Answer

Can you test the exclusion restriction of an instrumental variable?

Inspired by a question on Cross Validated · November 7, 2026

causal inferenceregression

The short answer

An instrumental variable (IV) analysis rests on two conditions. Relevance: the instrument must shift the treatment, which you can check with the first-stage regression. Exclusion: the instrument must affect the outcome only through the treatment, and must not share unmeasured causes with the outcome. The second condition is about paths you cannot observe, and in the common case of a single instrument and a single endogenous treatment (an exactly identified model) no statistical test can confirm it.

That does not leave you empty-handed. You can make the restriction more believable with subject-matter reasoning and falsification checks, and you can show readers how much the conclusion depends on it. The rest of this page explains why the data are silent, then demonstrates a placebo test and a sensitivity analysis on simulated data.

Why the data cannot check it

With one instrument Z, treatment X and outcome Y, the IV estimate is the ratio cov(Z, Y) / cov(Z, X): the effect of the instrument on the outcome, scaled by its effect on the treatment. That ratio is exactly the value that makes the residuals uncorrelated with Z. So the natural check, "are the residuals related to the instrument?", always returns zero, whether or not the instrument has a direct effect on Y. Any direct effect is silently absorbed into the estimate, which is biased by roughly the direct effect divided by the first-stage coefficient. A weak first stage therefore magnifies even a small violation.

Overidentification tests (Sargan or Hansen J) need more instruments than endogenous variables, so they are unavailable here. Even when you can run them, they only test whether the instruments agree with each other. If every instrument is invalid in the same direction, they agree and the test passes. A control-function approach does not escape the problem either: in a linear model with one instrument it gives the same estimate as two-stage least squares and relies on the same exclusion restriction.

See it in R and Python

The simulation creates 10,000 units with an unobserved confounder u, a binary instrument z and a treatment x whose true effect on y is 1. Half the units are eligible: only for them can the instrument change the treatment. Two outcomes are built from the same draws: in y_valid the instrument works only through x; in y_invalid it also raises y directly by 0.3. The code (1) compares OLS with IV, (2) runs a placebo test among ineligible units, and (3) runs a sensitivity analysis over assumed direct effects.

set.seed(677522)
n <- 10000
eligible <- rbinom(n, 1, 0.5)           # z can shift x only for eligible units
z <- rbinom(n, 1, 0.5)                  # binary instrument
u <- rnorm(n)                           # unobserved confounder
x <- 0.6 * z * eligible + u + rnorm(n)  # treatment; its true effect on y is 1
y_valid   <- x + u + rnorm(n)           # exclusion restriction holds
y_invalid <- y_valid + 0.3 * z          # same data, but z also affects y by 0.3

# Just-identified IV (2SLS) estimate with a conventional 95% CI
iv_fit <- function(y, x, z) {
  b  <- cov(z, y) / cov(z, x)
  e  <- (y - mean(y)) - b * (x - mean(x))   # structural residuals
  se <- sqrt(sum(e^2) / (length(y) - 2) * sum((z - mean(z))^2)) /
        abs(sum((z - mean(z)) * (x - mean(x))))
  c(est = b, lo = b - 1.96 * se, hi = b + 1.96 * se, cor_resid_z = cor(e, z))
}

# 1. OLS is confounded; IV recovers 1 only when the exclusion holds
coef(lm(y_valid ~ x))["x"]
round(rbind(valid = iv_fit(y_valid, x, z), invalid = iv_fit(y_invalid, x, z)), 3)

# 2. Zero-first-stage placebo: among ineligible units z cannot move x,
#    so any association between z and y must come through another path
in0 <- eligible == 0
sum(in0)
round(summary(lm(x[in0] ~ z[in0]))$coefficients[2, c(1, 2, 4)], 3)
round(rbind(valid   = summary(lm(y_valid[in0] ~ z[in0]))$coefficients[2, c(1, 2, 4)],
            invalid = summary(lm(y_invalid[in0] ~ z[in0]))$coefficients[2, c(1, 2, 4)]), 3)

# 3. Sensitivity: assume a direct effect gamma of z on y, remove it, re-estimate
gammas <- seq(0, 0.4, by = 0.1)
round(t(sapply(gammas, function(g) c(gamma = g, iv_fit(y_invalid - g * z, x, z)[1:3]))), 3)

import numpy as np
from scipy import stats

rng = np.random.default_rng(677522)
n = 10_000
eligible = rng.binomial(1, 0.5, n)            # z can shift x only for eligible units
z = rng.binomial(1, 0.5, n)                   # binary instrument
u = rng.normal(size=n)                        # unobserved confounder
x = 0.6 * z * eligible + u + rng.normal(size=n)  # treatment; true effect on y is 1
y_valid = x + u + rng.normal(size=n)          # exclusion restriction holds
y_invalid = y_valid + 0.3 * z                 # same data, but z also affects y by 0.3

# Just-identified IV (2SLS) estimate with a conventional 95% CI
def iv_fit(y, x, z):
    zc, xc, yc = z - z.mean(), x - x.mean(), y - y.mean()
    b = np.sum(zc * yc) / np.sum(zc * xc)
    e = yc - b * xc                           # structural residuals
    se = np.sqrt(np.sum(e**2) / (len(y) - 2) * np.sum(zc**2)) / abs(np.sum(zc * xc))
    return np.round([b, b - 1.96 * se, b + 1.96 * se, np.corrcoef(e, z)[0, 1]], 3)

# 1. OLS is confounded; IV recovers 1 only when the exclusion holds
print("OLS slope:", round(stats.linregress(x, y_valid).slope, 3))
print("est, lo, hi, cor_resid_z")
print("valid  ", iv_fit(y_valid, x, z))
print("invalid", iv_fit(y_invalid, x, z))

# 2. Zero-first-stage placebo among ineligible units
in0 = eligible == 0
print("ineligible n:", in0.sum())
for name, v in [("first stage", x), ("valid", y_valid), ("invalid", y_invalid)]:
    r = stats.linregress(z[in0], v[in0])
    print(name, round(r.slope, 3), round(r.stderr, 3), round(r.pvalue, 3))

# 3. Sensitivity: assume a direct effect gamma of z on y, remove it, re-estimate
for g in np.arange(0, 0.41, 0.1):
    print("gamma", round(g, 1), iv_fit(y_invalid - g * z, x, z)[:3])

       x 
1.474639 
          est    lo    hi cor_resid_z
valid   0.892 0.687 1.097           0
invalid 1.943 1.751 2.136           0
[1] 5022
  Estimate Std. Error   Pr(>|t|) 
     0.017      0.039      0.669 
        Estimate Std. Error Pr(>|t|)
valid      0.062      0.068    0.362
invalid    0.362      0.068    0.000
     gamma   est    lo    hi
[1,]   0.0 1.943 1.751 2.136
[2,]   0.1 1.593 1.422 1.764
[3,]   0.2 1.242 1.066 1.418
[4,]   0.3 0.892 0.687 1.097
[5,]   0.4 0.541 0.292 0.791

All figures come from one seeded run of the R code. Python's random numbers differ from R's, so its figures differ (a single IV estimate is noisy: 1.16 instead of 0.89 for the valid instrument, with a 95% CI that still contains 1), but they show the same pattern: residual correlations of zero in both cases, a placebo effect only for the invalid instrument, and a steady drop in the estimate as the assumed direct effect grows.

What the numbers show

• OLS is biased by the confounder. The OLS slope is 1.47, well above the true effect of 1.
• The valid instrument works. Its IV estimate is 0.89, 95% CI [0.69, 1.10], which contains the true value.
• The invalid instrument looks just as healthy. Its estimate is 1.94, 95% CI [1.75, 2.14], nearly double the truth, yet its residuals are also uncorrelated with the instrument (correlation 0 in both rows). Nothing in the main analysis warns you.
• The placebo test does warn you. Among the 5,022 ineligible units the instrument has no effect on the treatment (0.017, p = .669), so it should have no effect on the outcome either. With the valid instrument it does not (0.062, p = .362); with the invalid one it does (0.362, p < .001), which reveals the direct path.
• Sensitivity analysis shows what is at stake. Each 0.1 of assumed direct effect lowers the estimate by about 0.35, because the bias is the direct effect divided by the first-stage coefficient (about 0.3 here). Removing the true direct effect of 0.3 recovers the valid estimate exactly (0.892), and a direct effect of 0.4 would push it down to 0.54.

What to do instead of a test

• List the alternative paths. Write down every way the instrument could reach the outcome other than through the treatment (for example, a membership organisation that also offers credit or market access) and explain why each is absent or blocked. Adjusting for measured variables on those paths can help if the treatment does not affect them.
• Run zero-first-stage placebo tests. Find units for whom the instrument cannot change the treatment, confirm the first stage is near zero there, and test whether the instrument still predicts the outcome. A clear association is evidence against exclusion. A null result reassures but proves nothing: the test may lack power, and the direct path might act only where you cannot check.
• Use placebo outcomes. Test the instrument against outcomes measured before the treatment, or outcomes the treatment cannot plausibly affect. An association suggests the instrument is tied to something else.
• Check balance. Compare pre-treatment characteristics across instrument values. Imbalance does not prove a violation, but it means the instrument is not as-good-as-random.
• Report a sensitivity analysis. The plausibly exogenous approach of Conley, Hansen and Rossi (2012) works like part 3 of the code: assume a range of direct effects, re-estimate for each, and report the union of the confidence intervals (or put a prior on the direct effect). Readers can then judge whether a plausible violation would change the conclusion.
• Look for a second, different instrument. If two instruments that work through unrelated mechanisms give similar estimates, that is more convincing than either alone, even though agreement still cannot rule out a shared bias.

For background on why confounding needs a design solution in the first place, see what it means to control for a variable. Other quasi-experimental designs are compared in interrupted time series vs event study vs difference-in-differences, and the DASS blog covers planning and reporting analyses more broadly.

How to report an instrumental variable estimate in APA style (7th edition)

Report the IV estimate with its confidence interval, describe the first stage, and state the exclusion restriction as an assumption together with the checks you ran. Using the valid-instrument results above:

"Using z as an instrument for x in a two-stage least squares model, the estimated effect of x on y was b = 0.89, 95% CI [0.69, 1.10]. The estimate assumes that z affects y only through x. Consistent with this assumption, among the 5,022 units for whom z could not affect x, z was not associated with y (b = 0.06, SE = 0.07, p = .362)."

Add the sensitivity analysis in a sentence or a table: "Allowing z a direct effect on y of up to x.xx reduced the estimate to x.xx, 95% CI [LL, UL]." Give the first-stage coefficient and its F statistic in the method section or a table, and say which software and estimator you used.

Related tools and guides

• APA 7 formatter for regression
• What does it mean to control for a variable in regression?
• Interrupted time series vs event study vs difference-in-differences
• Which statistical test should I use?
• Instrumental variables estimation (Wikipedia)

More answered questions

• What does it mean to control for a variable in regression?
• How do you simulate a prediction interval from a fitted regression model?
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Written by AskStats with AI assistance. This is general information, not advice for your specific data or study. When the results matter, check your approach with a qualified statistician.