Tutorial 4 - Frisch-Waugh Theorem, Omitted Variables, Instrumentral Variables Estimation

Question 1

Say we are interested in β₂: the effect of x on y, after controlling for w (=holding w constant).

How can we obtain the estimator ^β₂ according to Frisch-Waugh?

Answer 1

1. Regress y on a constant and w. Take the residuals from this regression -> y_res,w (the variation in y that cannot be explained by w)
2. Regress x on a constant and w. Take the residuals from this regression -> x_res,w (the variation in x that can not be explained by w)
3. Regress y_res,w on x_res,w. The coefficient of x_res,w is the same as the coefficient of x in the multivariate regression

This procedure illustrates the idea of filtering out the effect of w when estimating the effect of x on y.

Question 2

Which conditions are met when the OLS estimator ^β is unbiased?

Answer 2

Unbiased: E(^β) = β

Note: If it is unbiased, that also means it is consistent:

Question 3

What does this mean?

Answer 3

The OLS estimator is consistent:

Question 4

How can you apply the Frisch Waugh theorem here?

Answer 4

1. Regress log(invest) on a constant and trend and take residuals (“de-trended housing investment”)
2. Regress log(price) on a constant and trend and take residuals (“de-trended house prices”)
3. Regress the de-trended investment on de-trended prices.

Question 5

Show that under the assumption of mean independence of the error term given the regressors, the OLS estimator is unbiased!

Answer 5

Question 6

Show that the OLS estimator is consistent (under the assumption that the regressors and the error term are uncorrelated)!

Answer 6

Question 7

When is the OLS estimator unbiased?

Answer 7

if E(ϵ|X) = 0

i.e. if the error term and the regressors are mean independent.

Question 8

When is the the OLS estimator consistent?

Answer 8

if Cov(x, ϵ) = 0,

i.e. if the error term and the regressors are uncorrelated.

Question 9

How are unbiasedness and consistency related with each other?

Answer 9

Note that mean independence implies uncorrelatedness, but not vice versa.

Thus: if the OLS estimator is unbiased, it is also consistent. But if the OLS estimator is consistent, it need not be unbiased.

Question 10

How do you call regressors that are correlated with the error term, Cov(x, ϵ) ≠ 0, i.e. where the OLS estimator will be inconsistent and biased?

Answer 10

the regressors x are endogenous

Question 11

Why can regressors be endogenous?

Answer 11

• Omitted variables: x and y are both driven by a third, unobserved, variable
• Simultaneity: y causes x, rather than vice versa
• Measurement Error: x is measured with error
• Dynamic Models with Serially Correlated Errors

Question 12

Which conditions does an instrumental variable z have to fulfill?

Answer 12

1. Instrument relevance
2. Instrument exogeneity

Question 13

What is “instrument relevance”?

Answer 13

The instrument has to be correlated with the endogenous variable Cov(zₖ, xₖ ) ≠ 0.

In the present case with multiple covariates, a stronger condition is that in the regression below, the parameter 𝛿ₖ ≠ 0, i.e. the instrument zₖ has to be correlated with xₖ also when controlling for all exogenous covariates!

Question 14

What is instrument exogeneity?

Answer 14

The instrument has to be uncorrelated with the error term. It has to affect the outcome only via the endogenous variable, i.e. it should not have a direct effect on the outcome:

Question 15

How can you test relevance of the instrument with instrumental variables?

Answer 15

It can be tested by estimating the first stage regression (below) and use a usual t-test for H₀: 𝛿ₖ = 0. -> the more significant, the better.

For multiple instruments z₁, …, zₗ: use a test of joint significance of the instruments in the first stage. Rule of thumb: F-statistic >10 to have a “strong” instrument

Question 16

How can you test exogeneity of the instrument with instrumental variables?

Answer 16

It is not testable, has to be justified based on economic theory.

Question 17

What is a Two-Stage Least Squares (2SLS) Estimator? What is the first stage?

Answer 17

It is a way to carry out IV estimation:

1. First Stage: Regress endogenous variable xₖ on all exogenous covariates 1, x₂, …, xₖ₋₁ and instrument zₖ, and form fitted values ^xₖ:

• ^xₖ contains only exogenous variation!
• It is important to include the exogenous covariates as well because the instrument has to affect the endogenous variable, conditional on the exogenous covariates

Question 18

With a Two-Stage Least Squares (2SLS) Estimator: What is the second stage?

Answer 18

Second Stage: Regress the outcome variable y on all exogenous covariates and the fitted values^xₖ from the first stage -> coefficient on ^xₖ is the IV Estimator

Question 19

What are practical issues with IV?

Answer 19

• Often standard errors for IV are much higher than for OLS (bias-variance tradeoff), in particular if there is a weak instrument problem, i.e. only a weak correlation between the instrument and the endogenous variable
  
  • Always check the first stage regression!
• If the instrument is correlated with the error term, the IV estimator is not consistent. The inconsistency of the IV estimator might be even larger than for the OLS estimator…

Question 20

Which test can you use to test for endogeneity of regressors?

Answer 20

Hausman test for endogeneity of regressors

Question 21

What are the hypotheses with the Hausman test for endogeneity of regressors?

Answer 21

Question 22

What is the idea of the Hausman test (null-hypothesis)?

Answer 22

Under the null hypothesis (xₖ is exogenous), the OLS and 2SLS estimators are both consistent, and should differ only by sampling error. A statistically significant difference between the two estimators would be seen as evidence against exogeneity of xₖ .

Question 23

What is the test statistic and distribution for the Hausman test?

Answer 23

Under the H₀ of exogeneity, the Hausman statistic is asymptotically Xₖ² -distributed.

Question 24

What are the limitations of the Hausman test?

Answer 24

• It assumes that the instrument is valid. Otherwise, a significant difference between OLS and IV could be driven by the fact that IV is inconsistent, which will not necessarily imply that xₖ is endogenous!
• Even in the case of a valid instrument, the test might have low power if the IV estimates are very imprecise (standard errors are high). In this case, the test might fail to reject the H₀ of exogeneity even if xₖ is endogenous.

Question 25

What does the “Sargan test” test?

Answer 25

Overidenti cation test for validity of instruments

Question 26

What can you do with an overidentified model?

Answer 26

overidentification test for validity of instruments

Question 27

What are the hypotheses and idea behind the Sargan test?

Answer 27

Idea: under H0, the instruments should be uncorrelated with the 2SLS residuals.

Question 28

What is the procedure of the Sargan test?

Answer 28

1. Estimate the model using 2SLS and take residuals.
2. Regress 2SLS residuals on all exogenous covariates plus instruments. Compute the test statistic, N * R²
3. Under H0, the test statistic is asymptotically X² with q degrees of freedom (q= # instruments - # endogenous variables)

Question 29

What could be problems with the Hausman test?

Answer 29

It is based on the assumption that at least one of the instruments is exogenous! (e.g., if all instruments are biased in the same direction, the test might not reject H0!)

Question 30

What is a spurious correlation?

Answer 30

a statistical relationship between two variables appears to be causally related, but only appears so by coincidence or due to the role of a third, intermediary variable