L22 - Modern approaches to Exogeneity

Question 1

What used to be the accepted definition of exogeneity?

Answer 1

If the Cov(X_t,u_t) = 0

• However, it is very hard to test this hypothesis as the errors are not observables
  
  • Therefore we used the residuals as an estimation of the regression errors
  • But the Cov(X_t,u_t(hat)) = 0 by construction
  • Therefore this cannot be used to test for exogeneity

Question 2

What is the formal definition of Weak Exogenity?

Answer 2

• regressors are weakly exogenous if we can use them to construct consistent estimates of the unknown parameter in the model
• The concept of weak exogeneity was in Engle, Hendry and Richard ‘Exogeneity’, Econometrica (March 1983)

The equation expresses the probability density or relationship between two random variables - (LHS)

• The right-hand side is saying that we can decompose a joint probability into a conditional probability multiplied by a marginal probability –> P(AnB) = P(A|B) * P(B), therefore f(X,Y) = f(Y|X)*f(X)

X can be treated as exogenous if the β (the parameter we are interested in) is a function of Λ₁ - (the parameter from the conditional distribution of Y|X and Λ₁ and Λ₂ are free to vary independently of each other

Question 3

Example of Weak form exogeneity?

Answer 3

orthogonal –> statistically independent

Question 4

How do you test for weak exogeneity?

Answer 4

• With the Hausman Specification Test
• This is a test for endogeneity of the regressor based on the difference between the OLS and the IV estimators. If X is weakly exogenous then the use of the instrumental variable estimator should make little difference to the results - should not be significantly different

If this test is not significant you would choose the OLS estimator as it is consistent and has a lower variance than the IV estimator

Question 5

What is an easier way to apply the Hausman Specification Test?

Answer 5

• This is the way most people apply this test in practice