{"id":158,"date":"2024-08-05T08:43:28","date_gmt":"2024-08-05T08:43:28","guid":{"rendered":"https:\/\/www.endogeneity.net\/?page_id=158"},"modified":"2026-04-04T14:37:17","modified_gmt":"2026-04-04T14:37:17","slug":"control-function-approach","status":"publish","type":"page","link":"https:\/\/www.endogeneity.net\/?page_id=158","title":{"rendered":"Control Function Approach"},"content":{"rendered":"

Control Function<\/strong><\/span><\/p>\n

The control function approach addresses endogeneity from omitted variables, simultaneity, and measurement error. Like two-stage least squares, it relies on a strong and valid instrumental variable to isolate the exogenous part of the predictor. But instead of replacing the predictor with its predicted values, the control function approach adds the residual from the first stage as an extra control variable.<\/p>\n

How It Works<\/strong><\/p>\n

Step 1: First-Stage Regression<\/span><\/em><\/strong><\/p>\n

Regress the endogenous predictor on the instrumental variable(s) and all exogenous controls and fixed effects from the outcome equation. Save the fitted residuals \u2014 these capture the endogenous component of the predictor.<\/p>\n

Step 2: Second-Stage Regression<\/strong><\/em><\/p>\n

Estimate the outcome equation as usual, but include the first-stage residuals as an additional control variable. The instrument is excluded from this stage.<\/span>
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\nThe residual absorbs the correlation between the predictor and the error term. If the residual is statistically significant, this signals that endogeneity is present and that the correction is doing meaningful work.<\/span><\/p>\n

Standard Errors<\/strong><\/p>\n

Because the second-stage regression uses a generated variable (the first-stage residual), conventional standard errors are incorrect. Bootstrapping is required: repeatedly re-estimate both stages together and compute standard errors from the distribution of bootstrap estimates.<\/p>\n

When to Use the Control Function over 2SLS<\/strong><\/p>\n

In linear models with continuous variables, the control function and two-stage least squares produce identical results. They diverge \u2014 and the control function becomes particularly useful \u2014 in several settings:<\/p>\n