L9 - The Regression Model in Matrix Form

Question 1

What does the value of β give you in Multivariate regression?

Answer 1

• the line means partial differentiation holding that variable constant

Question 2

When is Multivariate regression equilvalent to Bivariate regression?

Answer 2

• Based on purged data
• So instead of running a multivariate regression model with 3 parameters, you can run a bivarate model 3 different times and get the same answer
• This is proved by the Frisch-Waugh-Lovell theorem

Question 3

What is Least Square Model of regression in matrix form?

Answer 3

y(Ubar)= X(Ubar)β +u

The Guass-Markov assumptions are:

E(u) = 0

E(uu^T)=σ_u²I_N

X is fixed in repeated samples

The OLS estimator is:

β(hat)=(X(UBar)^TX(Ubar))^-1X(Ubar)^Ty(Ubar)

Question 4

What is the prooft that the OLS is unbiased unfer the Gauss-Markov assumptions?

Answer 4

• (LAST LINE) If we take the expectation of both sides you will see that the expect value of β(hat) equal the expected side of the other side of the equation
• The reasons why β is not in the expectation function is because its constant and the expected value of a constant is itself
• as from the GM assumptions we know the X values are fixed so we can take them out of the expectation operation so it is (X(Ubar)^TX(Ubar))^-1X(Ubar)^T*E(u) but as E(u)=0 we get the final line proving the OLS is unbiased

Question 5

How do you calculate the variance of the OLS estimator in matrix form?

Answer 5

• Line 2 –> shows both β(hat)-E(β(hat) and the transposed version

Question 6

What is a variance-covariance matrix?

Answer 6

• Variance is a measure of the variability or spread in a set of data. Mathematically, it is the average squared deviation from the mean score.
• Covariance is a measure of the extent to which corresponding elements from two sets of ordered data move in the same direction.
• Variance and covariance are often displayed together in a variance-covariance matrix, (aka, a covariance matrix). The variances appear along the diagonal and covariances appear in the off-diagonal elements

Question 7

What is the distribution of the OLS estimator in matrix form?

Answer 7

Question 8

How do you find the sample-covariance matrix?

Answer 8

• just like the bivariate model as we dont know the actual value of the variance we can only use the estimator of the variance
• σ_u(hat)²–> is the residual sum of squares divided by its degrees of freedom
• k is the number of parameters we estimate, or restrictions used

Question 9

How would you set up a hypothesis test of a multivariate regression model?

Answer 9

Question 10

What does the Guass-Markov Theorem tell you about the variance-covariance matrix of any linear unbiased estimator?

Answer 10

The Gauss-Markov Theorem shows that, under the GM assumptions, the variance-covariance matrix of any linear unbiased estimator differs from the OLS v-cov matrix by a positive semi-definite matrix.

Hence OLS is BLUE.