Linear Regression and Causality

• we can always use regression to answer descriptive questions, but only sometimes to answer causal questions
• Orthogonality: cov(X,u)=0 or E(Xu)=0 - all other determinants of Y should be uncorrelated with X - OR means this plus Eu=0
• If orthogonality holds, the population regression coefficients identify the coefficients of the causal model
• If orthogonality holds, the OLS estimate for Beta1/Beta2 converges in probability to the population coefficient (is consistent) so causal interpretations based on OLS estimates are valid
• If orthogonality not satisfied in the causal model, population regression coefficient no longer agrees with causal model coefficient - OLS estimates only have a descriptive interpretation in terms of rho1
• If OR fails, we can say 'on average, a unit increase in X is associated with a rho1 increase in Y'
• Causation can only come from a causal model - from economic theory
• Mean independence implies orthogonality
• Mean independence is needed only for unbiasedness of Beta1hat, orthogonality only is needed for consistency
• Multiple regression: if OR fails, rho1 can only be interpreted descriptively 'on average, a unit increase in X1 is associated with a rho1 increase in Y, holding each of X2,...,Xk constant'
• A good proxy is a 'close correlate' or 'good predictor' of an unobserved determinant of Y