9. Instrumental Variables Flashcards
What is the problem of endogenity?
The problem of endogeneity occurs when an explanatory variable (X) is correlated with the error term (ϵ) in a regression model (OLS). This violates critical assumptions such as the zero mean of errors (E(u)=0) and the zero conditional mean (E(u∣X)=0). This violation biases the estimated effect of X on the outcome (Y).
This correlation arises when unobserved factors (u) affect both the treatment (X) and the outcome (Y), creating a backdoor path (X←u→Y) that biases the estimated effect of X on Y. As a result, the regression incorrectly attributes part of the effect of u to X, making the coefficient estimates biased and unreliable.
Endogeneity can stem from unobserved confounders, omitted variable bias, or reverse causality, where unobserved confounders are the focus of IV estatmation.
What is IV estimation and its purpose?
Instrumental variable (IV) estimation is a statistical method used to estimate causal relationships when the treatment variable (D) is endogenous, meaning it is influenced by unobserved factors (U) that also affect the outcome (Y).
The purpose of IV estimation is to address this endogeneity. By introducing an instrumental variable (Z), IV isolates exogenous variation in D that is unrelated to U, enabling researchers to estimate the true causal effect of D on Y.
What are the three main conditions for a valid instrument variable (Z)?
For Z to be a good instrument for D, it must meet three main conditions:
1. Correlation with treatment (First-stage condition): The instrument (Z) must be correlated with the endogenous explanatory variable (D).
2. Independence: The instrument (Z) must be independent of unobserved factors (u) that affect the outcome (Y).
3. Exclusion restriction: The instrument (Z) must affect the outcome (Y) only through the treatment (D), not directly or through other pathways –> Z does not directly impact Y or interact with other factors that influence Y.
What are homogeneous and heterogeneous treatment effects in IV estimation?
Homogeneous treatment effects: The treatment effect (δ) is constant across all individuals in the population. This means everyone responds to the treatment in the same way. Under this assumption, IV estimates the average treatment effect (ATE), which applies to the entire population.
Heterogeneous treatment effects: The treatment effect (δi) varies across individuals, meaning the impact of the treatment depends on individual characteristics. In this case, IV estimates the local average treatment effect (LATE), which applies specifically to compliers—individuals whose treatment status is influenced by the instrument (Z).
The estimation process is the same (next flashcard), but the interpretation is different.
How is the causal effect estimated when using Instrumental Variables (IV)?
Covariance approach:
δ^=Cov(Y,Z)/Cov(D,Z)=(effect of Z on Y)/(effect of Z on D)
* Numerator Cov(Y,Z): Measures the relationship between the instrument Z and the outcome Y (the “reduced form”).
* Denominator Cov(D,Z): Measures the relationship between the instrument Z and the treatment D (the “first stage”).
* Interpretation: The formula calculates how much of the variation in Y caused by Z flows through D. This isolates the variation in D induced by Z, effectively removing the endogeneity caused by U.
* Assumptions: 1. Z is uncorrelated with U, Cov(U,Z)=0 (Independence). 2. Z is uncorrelated with the error term 𝜖, Cov(ϵ,Z)=0 (Exclusion restriction). 3. Z is highly correlated with D (non-zero-first-stage)
Two-stage-least-squares (2SLS):
2SLS is a regression-based implementation of IV that works in two steps:
1. First stage: Regress the endogenous variable D on the instrument Z (and any control variables). This isolates the part of D that can be predicted by Z.
Di=γ+βZi+νi
The fitted values (𝐷i^) from this regression represent the exogenous variation in D that is driven by Z, free from the bias introduced by unobserved factors.
2. Second stage: Regress the outcome variable Y on the fitted values 𝐷i^ from the first stage.This isolates the variation in Y due to changes in D driven by Z.
Yi=α+δDi^+ϵi
* Assumptions: Same as covariance approach
The 2SLS estimator can be algebraically shown to be equivalent to the covariance-based IV estimator. 2SLS and the covariance formula provide the same result, but 2SLS offers a regression-based implementation suitable for complex models with multiple instruments or control variables.
What is the problem of weak instruments?
Weak instruments occur when the correlation between the instrument (Z) and the endogenous variable (D) is low. This violates the non-zero first stage assumption, which requires Z to strongly predict D.
Weak instruments make IV estimates unreliable and biased. Even if the exclusion restriction is satisfied, weak instruments can undermine the validity of IV estimation.
* A weak instrument poorly isolates the exogenous variation in D. The first-stage fitted values (D^) are dominated by noise, and this noise interacts with the error term (ϵ) in the second stage, introducing bias.
* When the instrument is weak, the second-stage regression begins to rely on endogenous variation in D, making the 2SLS estimate biased. The bias approaches the same level as the OLS estimate as the first-stage
F-statistic approaches zero.
What assumptions for IV estimation are common and different when dealing with heterogenous treatment effects compared to homogenous treatment effects?
Assumptions common to both homogeneous and heterogeneous treatment effects:
1. Stable unit treatment value assumption (SUTVA): The treatment status of one unit does not affect the potential outcomes of another (no spillover effects). The potential outcome for each unit depends only on its own treatment assignment.
2. Independence assumption (“As good as random”): The instrument (Z) is independent of both potential outcomes (Y0, Y1) and treatment assignments (D0, D1).
3. Exclusion restriction: The instrument Z affects the outcome Y only through its effect on the treatment D.
4. First-stage condition: The instrument Z must have a statistically significant effect on the probability of receiving treatment D. Under this is also the non-zero first stae condition.
Additional assumption for heterogeneous treatment effects:
5. Monotonicity assumption: The instrument (Z) influences the treatment (D) in the same direction for all individuals. There are no defiers (individuals who respond to Z in the opposite direction of its intended effect).
Explain the LATE framework for heterogenous treatment effect
The Local Average Treatment Effect (LATE) represents the causal effect of the treatment (D) on the outcome (Y) for individuals whose treatment status changes due to the instrument (Z). It is estimated under heterogeneous treatment effects and applies only to “compliers”, a specific subpopulation influenced by Z.
The population in the LATE framework is divided into four groups based on how their treatment status responds to Z:
* Compliers: This is the subpopulation whose treatment status is affected by the instrument in the correct direction. Thatis, D1i=1 and D0i =0.
* Defiers: This is the subpopulation whose treatment status is affected bytheinstrumentinthewrongdirection. Thatis, D1i=0 and D0i=1.
* Never-takers: This is the subpopulation of units that never take the treatment regardless of the value of the instrument. So, D1i=D0i =0.
* Always-takers: This is the subpopulation of units that always take the treatment regardless of the value of the instrument. So, D1i=D0i=1.
LATE is estimated only for compliers, making it highly specific to those influenced by Z. LATE does not generalize to never-takers, always-takers, or defiers, meaning the results may not represent the treatment effect for the broader population.
