L16 - Stochastic Regressors

Question 1

If the variable X is no longer non-stochastic how does this affect our unbiasness assumption?

Answer 1

• makes it difficult to prove unbiasedness

Question 2

If the variable X is no longer non-stochastic how does this affect our ability to take expectations?

Answer 2

• we need to find the expectation of the numerator and the denomiator - but expectations of the entire equation does not equal the ratio of expectations of the two.
• Similarily the expected value of the product of the variables is not equal to the product of the expectations
• it still possible to prove unbiasedness, but the X variable would need to be uncorrelated with all the errors in a data set (not just the current error like our previous assumption, thus making this a rather strong assumption)

Question 3

if E(X_tu_t-k)=0 what is the proof of unbiasedness?

Answer 3

• in many cases this is a unrealistic assumption

Question 4

What is consistency?

Answer 4

• Consistency of an estimator means that as the sample size gets large the estimate gets closer and closer to the true value of the parameter
• in general this is a weaker assumption than biasedness as it can be applied to all sample size
• would write the estimator as following:
• Let β_T(hat) be an estimator of the slope coefficient bsed on a sample of size T
• Does β_T(hat) coverge to the true population parameter as T becomes large?

Question 5

What is the mathematical defintion of Consistency?

Answer 5

• the equation shows that the limit of the probability that the difference between θ(hat) and the true value θ being greater than ε tends to zero when T tends to infinity
  
  • the probability that you will get a large gap between the estimator and the true value tends to zero as the sample size become large

Question 6

What are the conditions for an estimator to be consistent?

Answer 6

• This essential means that the PDF of an estimator needs to collapse on a single point, which should be the true estimator if it is consistent

Question 7

Example of how an estimator can be biased in small samples but still be consistent in large ones?

Answer 7

• uses the estimator of the mean of X(tilde)
• when we take expectation of a sum of T lots of X we get the value Tμ_X