This article considers the question of how to cope with heterogeneity when studying causal effects. The standard approach in empirical economics is still to use a linear model and interpret the coefficients as the average returns or effects. Nowadays, instrumental variables (IV) are quite popular to account for (unobserved) heterogeneity when estimating these parameters. First the inadequacy of these standard methods is illustrated. Then it is shown why varying-coefficient models have a strong natural potential to model heterogeneity in many interesting regression problems. Moreover, it is straight forward to develop alternative IV specifications in the varying-coefficient models framework. The corresponding modeling and implementation facilities that are nowadays available in R are studied
Disaggregated data are characterized by a high degree of diversity. Nonparametric models are often flexible enough to capture it but they are hardly interpretable. A semiparametric specification that models heterogeneity directly creates the preconditions to identify causal links. Certainly, the presence of endogenous variables can destroy the ability of the model to distinguish correlation from causality. Triangular varying coefficient models that consider the returns as non-random functions, and at the same time exogeneize the problematic regressors are able to add to the flexibility of a semiparametric specification the causal interpretability. Moreover, they make the necessary assumptions much more credible than they typically are in the standard linear models. The Causality Problem in the Presence of Heterogeneous ReturnsDisentangling causality from correlation is one of the fundamental problems of data analysis. Every time the experimental methodology -typical in some hard sciences -is not applicable, it becomes almost impossible to separate causality from observed correlations using non-simulated data. The only available alternative is to find a set of non testable assumptions that allow to express the causal links as parameters or as functions, and to subsequently find consistent estimators for the conditional moments or distributions that describe the parameters (or functions) of interest. In particular, consider a response Y to be regressed on an explanatory variable W . The assumption that transforms a simple (cor)relation into a causal effect of W on Y , is often called 'exogeneity'. Definition 1. A variable W is weakly exogenous for the parameter of interest ψ, if and only if there exists a re-parametrization λ for the joint density with parameter λ = (λ 1 , λ 2 ) such that 1 We thank an anonymous referee and the participants of the ISNPS 2014 meeting in Cadiz for helpful comments and discussion.
We propose a novel penalized splines method to estimate a stochastic frontier model in which the frontier is linear and the inefficiency has a single index structure with unknown link function and a linear index. The approach is more flexible than the traditional methodology requiring a parametric link function and, at the same time, it does not incur the curse of dimensionality as a fully non-parametric approach. The procedure can be easily implemented using existing software. We give conditions for the model to be identified and provide some asymptotic results. We also use Monte Carlo simulations to show that the approach works well in finite samples in many situations when compared to the well specified maximum likelihood estimator. An application to the residential energy demand of US states is considered. In this case, the penalized splines approach estimates inefficiency functions that deviate substantially from those resulting from parametric maximum likelihood methods previously implemented.
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