Week 12: Chapter 12

Question 1

Write out the adjusted VarB^1 formula

Answer 1

notes

Question 2

OLS is only inconsistent with serial correlation and a lagged dependent variable if we
assume …

Answer 2

if we assume ut follows an AR(1) model

Question 3

et is uncorrelated with yt−1, Cov(yt−1, ut ) = ρCov(yt−1, ut−1) which is not zero unless ρ = 0, what happens when p /= 0

Answer 3

OLS estimators of 𝛽0 and 𝛽1 are inconsistent

Question 4

OLS is only inconsistent with serial correlation and a lagged dependent variable if we
assume ut follows an AR(1) model, write the formula for an AR(1) MODEL with E(….)

Answer 4

E(et|ut−1,ut−2,…) = E(et|yt−1,yt−2,…) = 0

Question 5

What does the adjusted VAR(B^j) assume?

Answer 5

• Estimated errors following AR(1) process
• Homoskedasticity

Question 6

Whats the serial correlation robust standard error of 𝛽^1

Answer 6

see notes

Question 7

write the formula for the truncation lag

Answer 7

notes

Question 8

what does g stand for in the truncation lag equation?

Answer 8

g is the truncation lag which controls for how much serial correlation is allowed in the model

Question 9

What does HAC stand for?

Answer 9

Heteroskedastic and autocorrelation consistent standard errors

Question 10

What are the two standard AR(1) model assumptions?

Answer 10

E(et|ut−1,ut−1,…) = 0 and Var(et|ut−1) = Var(et) = 0

Question 11

A t-test for serial correlation with strictly exogenous regressors, what is the equation and what is the null?

Answer 11

ut = p(ut-1) + et
H0: p = 0

Question 12

What are the steps for testing serial correlation with strictly exogenous regressors?

Answer 12

1. Estimate the OLS regression on yt and obtain the residual, uˆt
2. Regress ut on ut−1 to obtain pˆ and tpˆ
3. Use tpˆ to test H0 : ρ = 0. Serial correlation is a problem if H0 is rejected at the 5% level.

Question 13

Write the equation for the DW test (Durbin-Watson)

Answer 13

see notes

Question 14

What are the conditions for the DW test to reject/not reject the hypotheses?

Answer 14

H0: ρ=0, H1: ρ>0 and for ρˆ≈ 0, DW ≈ 2
If DW < dL, we can reject H0
* DW > dU, we cannot reject H0
* dL ≤ DW ≤ dL, the test is inconclusive

Question 15

In correcting for serial correlation, what can be done AND what is required?

Answer 15

Use GLS and strict exogeneity is required

Question 16

Transform the model to get rid of the serial correlation in ut

Answer 16

see in notes

Question 17

what are y ̃ and x ̃ called?

Answer 17

quasi-differenced data

Question 18

Whats the problem with the FGLS in AR(1)

Answer 18

p is rarely known, but we can estimate p^ and use that instead

Question 19

FGLS has no finite sample properties, which has two major implications

Answer 19

• FGLS estimators are biased but they can be consistent if the data is weakly dependent
• t and F statistics are only approximately t and F distributed due to the estimation error in
  ρˆ

Question 20

HOW is FGLS still useful?

Answer 20

FGLS is asymptotically more efficient than OLS when the errors are AR (1) and the explanatory variables are strictly exogenous

Question 21

there are two variants of FGLS that differ in their treatment of ρ, name them and which one is used more

Answer 21

• Cochrane-Orcutt (CO) → omits the first observation
• Prais-Winsten (PW) → uses a transformed first observation
• Asymptotically, CO and PW are identical, but in small samples, we want to use PW

Question 22

What are the benefits of differencing in time series?

Answer 22

• it will often eliminate most of the serial correlation in the errors if in the AR error process, ρ is positive and large

Question 23

What happens to Var(ut) without differencing?

Answer 23

It grows overtime
BUT differenced, it has a zero mean, constant variance and is serially uncorrelated

Question 24

What are two things you need to note before testing for heteroskedasticity

Answer 24

• Errors are uncorrelated (first test/correct for serial correlation than test for hetersokedasticity)
• Errors are homoskedastic

Question 25

What does ARCH stand for

Answer 25

Autoregressive conditional heteroskedasticity
Var(ut |Z)

Question 26

first order ARCH model

Answer 26

E(ut^2|ut−1, ut−2,…) = E(ut^2|ut−1) = α0 +α1u^2t-1