Proxy Variables And Measurement Error

Question 1

Why are proxy variables used?

Answer 1

They are used in place of explanatory variables we cannot observe and/or don’t have sufficient data on

Question 2

What would we expect from proxy variable?

Answer 2

Would expect proxy variable to be highly correlated with the omitted variable

Question 3

Assumptions necessary for proxy variable method?

Answer 3

• u uncorrelated with x1, x2 and x3*.
• corr(x3,u)=0
• corr(x1,v3)=corr(x2,v3)=0

Question 4

What does it mean if corr(x3,u) is not =0

Answer 4

If the original error and proxy variable are correlated it would mean that the proxy variable should be included in the original regression in its own right
- when corr(x3,u)=0 means if x3* was available it should not be included in the regression
- I.e x3 should not have a separate influence on y

Question 5

What does it mean if the error term in the x3* regression (v) is correlated with x1,x2?

Answer 5

If v is correlated with x1 and x2 they would have to be included in the regression of x3* for the omitted variable
- Note: v also has to be uncorrelated with x3 to satisfy ZCM
- if v is correlated with x1 and x2 it will lead to biased estimators
- hopefully the bias is smaller than OBV from not including proxy

Question 6

If all additional assumptions hold what does this mean for regression model?

Answer 6

• combining the two regressions (original + x3* regression)
• in the new regression model the error term is uncorrelated with all explanatory variables
• coefficients will be correctly estimated using OLS
• coefficient will correctly identified for x1 and x2

Question 7

What else can be used as proxy variables?

Answer 7

• omitted observed factors may be proxied by the values of the dependent variables from an earlier time period
• I.e including crime rate from previous period as an explanatory variable in regression for crime in present period

Question 8

What does measurement error in dep. variable equal?

Answer 8

e0 = y - y*
I.e measurement error = mismeasured (observed) - actual value
Note: we don’t know actual value only see observed/mismeasured value and assume measurement error

Question 9

What happens to regression when measurement error is included?

Answer 9

y=y*+e0
y=b0+b1x1+b2x2+…….+bkxk+ (u + e0)

Question 10

Along with MLR1-4 what else must hold in order for estimated regression with measurement error to give unbiased and consistent estimators?

Answer 10

• cov(e0,Xj) = 0 - measurement error must be uncorrelated with all explanatory variables
• cov(e0,u)=0-measurement error uncorrelated with error term
• E(e0)=0 - expected value of measurement error = 0

Question 11

What happens to overall variance in estimated regression with measurement error?

Answer 11

-overall variance is increased

Question 12

What does mismeasured explanatory variable equal?

Answer 12

x1=x1*+e1
- mismeasured explanatory variable = actual value + measurement error

Question 13

What happens to regression equation when measurement error for an explanatory variable is included?

Answer 13

y=b0+b1x1+…..+bkxk+u
x1=x1-e1
y=b0+b1x1+…..+bkxk + (u-b1e1)

Question 14

What must be assumed in order for estimators to be unbiased and consistent when measurement error for explanatory is included in the regression?

Answer 14

• MLR 1-4 must hold
• E(e1)=0
• E(u)=0
• cov(x1,u)=0
• cov(x1*,u)=0
• cov(x1,e1)=0

Question 15

What is the Classical Errors-in-Variables (CEV) Assumption?

Answer 15

Cov(e1,x1*)=0

Question 16

What happens when CEV assumption is made instead of Cov(e1,x1)=0?

Answer 16

Estimator on b1 will biased and inconsistent
- because of correlation across explanatory variables other estimators are also likely to be biased/inconsistent

Question 17

If all assumptions necessary hold what happens to to error variance?

Answer 17

Overall error variance increases
- estimators unbiased and consistent

Question 18

What is attenuation bias?

Answer 18

-magnitude of effect is attenuated towards 0
- estimated coefficient will tend to underestimate the true coefficient