WLS + FGLS+IV+TSLS

Question 1

Why is WLS more efficient than OLS on original model when form of heteroskedasticity is known?

Answer 1

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

Question 2

When would Feasible Generalised Least Squares (FGLS) be useful?

Answer 2

Won’t normally know form of heteroskedasticity so FGLS is used

Question 3

How does method of FGLS work?

Answer 3

• 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

Question 4

What are the benefits of FGLS?

Answer 4

• 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

Question 5

What is downfall of FGLS?

Answer 5

• only valid asymptotically - on large samples
• since we’ve estimated hi it is no longer unbiased

Question 6

What is the purpose of an Instrument Variable?

Answer 6

Instrument variable is a variable we use to replace an endogenous explanatory variable to allow for CONSISTENT but BIASED ESTIMATORS

Question 7

What are endogenous explanatory variables?

Answer 7

Explanatory variables that are correlated with the error term
- violation of MLR 4

Question 8

What are exogenous variables?

Answer 8

Variables that are uncorrelated with the error term

Question 9

What does violation of ZCM assumption lead to?

Answer 9

Coefficient estimators are biased

Question 10

What are the 3 possible reasons for this bias?

Answer 10

• Omitted Variables (OVB)
• CEV, measurement error
• simultaneous causality

Question 11

What properties do we want our instrument to have?

Answer 11

-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

Question 12

What can we note about our two assumption?

Answer 12

• Instrument Exogeneity cannot be tested - we don’t know population error term
• Instrument relevance can be tested

Question 13

What is true population value of B1 equal to?

Answer 13

B1=cov(z,y)/cov(z,x)
- since sample covariances are consistent estimators of population covariances then B1 is correctly identified

Question 14

What can we say about standard errors from IV compared to standard error in OLS?

Answer 14

S.e(OLS) < S.e(IV)

Question 15

What is R-Squared(x,z)?

Answer 15

R-Squared(x,z) = the R-squared from the regression of our endogenous explanatory variable on the instrument

Question 16

What is a weak/poor instrument?

Answer 16

An instrument with low R-Squared(x,z)

Question 17

Problems with weak/poor instruments?

Answer 17

• 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

Question 18

What is original equation (that includes endogenous explanatory variable) also known as?

Answer 18

Structural equation

Question 19

What is TSLS used for?

Answer 19

Developed to allow estimation where there is one endogenous explanatory variable, but potentially more than one instrument

Question 20

How do we apply TSLS?

Answer 20

• 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

Question 21

What is the order condition?

Answer 21

For each endogenous explanatory variable there must be at least one valid instrument

Question 22

What happens when number of explanatory variables = number of instruments ?

Answer 22

• order condition satisfied
  -TSLS equivalent to IV

Question 23

What happens when number of endogenous explanatory variables < instruments

Answer 23

• second stage is OVERIDENTIFIED
• must use TSLS

Question 24

What happens when number of endogenous explanatory variables > number of instruments?

Answer 24

• second stage regression is UNDER-IDENTIFIED
• Can’t get unique estimates