Inverse problems and parsimony for econometric modeling and applications – IPANEMA

To reach the objectives described above our research consortium will develop methods and applications in the directions listed below.
1) We plan to study estimation methods for non-parametric panel data models with endogenous regressors. We will then proceed with the analysis of the probabilistic framework for structural dynamic models with endogenous regressors. Estimators for dynamic models will also be designed and applied to real data-sets.
2) We will estimate heterogeneous consumption models in a dynamic setting.
3) We plan to focus on inference for finite dimensional parameters that are functionals of infinite-dimensional parameter (the latter may be non identified).
4) Our research project will develop non-parametric estimation and inference under different types of constraints. In particular, we will use Bayesian procedure to make asymptotically valid frequentist inference for set-parameters characterized by those constraints.
5) Nonparametric Gaussian process priors will be used to study Bayesian GMM.
6) We will analyze prior distributions which allow to construct data-driven methods for choosing the regularization parameter for inverse problems.
7) We will study identification of non-separable models with endogeneity by making the link to the theory of non-linear inverse problems. From an applied point of view, we are interested in estimating auction models and two-sided market models.
8) New Bootstrap procedures for inverse problems will be studied (for separable and non-separable models).
9) Our consortium will study high-dimensional techniques that allow to make inference in structural models with error in variables, structured sparsity, nonlinear and partially identified models.
10) We want to develop user friendly software for: different types of Inverse Problems-based estimators (both frequentist and Bayesian) and for bootstrap procedures.

The first results we have obtained are the following.
1) Structural estimation of dynamic and panel data models: (i) study of nonparametric models with endogeneity and panel data structure. (ii) nonpararmetric estimation and identification of dynamic treatment models.
2) We have focused on the study of asymptotic normality of (linear functionals of) the solution of linear inverse problems.
3) We have developed non- and semi-parametric estimators for the regression function of a selectively observed variable. (ii) We have studied Bayesian estimation of partially-identified models resulting from shape and moment restrictions.
4) We have studied a semiparametric Bayesian moment estimation method that allows to incorporate overidentifying moment restrictions and that is based on a linear inverse problem.
5) (i) Analysis of non-separable models as an application of non-linear inverse problems. (ii) Preliminary results on the study of the solution of non-linear inverse problems (development of efficient algorithms for computing the solution, study of statistical properties of the solution). (iii) Quantile estimation of game theory models.
6) Identification study of the probability density function of the heterogeneity parameter in structural econometric models and of its linear functionals.
7) Development of a new method of estimation in high-dimensional linear regression models that allows for very weak distributional assumptions including heteroscedasticity, and does not require the knowledge of the variance of random errors.
8) (i) Development of a Bayesian maximum a posteriori estimator for the regularization parameter. (ii) Development of a Bayesian framework where the regularization parameter is integrated out from the linear inverse problem (by using a suitable prior distribution).

We believe that this project will contribute theoretically to both the inverse problem and high-dimensional approach to inference in structural econometric models.
In our opinion the gain, in terms of economic reliability of estimation results, which we can have by using our non-parametric structural Inverse Problems-based methods, is really big and this is why we want to put very much effort in making these techniques largely used in empirical studies.
Our project will develop also several applications by using the inference methods theoretically developed. Applications will range from consumer demand, demand for differentiated products, frontier estimations, two-sided markets, auction models, finance models, etc.
Further, we will make accessible our methods to empirical economists. This will be achieved by developing computer programs and by carrying out novel empirical studies.

1. Gautier, E., Tsybakov, A.B. (2013). Pivotal estimation in high-dimensional regression via linear programming. In: Empirical Inference -- Festschrift in Honor of Vladimir N. Vapnik, B.Schölkopf, Z. Luo, V. Vovk eds., 195 - 204. Springer, New York.
2. Florens, J.-P., Simar, L. and Van Keilegom, I. (2014). Frontier estimation in nonparametric location-scale models. J. Econometrics, 178, 456-470.
3. Dunker, F., Florens, J-P., Hohage, T., Johannes, J., Mammen, E. (2014). Iterative Estimation of Solutions to Noisy Nonlinear Operator Equations in Nonparametric Instrumental Regression. J. Econometrics, 178 (2014), 444-455.
4. Carrasco, M., Florens, J-P., Renault, E. Asymptotic Normal Inference in Linear Inverse Problems. Forthcoming in “Handbook of Applied Nonparametric and Semiparametric Econometrics and Statistics”, Oxford University Press.
5. Florens, J-P. and A. Simoni, “Regularizing Priors for Linear Inverse Problems”, Econometric Theory, (2014, First View)

The IPANEMA project is devoted to the development of inference for non-, semi-parametric and high-dimensional structural economic models. It aims at producing reliable results that are based on inference on model that incorporate features of the unknown parameter/function that is coming from economic theory only. Non-parametric and semi-parametric inference for structural models usually gives rise to inverse problems. High-dimensional methods aim at computationally tractable model selection and inference in models with many more possible unknowns than observations when there is some parsimony or approximate parsimony. Parsimony can correspond to the fact that many coefficients are zero, that there are piecewise constant over time, the model is a discrete mixture of regression models, etc. Approximate parsimony corresponds to the case where the model can be well approximated by a parsimonious model. In that case, a high-dimensional method should select the best sub-model without knowing it in advance or using knowledge on the unknown. It should also account for the approximation error in the inference.
Inverse problem techniques for non- and semi-parametric models have been introduced in structural econometrics since the beginning of the 2000’s. High-dimensional methods have been introduced in the econometrics in the last three years only. Though they already had a high impact in theoretical econometrics, we strongly believe that this is just the premises and that it will play an increasing role in empirical applications. The group that we form through the IPANEMA proposal consists of specialists of inverse problems and high-dimensional estimation, of economists and statisticians interested in economics. The objectives of our proposal are twofold. We will pursue the theoretical developments for inference based on non- and semi-parametric / high-dimensional structural economic models, as for instance: minimax theory, the development of data-driven methods for the choice of the regularization parameter that allow to achieve the minimax lower bounds, the construction of confidence sets -- possibly robust to identification -- and testing procedures, using bootstrap techniques or proper self-normalization, inference on functionals or in the presence of nuisance parameters (finite or infinite dimensional) - possibly under partial identification, and inference under shape restrictions. We will study structural models from various fields from economics including: evaluation of public policies, industrial organizations, labor economics, game theory, auctions and finance, among others. Importantly, we will consider both frequentist and Bayesian procedures. Interestingly, our group has strengths in both frequentist and Bayesian statistics. Bayesian procedures are appealing for designing data-driven methods and for incorporating economic restrictions. Incorporating prior knowledge can prove particularly useful when sample size is small. We also plan to go beyond cross-sections and propose methods for panel data or time series models. Handling dependence is very challenging and barely untouched for such models, even in the statistics literature.
The second objective of our proposal is to make our techniques: (1) easy to implement and (2) accessible. To handle point (1) we will bear attention on algorithmic issues. This is for example the motivation for the recent advances in the high-dimensional literature in comparison to earlier model selection methods. Algorithmic issues have been absent of econometrics until very recently and become increasingly important to handle rich new data configurations. Our proposal addresses point (2) through: systematic development of purely data-driven methods that achieve optimal theoretical bounds, carrying empirical applications, developing computer programs – written either in Matlab or in R - and making them accessible to the general public.