instrumental variable regression with single X and Z

Question 1

what are the three important threats to internal validity that instrumental variables can solve?

Answer 1

omitted variable bias from a variable that is correlated with X but is unobserved
simultaneous causality bias (X causes Y, Y causes X)
errors in variable bias (X is measured with error)
all three relate to E(u|X) not equal to 0

Question 2

what is an instrumental variable?

Answer 2

an instrumental variable Z is correlated with X but uncorrelated with u

Question 3

what is the purpose of instrumental variable regression?

Answer 3

it breaks X into two parts. a part that might be correlated with u and a part that is not. by isolating the part that is not correlated with u, it is possible to estimate B1

Question 4

what is an endogenous variable?

Answer 4

it is one that is correlated with u

Question 5

what is an exogenous variable??

Answer 5

it is one that is uncorrelated with u

Question 6

for an instrumental variable to be valid what are the two conditions that need to be satisified?

Answer 6

1) instrumental relevance : corr(Z,X) not equal to zero
2) instrumental exogeneity : corr(Z,u)=0

Question 7

what are the two stages of the least squares>

Answer 7

1) regress X_i on Z_i (including an intercept), obtain the predicted values of X_i^
2) regress Y_i on estimated X_i^ (including an intercept); the coefficient on X_i^ is the TSLS estimator B1_TSLS^ which is a consistent estimator of B1

Question 8

what is the equation of B1 when the instrumental variable is healthy?

Answer 8

B1= Cov(Y_i, Z_i)/ Cov(X_i,Zi)

Question 9

what are the steps for two stage least squares?

Answer 9

TSLS proceeds by first regressing X on Z to get estimator of X then regressing Y on estimator of X. the first stage isolates the part of the variation in X that is uncorrelated with u. if the instrument is valid then the large sample sampling distribution of the TSLS estimator is normal so inference proceeds as usual