Stefan Hoderlein

Analyzing The Random Coefficient Model Nonparametrically

Linearity in a causal relationship between a dependent variable and a set of regressors is a common assumption throughout economics. In this paper we consider the case when the coefficients in this relationship are random and distributed independently from the regressors. Our aim is to identify and estimate the distribution of the coefficients nonparametrically. We propose a kernel-based estimator for the joint probability density of the coefficients. Although this estimator shares certain features with standard nonparametric kernel density estimators, it also differs in some important characteristics that are due to the very different setup we are considering. Most importantly, the kernel is nonstandard and derives from the theory of Radon transforms. Consequently, we call our estimator the Radon transform estimator (RTE). We establish the large sample behavior of this estimator—in particular, rate optimality and asymptotic distribution. In addition, we extend the basic model to cover extensions, including endogenous regressors and additional controls. Finally, we analyze the properties of the estimator in finite samples by a simulation study, as well as an application to consumer demand using British household data.

Revealed Preferences in a Heterogeneous Population

This paper explores the empirical content of the weak axiom of revealed preference (WARP) for repeated cross-sections. In a heterogeneous population, the fraction of consumers who violate WARP is not point identified but can be bounded. These bounds, as well as some nonparametric refinements, correspond to intuitive behavioral assumptions if there are two goods. With three or more goods, such intuitions break down, and plausible assumptions can have counterintuitive implications. We also provide estimators and confidence regions. The empirical application reveals that in the British Family Expenditure Survey, upper bounds are frequently positive but lower bounds are not significantly so.

Identification and Estimation of Local Average Derivatives in Non-Separable Models Without Monotonicity

In many structural economic models there are no good arguments for additive separability of the error. Recently, this motivated intensive research on non-separable structures. For instance, in Hoderlein and Mammen (2007) a non-separable model in the single equation case was considered, and it was established that in the absence of the frequently employed monotonicity assumption local average structural derivatives (LASD) are still identified. In this paper, we introduce an estimator for the LASD. The estimator we propose is based on local polynomial fitting of conditional quantiles. We derive its large sample distribution through a Bahadur representation, and give some related results, e.g. about the asymptotic behaviour of the quantile process. Moreover, we generalize the concept of LASD to include endogeneity of regressors and discuss the case of a multivariate dependent variable. We also consider identification of structured non-separable models, including single index and additive models. We discuss specification testing, as well as testing for endogeneity and for the impact of unobserved heterogeneity. We also show that fixed censoring can easily be addressed in this framework. Finally, we apply some of the concepts to demand analysis using British Consumer Data. Copyright The Author(s). Journal compilation Royal Economic Society 2009

DEMAND ANALYSIS AS AN ILL-POSED INVERSE PROBLEM WITH SEMIPARAMETRIC SPECIFICATION

In this paper we are concerned with analyzing the behavior of a semiparametric estimator that corrects for endogeneity in a nonparametric regression by assuming mean independence of residuals from instruments only. Because it is common in many applications, we focus on the case where endogenous regressors and additional instruments are jointly normal, conditional on exogenous regressors. This leads to a severely ill-posed inverse problem. In this setup, we show first how to test for conditional normality. More importantly, we then establish how to exploit this knowledge when constructing an estimator, and we derive the large sample behavior of such an estimator. In addition, in a Monte Carlo experiment we analyze its finite sample behavior. Our application comes from consumer demand. We obtain new and interesting findings that highlight both the advantages and the difficulties of an approach that leads to ill-posed inverse problems. Finally, we discuss the somewhat problematic relationship between endogenous nonparametric regression models and the recently emphasized issue of unobserved heterogeneity in structural models.

Non‐parametric models in binary choice fixed effects panel data

In this paper we extend the fixed effects approach to deal with non‐linear panel data with non‐parametric components. Specifically, we propose a non‐parametric procedure that generalizes the conditional logit approach. We develop an estimator based on non‐linear stochastic integral equations and provide the asymptotic property of the estimator and an iterative algorithm to implement the estimator. We analyse the small sample behaviour of the estimator through a Monte Carlo study.

Nonparametric Identification in Dynamic Nonseparable Panel Data Models

We consider the identification of covariate-conditioned and average partial effects in dynamic nonseparable panel data structures. We demonstrate that a control function approach is sufficient to identify the effects of interest, and we show how the panel structure may be helpful in finding control functions. We also provide new results for the nonparametric binary dependent variable case with a lagged dependent variable.

Regressor Dimension Reduction with Economic Constraints: The Example of Demand Systems with Many Goods

Microeconomic theory often yields models with multiple nonlinear equations, nonseparable unobservables, nonlinear cross equation restrictions, and many potentially multicolinear covariates. We show how statistical dimension reduction techniques can be applied in models with these features. In particular, we consider estimation of derivatives of average structural functions in large consumer demand systems, which depend nonlinearly on the prices of many goods. Utility maximization imposes nonlinear cross equation constraints including Slutsky symmetry, and preference heterogeneity yields demand functions that are nonseparable in unobservables. The standard method of achieving dimension reduction in demand systems is to impose strong, empirically questionable economic restrictions such as separability. In contrast, the validity of statistical methods of dimension-reduction such as principal components has not hitherto been studied in contexts like these. We derive the restrictions implied by utility maximization on dimension-reduced demand systems and characterize the implications for identification and estimation of structural marginal effects. We illustrate the results by reporting estimates of the effects of gasoline prices on the demands for many goods, without imposing any economic separability assumptions.