Does an instrument have to be randomized?

No, but nonrandom instruments require a credible identification argument.

Can an exclusion restriction be established from correlation alone?

No; substantive knowledge and design are essential.

AI Glossary / Statistics and probability / Advanced

Instrumental variables

Definition

Instrumental variables are variables used to isolate variation in an explanatory variable that can support causal estimation despite endogeneity. A valid instrument must be relevant to the explanatory variable and satisfy substantive restrictions linking it to the outcome. Instrumental-variable methods do not make an arbitrary correlation causal merely by adding another variable.

What it means

Comparing voluntary tool users with nonusers can mix the tool's effect with differences in motivation or experience. An instrument provides another source of variation in tool use, such as a randomized encouragement.
Random encouragement can address selection into encouragement, but it does not automatically establish that encouragement affects outcomes only through tool use. Extra coaching or attention could create another pathway.

How it works

In a simple linear model with one endogenous variable and one instrument, the instrumental-variable coefficient is the covariance of the instrument with the outcome divided by its covariance with the endogenous variable. With a binary instrument, this becomes a ratio of two differences in group means, often called the Wald estimator.
Two-stage least squares generalizes the calculation. First predict the endogenous variable from the instruments and included exogenous controls. Then use the instrument-based variation to estimate the outcome relationship. Appropriate IV uncertainty estimates are needed; ordinary second-stage standard errors from an improvised regression are not sufficient.
For binary treatment and instrument, a common causal interpretation requires relevance, independence, exclusion, monotonicity, and well-defined outcomes without relevant interference. Under these assumptions, the ratio identifies the average treatment effect for compliers: people whose treatment status changes because of the instrument.

Three practical examples

A randomized invitation encourages uptake of an optional tool. It may serve as an instrument if invitation affects the measured outcome only through uptake and the other assumptions are credible.
A researcher studies an explanatory variable measured with error using a separate valid instrument. The instrument must avoid both the measurement error and the structural equation's unobserved causes of the outcome.
An economic analyst investigates a supply or demand relationship using a plausible source of variation that shifts one side of the market. The causal justification is specific to the market and the proposed instrument.

Think of it this way

An instrument supplies a separate nudge to a control whose setting people otherwise choose for hidden reasons. Its connection to the output must follow the assumed pathway.

Worked example

Consider an invented population evaluating an optional tool. Z equals one for randomized encouragement and zero otherwise. D indicates tool use. Suppose uptake is 80 percent with encouragement and 20 percent without it. The first-stage difference is 0.8 minus 0.2 = 0.6.
Suppose the corresponding average outcomes are 12 and 9 points. The reduced-form difference is 3 points. The Wald ratio is 3/0.6 = 5 points.
To explain what that number can mean, construct a hypothetical population with 20 percent always-users, 60 percent compliers, and 20 percent never-users. Always-users use the tool either way. Compliers use it only when encouraged. Never-users do not use it either way. Nobody responds to encouragement by stopping use, so monotonicity holds in this construction.
Let every person's outcome without the tool be 8.8 points. Assign treatment effects of 1 point to always-users, 5 points to compliers, and 2 points to never-users. Assume encouragement has no separate effect and the groups are represented identically across randomized assignments.
Without encouragement, only always-users receive treatment. Their outcome is 9.8, while the other groups remain at 8.8. The population average is 0.2 times 9.8 plus 0.8 times 8.8 = 9.
With encouragement, always-users still have outcome 9.8, compliers now have 13.8, and never-users remain at 8.8. The average is 0.2 times 9.8 plus 0.6 times 13.8 plus 0.2 times 8.8 = 12.
The ratio therefore recovers the compliers' effect of 5. However, the population average treatment effect in this invented construction is 0.2 times 1 plus 0.6 times 5 plus 0.2 times 2 = 3.6. The IV estimate is not automatically the effect for everyone.
Now suppose encouragement also raises outcomes directly by 0.6 points, independent of tool use. The reduced-form difference becomes 3.6, while the uptake difference stays 0.6. The ratio becomes 6, no longer recovering the compliers' tool effect of 5. Randomization of encouragement alone does not repair this exclusion violation.
These are population calculations under stipulated assumptions. Real sample estimates would vary and require uncertainty analysis. The compliance types and their unobserved outcomes are specified here for teaching; an actual study does not observe both treatment choices or both potential outcomes for each person.

Illustrative case study

Fictional company Cedar Training considers randomly encouraging employees to try an optional assistant. Before analyzing tool uptake, researchers examine whether the invitation also contains useful training that could improve performance without tool use.
They revise the study question and document which assumptions would support an IV interpretation. The report distinguishes the effect of encouragement from the estimated effect of use among people influenced by encouragement. It also examines the first stage and suitable uncertainty estimates. This design discussion is fictional.

Common misconceptions and mistakes

A strong association with treatment establishes relevance, not validity. Independence and exclusion still need justification.
An instrument need not be uncorrelated with the outcome. Its intended effect on treatment can produce an outcome association.
A weak first stage can make estimation and conventional inference unreliable. Adding many weak instruments is not an automatic remedy.
Compliers are defined by their response to the particular instrument. A different instrument can identify a different group's effect.
IV estimators are not generally unbiased in finite samples, even when their identifying assumptions support consistency.

Business use

Use IV when a credible source of variation and a clearly defined causal question justify the assumptions. Explain why the instrument changes treatment, which alternative outcome pathways are excluded, and whose effect is identified. Report the first stage, reduced form, effect estimate, and appropriate uncertainty together rather than presenting only the final ratio.

Check your understanding

What is the example's first stage?

Answer: A 0.6 increase in uptake.

Whose effect does the ratio recover under the stated assumptions?

Answer: The compliers' five-point effect.

Why does a direct encouragement effect cause trouble?

Answer: It changes outcomes through a pathway other than tool use.

Related terms

Endogeneity
exclusion restriction
two-stage least squares
Wald estimator
weak instrument
local average treatment effect
complier

Learn next

Study causal diagrams, randomized encouragement designs, weak-instrument inference, and the distinction between local and population average effects.

Sources

Angrist and Krueger, Instrumental Variables and the Search for Identification, 2001: https://economics.mit.edu/sites/default/files/publications/Instrumental%20Variables%20and%20the%20Search%20for%20Identifi.pdf . Cunningham, Instrumental Variables: https://mixtape.scunning.com/07-instrumental_variables . Population calculations and company scenario are original illustrations.

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