Notes on IV Free Methods

Here’s a overview of instrument-free methods:

• Latent Instrumental Variable (LIV)

• Gaussian Copula (GC)

The goal is for dealing with endogeneity when no external instruments are available.

Motivation

Finding good instruments is hard. Around 2010, marketing researchers recognized that the “IV cure can be worse than the endogeneity disease,” and began seeking ways to exploit features of the data (rather than external IVs) to identify causal effects when valid instruments aren’t available .

Intuition

• LIV treats the endogenous regressor as driven by a discrete, unobserved “latent instrument” that captures its exogenous variation, while the remaining variation is deemed endogenous. One then estimates this latent class structure (akin to a mixture model) alongside the main outcome equation under normal-error assumptions.
• GC builds a joint distribution of the endogenous regressor and the structural error via a Gaussian copula. By assuming the regressor is non-normal and the error is normal, the copula decomposition isolates the exogenous component and yields consistent estimates without explicit instruments .

Data Structure

• Both methods were originally developed for cross-sectional datasets.
• LIV has been extended to dynamic settings (e.g., panel data with time-varying latent classes) and to nonlinear models (e.g., binary logit) .
• Applying GC in panel contexts typically requires a first-difference or within transformation to purge fixed effects, which alters the error covariance and complicates copula estimation.

How It Works

1. LIV Approach
   • Model setup: Decompose the regressor $X$ into two parts: a discrete latent instrument $Z^{\ast }$ (exogenous) and residual $u$ (endogenous).
   • Assumptions:
     • $Z^{\ast }$ has a finite number of states
     • Structural errors are normally distributed
     • $X$ is non-normal
   • Estimation: Use an EM algorithm (or maximum likelihood) to jointly recover $Z^{\ast }$, the mixture probabilities, and the outcome regression parameters.
2. Gaussian Copula
   • Model setup: Specify a joint distribution of $(X,;\epsilon )$ via a Gaussian copula linking marginal distributions.
   • Assumptions:
     • $X$ is non-normal
     • $\epsilon$ is normal
     • Dependence captured entirely by the copula correlation parameter
   • Estimation: Fit marginal distributions and copula correlation by maximization of the joint likelihood, then recover the causal effect from the structural equation conditional on the estimated copula.

Pros and Cons

Key takeaways

• Instrument-free methods can be powerful when valid external instruments are unavailable, but they substitute one set of strong (often untestable) assumptions for another.
• Careful diagnostic checks—testing residual normality, examining the distribution of $X$, and sensitivity analyses—are essential to ensure credible inference.

Reference

Park, Sungho and Sachin Gupta (2012), “Handling Endogenous Regressors by Joint Estimation Using Copulas,” Marketing Science, 31 (4), 567–86.

A Review of Copula Correction Methods to Address Regressor–Error Correlation