PW2

Question 1

What is the law of large numbers

Answer 1

When we have iid random variables, the sample mean converges in probability to the expected value of the random variable

Question 2

What are the properties of the plim(x) = K and plim(y) = L

Answer 2

• plim(c) = c
• plim (x + y) = K + L
• plim(x * y) = K * L
• plim(x / y) = K/L

Question 3

What other property do OLS estimators have under MLR.1 to MLR.4

Answer 3

• Consistency

Question 4

What is consistency

Answer 4

• plim(βj hat) = βj as n tends to ∞
• As we get more data, we can expect the sampling distribution of βj hat to become more tightly centred around βj

Question 5

Prove the consistency of β1

Answer 5

https://blackboard.soton.ac.uk/ultra/courses/_227910_1/outline/file/_7069915_1

Question 6

When is an OLS estimator consistent and inconsistent

Answer 6

• If cov(x,u) = 0 then the OLS estimator is consistent (MLR.4)
• If cov(x,u) =/ 0 then the OLS estimator is inconsistent

Question 7

What determines the direction of the bias in OLS estimates

Answer 7

• If x and u are positively correlated then the least squares overestimates the true parameter
• If x and u are negatively correlated then the least squares overestimates the true parameter