6- Instrumental Variables

Question 1

What is Endogeneity?

Answer 1

Predictor in a linear regression model is correlated to the error term

Question 2

How does Endogeneity change the result of OLS?

Answer 2

OLS estimate becomes biased and inconsistent

Question 3

What does plim mean?

Answer 3

Function of what happens when the number of observations approaches infinity

Question 4

How can you show an estimate is consistent using plim?

Answer 4

plim(p̂) = p

Question 5

What are the 4 main properties of plim?

Answer 5

plim(x+y)=plim(x)+plim(y)
plim(x’y)=plim(x)’plim(y)
plim(a)=a
plim(f(x))=f(plim(x))

Question 6

What are the 3 main sources of Endogeneity?

Answer 6

Omitted variables
Simultaneity
Measurement error

Question 7

What is Omitted Variable Bias?

Answer 7

When a variable Z correlated to both independent and dependent variables is not in the regression it instead leaks into the error term correlating u to X

Question 8

What is the effect of Omitted Variable Bias?

Answer 8

Estimates of the independent variable X become biased, X becomes endogenous

Question 9

What are the 2 conditions for a biased estimate?

Answer 9

-𝒁 is a determinant of 𝒚 i.e. 𝜷𝟐 ≠ 𝟎
-𝒁 is correlated with 𝑿 i.e 𝐶𝑜𝑣(𝑿, 𝒁) ≠ 0

Question 10

What will the omitted variable bias be in an estimation of beta?

Answer 10

The constant added onto Beta following an estimation of beta

Question 11

What is Measurement Error?

Answer 11

The difference between the observed and the true values of the variable

Question 12

How may Measurement error arise when observing ages?

Answer 12

People usually look at the integer value of years lived as opposed to full age with months and days

Question 13

What happens when there is measurement error in the dependent variable?

Answer 13

It doesn’t cause endogeneity because the unexplained measurement error is an exogenous variable

Question 14

What is the effect from measurement error on the estimation of beta?

Answer 14

Coefficient with the same sign but between 0 & 1 such that beta is always an underestimate

Question 15

What is Simultaneity bias/reverse causality?

Answer 15

X causes y and y causes X

Question 16

What is the rank of the matrix of instrumental variables Z?

Answer 16

The number of instruments (l)

Question 17

What are the 2 conditions by which the instruments are valid?

Answer 17

-Relevance assumption
-Exogeneity assumption

Question 18

What is the Relevance assumption for IVs?

Answer 18

Variables in Z are correlated to those in X
plim(Z’X/n)=𝚺𝒁𝑿 ≠ 0

Question 19

What is the Exogeneity assumption for IVs?

Answer 19

Variables in Z are uncorrelated with those in u
plim(Z’u/n) = 0

Question 20

How many instruments must there be?

Answer 20

There must be at least one instrument for each independent variable correlated with u (endogenous)

Question 21

What is exact identification?

Answer 21

Where the number of instruments (l) is equal to the number of endogenous independent variables

Question 22

How can you transform a regular model with Z?

Answer 22

Multiply each element by Z’, as it only contains exogenous variables Z & u aren’t correlated

Question 23

What are the 2 steps for two-stage least squares estimation?

Answer 23

1. Regress X on all instruments (Z) and all exogenous regressors to obtain predicted values X̂
2. Do the usual OLS with y but sub in X̂

Question 24

What is the Relevance assumption with additional exogenous regressors (W)?

Answer 24

At least one of the instruments must be correlated with X after accounting for W

Question 25

How can you measure the strength of the instruments?

Answer 25

Look at the F-test of their joint significance in the first stage: less than 10 indicates weak instrument

Question 26

How can you show an independent variable will not be endogenous?

Answer 26

Show that its covariance with u is 0

Question 27

How can you show the size of bias in the OLS estimator?

Answer 27

E(ˆβ) - β

Question 28

How can you prove an OLS estimator is not consistent?

Answer 28

Take the plim of the estimator and expand to show it’s not equal to beta

Question 29

Why is the 2SLS estimator consistent?

Answer 29

It approaches the normal distribution as the sample size approaches infinity

Question 30

Why is IV described as a special case of GLS?

Answer 30

Because the variance of the errors in the transformed model is σ²Z’Z

Question 31

When can independent variables serve as instruments for themselves?

Answer 31

When they are exogenous