ECMT6002 Chap.6 Two Stage Least Squares and Systems
Two Stage Least Squares and Systems
Two stage least squares extends the IV idea to multiple instruments and endogenous regressors. The first stage projects each endogenous variable onto all included exogenous variables and excluded instruments; the second stage uses the projected variation to estimate the structural equation. This chapter also introduces simultaneous equation systems, identification and overidentifying restrictions.
The emphasis is that the stages are an estimator construction, while correct uncertainty must account for the full procedure. Every exogenous regressor in the structural equation belongs to the instrument set along with any credible excluded instruments. Projection onto that full space produces fitted endogenous regressors containing only instrument-explained variation.
The second-stage coefficient can then be written as a single matrix IV estimator, which is why manually running a fitted-value regression gives the coefficient intuition but ordinary second-stage standard errors are not valid. Simultaneous systems make identification visible. A demand equation needs a shifter excluded from demand but present in supply, while a supply equation needs an analogous demand shifter.
Counting exclusions is a necessary order check, but the rank condition asks whether the excluded variables actually create independent movement in the endogenous regressors. With extra instruments, overidentifying restrictions offer a diagnostic of their joint compatibility, not proof that all are valid. Strength also matters: adding weak or dubious instruments can degrade inference rather than improve it.
The entire 2SLS argument therefore connects structural exclusions, first-stage rank, coefficient estimation and uncertainty.
What this chapter covers
- 01
First stage projections
- 02
Second stage structural equation
- 03
Included and excluded instruments
- 04
Order and rank identification
- 05
Simultaneous equations
- 06
Overidentification and inference
Worked example · free
Reading a two stage least squares design
- +1First regress p on an intercept, w and z to isolate instrument generated price variation.
- +1Obtain fitted price p hat from that projection.
- +1Regress demand quantity on an intercept, p hat and w in the structural second stage representation.
- +1Use a proper 2SLS covariance estimator rather than ordinary second stage OLS standard errors.
Key terms
- Two stage least squares
- An IV estimator using projections onto an instrument space before estimating structural coefficients.
- Excluded instrument
- An exogenous variable used for identification but omitted from the structural outcome equation.
- Identification
- The ability to recover a unique structural parameter from the population relationships and restrictions.
- Overidentified
- A setting with more valid instrument information than the minimum needed to identify the endogenous coefficients.
Two Stage Least Squares and Systems FAQ
Can I manually run two regressions?
The fitted value intuition is useful, but ordinary standard errors from a manually run second regression are generally wrong. Use a 2SLS routine or the correct matrix covariance calculation for inference.
Why include exogenous controls in the first stage?
The instrument set includes the structural equation exogenous variables as their own instruments. Omitting them from the projection changes the variation used and generally fails to implement standard 2SLS correctly.
Does overidentification prove validity?
No. A test of overidentifying restrictions examines compatibility among maintained instrument conditions. It can fail to detect shared invalidity and depends on having at least some valid restrictions as the maintained reference.
What is the difference between order and rank?
The order condition counts whether enough excluded instruments are available for the endogenous regressors. The rank condition asks whether those instruments generate independent movements needed to identify each structural coefficient. Passing the count is necessary but not sufficient: several instruments may all move the same endogenous direction and leave another coefficient unidentified.
Exam move
Label every variable as outcome, endogenous regressor, included exogenous variable or excluded instrument before writing an equation. Practise mapping a verbal simultaneous system into structural and reduced form equations. For calculations, track matrix dimensions. For interpretation, keep identification arguments distinct from overidentification diagnostics.
Take a small demand-and-supply example and colour-code variables by equation membership and exclusion. Write one first stage for every endogenous regressor, making sure all structural exogenous controls reappear. Then describe the projection geometry in words: fitted x lies in the column space of the instrument matrix.
Reproduce the compact 2SLS matrix expression and annotate the dimensions of each product so transposition mistakes become obvious. For system identification, perform the order condition and then ask the harder rank question about independent variation. When several instruments are available, prepare a table recording first-stage strength, exclusion rationale and possible shared invalidity.
Explain why an overidentification test can reject a set without naming the guilty instrument and why nonrejection cannot certify the maintained assumptions. Finally, practise interpreting a structural coefficient in the units of the original variables, not in units of fitted x. Close every answer by noting that reported uncertainty must come from a 2SLS covariance estimator.
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