Common mistake 1 with Instrumental variables (2SLS)

Weak instruments. The cure is worse than the disease - a weak IV is more biased than the OLS it replaces.

Common mistake 2 with Instrumental variables (2SLS)

Justifying the exclusion restriction with statistics. It can only be argued from economics and institutional detail.

Common mistake 3 with Instrumental variables (2SLS)

Running two OLS regressions by hand. The second-stage standard errors will be wrong.

Common mistake 4 with Instrumental variables (2SLS)

Reporting 2SLS as "the" effect when it is a local effect for compliers only.

EconStatsCraft › Instrumental variables (2SLS)

Instrumental variables (2SLS)

Two-stage least squares. Reports the first stage, the weak-instrument F, the Wu–Hausman endogeneity test and the Sargan over-identification test.

What it needs

Dependent variable
Endogenous regressor - The variable you suspect is correlated with the error term.
Excluded instruments - Must move the endogenous regressor but not affect the outcome directly.
Exogenous controls - Optional.

The intuition

You want the effect of schooling on wages, but people who choose more schooling differ in unobserved ways. An instrument is a lever that pushes schooling around for reasons unrelated to those unobservables - proximity to a college, a birth date that changes compulsory-schooling exposure. Use ONLY the variation the lever creates, and the confounding is left behind with the variation you discarded.

When to use it

Simultaneity (price and quantity determined together), omitted variables (ability affecting both schooling and wages), or measurement error in a regressor.

Where the estimator comes from

OLS solves E[x(y-x'\beta)]=0, which fails when E[xu]\ne 0. IV replaces it with the moment condition E[z(y-x'\beta)]=0, giving \hat{\beta}=(Z'X)^{-1}Z'y when instruments and regressors are equal in number. With more instruments, 2SLS projects X onto Z: \hat{\beta}=(\hat{X}'\hat{X})^{-1}\hat{X}'y with \hat{X}=P_Z X. Crucially the variance uses \hat{\sigma}^2(\hat{X}'\hat{X})^{-1} where \hat{\sigma}^2 comes from residuals computed with the ACTUAL X - which is exactly why running two OLS regressions by hand gives the wrong standard errors.

The model

\hat{\boldsymbol{\beta}}_{\text{2SLS}}=(\hat{\mathbf{X}}^{\prime}\hat{\mathbf{X}})^{-1}\hat{\mathbf{X}}^{\prime}\mathbf{y},\qquad \hat{\mathbf{X}}=\mathbf{P}_{\mathbf{Z}}\mathbf{X}=\mathbf{Z}(\mathbf{Z}^{\prime}\mathbf{Z})^{-1}\mathbf{Z}^{\prime}\mathbf{X}

What it assumes

Relevance: the instrument genuinely moves the endogenous regressor. Testable - first-stage F > 10.
Exclusion: the instrument affects y only through the endogenous regressor. NOT testable, and it is the assumption reviewers attack.
Exogeneity: the instrument is uncorrelated with the error.
Monotonicity, if you want to interpret the result as a local average treatment effect.

Statistical properties

IV is consistent but BIASED in finite samples, with bias towards OLS that grows as instruments weaken.
With weak instruments, 2SLS is worse than the OLS it replaces, and conventional standard errors understate uncertainty severely.
The first-stage F > 10 rule of thumb comes from Staiger & Stock (1997); Lee et al. (2022) argue it should really be much higher.
With heterogeneous effects, IV identifies a LATE - the effect on those whose behaviour the instrument actually changed, not the population average.
Sargan/Hansen tests over-identifying restrictions only, and assumes at least one instrument is valid. It cannot validate an exactly identified model.
2SLS standard errors always exceed OLS. That is the price of consistency.

How to read the output

First-stage F below 10 means weak instruments - 2SLS is then biased towards OLS and its standard errors are too small.
A large gap between OLS and 2SLS suggests OLS was genuinely contaminated.
Wu–Hausman: rejecting means the regressor really is endogenous and IV is needed.
Sargan (only possible when over-identified): rejecting means at least one instrument fails the exclusion restriction.
2SLS standard errors are always larger than OLS. That is the price you pay.

A worked reading

OLS puts the return to schooling at 0.2072; 2SLS, instrumenting with proximity to a college, puts it at 0.0924 (SE 0.0257). The first-stage F is 144.84 on (1, 896), far above the rule-of-thumb 10, and the Wu-Hausman test rejects exogeneity at F = 28.16. So OLS is more than twice the instrumented estimate - the direction omitted ability predicts, and a larger gap than most published applications find. The first stage is what makes the second stage readable: with a weak instrument the 2SLS estimate is biased towards OLS and its confidence interval is not to be trusted at all, which is why the F is reported before the coefficient rather than after it.

Symptom → diagnosis → what to do

If you see this	It means	Do this
First-stage F < 10	Weak instruments	Find stronger ones, or report Anderson–Rubin confidence sets
2SLS ≈ OLS and Wu–Hausman does not reject	Endogeneity may be minor	OLS is defensible and more efficient - but the test has low power
2SLS enormous with huge SEs	Weak instrument dividing by nearly zero	Do not report it as a point estimate
Sargan rejects	At least one instrument violates exclusion	Drop the suspect one and re-test
Multiple endogenous regressors	Per-equation F is the wrong test	Use Cragg–Donald / Kleibergen–Paap with Stock–Yogo values
Instrument plausibly affects y directly	Exclusion fails	No statistic can fix this - argue it or abandon it

Common mistakes

Weak instruments. The cure is worse than the disease - a weak IV is more biased than the OLS it replaces.
Justifying the exclusion restriction with statistics. It can only be argued from economics and institutional detail.
Running two OLS regressions by hand. The second-stage standard errors will be wrong.
Reporting 2SLS as "the" effect when it is a local effect for compliers only.

Where to read this properly

Angrist and Pischke (2009), Mostly Harmless Econometrics: An Empiricist's Companion, Princeton University Press, chapter 4
Staiger & Stock (1997); Stock & Yogo (2005) - weak instruments
Imbens & Angrist (1994) - LATE
Card (1995) - the college-proximity instrument
Lee, McCrary, Moreira & Porter (2022) - why F > 10 is too lenient

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Instrumental variables (2SLS)

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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