L21 - Two Stage Least Squares

Question 1

Under two-stage least squares, what happens when we have more instruments than we need?

Answer 1

• Any linear combination of Z₁ and Z₂ will also be a valid instrument –> there are an infinite number of weighted averages of the two instruments
• This suggests a strategy – instead of choosing one instrument or the other – form a linear combination which minimises the variance of the estimator.

Question 2

How do you define the Two-Stage Least Squares Estimator?

Answer 2

Question 3

What will be the properties of the two-stage least squares estimator?

Answer 3

• The more instruments we use the better fit we get in the first stage regression, and due to the increase in the number of parameters, R² will also increase –> more efficient
• The standard Instrument Variable estimator is a TSLS estimator with only one variable.

Question 4

What happens to R² when using two-stage least squares?

Answer 4

Question 5

What is the equation for TSS for the two-stage least squares estimator and why is it difficult to interpret R²?

Answer 5

• Unlike under OLS, there is no guarantee that Σ(X_t - X(bar))u_t(hat) will be equal to zero and the term will drop out