Identification in Nonseparable Models

Abstract

Weak nonparametric restrictions are developed, sufficient to identify the values of derivatives of structural functions in which latent random variables are nonseparable. These derivatives can exhibit stochastic variation. In a microeconometric context this allows the impact of a policy intervention, as measured by the value of a structural derivative, to vary across people who are identical as measured by covariates. When the restrictions are satisfied quantiles of the distribution of a policy impact across people can be identified. The identification restrictions are local in the sense that they are specific to the values of the covariates and the specific quantiles of latent variables at which identification is sought. The conditions do not include the commonly required independence of latent variables and covariates. They include local versions of the classical rank and order conditions and local quantile insensitivity conditions. Values of structural derivatives are identified by functionals of quantile regression functions and can be estimated using the same functionals applied to estimated quantile regression functions.

Instrumental variable models for discrete outcomes

“…For example, specifying recursive structural equations for endogenous explanatory variables and restricting all latent variates and instrumental variables to be jointly independently distributed produces a triangular system model which can be point identifying. 9 This is the control function approach studied in Blundell and Powell (2004), Chesher (2003) and Imbens and Newey (2003). The restrictions of the triangular model rule out full simultaneity (Koenker (2005), Section 8.8.2) such as arises in the simultaneous entry game model of Tamer (2003).…”

Local indirect least squares and average marginal effects in nonseparable structural systems

“…This is the "monotonicity of the endogenous regressor in the unobserved component" assumed in Imbens and Newey (2009) (see also Chesher (2003) and Matzkin (2003), for example); this always holds when X is separably determined. With index monotonicity, an explicit expression for q 1 can be given along the lines of Hoderlein (2005Hoderlein ( , 2007 …”

“…In the control variable literature, Imbens and Newey (2009) (see also Chesher (2003) and Matzkin (2003)) study nonseparable structures in which although X is nonseparably determined, it is strictly monotonic in a scalar unobserved cause. As we show, this structure also enables suitably constructed DR ratios to measure average marginal e¤ects based on IVs rather than control variables.…”

Identification of Nonparametric Simultaneous Equations Models with a Residual Index Structure

“…A high-level assumption like completeness implicitly places further restrictions on the model, although the nature of these restrictions is typically unclear. 4 Much recent work has focused on systems of equations with a triangular (recursive) structure (see, e.g., Chesher (2003), Imbens andNewey (2009), andTorgovitzky (2015)). A two-equation version of the triangular model takes the form…”