Tentamen

Question 1

What is the LS estimator of beta?

Answer 1

β hat = (X’X)^-1X’y

Question 2

What is the LS estimator of σ²?

Answer 2

s² = e’e/(n-k)

Question 3

How to show that beta hat (LS) is unbiased?

Answer 3

E(β hat) = E[(X’X)^-1X’y] = (X’X)^-1X’ E[y] = (X’X)^-1X’X β = β

Question 4

How to show that s² is unbiased, or how to derive it?

Answer 4

Let B, a positive semidefinite matrix. Then we can rewrite s² = y’By = (Xβ + u)’B(Xβ +u) = u’Bu + 2 β’X’Bu + β’ X’ BX β.

If we take the expectation of this:
E(s²) = E(u’Bu) + β’ X’ BX’ β = σ² tr(B) + β’ X’ BX β.

Since we know that BX = 0 is satisfied if B = M (residual maker), and tr(M) = n - k, s² = y’My/(n-k) = e’e/(n-k) is unbiased.

Question 5

How to derive the variance of beta hat?

Answer 5

Var(β hat) = Var[(X’X)^-1X’y] = (X’X)^-1X’ Var(y) ((X’X)^-1X)’ = (X’X)^-1X’ σ² I_n X’(X’X)^-1 = σ² (X’X)^-1

Question 6

Show that no other linear unbiased estimator of β has a lower variance.

Answer 6

Question 7

When to use the F-statistic for testing, and what is it?

Answer 7

The F-statistic should be used when testing multiple linear restrictions, where R is a matrix with the formula of the restriction, and r is the value.

Question 8

What is the statistic for testing a single restriction?

Answer 8

w is a vector of the restriction, r is the value.

Question 9

What is the statistic for testing if beta equals zero?

Answer 9

Question 10

How to derive a ML-estimator?

Answer 10

1. First create the Likelihood function (the product of n draws of the CDF)
2. Take the ln of this function
3. Derive it
4. Equate to zero
5. Possibly take the second derivative to show that it is in fact a minimum

Question 11

Show the Cramer-Rao inequality, what does this mean?

Answer 11

It means that if an unbiased estimator achieves the lower bound it’s efficient.

Question 12

What is the adjusted R², and what is R² (formulas)?

Answer 12

Note: SST = y’Ay

Question 13

What is the difference between a model and a data generating process?

Answer 13

A model tries to approximate the DGP, but is does not equal the DGP (generally).

Question 14

How can be shown that the normal distribution is a second order approximation around te mode?

Answer 14

Question 15

How is σ hat, and β hat derived (using the ML)?

Answer 15

Question 16

What is the formula for the covariance and the variance?

Answer 16

Cov(a, b) = E[(a - E(a))(b - E(b))’]

Var(a) = E[(a - E(a))(a - E(a))’]

Question 17

What is the formula of the SST? And A?

Answer 17

SST = y’Ay

A = I_n - ɩɩ’/n

Question 18

How to get the restricted ML values?

Answer 18

Question 19

On what is the Wald, the Lagrangian Multiplier and the Likelyhood ratio test based?

Answer 19

Wald: Based on the idea that Rβ hat should be close to r if the null hypothesis is true.

LM: Based on the fact that the gradient q(β hat) vanishes if the null hypothesis is true, thus q(β hat) should be close to zero (for the null hypothesis).

LR: Based on the ratio of the maximized likelihood with the restriction to the maximized likelihood without the restiction, which should be close to one if the null hypothesis is true.

Question 20

What is the Wald, LR, and LM test (when σ is given)?

Answer 20

(Rβ hat - r)’(R(X’X)^-1R’)^-1(Rβ hat - r)/σ², it is chi-squared (m) distributed, where m is the number of restrictions.

Question 21

How to show the order of the Wald, LM and LR tests?

Answer 21

Question 22

What is the matrix M and what does it do?

Answer 22

M is the residual maker matrix, s.t. e = y - Xβ hat = My = Mu,

var(e) = var(Mu) = σ²M.

M = I_n - X(X’X)^-1X’

Question 23

How to do confidence interval?

Answer 23

(β_j hat - t sqrt(s²v_jj), β_j hat + t sqrt(s²v_jj)), where t is the 95% confidence statistic (for two sided test).

Question 24

How to do prediction interval?

Answer 24

(y_*hat - A, y_*hat + A), with A = t s sqrt(x_*‘(X’X)^-1x_* + 1), t is the 95% test statistic

Question 25

What is the LS derivation?

Answer 25

Question 26

What are the advantages and disadvantages of the Least Absolute Estimation?

Answer 26

Question 27

What are the ideal 5 conditions/assumptions?

Answer 27

1. The model is linear
2. The n x k matrix X is nonrandom and has rank k (thus all columns are linearly independent)
3. The n x 1 vector u has mean zero and variance σ²I_n
4. The N x 1 vector u follows a multivariate normal distribution
5. If we let Q_n = X’X/n then Q_n –> Q as n –> inf, where Q is finite and positive definite k x k matrix. (Asymptotic, thus not very important for us)

Question 28

What is meant by significance and importance?

Answer 28

Often we can ignore small unimportant variables, although our estimation could then be slightly more biased, it often also gains precision.

Significance: how precisely we manage to estimate a parameter

Importance: the impact of the parameter on the total outcome.

Question 29

What does the Crame-Rao inequality mean?

Answer 29

It says that the varaince of an unbiased estimator achieves the Cramer-Rao lower bound it is efficient.