Week 1

Question 1

What is heteroskedasticity? Verbal and mathematical.

Answer 1

The amount of randomness may differ for each observation

E(Ɛi^2) = σi^2

Question 2

How do you detect heteroskedasticity graphically?

Answer 2

In x-y graph: points closer together and further apart at different places in x

In x-residual graph: residuals deviate from 0 more at certain points than others

Question 3

Consequences of heteroskedasticity

Answer 3

Unbiased
Consistent
No longer efficient (not BLUE)

Question 4

What do you need to do weighted least squares (WLS)?

Answer 4

σi^2 = σ^2vi, with vi known

Question 5

What is the procedure for WLS?

Answer 5

Standardize variables (y, x and ɛ) by 1/sqrt(vi)
Call standardized variables y*, x* and ɛ*

Question 6

What is the expected value and variance of the WLS estimator?

Answer 6

E(b) = β
Var(b) = σ^2(X*'X*)^(-1)

Question 7

What are the properties of the WLS estimator?

Answer 7

Unbiased
Consistent
BLUE

Question 8

How does WLS compare to OLS?

Answer 8

Same coefficient
Conclusions not affect
Different R^2

WLS estimator is more efficient

Question 9

WLS in practice

Answer 9

In practice, vi is unknown or unobserved -> need to estimate variances
Estimation methods:
- Two step feasible WLS
- Maximum likelihood

Question 10

Two cases of WLS

Answer 10

σi^2 = zi'γ
σi^2 = exp(zi'γ)

Question 11

Explain feasible WLS (FWLS)

Answer 11

1) Estimate variance parameters
a) Run normal OLS regression to obtain ei^2 (asymptotically unbiased estimators of σi^2)
b) Run regression ei^2 = zi’γ + ηi (or log(ei^2)
2) Apply WLS with estimated variances: σi^ = zi’γ^

Question 12

Properties of FWLS

Answer 12

Consistent (if γ estimated consistently)
- in linear form - always consistent
- in multiplicative form - only when correction is included
Asymptotically efficient (and equal to WLS)

Question 13

Maximum likelihood for WLS

Answer 13

θ^ ≈ N(θ0, (I^)-1), I^ = - second derivative of log likelihood function evaluated at θ^

Question 14

What are the tests for heteroskedasticity?

Answer 14

• Goldfeld-Quant
• White
• Breusch-Pagan
• Likelihood Ratio

(H0 of homoskedasticity)

Question 15

How to detect heteroskedasticity?

Answer 15

• Plot (sqaured) residuals against explanatory variables/time/etc.
• Compare OLS and White standard errors

Question 16

Steps of Goldfeld-Quant test?

Answer 16

1) Split data into 3 (or 2) groups with n1, n2 and n3 observations
- > Groups based on suspicion of heteroskedasticity
- > Make sure largest variance is expected in group 2
2) Apply OLS in groups n1 (obtain e1) and in n2 (obtain n2)
3) Compare estimated variances

ei’ei/σ^2 ~ χ^2(ni - k)

[(e2’e2/σ^2)/(n2 - k)]/[(e1’e1/σ^2)/(n1 - k)] = (s2/s1)^2 ~ F(n2 - k, n1 - k)
- > in Eviews sum of square residuals
Reject H0 if test statistic large

Question 17

What type of test are White and Breusch-Pagan tests? What is the idea?

Answer 17

Lagrange Multiplier (LM) tests
Test whether gradient ('score') is sufficiently close to 0 at the restricted estimate

Question 18

Steps of White and Breusch-Pagan test?

Answer 18

1) Estimate the restricted model and obtain residuals (OLS)
2) Auxiliary regression of squared residuals on specific set of regressors
3) Under H0 of homoskedasticity, nR^2 ~ χ^2(p-1) , where p is the number of coefficients in the auxiliary regression (including constant)

Question 19

What is the difference between the White and Breusch-Pagan test?

Answer 19

In the Breusch-Pagan test, some known variables are suspected of driving the variance
-> zi is a set of selected/known variables

Question 20

Likelihood Ratio Test

Answer 20

Test significant of γ in σi^2 = h(zi'γ)
Function h() very important
Procedure: Estimate model under H0 and H1 and compare restricted and unrestricted log-likelihood values
-2(logLr - logLu) ~ χ^2(k),    k = #restrictions

This is a general approach that can be used to test multiple forms of heteroskedasticity, including White and BP (assuming h()!) and GQ

Question 21

Equation for b(OLS) (both sum and matrix)

Answer 21

b = Σxiyi/Σxi^2

b = (X’X)^(-1)X’y

Question 22

Equation for b(WLS)

Answer 22

b = Σ(xi*xi'*)^(-1)xi*yi*   , xi* = xi/sqrt(vi)
b = (X'Ω^(-1)X)^(-1)X'Ω^(-1)y

Question 23

How to check for consistency? What is the necessary middle step? What are the two conditions that need to hold?

Answer 23

Take plim
Multiply by 1/n
Stability condition and orthogonality condition