ECMT3120 Chap.5 IV Diagnostics and Simultaneous Equations
IV Diagnostics and Simultaneous Equations
Instrumental variables analysis does not end when a coefficient appears. A credible answer separates relevance, exogeneity and exclusion, then selects diagnostics that address only the assumptions they can actually test. This chapter develops overidentification tests, explains why rejection is informative but non-rejection is not proof, and connects identification to simultaneous equation systems.
It also shows how order and rank reasoning identify which excluded exogenous variables can provide independent movement for a particular structural equation.
What this chapter covers
- 01
Relevance, exogeneity and exclusion as separate claims
- 02
Exactly identified and overidentified specifications
- 03
Sargan and Hansen overidentification logic
- 04
Interpreting rejection and non-rejection
- 05
Structural and reduced form equations
- 06
Order condition for equation identification
- 07
Rank condition and independent variation
- 08
Specification repair after a failed diagnostic
Worked example · free
Diagnose an overidentified demand equation
- +1Recognise two excluded instruments for one endogenous regressor, leaving one overidentifying restriction.
- +1Use the auxiliary regression formula J equals n times R squared.
- +2Calculate J equals 400 times 0.003 equals 1.2.
- +1Compare 1.2 with a chi squared reference distribution with one degree of freedom.
- +1State that the small statistic supplies no evidence against the joint instrument restrictions at conventional levels.
- +2Explain that non-rejection does not prove that either instrument is valid and cannot reveal which instrument is faulty after rejection.
Key terms
- Overidentification
- A setting with more excluded instruments than endogenous regressors, creating restrictions that can be checked jointly.
- Sargan statistic
- Under homoskedasticity, an auxiliary regression statistic used to test overidentifying restrictions.
- Hansen J statistic
- A heteroskedasticity robust test of the joint orthogonality conditions in an overidentified IV model.
- Structural equation
- An equation describing a behavioural relationship in which some right hand variables may be jointly determined.
- Reduced form
- An equation expressing an endogenous variable using exogenous variables from the system.
- Order condition
- A necessary counting condition comparing excluded exogenous variables with endogenous variables requiring instruments.
IV Diagnostics and Simultaneous Equations FAQ
Does passing an overidentification test prove every instrument is valid?
No. The test evaluates joint compatibility with the maintained model and has limited power in finite samples. Instruments can share the same violation, or the test may simply fail to detect a modest violation. Defend each exclusion restriction substantively even after non-rejection.
What should I conclude when the J test rejects?
Conclude that the full set of orthogonality restrictions is inconsistent with the data under the maintained specification. Do not automatically discard the instrument with the weakest first stage. Revisit timing, exclusion arguments, functional form and the instrument set using theory and design evidence.
Why is the order condition not enough for simultaneous equations?
Counting excluded variables is only necessary. Those variables must also move the included endogenous regressors in sufficiently independent directions. The rank condition captures this variation requirement, so an equation can satisfy the count yet remain unidentified.
Exam move
Organise every diagnostic response in three columns: assumption, observable implication and available test. Practise computing the number of overidentifying restrictions as excluded instruments minus endogenous regressors, then state the reference degrees of freedom before reading a statistic.
For simultaneous systems, circle the equation being identified, list its included endogenous variables and count the exogenous variables excluded from that equation. Treat the order condition as a first screen and then explain the rank requirement in words.
End every diagnostic answer with a calibrated conclusion: rejection is evidence against the joint restrictions, while non-rejection only means the sample did not reveal a contradiction. Build a null-hypothesis card for each diagnostic.
The first-stage test concerns a block of excluded coefficients, residual inclusion concerns the disputed regressor under instrument validity, and an overidentification statistic concerns extra orthogonality restrictions jointly. Write what rejection changes and what it leaves unresolved.
For the Sargan calculation, count restrictions before multiplying sample size by the auxiliary R squared, and name the homoskedastic reference conditions. When a result rejects, practise proposing an evidence-led response rather than deleting whichever instrument has the weakest t statistic.
In a supply-and-demand system, mark every endogenous variable and every exogenous shifter, then identify the demand and supply equations separately. A final verbal check should explain why satisfying the count does not guarantee that the shifters produce independent movements needed for rank.
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