Two-stage least squares (2SLS) instrumental variable estimator
Source:R/causal_iv.R
causal_iv_fit_2sls.RdEstimates the causal effect of an endogenous exposure exposure on
an outcome outcome using one or more instruments, optionally
conditional on exogenous controls. Implements the classical 2SLS
estimator with the Wooldridge (2010) asymptotic variance – the
naive lm() SE of the second stage is biased because it treats
the generated regressor as observed, a common mistake.
Arguments
- data
A data frame.
- exposure
Character; name of the endogenous exposure column.
- outcome
Character; name of the outcome column.
- instruments
Character vector; names of instrument columns. More instruments than exposures gives an over-identified model on which the Sargan test is applicable.
- covariates
Optional character vector of exogenous-control column names included in both first and second stage.
Value
An edaphos_causal_iv object with components effect,
se, ci, stage1_F, stage1_R2, sargan_p (NULL if exactly
identified), n, plus auxiliary fits for inspection.
Details
Formal setup
Let
Xbe the endogenous exposure (one column),Zthe matrix of instruments,Wthe matrix of exogenous controls (included in both stages).
Define the augmented matrices X_all = [W, X] and
Z_all = [W, Z] and the projection P = Z_all (Z_all' Z_all)^-1 Z_all'. The 2SLS estimator is
$$\hat{\beta} = (X_{\text{all}}^\top P X_{\text{all}})^{-1} X_{\text{all}}^\top P Y$$
with residuals computed using the ORIGINAL X_all, not the
first-stage fitted values, so that
$$\hat{\sigma}^2 = (Y - X_{\text{all}} \hat{\beta})^\top (Y - X_{\text{all}} \hat{\beta}) / (n - k)$$ $$\widehat{\mathrm{Var}}(\hat{\beta}) = \hat{\sigma}^2 (X_{\text{all}}^\top P X_{\text{all}})^{-1}$$
Identification conditions (Wooldridge 2010, section 5.1):
Relevance: \(\mathrm{rank}(Z' X) = \dim(X)\); the first-stage F-statistic should exceed 10 (Stock & Yogo 2005).
Exclusion: \(Z\) affects \(Y\) only through \(X\).
Unconfoundedness: \(Z \perp U\) where \(U\) is any unobserved confounder.