Week 1 Flashcards
What is heteroskedasticity? Verbal and mathematical.
The amount of randomness may differ for each observation
E(Ɛi^2) = σi^2
How do you detect heteroskedasticity graphically?
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
Consequences of heteroskedasticity
Unbiased
Consistent
No longer efficient (not BLUE)
What do you need to do weighted least squares (WLS)?
σi^2 = σ^2vi, with vi known
What is the procedure for WLS?
Standardize variables (y, x and ɛ) by 1/sqrt(vi) Call standardized variables y*, x* and ɛ*
What is the expected value and variance of the WLS estimator?
E(b) = β Var(b) = σ^2(X*'X*)^(-1)
What are the properties of the WLS estimator?
Unbiased
Consistent
BLUE
How does WLS compare to OLS?
Same coefficient
Conclusions not affect
Different R^2
WLS estimator is more efficient
WLS in practice
In practice, vi is unknown or unobserved -> need to estimate variances
Estimation methods:
- Two step feasible WLS
- Maximum likelihood
Two cases of WLS
σi^2 = zi'γ σi^2 = exp(zi'γ)
Explain feasible WLS (FWLS)
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^
Properties of FWLS
Consistent (if γ estimated consistently)
- in linear form - always consistent
- in multiplicative form - only when correction is included
Asymptotically efficient (and equal to WLS)
Maximum likelihood for WLS
θ^ ≈ N(θ0, (I^)-1), I^ = - second derivative of log likelihood function evaluated at θ^
What are the tests for heteroskedasticity?
- Goldfeld-Quant
- White
- Breusch-Pagan
- Likelihood Ratio
(H0 of homoskedasticity)
How to detect heteroskedasticity?
- Plot (sqaured) residuals against explanatory variables/time/etc.
- Compare OLS and White standard errors
Steps of Goldfeld-Quant test?
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
What type of test are White and Breusch-Pagan tests? What is the idea?
Lagrange Multiplier (LM) tests
Test whether gradient ('score') is sufficiently close to 0 at the restricted estimateSteps of White and Breusch-Pagan test?
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)
What is the difference between the White and Breusch-Pagan test?
In the Breusch-Pagan test, some known variables are suspected of driving the variance
-> zi is a set of selected/known variables
Likelihood Ratio Test
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
Equation for b(OLS) (both sum and matrix)
b = Σxiyi/Σxi^2
b = (X’X)^(-1)X’y
Equation for b(WLS)
b = Σ(xi*xi'*)^(-1)xi*yi* , xi* = xi/sqrt(vi) b = (X'Ω^(-1)X)^(-1)X'Ω^(-1)y
How to check for consistency? What is the necessary middle step? What are the two conditions that need to hold?
Take plim
Multiply by 1/n
Stability condition and orthogonality condition
