ECMT3120 Chap.4 Two-Stage Least Squares and Weak Instruments
Two-Stage Least Squares and Weak Instruments
Two stage least squares extends simple IV to models with controls, several instruments and several endogenous regressors. This chapter explains why included exogenous variables belong in the instrument matrix, how the projection matrix creates fitted regressors, and how the second stage uses their predicted variation.
It also develops the joint first stage F test, rank conditions, precision costs and the distinction between a cleaner design and a smaller standard error.
What this chapter covers
- 01
Included exogenous variables as instruments for themselves
- 02
First stage projection of endogenous regressors
- 03
Second stage structural regression
- 04
Projection matrix and matrix 2SLS
- 05
Joint first stage relevance tests
- 06
Rank with multiple endogenous regressors
- 07
Weak instruments and 2SLS variance
- 08
Stata syntax for endogenous and excluded variables
Worked example · free
Explain a transport 2SLS design
- +1Place latent demand in the structural error and explain why operators increase frequency when demand is expected to be high.
- +2Regress frequency on both controls and both excluded scheduling instruments in the first stage.
- +1Test the excluded scheduling instruments jointly after the controls.
- +2Regress ridership on predicted frequency and the original controls in the second stage.
- +1Interpret the coefficient using scheduling induced variation in frequency.
- +1Explain that 2SLS uses a narrower fitted component, so less variation can yield a larger standard error without making the design worse.
Key terms
- Two stage least squares
- An IV estimator that projects endogenous regressors onto instruments and uses the fitted components in the structural equation.
- Included instrument
- An exogenous regressor that appears in the structural equation and instruments its own coefficient.
- Excluded instrument
- An exogenous variable used to predict an endogenous regressor but omitted from the structural equation under exclusion.
- Projection matrix
- The matrix that maps a variable to its least squares fitted value from the full instrument set.
- First stage F test
- A joint test of whether excluded instruments add predictive power for an endogenous regressor after controls.
- Rank condition
- The requirement that instruments generate enough distinct fitted variation to identify each endogenous regressor separately.
Two-Stage Least Squares and Weak Instruments FAQ
Why do included controls appear in both stages?
An included control is exogenous and therefore instruments its own coefficient. The first stage asks whether excluded instruments predict the endogenous regressor after conditioning on those controls. The second stage retains the original controls so the structural comparison matches the stated model.
Does a first stage F statistic above ten prove strong identification?
No. The familiar threshold is a rough warning rule in simple settings. It does not prove instrument exogeneity, and with multiple endogenous regressors separate rank problems can remain even when individual first stages appear significant. Report the statistic, tested set and design context.
Why can 2SLS have a larger standard error than OLS?
OLS uses all observed regressor variation, including variation that may be endogenous. 2SLS identifies from the instrument predicted portion. If that portion varies little or is highly collinear with controls, the estimator has less information and a larger sampling variance.
Exam move
Write every 2SLS problem as a role table before algebra: outcome, endogenous regressors, included exogenous controls and excluded instruments. Build the first stage for each endogenous regressor, state the joint relevance null, and inspect whether fitted movements are distinct enough for rank. Then write the second stage and interpret the coefficient through the instrument induced variation.
Practise the restricted and unrestricted sum of squares formula for the first stage F statistic, including numerator and denominator degrees of freedom. Finish by separating precision from identification: a smaller standard error is valuable only among estimators that consistently target the intended effect under credible assumptions.
Rehearse the software notation only after the role table is correct, reading every parenthesis as an economic statement about what is endogenous and what is excluded. For several instruments, calculate the joint F statistic from restricted and unrestricted sums of squares and label L, N and the unrestricted parameter count before substitution.
With several endogenous regressors, sketch the matrix of first-stage coefficients and ask whether the predicted movements differ enough to separate structural effects. Compare OLS and 2SLS standard errors by naming the variation each estimator uses, not by treating the smaller number as automatically superior.
In design questions, give every excluded instrument its own exclusion defence; one plausible instrument does not validate its companions. Close by stating what redesign would help if rank is poor: a distinct source of movement, a narrower structural question, or an honest acknowledgment that separate coefficients are not identified.
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