Endogeneity

• If OR holds, we say X1 and X2 are exogenous
• If OR fails, we say at least one of the variables is endogenous
• causes: omitted variable, measurement error, simultaneity/reverse causality
• solutions to endogeneity: try to say something definite about the likely biases of OLS, include proxies for omitted variables, use RCTs/natural experiments, instrumental variables
• No omitted variable bias if either the variables are uncorrelated or one of the variables is irrelevant to the determination of Y
• Proxying for omitted variables typically motivates the inclusion of demographic variables like family characteristics, and race and region dummies etc.
• proxies cannot be given a causal interpretation as they are not included in the causal model
• the coefficient on a mismeasured regressor is subject to attenuation bias (biased towards 0), coefficients on other regressors may be biased in either direction
• RCT characterised by: a sample of individuals under study who have elected/been compelled to participate, assignment of Xi being under the control of the researcher, for the individuals in that sample
• actual treatment equals assigned treatment if there is perfect compliance
• RCTs make X and u independent by construction
• Limitation of RCTs: needs to be feasible and ethical, so only possible for some kinds of 'treatment'
• population linear regression coefficient is the difference of population group means, if treatment is binary
• control variables used for RCTs must be pre-treatment characteristics
• can test for balancedness: the treated and the untreated should have approximately similar pre-treatment characteristics
• a shortcoming of average treatment effect (ATE) is that it averages effect of treatment over people who may never receive it in reality
• internal validity: are inferences on causal effects credible for the population studied
• external validity: can inferences be credibly generalised to other populations?
• threats to internal validity: imperfect compliance, small samples, attrition (people may drop out), Hawthorne effects
• problems for external validity arise when: populations differ in a way that matters for the determination of Y and which is not accounted for by the model
• a quasi-/natural experiment is an observational study where 'nature' partly replicates an RCT - X is 'as if' randomly assigned
• instrumental variables must satisfy: 1. Z is correlated with X 2. Z is uncorrelated with u (is exogenous) 3. Z does not enter the structural equation
• exogeneity and exclusion of IVs mean changes in Z do not affect Y directly, relevance means a change in Z affects Y by 'shifting' X
• source of endogeneity in X is not important for IVs, as long as we have a Z that fulfils the criteria it can be used as an IV
• under homoskedasticity 2SLS if less efficient than OLS
• weak instruments arises when the relevance condition may technically hold, but with a coefficient too small relative to the sample size for normal approximation to be reliable
• Can use F test to detect weak instruments with a larger rule of thumb value of c=10
• If you have weak instruments, standard inferences cannot be drawn for Beta1 and usual confidence intervals may be misleadingly narrow
• if random assignment fails, can randomly assign an inducement - if the inducement affects uptake of treatments and is independent of individual characteristics it is an instrumental variable
• Local average treatment effect (LATE) = a population average of individual-level treatment effects, weighted by responsiveness to the instrument