WLS + FGLS+IV+TSLS Flashcards
Why is WLS more efficient than OLS on original model when form of heteroskedasticity is known?
Observations with large variance are less informative than observations with a small variance and therefore should get less weight
- WLS estimates tend to have considerably smaller s.e’s
When would Feasible Generalised Least Squares (FGLS) be useful?
Won’t normally know form of heteroskedasticity so FGLS is used
How does method of FGLS work?
- estimate h(xi) using regression data at the same time as we estimate the parameters of the model
- we use each estimate instead of the assumed weights in WLS
What are the benefits of FGLS?
- is consistent asymptotically and more efficient than OLS
- can use asymptotic F-tests and t-tests
- as long as G-M assumptions hold FGLS estimates are unbiased
What is downfall of FGLS?
- only valid asymptotically - on large samples
- since we’ve estimated hi it is no longer unbiased
What is the purpose of an Instrument Variable?
Instrument variable is a variable we use to replace an endogenous explanatory variable to allow for CONSISTENT but BIASED ESTIMATORS
What are endogenous explanatory variables?
Explanatory variables that are correlated with the error term
- violation of MLR 4
What are exogenous variables?
Variables that are uncorrelated with the error term
What does violation of ZCM assumption lead to?
Coefficient estimators are biased
What are the 3 possible reasons for this bias?
- Omitted Variables (OVB)
- CEV, measurement error
- simultaneous causality
What properties do we want our instrument to have?
-IV uncorrelated with the error term - also known as Instrument Exogeneity
- IV (highly) correlated with the endogenous explanatory variable - known as Instrument relevance
- IV should also not already be in original regression
What can we note about our two assumption?
- Instrument Exogeneity cannot be tested - we don’t know population error term
- Instrument relevance can be tested
What is true population value of B1 equal to?
B1=cov(z,y)/cov(z,x)
- since sample covariances are consistent estimators of population covariances then B1 is correctly identified
What can we say about standard errors from IV compared to standard error in OLS?
S.e(OLS) < S.e(IV)
What is R-Squared(x,z)?
R-Squared(x,z) = the R-squared from the regression of our endogenous explanatory variable on the instrument
What is a weak/poor instrument?
An instrument with low R-Squared(x,z)
Problems with weak/poor instruments?
- leads to noticeably larger s.e’s which can lead to estimators becoming insignificant
- IV estimators although consistent are likely to be biased even in large samples
- even when corr(z,u) < corr(x,u) if R-Squared(x,z) is low IV might lead to more bias
What is original equation (that includes endogenous explanatory variable) also known as?
Structural equation
What is TSLS used for?
Developed to allow estimation where there is one endogenous explanatory variable, but potentially more than one instrument
How do we apply TSLS?
- start with structural equation as before
- assume now there are two possible relevant instruments (1 endogenous explanatory variable)
- First stage: run regression of endogenous variable on all exogenous variable (original exogenous + IV)
- Second stage: replace endogenous variable in structural equation with its fitted values from regression
What is the order condition?
For each endogenous explanatory variable there must be at least one valid instrument
What happens when number of explanatory variables = number of instruments ?
- order condition satisfied
-TSLS equivalent to IV
What happens when number of endogenous explanatory variables < instruments
- second stage is OVERIDENTIFIED
- must use TSLS
What happens when number of endogenous explanatory variables > number of instruments?
- second stage regression is UNDER-IDENTIFIED
- Can’t get unique estimates
