Skip to content

Lesson M10.L03: Two-Stage Least Squares (2SLS)

Field Detail
Module M10 — Instrumental Variables
Level Graduate (ECMT5001 S1 2027)
Duration 35 min
Learning Objective Implement 2SLS with multiple instruments; understand over-, under-, and exact-identification; correctly interpret first- and second-stage output; avoid the SEs mistake from naive two-step OLS
Provenance Wooldridge, Introductory Econometrics, Ch 15.3–15.4

Explanation

Why 2SLS?

The simple IV estimator from M10.L02 handles one endogenous variable with one instrument. In practice, we often have:

  • Multiple instruments for one endogenous variable (over-identified)
  • Multiple endogenous variables requiring multiple instruments
  • Exogenous control variables that must be included in both stages

Two-stage least squares (2SLS) is the general framework that handles all these cases efficiently, combining all instruments optimally.

The 2SLS Procedure

Setup: Structural model with one endogenous regressor \(x\), exogenous controls \(\mathbf{w}\), and instruments \(\mathbf{z}\):

\[y = \beta_0 + \beta_1 x + \boldsymbol{\beta}_w' \mathbf{w} + u\]

where \(\text{Cov}(x, u) \neq 0\), \(\text{Cov}(\mathbf{w}, u) = 0\), \(\text{Cov}(\mathbf{z}, u) = 0\).

Stage 1: First-Stage Regression

Regress the endogenous variable \(x\) on all instruments \(\mathbf{z}\) and all exogenous controls \(\mathbf{w}\):

\[x = \pi_0 + \boldsymbol{\pi}_z' \mathbf{z} + \boldsymbol{\pi}_w' \mathbf{w} + v\]

Save the fitted values \(\hat{x}\) from this OLS regression. The fitted values represent the "clean" (exogenous) variation in \(x\) — the part driven by instruments.

Important: Include all exogenous controls \(\mathbf{w}\) in Stage 1, even though they're already exogenous. Omitting them leads to misspecification.

Stage 2: Second-Stage Regression

Regress \(y\) on \(\hat{x}\) (from Stage 1) and the exogenous controls \(\mathbf{w}\):

\[y = \beta_0 + \beta_1 \hat{x} + \boldsymbol{\beta}_w' \mathbf{w} + \text{error}\]

The OLS coefficients from this second regression are the 2SLS estimates \(\hat{\boldsymbol{\beta}}^{\text{2SLS}}\).

Intuition: Stage 2 uses only the variation in \(x\) that is explained by the instruments — the part uncorrelated with \(u\). The endogenous variation (correlated with \(u\)) is "projected out" in Stage 1.

Equivalence to Simple IV

With one instrument and one endogenous variable, 2SLS = IV.

Stage 1: \(\hat{x} = \hat{\pi}_0 + \hat{\pi}_z z\)
Stage 2 coefficient on \(\hat{x}\): \(\hat{\beta}_1^{\text{2SLS}} = \hat{\beta}_1^{\text{IV}} = \widehat{\text{Cov}}(z,y)/\widehat{\text{Cov}}(z,x)\) ✓

The advantage of 2SLS is that it generalises to multiple instruments by projecting \(x\) onto the instrument subspace optimally.

Identification: Under-, Exact-, and Over-Identification

Let: - \(m\) = number of endogenous regressors
- \(\ell\) = number of instruments (excluded from structural equation)

Case Condition Implication
Under-identified \(\ell < m\) Cannot identify all endogenous coefficients; 2SLS not feasible
Exactly identified \(\ell = m\) Unique 2SLS solution; cannot test instrument validity
Over-identified \(\ell > m\) Multiple instruments; 2SLS uses them efficiently; can test exogeneity

Over-identification is a gift — it provides testable restrictions (Sargan-Hansen \(J\)-test) on instrument validity, and the extra instruments improve efficiency.

The Sargan-Hansen J-Test (Over-Identified Case)

When \(\ell > m\), we can test whether the instruments are jointly exogenous.

\[H_0: \text{all } \ell \text{ instruments are exogenous (i.e., } \text{Cov}(\mathbf{z}, u) = 0\text{)}\]

Steps:

  1. Estimate by 2SLS and save residuals \(\hat{u}_t^{\text{2SLS}}\)
  2. Regress \(\hat{u}_t^{\text{2SLS}}\) on all instruments \(\mathbf{z}\) and exogenous controls \(\mathbf{w}\)
  3. \(J = n \cdot R^2_{\text{aux}} \xrightarrow{d} \chi^2_{\ell - m}\) under \(H_0\)
  4. Reject \(H_0\) if \(J > \chi^2_{\ell-m, \alpha}\)

Limitation: The \(J\)-test requires the model to be over-identified (\(\ell > m\)). In the exactly-identified case (\(\ell = m\)), \(J = 0\) identically — you cannot test instrument validity. This is why finding multiple valid instruments is valuable.

Caution: Failing to reject \(H_0\) does NOT confirm instrument validity — it only says the over-identifying restrictions are not rejected. All instruments could be endogenous if they are endogenous in the same direction.

2SLS Standard Errors

2SLS SEs are always larger than OLS SEs — this is the efficiency cost of using IV.

The asymptotic variance of \(\hat{\beta}_1^{\text{2SLS}}\) (simplified for one endogenous variable):

\[\text{Var}(\hat{\beta}_1^{\text{2SLS}}) \approx \frac{\sigma_u^2}{n \cdot \text{Var}(\hat{x})} = \frac{\sigma_u^2}{n \cdot R^2_{1} \cdot \text{Var}(x)}\]

where \(R^2_1\) is the R² from the first-stage regression. Since \(R^2_1 \leq 1\), \(\text{Var}^{\text{2SLS}} \geq \text{Var}^{\text{OLS}}\).

Use robust (heteroskedasticity-consistent) SEs in 2SLS — most software does this by default. Stata: ivregress 2sls ... , robust.

Worked Example

Setup: Estimate the return to education with two instruments.

\[\log(\text{wage}_i) = \beta_0 + \beta_1 \text{educ}_i + \beta_2 \text{exper}_i + u_i\]

where exper (work experience) is exogenous. Two instruments: - \(z_1\): proximity to 4-year college (binary) - \(z_2\): proximity to 2-year college (binary)

Since \(m = 1\) (one endogenous variable: education) and \(\ell = 2\) (two instruments), this is over-identified (\(\ell - m = 1\) degree of freedom for J-test).

Stage 1: Regress educ on exper, \(z_1\), \(z_2\):

\[\text{educ}_i = \pi_0 + \pi_1 \text{exper}_i + \pi_2 z_{1i} + \pi_3 z_{2i} + v_i\]
Variable \(\hat{\pi}\) SE \(t\)
Intercept 14.32 0.58 24.7
exper −0.073 0.018 −4.1
near4yr (\(z_1\)) 0.442 0.107 4.13
near2yr (\(z_2\)) 0.312 0.098 3.18

First-stage F-statistic (joint test of \(\pi_2 = \pi_3 = 0\)): \(F = 12.4 > 10\) → instruments are relevant (not weak).

Stage 2: Regress \(\log(\text{wage})\) on \(\hat{\text{educ}}\), exper:

\[\log(\text{wage}_i) = \beta_0 + \beta_1 \hat{\text{educ}}_i + \beta_2 \text{exper}_i + \text{error}\]
Variable \(\hat{\beta}^{\text{2SLS}}\) SE (robust) \(t\)
Intercept 3.41 0.72 4.74
educ 0.129 0.047 2.74
exper 0.032 0.011 2.91

Second-stage interpretation: Each additional year of education raises wages by approximately 12.9% (2SLS), compared to an OLS estimate of 4.0%. The 2SLS SE (0.047) is larger than the hypothetical OLS SE, reflecting the efficiency cost.

Sargan-Hansen J-test:

Regress 2SLS residuals on all instruments and controls; \(n \cdot R^2_{\text{aux}} = 2.14\). Critical value \(\chi^2_{1,0.05} = 3.84\).

Since \(2.14 < 3.84\) → fail to reject \(H_0\). Over-identifying restriction not rejected — instruments appear jointly exogenous.

Common Misconception

"2SLS is just running two separate OLS regressions and plugging in predicted values from the first — you can do it manually in two steps."

This gives wrong standard errors. If you manually run Stage 2 with \(\hat{x}\) as a regressor, the Stage 2 residuals are computed as:

\[\hat{e}_i = y_i - \hat{\beta}_0 - \hat{\beta}_1 \hat{x}_i - \boldsymbol{\hat{\beta}}_w' \mathbf{w}_i\]

These residuals are based on \(\hat{x}\), not \(x\). The Stage 2 OLS formula uses \(\hat{e}\) to estimate \(\sigma^2\), but \(\hat{e}\) is not the correct structural residual \(u = y - \beta_0 - \beta_1 x - \boldsymbol{\beta}_w' \mathbf{w}\). The proper 2SLS variance formula corrects for this substitution — manual two-step OLS does not. Always use a proper 2SLS command (ivregress 2sls in Stata, IV2SLS in statsmodels or linearmodels.IV2SLS in Python).

The coefficient estimates from manual two-step OLS and proper 2SLS are identical; only the SEs differ.

Practice Prompts

  1. A researcher has two endogenous variables (education and experience) and three instruments (proximity to college, quarter of birth, and state compulsory schooling laws). Is the model under-, exactly-, or over-identified? How many degrees of freedom does the Sargan-Hansen J-test have?

  2. In the worked example, the Sargan-Hansen J-statistic is 2.14. Interpret this result carefully: does it confirm that both instruments are valid? If one instrument were endogenous but both instruments were endogenous in the same direction, would the test detect this?

  3. Explain why 2SLS standard errors are always at least as large as OLS standard errors, using the formula \(\text{Var}(\hat{\beta}_1^{\text{2SLS}}) \approx \sigma_u^2 / (n \cdot R^2_1 \cdot \text{Var}(x))\). Under what limiting condition would 2SLS be equally efficient as OLS, and why can that condition never hold in practice for a truly endogenous regressor?

Further Resources

  • Wooldridge, Introductory Econometrics (7th ed.), Ch 15.3–15.4
  • Angrist, J.D. & Pischke, J.-S. (2009), Mostly Harmless Econometrics, Ch 4.6
  • Sargan, J.D. (1958), "The Estimation of Economic Relationships Using Instrumental Variables," Econometrica
  • Python: linearmodels.iv.model.IV2SLS; Stata: ivregress 2sls y (x = z1 z2) w, robust