6- Instrumental Variables Flashcards
What is Endogeneity?
Predictor in a linear regression model is correlated to the error term
How does Endogeneity change the result of OLS?
OLS estimate becomes biased and inconsistent
What does plim mean?
Function of what happens when the number of observations approaches infinity
How can you show an estimate is consistent using plim?
plim(p̂) = p
What are the 4 main properties of plim?
plim(x+y)=plim(x)+plim(y)
plim(x’y)=plim(x)’plim(y)
plim(a)=a
plim(f(x))=f(plim(x))
What are the 3 main sources of Endogeneity?
Omitted variables
Simultaneity
Measurement error
What is Omitted Variable Bias?
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
What is the effect of Omitted Variable Bias?
Estimates of the independent variable X become biased, X becomes endogenous
What are the 2 conditions for a biased estimate?
-𝒁 is a determinant of 𝒚 i.e. 𝜷𝟐 ≠ 𝟎
-𝒁 is correlated with 𝑿 i.e 𝐶𝑜𝑣(𝑿, 𝒁) ≠ 0
What will the omitted variable bias be in an estimation of beta?
The constant added onto Beta following an estimation of beta
What is Measurement Error?
The difference between the observed and the true values of the variable
How may Measurement error arise when observing ages?
People usually look at the integer value of years lived as opposed to full age with months and days
What happens when there is measurement error in the dependent variable?
It doesn’t cause endogeneity because the unexplained measurement error is an exogenous variable
What is the effect from measurement error on the estimation of beta?
Coefficient with the same sign but between 0 & 1 such that beta is always an underestimate
What is Simultaneity bias/reverse causality?
X causes y and y causes X
What is the rank of the matrix of instrumental variables Z?
The number of instruments (l)
What are the 2 conditions by which the instruments are valid?
-Relevance assumption
-Exogeneity assumption
What is the Relevance assumption for IVs?
Variables in Z are correlated to those in X
plim(Z’X/n)=𝚺𝒁𝑿 ≠ 0
What is the Exogeneity assumption for IVs?
Variables in Z are uncorrelated with those in u
plim(Z’u/n) = 0
How many instruments must there be?
There must be at least one instrument for each independent variable correlated with u (endogenous)
What is exact identification?
Where the number of instruments (l) is equal to the number of endogenous independent variables
How can you transform a regular model with Z?
Multiply each element by Z’, as it only contains exogenous variables Z & u aren’t correlated
What are the 2 steps for two-stage least squares estimation?
- Regress X on all instruments (Z) and all exogenous regressors to obtain predicted values X̂
- Do the usual OLS with y but sub in X̂
What is the Relevance assumption with additional exogenous regressors (W)?
At least one of the instruments must be correlated with X after accounting for W
How can you measure the strength of the instruments?
Look at the F-test of their joint significance in the first stage: less than 10 indicates weak instrument
How can you show an independent variable will not be endogenous?
Show that its covariance with u is 0
How can you show the size of bias in the OLS estimator?
E(ˆβ) - β
How can you prove an OLS estimator is not consistent?
Take the plim of the estimator and expand to show it’s not equal to beta
Why is the 2SLS estimator consistent?
It approaches the normal distribution as the sample size approaches infinity
Why is IV described as a special case of GLS?
Because the variance of the errors in the transformed model is σ²Z’Z
When can independent variables serve as instruments for themselves?
When they are exogenous
