ECMT6002 Chap.5 Stochastic Regressors and Instruments
Stochastic Regressors and Instruments
Stochastic regressors become a problem when their sample variation is connected to the structural disturbance. This chapter traces endogeneity to omitted variables, simultaneous causality and measurement error, then introduces instrumental variables as a source of exogenous variation. A valid instrument must be relevant and exogenous, and those requirements answer different questions.
The ratio estimator is developed through reduced form and first stage covariances so that its identification logic remains visible. Omitted factors cause trouble when they affect the outcome and covary with an included regressor. Simultaneity arises when the outcome and explanatory variable are jointly determined, so a single equation cannot treat the regressor as externally assigned.
Classical measurement error can attenuate a slope by loading the observed regressor with noise. Each mechanism points toward a different design response. An excluded instrument addresses endogeneity by isolating variation in the regressor that is unrelated to the structural disturbance.
Relevance asks whether that instrument moves the endogenous regressor after controls; exogeneity and exclusion ask whether it has any forbidden connection to outcome shocks or a direct outcome path. A first stage can measure the first claim, while economic and institutional reasoning must defend the second.
In the just identified case, the IV slope is the instrument-outcome covariance divided by the instrument-regressor covariance. A small denominator exposes weak identification, making the ratio unstable and ordinary normal approximations unreliable. The estimate remains conditional on the instrument design rather than becoming causal merely because software labels it IV.
What this chapter covers
- 01
Random regressors and exogeneity
- 02
Sources of endogeneity
- 03
Instrument relevance
- 04
Instrument exogeneity
- 05
Reduced form and first stage
- 06
Single instrument IV estimator
Worked example · free
Computing a just identified IV estimate
- +1Write the just identified ratio as Cov(z,y) divided by Cov(z,x).
- +1Insert the sample covariances: 8 divided by 4.
- +1The IV slope is 2, conditional on the instrument validity assumptions.
Key terms
- Endogeneity
- Correlation between an explanatory variable and the structural disturbance.
- Instrument
- A variable supplying variation in an endogenous regressor while remaining excluded from the structural disturbance.
- Relevance
- Nonzero predictive content of the instrument for the endogenous regressor, conditional on controls.
- Reduced form
- A relationship expressing an endogenous variable as a function of exogenous variables and disturbances.
Stochastic Regressors and Instruments FAQ
Can relevance be tested directly?
The first stage provides evidence about sample relevance and instrument strength. It does not establish population exogeneity, and weak instrument diagnostics must be interpreted in the setting of the chosen estimator and test.
Why is an instrument not just another control?
A control enters the structural outcome equation to condition the relationship. An excluded instrument instead shifts the endogenous regressor and identifies its structural effect while being omitted from the outcome equation.
What happens with a weak instrument?
The denominator connecting the instrument to the endogenous regressor is small, so IV estimates become unstable and conventional approximations can be poor. More observations do not automatically repair weak identification.
How should I defend an exclusion restriction?
Trace every plausible route from the instrument to the outcome. Explain why the intended path operates through the endogenous regressor, then address direct effects and common causes using institutional detail or design evidence. A significant first stage supports relevance only; it cannot demonstrate that the instrument is absent from the structural disturbance.
Exam move
For every proposed instrument, write two separate paragraphs headed relevance and exogeneity. Draw arrows for the instrument, endogenous regressor, outcome and omitted causes. Only after the economic argument is clear should you calculate the first stage, reduced form or IV ratio. Keep a list of common endogeneity sources and their expected OLS consequences.
Practise classifying short scenarios as omitted-variable, simultaneity or measurement-error problems before proposing a remedy. For omitted variables, write the two signs that determine the likely OLS bias; for simultaneity, write both structural equations; for measurement error, specify which variable is measured badly.
Create an instrument audit with columns for the first-stage mechanism, every plausible direct outcome path, shared causes and the population whose behaviour the instrument changes. Calculate the reduced-form effect and first-stage effect separately before taking their ratio, and verify that units cancel to the structural slope units. Compare OLS and IV without automatically treating disagreement as proof that IV succeeded.
Weak instruments, heterogeneous effects and exclusion failures can also create differences. In written answers, place the strongest threat immediately after the relevance argument and state what evidence would reduce it. That adversarial habit is more useful than searching for a convenient variable that merely predicts x.
Working through Stochastic Regressors and Instruments in ECMT6002? Sia is AskSia’s AI Economics tutor — ask any ECMT6002 Stochastic Regressors and Instruments question and get a clear, step-by-step explanation grounded in how ECMT6002 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
