• Abstract: In recent years high dimensional sparse models have gained considerable importance in several areas of economics and finance, which have emerged to deal with many new applications. In an ultra high dimensional sparse model, the number of regressors and candidate moment conditions can be possibly much larger than the sample size, but only a relatively small number of these regressors are important that interpret the main features of the regression function. The goal is to achieve the oracle property: identifying the important variables with high probability, when both the important and unimportant regressors are possibly endogenous. We derive sufficient and necessary conditions for a general penalized minimization to achieve the oracle property, using a general form of penalty functions. We then show that the penalized GMM and penalized empirical likelihood are consistent in both estimation and selection when (i) the unimportant covariates are endogenous but the important ones are not, or (ii) the important covariates are also possibly endogenous and a set of valid instrumental variables are available. However, the penalized OLS is not. Finally, we develop new results for estimating the optimal instruments in the conditional moment restricted model using ultra high dimensional sieves with the number of instruments growing exponentially fast. This extends Belloni et al (2010) to the possibly nonlinear models as well as more general penalty that allows for SCAD, Lasso, and many other penalty functions.

• Paper: pdf file

• Slides: pdf file

• Related literature:

  Belloni, Chen, Chernozhukov and Hansen (2010).
  Sparse models and methods for optimal instruments with an application to eminent domain.

  Fan and Lv (2011)
  Non-concave penalized likelihood with NP-dimensionality.
  IEEE Transactions on Information Theory, 57 5467-5484.

• Presentations:

  2011

  • Midwest Econometric Group Meeting, Chicago
  • University of Maryland
  • JSM, Miami
  • Econometric Society Summer Meeting, St. Louis