MFIN6201 Chap.7 Omitted Variables and Model Specification
Omitted Variables and Model Specification
This chapter gives a two-condition test for omitted-variable bias and a sign rule for its direction. A missing factor must affect the outcome and correlate with the regressor for the standard bias channel to operate. Control variables, conditional mean independence, mediators and sensitivity across plausible specifications are evaluated from the target estimand rather than model fit.
Sign analysis proceeds in two steps: determine how X moves with the omitted W, then determine how W changes Y while X is held fixed. Multiplying the signs gives the direction in which the original X coefficient is pulled. If either link is zero, the omission may reduce precision without biasing that slope through the standard channel.
Observed common causes can be included as controls when they create a credible conditional comparison. Variables caused by treatment can instead block part of the effect, and conditioning on a collider can introduce association. Specification therefore begins with the intended total or direct effect and the timing of candidate variables.
Comparing a benchmark with plausible alternatives exposes fragility, but neither coefficient stability nor a higher R-squared proves that the identifying assumption is true.
What this chapter covers
- 01
Two OVB conditions
- 02
Bias direction rule
- 03
Control variables
- 04
Conditional mean independence
- 05
Bad controls and mediators
- 06
Specification sensitivity
Worked example · free
Signing an omitted-variable bias
- +1Set sign of correlation between X and W as positive.
- +1Set sign of W's effect on Y as negative.
- +1Positive times negative gives downward bias in the X coefficient.
Key terms
- Omitted-variable bias
- Coefficient distortion caused by a missing factor linked to both regressor and outcome.
- Control variable
- An included regressor used to improve the conditional comparison for the target coefficient.
- Mediator
- A variable that transmits part of the effect from treatment to outcome.
Omitted Variables and Model Specification FAQ
When does an omitted variable create bias?
The omitted factor must influence the outcome after accounting for the regressor and must be correlated with that regressor. If either link is absent, that factor does not create the standard omitted-variable bias in the target slope.
How do I determine bias direction?
Sign the relationship between the included regressor and omitted factor, then sign the omitted factor's conditional effect on the outcome. Multiply those signs. State the mechanism for each link instead of guessing from the observed coefficient.
Can adding controls make a model worse?
Yes. A control caused by the treatment can block part of the effect of interest, while a collider can create a spurious association. Controls should follow the estimand, timing and causal structure rather than a desire to maximise fit.
What if only one OVB condition holds?
If a missing factor affects Y but is independent of X, it may raise residual variance without biasing the X slope through the standard channel. Correlation with X alone is also insufficient when the factor has no conditional effect on Y.
How should sensitivity be reported?
Name the benchmark and substantive reason for each alternative, then compare the target estimate and uncertainty. Stability supports robustness without proving exogeneity, while movement shows dependence on modelling choices without identifying an automatic winner.
Why must the target effect be stated first?
A variable can be a confounder when estimating one effect and a mediator when estimating another. Naming whether the goal is a total or direct effect determines which pathways should remain open and which variables belong in the conditioning set.
Does a stable coefficient prove there is no confounding?
No. Several omitted influences can offset each other, or all plausible specifications can share the same missing factor. Stability is useful robustness evidence, but the identifying argument still depends on substantive knowledge of the data-generating process. Treat stability as evidence, not a logical guarantee.
Exam move
Draw a three-node diagram for every OVB question and label both relevant signs. State the target effect before evaluating a control, because the same variable can be a confounder for one estimand and a mediator for another. Compare only plausible specifications and explain coefficient movement as evidence about the comparison, not automatic proof of correction.
Practise all four sign combinations until upward and downward bias follow from the arrows rather than memory. For each candidate control, mark whether it occurs before treatment, after treatment or through selection. Write the benchmark specification and the reason for every alternative before calculating fit. When a coefficient changes, describe what comparison changed and what new assumption was added.
Include one example where omission affects precision but not bias because the missing variable is independent of X. Finish with a sensitivity paragraph naming the strongest unmeasured common cause and the direction it would need to move the result. This workflow keeps model choice connected to the empirical question instead of the preferred estimate.
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