Econ B: L8

Question 1

See (extending instrumental variables to MRM)

Answer 1

top of notes lecture 8

Question 2

Top of page 8 equation; what is it called and what do we expect it to measure?

Answer 2

It is called a structural equation to emphasize our interest in beta(j), meaning that we expect it to measure a causal relationship

Question 3

See example in notes at top; what would q be (suspected to be correlated with u), and what happens if we ignore this and estimate the equation?

Answer 3

q=educ since might be correlated with u, since ability is in u
if estimated by OLS, all estimator will be biased and inconsistent!

Question 4

What does the variable z need to satisfy to be an appropriate instrumental variable? (2)

Answer 4

1) not correlated with u (instrument exogeneity)

2) BUT correlated with q (instrument relevance)

Question 5

Variance of q is split into ‘good’ and ‘bad’. What is the equation for this?

Answer 5

variance of q=q-specific info+co-movement with x+co-movement with u

Question 6

What do we need z to be correlated with?

Answer 6

q specific info

Question 7

How do you test for instrument relevance in z?

Answer 7

Given: q=π0+π1x+π2z+v
Estimate this equation by OLS
T-test it using null: H0:π2=0
If the null is rejected (π2 not equal to 0), we know that q and z are still correlated (after partialling out x)

Question 8

What can we not test for?

Answer 8

Cannot test if x and z are uncorrelated with u; this must be taken on faith

Question 9

What is a reduced form equation?

Answer 9

An equation where an endogenous variable is written in terms of exogenous variables (eg: q=π0+π1x+π2z+v)

Question 10

How would you add more exogenous variables to the model? (both equations and test)

Answer 10

Very easily:
y=β0+β1q+β2x1+β3x2+u
Tf:
q=π0+π1x+π2x2+π3z+v
Test if: H0: π=0
Note: not same as adding more ENDOGENOUS variables!

Question 11

2SLS: Say we have 2 instrumental variables for q: z1 and z2. Why don’t we use the best of the 2 instruments (ie. higher R^2)?

Answer 11

Since we want to capture ‘q-specific’ information, but since neither z1 or z2 is perfectly correlated with it, dropping either would lead to a loss of information!

Question 12

2SLS: Solution to problem?

Answer 12

Since each of x, z1 and z2 is uncorrelated with u, any LINEAR COMBINATION of them (all exogenous) is also uncorrelated with u tf to find the best instrumental variable, choose the linear combination that’s most closely correlated with q!

Question 13

2SLS: How do you find the linear combination for q with z1 and z1 both being instrumental variables? How do we then test if this best linear instrumental variable is valid?

Answer 13

See equation in notes (q(hat) one)
For IV estimator to be valid need at least: π2 or π3 not equal to 0
Test:
H0: π2=π3=0 H1: π2/=0 OR π3/=0 USING AN F-TEST
with multiple instruments the IV estimator then becomes the 2SLS estimator!

Question 14

2SLS: note regarding OLS estimators?

Answer 14

Using algebra of OLS it can be shown that when we use q(hat) as the instrumental variable for q, the IV estimates of β(hat)0, β(hat)1 and β(hat)2 are identical to the OLS estimators from the regression of y on q(hat) and x

Question 15

2SLS: once we have rejected H0 (ie. at least one of z1 and z2 are significant), what are the final 2 stages?

Answer 15

1) Run regression 2 in notes to obtain fitted values for q(hat)
2) Run regression of y on q(hat) and x (q(hat) in place of q)
- >differences between 2SLS and OLS estimates

Question 16

Learn

Answer 16

Bias table at end of L8 notes!