Two-sample least squares projection

ABSTRACT

This article investigates the problem of making inference about the coefficients in the linear projection of an outcome variable y on covariates (x,z) when data are available from two independent random samples; the first sample contains information on only the variables (y,z), while the second sample contains information on only the covariates. In this context, the validity of existing inference procedures depends crucially on the assumptions imposed on the joint distribution of (y,z,x). This article introduces a novel characterization of the identified set of the coefficients of interest when no assumption (except for the existence of second moments) on this joint distribution is imposed. One finding is that inference is necessarily nonstandard because the function characterizing the identified set is a nondifferentiable (yet directionally differentiable) function of the data. The article then introduces an estimator and a confidence interval based on the directional differential of the function characterizing the identified set. Monte Carlo experiments explore the numerical performance of the proposed estimator and confidence interval.

KEYWORDS:

• Data combination
• identification
• least squares projection

JEL CLASSIFICATION:

• C21
• C26

Acknowledgments

I wish to thank Thierry Magnac, Stephane Bonhomme, Christian Bontemps, Andrew Chesher, Fabiana Gomez, Gregory Jolivet, Pascal Lavergne, Nour Meddahi, Jorge Ponce, Christoph Rothe, Senay Sokullu, Frank Windmeijer the Associate Editor, two anonymous referees, and seminar participants at the Workshop on Set Identified Models in Toulouse ’13, University of Bristol, the European Winter Meeting of the Econometric Society ’11, Banco Central del Uruguay, the EC² ’10 meeting, Universidad Carlos III/Madrid, the NESG ’10 meeting, the ENTER Jamboree ’10 meeting, and Toulouse School of Economics for useful comments and suggestions. All remaining errors are my responsability.

Notes

¹Examples include Japelli et al. (1998); Meghir and Palme (1999), Carroll, Dynan, and Krane (2003); Fang et al. (2008); Bostic et al. (2009); and Brzozowski et al. (2010). Additional examples are discussed in the text (see Section 2.1) and in the survey articles by Chesher and Nesheim (2006), and Ridder and Moffitt (2007).

²The support function of a convex set is equal to the signed distance of supporting hyperplanes of the set from the origin (see, e.g., Hiriart-Urruty and Lemarechal (2004).

³For a random vector z and a random variable y, the kth component of the vector of correlation coefficients ρ_zy is $\left[𝔼\right(z_{k}y)-𝔼(z_k\left)𝔼\right(y\left)\right]∕{\left(𝕍\right(z_k\left)𝕍\right(y\left)\right)}^{1∕2}$.

⁴The statistic $T_n\left({𝜃}_k\right)$ is of the type considered by Fan and Park (2014) in the context of nonparametric inference for counterfactual means

⁵We find increasing the number of replications computationally costly, especially for the largest sample size in consideration (i.e., $n_A=n_B=1,000$). For the smallest sample size (i.e., $n_A=n_B=250$), we have also produced results (available upon request) for 1,000 Monte Carlo replications. The qualitative conclusions obtained from 250 replications are not affected.

⁶ An alternative to the MSEI is a loss function weighting coverage and length of the interval estimators. We are not aware, however, of the use of this type of loss functions in the context of set identifying models. For the sake of completeness, we report the Monte Carlo coverage and average length as well.

⁷All the experiments were carried out in the program R using the libraries “splines" (to generate cubic spline basis) and “quantreg" (to estimate $\tau ,z\mapsto Q_{x|z}^o\left(\tau \right|z)$).

⁸$c(ỹ,{\tilde{x}}_k)-c(y,{\tilde{x}}_k)-c(ỹ,x_k)+c(y,x_k)=cỹ{\tilde{x}}_k-y{\tilde{x}}_k-ỹx_k+y{x}_k=(ỹ-y)({\tilde{x}}_k-x_k)$, which is non-negative whenever $\tilde{x}\ge x$ and ỹ≥y as required by the Monge Condition.