Worked Example With A Policy Instrument

From a Policy Rule to an IV Estimate

A policy can create variation in an endogenous treatment: for example, a change in compulsory-schooling rules may affect how long students remain in school. The resulting policy exposure can be an instrument for schooling when it shifts schooling and affects later earnings only through schooling. The second claim is demanding. A policy rule may be clearly relevant yet still fail as an instrument if it also changes earnings through another channel.

This tutorial works through a just-identified example with one policy instrument and one endogenous treatment. The goal is to calculate the first stage, the reduced form, and their ratio, then evaluate what the numbers do—and do not—establish about the instrument.

For a simple constant-effect model, write \(Y_i=\beta D_i+u_i\), where \(\beta\) is the treatment effect of interest. A valid instrument must satisfy three central conditions. Relevance means that exposure changes treatment, so the population first stage is nonzero. Independence means that instrument assignment is independent of the unobserved determinants of the outcome, either as designed or conditional on the included controls. Exclusion means that the instrument affects the outcome only through the treatment; in the model, there is no separate direct effect of \(Z_i\) on \(Y_i\). These conditions concern the assignment mechanism and causal pathways, not just the arithmetic of an estimate.

For a binary \(Z_i\), define the sample first stage and reduced form as

The just-identified IV ratio is

The ratio has the units of \(Y\) per unit of \(D\). Its causal interpretation depends on instrument validity; calculating a ratio alone does not establish that validity.

Worked Policy Example

Why the Ratio Works—and What Relevance Means

The ratio follows from the model’s implications for the two policy contrasts. Suppose \(Y_i=\beta D_i+u_i\), and suppose the instrument is independent of \(u_i\) with no direct effect on \(Y_i\). For a binary instrument, the population difference in mean outcomes is then the treatment effect multiplied by the population difference in mean treatment. The sample ratio replaces those population contrasts with their observed counterparts.

The second equality uses the assumption that the instrument is independent of the unobserved outcome determinants, so the conditional means of \(u_i\) are equal. If the population first stage is nonzero, dividing both sides by it gives the population IV ratio \(\beta\). In a sample, the calculated ratio is an estimate, not an identity guaranteeing that the assumptions hold.

Relevance has both a substantive and an empirical aspect. Substantively, a policy must plausibly change treatment—for instance, a binding minimum-schooling rule should affect schooling for at least some students. Empirically, the estimated first stage should be examined, along with its uncertainty. A positive point estimate is not enough to conclude that an instrument is informative. Conversely, a policy’s intended mechanism is not proof that it changed treatment in the analyzed sample.

When the instrument is binary and treatment effects vary across people, the ratio does not generally equal the average treatment effect for everyone. Under instrument independence, exclusion, and the additional condition that the policy does not cause some people to take less schooling while causing others to take more, the ratio can instead have a local interpretation for people whose schooling is changed by the policy. Thus, the population represented by an IV estimate depends on the policy-induced treatment changes.

Making the Exclusion Argument Explicit

Exclusion is often the most difficult condition to defend for a policy instrument. The question is not simply whether the policy was designed to change schooling. It is whether, in the outcome model and study population, policy exposure could affect adult wages through any route other than schooling. A credible discussion should identify the policy assignment rule, its intended treatment mechanism, plausible alternative pathways, and design features that make those alternatives less likely.

For a reform tied to a birth-date cutoff, useful considerations may include whether other policies changed at the same cutoff, whether people on either side differ in ways that independently predict wages, and whether the reform changed schooling quality or labor-market experience as well as years of schooling. Restricting comparisons to nearby birth cohorts and accounting for relevant cohort patterns may address some concerns, but no adjustment automatically proves exclusion. If controls are included, the first-stage and reduced-form contrasts must be computed under the same control specification; unadjusted group differences need not produce the adjusted IV estimate.

A direct policy effect shows exactly why exclusion matters. Suppose the true outcome equation includes a direct effect \(\delta\) of the instrument:

If the instrument is otherwise independent of \(u_i\), the reduced-form contrast equals \(\beta\) times the first-stage contrast plus \(\delta\). Dividing by the first stage therefore adds \(\delta/\pi_1\) to the desired treatment effect. This relationship also gives a transparent way to describe sensitivity to a plausible direct policy pathway.

Common Mistakes and Practical Checks

• Calling a nonzero first stage proof of validity. A policy can shift schooling and still affect wages directly or be related to unobserved wage determinants. Relevance is only one instrument condition.
• Reversing the ratio. The IV estimate is reduced form divided by first stage: change in outcome per policy-induced change in treatment. Reversing it gives units of treatment per unit of outcome, not the treatment effect.
• Interpreting the ratio as an ordinary group comparison. The policy groups may differ in schooling because of the policy, but the IV interpretation also requires the instrument assumptions. State the causal interpretation conditionally.
• Treating the exclusion argument as a statistical test. A first-stage calculation cannot establish that the policy has no direct outcome pathway. Exclusion requires a design-based and substantive argument.
• Reporting an estimate without its units. Here the ratio is log points per year of schooling, not simply “0.200.” Units clarify the scale of the effect and the direction of the division.

In an applied analysis, report the policy rule and comparison population, the estimated first stage and reduced form, and the IV ratio with units. Then discuss relevance, independence, and exclusion separately. Be clear about what the design makes plausible, what the data show, and what remains an assumption.