L3 - Serial Correlation

Question 1

Under serial correlation, OLS is … ?

Answer 1

Consistent, but becomes inefficient so standard errors need to be adjusted

Question 2

How do we test for serial correlation under strong exogeneity?

Answer 2

(1) Perform OLS regression of yₜ on xₜ and obtain the estimated OLS coefficient B̂.

(2) Form the estimated residual ûₜ as ûₜ= yₜ − xₜ’B̂ , for all t = 1, . . . , T.

(3) Perform OLS regression of ûₜ on ûₜ₋₁ for all t = 2, . . . , T, and obtain the estimated OLS
coefficient ρˆ

(4) Form the t stat for ρˆ to test the null hypothesis H0 : ρ = 0 against the alternative
H1 : ρ ̸= 0.

Question 3

What are the limitations of testing for serial correlation under strong exogeneity?

Answer 3

Say ûₜ and ûₜ₋₁ may not be correlated, but ûₜ and ûₜ₋₂ maybe. Hence we need to test for higher-order correlation. This will also violate strong exogeneity.

Question 4

What is the effect on the OLS estimator if there is serial correlation without strong exogeneity

Answer 4

It will make OLS biased due to endogeneity

Question 5

How do we test for serial correlation without strong exogeneity?

Answer 5

(1) Perform OLS regression of yₜ on xₜ and obtain the estimated OLS coefficient B̂.

(2) Form the estimated residual ûₜ as ûₜ= yₜ − xₜ’B̂ , for all t = 1, . . . , T.

(3) Perform OLS regression of ûₜ on x₁ₜ, x₂ₜ … and ûₜ₋₁ for all t = 2, . . . , T, and obtain the estimated OLS coefficient ρˆon ûₜ₋₁

(4) Form the t stat for ρˆ to test the null hypothesis H0 : ρ = 0 against the alternative
H1 : ρ ̸= 0.

Question 6

What is the main limitation of the test for serial correlation without strong exogeneity?

Answer 6

Test does not pick up correlation of error terms over periods more than 1 time period apart. We thus need a test for higher-order serial correlation

Question 7

What is the procedure for testing for higher-order serial correlation?

Answer 7

(1) Perform OLS regression of yₜ on xₜ and obtain the estimated OLS coefficient B̂.

(2) Form the estimated residual ûₜ as ûₜ= yₜ − xₜ’B̂ , for all t = 1, . . . , T.

(3) Perform OLS regression of ûₜ on x₁ₜ, x₂ₜ … and ûₜ₋₁, ûₜ₋₂, ûₜ₋₃…., for all t = 2, . . . , T, and obtain the estimated OLS coefficient ρ₁ˆ= ρ₂ˆ= ρ₃ˆ = … on ûₜ₋₁, ûₜ₋₂, ûₜ₋₃…., in the auxiliary regression

(4) Form the t stat for ρˆ to test the null hypothesis H0 : ρ = 0 against the alternative
H1 : ρ ̸= 0.

Question 8

At surface level, how do we deal with the effect of serial correlation once we have detected it?

Answer 8

Estimate the variance-covariance matrix of the errors, reweight the data to obtain an efficient estimator, and use corrected standard errors to perform valid inference.

Question 9

Quasi-differentiate AR(1)

Answer 9

Start with standard: yₜ = β₀ + β₁xₜ + uₜ
denote: eₜ = uₜ − ρuₜ₋₁

yₜ₋₁ = β₀ + β₁xₜ₋₁ + uₜ₋₁
ρ(yₜ₋₁) = ρ( β₀ + β₁xₜ₋₁ + uₜ₋₁)

yₜ - ρ(yₜ₋₁) = (β₀ + β₁xₜ + uₜ) - ρ( β₀ + β₁xₜ₋₁ + uₜ₋₁)
yₜ - ρyₜ₋₁ = (1- ρ)β₀ + (xₜ - xₜ₋₁ρ)β₁ + eₜ

Let ỹₜ = yₜ - ρyₜ₋₁
Let x˜ₜ = xₜ - ρxₜ₋₁

ỹₜ = (1 − ρ)β₀ + β₁x˜ₜ + eₜ

Question 10

What is the procedure of Feasible Generalised Least Squares in the presence of AR(1) serial correlation?

Answer 10

(1) Perform OLS regression of yₜ on xₜ and obtain the estimated OLS coefficient B̂.

(2) Form the estimated residual ûₜ as ûₜ= yₜ − xₜ’B̂ , for all t = 1, . . . , T.

(3) Perform OLS regression of ûₜ on ûₜ₋₁ for all t = 2, . . . , T, and obtain the estimated OLS coefficient ρˆ

(4) Form the quasi-differenced data, and perform OLS regression on the quasi-differenced model ỹₜ = β₀ x˜₀ₜ + β₁x˜1ₜ + . . . + βₖx˜ₖₜ + vₜ

Usual OLS inference is then valid

Question 11

Is FGLS BLUE?

Answer 11

Strictly speaking. FGLS is not BLUE. It is however more efficient than OLS.