On the structure of IV estimands
Introduction
A wide variety of estimators have been proposed for the constant-effect linear instrumental variables (IV) model, all of which converge to the true parameter value when the model is correctly specified and an instrument relevance condition holds. When the IV model is misspecified, on the other hand, common IV estimators typically converge to different probability limits.
The goal of this paper is to characterize the behavior of commonly-used estimators under model misspecification in linear IV models with a single endogenous regressor. In particular, the paper considers two-stage least squares (TSLS), two-step generalized method of moments (TSGMM), limited information maximum likelihood (LIML), and continuous updating generalized method of moments (CUGMM). The probability limits (estimands) of TSLS, TSGMM, and LIML are characterized as functions of the estimands in the just-identified models that use one instrument at a time, holding other features of the data generating process fixed. More limited results are derived for the CUGMM estimand.
As is well understood, the TSLS estimand is linear in the single-instrument estimands with linear combination weights summing to one. By contrast, the TSGMM estimand is generally nonlinear, though continuous, in the single-instrument estimands. More surprisingly the LIML estimand is highly nonlinear in the single-instrument estimands and is discontinuous on a set of dimension equal to the number of instruments minus one. If the controls include a constant, I show that the LIML estimand is discontinuous if and only if the vector of single-instrument estimands is such that (a) the TSLS estimand coincides with the ordinary least squares (OLS) estimand and (b) the from the reduced-form regression of the outcome on the instruments is greater than the from the first-stage regression of the endogenous regressor on the instruments. As the TSLS estimand approaches the OLS estimand from above the LIML estimand diverges to positive infinity, while as the TSLS estimand approaches the OLS estimand from below the LIML estimand diverges to negative infinity. Moreover, when the TSLS and OLS estimands coincide and the reduced-form is equal to the first-stage , the population LIML objective function does not depend on the structural parameter value considered, so the minimizer is the full parameter space.
Analytical results for the CUGMM estimand are more elusive, but the level sets of this estimand (viewed as a function of the vector of single-instrument estimands) have a structure similar to those of LIML, and I find similar behavior for the LIML and CUGMM estimands in a calibration to data from Yogo (2004).
The approach taken in this paper is distinct from that in the literature on heterogeneous treatment effects. A large literature originating with Imbens and Angrist (1994) characterizes the probability limits of IV estimators as combinations of heterogenous treatment effects under exogeneity and monotonicity assumptions. By contrast my approach, based on single-instrument IV estimands, is agnostic about the source and form of misspecification and so can accommodate heterogeneous treatment effects, invalidity of the instruments, or misspecification of the linear functional form. Further, my results apply directly to IV applications which are difficult to cast into the treatment effects framework, for example Yogo (2004). At the same time, however, my results only relate IV estimands to the single-instrument estimands and other statistical objects, rather than to the causal or structural parameters of interest. Hence, by remaining agnostic about the source of misspecification my approach accommodates models beyond the scope of the heterogeneous treatment effect literature but obtains correspondingly weaker results.
Two papers from the literature on heterogeneous treatment effects of particular relevance to my results are Kolesar (2013) and Mogstad et al. (2018). Kolesar (2013) shows that the LIML estimand can lie outside the convex hull of the individual treatment effects in a heterogeneous treatment effect model. Kolesar’s results do not imply the discontinuity of LIML estimand but do suggest peculiar behavior for this quantity, which my results strongly confirm. Mogstad et al. (2018) derive expressions for a wide variety of estimands in terms of the potential outcomes in the treated and untreated states in a heterogenous treatment effect model with a binary treatment. Their results could be used to link the expressions in the present paper to causal effects in that setting, though further exploration of this possibility is left for future work. Other related work includes Hall and Inoue (2003), who examine the large-sample behavior of GMM estimators under misspecification, and Lee (2017), who proposes an asymptotic variance estimator for TSLS in models with heterogenous treatment effects.
In the next Section I formally introduce the IV model and define the IV estimands. Section 3 then presents analytical results on the structure of IV estimands, while Section 4 illustrates these results in a calibration to data from Yogo (2004). All proofs are given in the Appendix.
Section snippets
The linear IV model and estimands
Suppose we observe a sample of observations drawn from distribution , where is an outcome variable, is a potentially endogenous regressor, and is a vector of instrumental variables. Let us stack these observations into vectors and with row equal to and respectively, and a matrix with row equal to . Suppose the data obey the linear model where is the scalar parameter of interest. Conventional IV methods impose two further
The structure of IV estimands
As noted in the previous section, all IV estimands coincide in just-identified models, provided the instrument relevance condition holds. In over-identified models, by contrast, each instrument implies a corresponding IV estimand and the question is how to combine the single-instrument estimands into an overall estimate. The different IV estimators discussed above imply different answers to this question, and the goal of this section is to characterize the behavior of the IV estimands as
IV estimands in an example
To illustrate the analytic results above, Fig. 1, Fig. 2, Fig. 3, Fig. 4 plot the contours of the IV estimands as functions of single-instrument estimands in a calibration based on Yogo (2004). Yogo studies the effect of weak instruments on estimation of the elasticity of intertemporal substitution using a linear Euler equation model and data from a number of countries. Here I calibrate all elements of other than to values estimated from the quarterly US data series used by Yogo,
Conclusion
When the over-identifying restrictions of the classical IV model fail, common IV estimators converge to distinct probability limits. Characterizing these limits as a function of the single-instrument IV estimands, I find that the LIML estimand is discontinuous and, further, is sometimes equal to the full parameter space. If the set of controls includes a constant, these issues arise when the OLS and TSLS estimands are equal and the reduced-form is weakly larger than the first-stage . While
Acknowledgments
I am grateful to Jushan Bai, Anna Mikusheva, Whitney Newey, Christoph Rothe, Miikka Rokkanen, Jim Stock, participants in the Fall 2015 Conference in Honor of Jerry Hausman, and two anonymous referees for helpful comments. Support from the Silverman (1978) Family Career Development Chair at MIT and from the National Science Foundation under grant number 1654234 is gratefully acknowledged.
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