How can you identify causal effects with instrumental variables?

1 What are instrumental variables?

An instrumental variable (IV) is a variable that affects the treatment, but not the outcome, except through the treatment. In other words, an IV is a source of exogenous variation in the treatment that is independent of the confounding factors and the outcome. For example, suppose you want to estimate the causal effect of smoking on lung cancer, but you know that smoking is correlated with other factors that may affect lung cancer risk, such as genetics, diet, or exposure to pollution. A possible IV could be the distance from the nearest tobacco store, which affects the likelihood of smoking, but not lung cancer, except through smoking.

2 How do instrumental variables work?

The idea behind instrumental variables is to use the variation in the IV to create a quasi-experimental setting, where the treatment is as good as randomly assigned among the units that are affected by the IV. This way, the confounding factors are balanced between the treatment and control groups, and the difference in outcomes can be attributed to the treatment. For example, suppose you use the distance from the nearest tobacco store as an IV for smoking. Then, among the people who live close to or far from a tobacco store, smoking is essentially random, and you can compare the lung cancer rates between smokers and non-smokers within these groups.

3 How do you find instrumental variables?

Finding valid and relevant instrumental variables is often the hardest part of using this technique. A valid IV must satisfy two conditions: relevance and exogeneity. Relevance means that the IV must have a strong and significant effect on the treatment. If the IV is weak or unrelated to the treatment, then it will not generate enough variation in the treatment to identify the causal effect. Exogeneity means that the IV must be independent of the confounding factors and the outcome, except through the treatment. If the IV is correlated with the confounding factors or the outcome, then it will introduce bias and inconsistency in the causal effect estimate.

4 How do you test instrumental variables?

Once you have a candidate IV, you need to test whether it meets the relevance and exogeneity conditions. To test relevance, you can perform a first-stage regression, where you regress the treatment on the IV and other covariates, and check the coefficient and significance of the IV. A rule of thumb is that the IV should have a F-statistic of at least 10 in the first-stage regression, indicating that it explains a substantial amount of the variation in the treatment. To test exogeneity, you can perform a Hausman test, where you compare the estimates of the causal effect obtained with and without the IV, and check whether they are significantly different. A significant difference suggests that the IV is not exogenous and that it affects the outcome through other channels than the treatment.

5 How do you estimate causal effects with instrumental variables?

To estimate the causal effect of the treatment on the outcome using an IV, you can use a two-stage least squares (2SLS) regression, which consists of two steps. In the first step, you regress the treatment on the IV and other covariates, and obtain the predicted values of the treatment. In the second step, you regress the outcome on the predicted values of the treatment and other covariates, and obtain the coefficient and significance of the predicted treatment. This coefficient is the estimate of the causal effect of the treatment on the outcome, also known as the local average treatment effect (LATE), which is the average effect among the units whose treatment status is influenced by the IV. In most statistical software, you can perform 2SLS regression using a ivregress or ivreg command.

6 Here’s what else to consider

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