Week 2

Question 1

What is autocorrelation? Verbal and mathematical

Answer 1

Relationship between variable and its lag
E(εiεj) = σij, i!=j
E(εε’) = Ω = matrix of all different σij -> symmetric and positive definite

Also possible heteroskedasticity

Can see in plot of residual against residual lag - correlated

Question 2

What are the properties of OLS under autocorrelation?

Answer 2

• Unbiased
• Consistent
• Inefficient
• Incorrect SE

Question 3

What estimators should we use if autocorrelation?

Answer 3

Newey-West Estimator Covariance Matrix

Question 4

Where do the weights in the Newey-West estimator come from?

Answer 4

The Kernel function

Question 5

What are the properties of Newey-West SE?

Answer 5

HAC (Heteroskedastic and autocorrelation consistent)

Question 6

What is the idea of the Cochrane-Orcutt procedure?

Answer 6

Errors are from an autoregressive model of order 1
You have the normal regression, the lag regression, and the regression of error term with lag
Find value of y(i) - py(i-1)

Question 7

What are the alternatives to Cochrane-Orcutt procedure?

Answer 7

NLS for y(i) = py(i-1) + B1(1-p) + B2(xi-px(i-1)) + n(i)

In Eviews, AR(1)

Question 8

What is the idea of GLS?

Answer 8

Transform data s.t. the conditions for efficient OLS hold

Question 9

What is the Choleski decomposition?

Answer 9

PP’ = Ω

Transformed data: y* = P^(-1)y ; X* = P^(-1)X ; ε* = P^(-1)ε

Question 10

Properties of GLS disturbances

Answer 10

Homoskedastic and no autocorrelation

=> OLS for transformed model efficient estimator for β

Question 11

Compare GLS to Cochrane-Orcutt

Answer 11

• In GLS 1st observation is included

- In GLS scaling factor 1/σ(n)

Question 12

GLS estimator + expected value and variance

Answer 12

b = (X'Ω^(-1)X)^(-1)X'Ω^(-1)y
E(b) = β
Var(b) = (X'Ω^(-1)X)^(-1)

Question 13

Why do we need feasible GLS?

Answer 13

In practice often Ω unknown and have to estimate it

Question 14

What are the steps of FGLS?

Answer 14

1) Estimate the covariance matrix
a) Apply OLS in y=Xβ + ε -> b consistent
b) Estimate Ω using residuals e = y - Xb : Ω^ = ee’
2) Apply OLS on the transformed data
a) Use Ω^ to determine P^
b) Transform data with P^^(-1) : y=P^^(-1)y and X=P^^(-1)X
c) Estimate β with OLS in the model for the transformed data: y* = Xβ + ε -> b(FLGS)

Question 15

What is the null hypothesis of the autocorrelation tests?

Answer 15

No autocorrelation

Question 16

What is the equation for the autocorrelation of residuals?

Answer 16

r(k) = Σe(i)e(i-k)/Σe(i)^2 : first sum is from i=k+1 to n; second sum i=1 to n

Question 17

Durbin-Watson test

Answer 17

DW = Σ(i=2->n) (e(i) - e(i-1))^2/Σe(i)^2 ≈ 2(1 - r(1))
0 (r(1) = 1 => perfect correlation)
4 (r(1) = -1 => perfect negative correlation)
H0: Value should be around 2

Question 18

What are the disadvantages of the Durbin-Watson test?

Answer 18

• Distribution under H0 depends on the properties of regressors
• Not applicable when lagged dependent variables are included as regressors

Question 19

Box-Pierce Test

Answer 19

H0: No autocorrelation

BP = nΣ(k=1 -> p) r(k)^2 ≈ χ2(p)

Question 20

Ljung-Box Test

Answer 20

LB = nΣ(k=1 -> p) (n+2)/(n-k) r(k)^2 ≈ χ2(p)

Question 21

What type of test is a Breusch-Godfrey test?

Answer 21

Lagrange Multiplier (LM) test

Question 22

Which is the procedure for the Breusch-Godfrey test?

Answer 22

1) OLS on y(i) = x(i)’β + ε(i)
2) Run auxiliary regression
3) Under H0 (no autocorrelation) have nR^2 ≈ χ2(p)

Question 23

What is the main difference between the Box-Pierce and Ljung-Box test?

Answer 23

The Box-Pierce test is an approximated version of the Ljung-Box test

Question 24

What happens to the significance of the parameter of the independent variable with NW SE?

Answer 24

The significance may change

Question 25

What happens to the marginal effect of the independent variables with NW SE?

Answer 25

Doesn’t change

Question 26

Do NW SE automatically correct for possible heteroskedasticity?

Answer 26

Yes

Question 27

Do NW SE not harm i.e. should they always be used?

Answer 27

True - Ω = σ^2*I

False - We add uncertainly, less efficient