Lecture notes 11 errors in variables

Question 1

What is a measurement error in regressions?§

Answer 1

Instead of observing true X you observe X*

Question 2

How is measurement error calculated?

What are the assumptions?

Answer 2

U = X* - X

Cov(U,X) = 0
Cov(u, epsilon) =0

Question 3

When would mismeasurement occur intentionally?

Answer 3

When people are asked about something they have an incentive to overreport eg earnings.

Question 4

How do you do a regression with mismeasurement error?

Answer 4

U= X* - X
Y = B0 + B1X + epsilon
sub in for X
Y = B0 + B1(X*-U) + epsilon

Y= B0 + B1X* - B1U + epsilon

This gives OLS estimate of

B1 + COV(X,V)/ Var(X)

Then to check the
b1 is approx B1 Var(X)/ Var(X) + Var(U)

As U increases the varianced term decreases so as mismeasurement increases b1 becomes biased

Question 5

What is the direct impact of mismeasurement on OLS estimate of B1

Answer 5

Bigger mismeasurement b1 is smaller and biased.

Question 6

What is the impact of mismeasurement error in Y

Answer 6

under the asusmption that Cov(u,X) = 0
it causes no issues on b1

But the variance changes from normally

sigma^2/ sum of (xi-xbari)^2 vs sigma squared + var(u)/ sum of i=1 to n (Xi -Xibarn)