors into a single‐index variable where the weights are identified up to scale, using the average derivative estimator proposed by T. Stoker (1986). We then use the Nadaraya–Watson kernel estimator to calculate (and visually depict) the relationship between the estimated index and the expected futures returns. We discuss implications of this finding for a noninfinitely risk‐averse hedger. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:1040–1065, 2008

Inference of change-point in single index models

Single index models are widely used in medicine, econometrics and some other fields. In this paper, we consider the inference of a change point problem in single index models. Based on density-weighted average derivative estimation (ADE) method, we propose a statistic to test whether a change point exists or not. The null distribution of the test statistic is obtained using a permutation technique. The permuted statistic is rigorously shown to have the same distribution in the limiting sense under both null and alternative hypotheses. After the null hypothesis of no change point is rejected, an ADE-based estimate of the change point is proposed under assumption that the change point is unique. A simulation study confirms the theoretical results.

Small Bandwidth Asymptotics for Density-Weighted Average Derivatives

This paper proposes (apparently) novel standard error formulas for the density-weighted average derivative estimator of Powell, Stock, and Stoker (1989). Asymptotic validity of the standard errors developed in this paper does not require the use of higher-order kernels and the standard errors are robust in the sense that they accommodate (but do not require) bandwidths that are smaller than those for which conventional standard errors are valid. Moreover, the results of a Monte Carlo experiment suggest that the finite sample coverage rates of confidence intervals constructed using the standard errors developed in this paper coincide (approximately) with the nominal coverage rates across a nontrivial range of bandwidths.