6.1 Instrumental variables (IV) assumptions

Instrumental variables (IVs) are a powerful tool in causal inference, used to estimate treatment effects when unmeasured confounding is present. They rely on variables associated with the treatment but not directly affecting the outcome, except through the treatment itself.

To use IVs effectively, several key assumptions must be met: relevance, exclusion restriction, exchangeability, and monotonicity. Understanding these assumptions is crucial for obtaining valid causal estimates and interpreting the results correctly.

Definition of instrumental variables

• Instrumental variables (IVs) are a powerful tool in causal inference used to estimate the causal effect of a treatment on an outcome in the presence of unmeasured confounding
• IVs are variables that are associated with the treatment but do not directly affect the outcome except through their effect on the treatment
• The use of IVs relies on several key assumptions that must be satisfied to obtain valid causal estimates

Relevance assumption

Instrument correlated with treatment

Top images from around the web for Instrument correlated with treatment

• 

  Frontiers | Illustrating Instrumental Variable Regressions Using the Career Adaptability – Job ... View original

  Is this image relevant?

• 

  Instrumental variables estimation - Wikipedia View original

  Is this image relevant?

• 

  Frontiers | Mendelian Randomization With Refined Instrumental Variables From Genetic Score ... View original

  Is this image relevant?

• 

  Frontiers | Illustrating Instrumental Variable Regressions Using the Career Adaptability – Job ... View original

  Is this image relevant?

• 

  Instrumental variables estimation - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Instrument correlated with treatment

• 

  Frontiers | Illustrating Instrumental Variable Regressions Using the Career Adaptability – Job ... View original

  Is this image relevant?

• 

  Instrumental variables estimation - Wikipedia View original

  Is this image relevant?

• 

  Frontiers | Mendelian Randomization With Refined Instrumental Variables From Genetic Score ... View original

  Is this image relevant?

• 

  Frontiers | Illustrating Instrumental Variable Regressions Using the Career Adaptability – Job ... View original

  Is this image relevant?

• 

  Instrumental variables estimation - Wikipedia View original

  Is this image relevant?

1 of 3

• The relevance assumption requires that the instrumental variable is correlated with the treatment variable
• This correlation ensures that the instrument provides information about the variation in the treatment
• The strength of the correlation between the instrument and the treatment is crucial for the precision of the IV estimates
  • Weak instruments that are only weakly correlated with the treatment can lead to imprecise and biased estimates

Strength of instrument

• The strength of an instrument is often assessed using the F-statistic or the partial $R^2$ from the first-stage regression of the treatment on the instrument
• A commonly used rule of thumb is that an F-statistic greater than 10 indicates a sufficiently strong instrument
• Weak instruments can result in large standard errors and confidence intervals for the IV estimates, making it difficult to draw precise conclusions about the causal effect

Exclusion restriction

Instrument affects outcome only through treatment

• The exclusion restriction assumes that the instrumental variable affects the outcome only through its effect on the treatment
• This assumption implies that there are no direct paths or backdoor paths from the instrument to the outcome, other than through the treatment
• Violations of the exclusion restriction can occur if the instrument has a direct effect on the outcome or if there are unmeasured common causes of the instrument and the outcome

Violations of exclusion restriction

• Violations of the exclusion restriction can lead to biased IV estimates and invalid causal conclusions
• For example, if an instrument (Z) directly affects the outcome (Y) in addition to its effect through the treatment (X), the IV estimate will capture both the direct effect of Z on Y and the indirect effect of Z on Y through X
• Assessing the plausibility of the exclusion restriction often relies on subject matter knowledge and careful consideration of potential violations

Exchangeability

Instrument independent of potential outcomes

• Exchangeability in the context of instrumental variables requires that the instrument is independent of the potential outcomes
• This assumption implies that the instrument does not have any direct effect on the outcome and is not associated with any unmeasured confounders that affect both the treatment and the outcome
• Exchangeability ensures that the distribution of potential outcomes is the same across different levels of the instrument

Instrument independent of measured confounders

• The instrumental variable should also be independent of measured confounders that are associated with both the treatment and the outcome
• This assumption can be assessed by examining the balance of measured covariates across different levels of the instrument
• If there are systematic differences in measured confounders across instrument levels, it suggests that the instrument may be associated with unmeasured confounders as well

Instrument independent of unmeasured confounders

• Exchangeability also requires that the instrument is independent of unmeasured confounders that affect both the treatment and the outcome
• This assumption is not directly testable, as unmeasured confounders are, by definition, not observed
• The plausibility of this assumption relies on subject matter knowledge and careful consideration of potential unmeasured confounders

Monotonicity assumption

Concept of defiers

• The monotonicity assumption in instrumental variables relates to the concept of defiers
• Defiers are individuals who always do the opposite of what the instrument suggests
  • For example, if the instrument encourages treatment, defiers would always refuse treatment, and if the instrument discourages treatment, defiers would always take the treatment
• The presence of defiers can lead to biased IV estimates and violation of the monotonicity assumption

Absence of defiers

• The monotonicity assumption requires that there are no defiers in the population
• This assumption implies that the instrument affects the treatment in a monotonic way, meaning that the instrument only encourages or discourages the treatment in one direction
• The absence of defiers ensures that the IV estimate can be interpreted as a local average treatment effect (LATE) for the subpopulation of compliers who respond to the instrument as intended

Consequences of violated assumptions

Bias in IV estimates

• Violations of the key assumptions (relevance, exclusion restriction, exchangeability, monotonicity) can lead to biased IV estimates
• The direction and magnitude of the bias depend on the nature and severity of the violation
  • For example, violations of the exclusion restriction can result in estimates that are biased towards the confounded association between the treatment and the outcome
• Bias in IV estimates can lead to incorrect causal conclusions and misleading policy implications

Sensitivity analyses for violations

• Sensitivity analyses can be conducted to assess the robustness of IV estimates to potential violations of assumptions
• These analyses involve simulating different scenarios of assumption violations and examining how the IV estimates change under these scenarios
• Sensitivity analyses can provide insights into the potential impact of assumption violations on the causal conclusions
  • For example, researchers can assess how large a direct effect of the instrument on the outcome would need to be to substantially alter the IV estimate

Considerations for choosing instruments

Subject matter knowledge

• Choosing appropriate instrumental variables often relies on subject matter knowledge and understanding of the causal relationships in the specific context
• Researchers should carefully consider the plausibility of the key assumptions (relevance, exclusion restriction, exchangeability, monotonicity) based on their knowledge of the problem domain
• Collaboration with experts in the field can help identify potential instruments and assess their suitability

Data-driven approaches

• Data-driven approaches can also be used to identify potential instrumental variables
• These approaches involve using statistical methods to search for variables that are strongly associated with the treatment but not directly associated with the outcome
  • For example, researchers can use machine learning algorithms to identify variables that predict treatment assignment but have no direct effect on the outcome
• However, data-driven approaches should be used cautiously and in combination with subject matter knowledge to ensure the validity of the selected instruments

Interpretation of IV estimates

Local average treatment effect (LATE)

• IV estimates are often interpreted as the local average treatment effect (LATE) for the subpopulation of compliers
• Compliers are individuals whose treatment status is affected by the instrument in the intended way
  • For example, if the instrument encourages treatment, compliers are those who would take the treatment when encouraged and not take the treatment when not encouraged
• The LATE represents the average causal effect of the treatment for the subpopulation of compliers

Generalizability of IV estimates

• The generalizability of IV estimates to the entire population depends on the similarity between the compliers and the overall population
• If the compliers are systematically different from the non-compliers in terms of their characteristics or their response to the treatment, the LATE may not be representative of the average treatment effect for the entire population
• Researchers should be cautious when extrapolating IV estimates to different populations or settings, as the LATE is specific to the subpopulation of compliers in the given study

Extensions of IV methods

Multiple instruments

• In some cases, researchers may have access to multiple instrumental variables for the same treatment
• Multiple instruments can be used to increase the efficiency of IV estimates and test the validity of the exclusion restriction
  • Overidentification tests can be used to assess whether the multiple instruments lead to consistent estimates of the causal effect
• Combining multiple instruments requires additional assumptions, such as the independence of the instruments from each other and the homogeneity of the LATEs across the different instruments

Nonlinear models with IV

• While the standard IV framework assumes a linear relationship between the treatment and the outcome, IV methods can be extended to handle nonlinear models
• Nonlinear IV methods, such as two-stage least squares (2SLS) with nonlinear transformations of the treatment or outcome, can be used to estimate causal effects in the presence of nonlinearities
  • For example, researchers can use IV methods to estimate the causal effect of a binary treatment on a binary outcome using probit or logit models
• Nonlinear IV methods require additional assumptions and considerations, such as the correct specification of the functional form and the validity of the instruments in the nonlinear setting