1.2 OLS Estimation

Question 1

What does OLS estimation do?

Answer 1

If chooses values for beta that minimises the residual sum of squares. These values are denoted beta hat

Question 2

In matrix form how does choosing beta hat to minimise the residual sum of squares translate?

Answer 2

Choose beta hat to minimise RSS= u hat’ x u hat

Question 3

What is u hat equal to?

Answer 3

U hat = y-x(beta hat)

Question 4

When we solve for beta hat what do we get?

Answer 4

Beta hat = (x’x)^(-1) x’y

Question 5

Properties of OLS estimator beta hat

Answer 5

• beta hat= beta + (x’x)^(-1) x’u
• E(beta hat) = beta (unbiased estimator)
• V(beta hat) = ó^2 (x’x)^(-1)

Question 6

What is M and how can we define it?

Answer 6

M is a symmetric idempotent matrix
M= I- x(x’x)^(-1) x’

Question 7

How can we measure how well the sample regression fits the data?

Answer 7

Using R^2= ESS/TSS= 1- RSS/TSS

Question 8

When adding a new explanatory variable when will the R^2 value increase?

Answer 8

Always regardless of how much the new variables contributes in terms of explanation

Question 9

How is adjusted R^2 different from R^2?

Answer 9

It is exactly the same apart from there is a penalty for including additional varaibles