Week 2: Chapter 8 Flashcards
Heteroskedasticity
Var(u|x) = σ^2x
What Hetero DOES NOT effect
- Whether OLS estimators are unbiased or inconsistent
- Goodness of fit measures
What Hetero DOES effect
- OLS no longer BLUE.
- A bias in Var (Bj) which invalidates t-tests, F-tests and confidence intervals.
- OLS no longer being asymptotically efficient.
Var(Bj) under SLR
Look @ notes
Var(Bj) under MLR
Look @ notes
Reasons to use WLS over OLS
If variance is correctly specified, WLS is more efficient than OLS.
WLS formula: The form of Hetero is known
Var(u|x) = σ^2h(x)
Transform model with heteroskedastic errors
notes
What do we divide by root hi?
To transform the heteroskedastic errors into a constant homoskedastic result
GLS, why do we use it?
Technique when correlation is suspected between the residuals.
GLS: Interpret B1 after transformation
β1 is the change in yi/ √ hi given a one-unit change in (xi1/ √ hi), ceteris paribus.
When do use GLS/WLS?
- When the errors are dependent, we can use generalized least squares (GLS).
- When the errors are independent, but not identically distributed, we can use weighted least squares (WLS), which is a special case of GLS
Feasible GLS (FGLS) estimator
When the form of heteroskedasticity is unknown, weight residuals by hi^
FGLS
Var(u│x)=σ^2 exp (δ0+ δ1 x1+⋯+δk xk)v
Why do we use an exponential function in FLGS?
It is required that our estimated variances be positive to use WLS. However, linear models do not guarantee that the predicted values produced are positive. Using a non-linear model ensures that we have strictly positive predicted values
Why do we use an exponential function in FLGS?
It is required that our estimated variances be positive to use WLS. However, linear models do not guarantee that the predicted values produced are positive. Using a non-linear model ensures that we have strictly positive predicted values
What does the v mean in FGLS?
Mean unity, conditional on x
Assume v is independent of x, rewrite the equation using log form to linearise model
log (u^2) = … look @ notes
What does the Breusch-Pagan test assume about U^2
Test assumes u^2 is a linear function of the independent variables.
Write out the steps for the BP test
Find in notes
White test assumption
MLR5 can be replaced with a WEAKER assumption that u^2 is uncorrelated with the explanatory variables, the squared independent variables and their cross-products.
Weakness of white test?
Uses too many degrees of freedom, fix this by using the modified white test.
Special Case white test
Uses residuals and fitted values, WILL ALWAYS HAVE TWO restrictions.
If you reject the null, what does this mean?
There is evidence of heteroskedasticity within the model
What does the LPM always violate?
Always violates homoskedasticity assumption, unless all slope parameters are 0
Why does the LPM always violate MLR5? How to overcome this
Due to Var(y|x) = Var (u|x) = p(x)[1-p(x)]
Overcome this by estimating the LMP using FGLS
For LPM, need to weight each observation i by 1/h^i, what does h^i entail?
h^i = y^i(1-y^i), NEED TO ENSURE 0 < y^i < 1
How do we ensure 0 < y^i < 1 in LPM?
- Throw away observations such that fitted values 0 < y^i < 1
- Use OLS with hetero robust standard errors
