L20 - The Errors in Variables Model

Question 1

What is the Mean Square Error criterion?

Answer 1

• For an unbiased estimator :
  
  • The MSE = V(β(hat)) because E(β(hat)) = β

We may want to choose the biased but more efficient estimator (blue curve) as it will give us more of a chance to get closer to the true value rather than the unbiased red pdf

• Biased estimator is still superior if it have a lower MSE

Question 2

What are Method of Moments Estimators?

Answer 2

This means that we define the estimator by equating unknown population moments (unknown mean and variance) with sample moments (arthimetic mean and sample variance)

• The instrumental variable estimator can be interpreted as a method of moments estimator

Question 3

How do we derive the normal IV estimator equations based on unknown parameters?

Answer 3

• for the mean we take expectations
• for the variance equation we multiply through by Z and then take expectations

By using the sample moments we can get the IV estimators for beta and alpha like in the previous lecture

Question 4

What should the sample covariances be divided by?

Answer 4

Question 5

What is the Errors in Variables Model?

Answer 5

• Another way we may get correlated between RHS variables and the errors term
  
  • This produces a variable that is biased and inconsistent
• The X variable here is referred to as a proxy variable - its an observable variable that is not exactly the same variable that we want to test in our model but a measurement or estimator of our variable that is subject to some degree of error

Question 6

What is the Proof that Proxy Variables cause inconsistent estimators?

Answer 6

• As X* is the variable we want to obtain thus is the signal
• the errros are the noise - they are inevitable but are unavoidable

It follows that the higher is the signal to noise ratio, the closer is the probability limit of the OLS estimator to the unknown population parameter β.

Similarly, a ‘noisy’ proxy variable i.e. one with a low signal to noise ratio will mean that the plim of the OLS estimator is further from the true value.

Question 7

So what makes a good proxy variable?

Answer 7

one with a high signal to noise ratio

• even in a large sample, a poor signal to noise ratio and measurement errors, will still give us inconsistent parameter estimates

Question 8

What is the Rational Expectations Models?

Answer 8

• Just like in macro, agents use all possible information available to form the best possible estimator or make the best possible decision they can

Ω_t​ = a set of information

• ε_t+1 is a forecast error

Question 9

What is the inconsistency in the rational expectation model based on?

Answer 9

• We need to use an alternate estimator to deal with this inconsistency like IV estimator