Week 2: Chapter 8

Question 1

Heteroskedasticity

Answer 1

Var(u|x) = σ^2x

Question 2

What Hetero DOES NOT effect

Answer 2

• Whether OLS estimators are unbiased or inconsistent
• Goodness of fit measures

Question 3

What Hetero DOES effect

Answer 3

• OLS no longer BLUE.
• A bias in Var (Bj) which invalidates t-tests, F-tests and confidence intervals.
• OLS no longer being asymptotically efficient.

Question 4

Var(Bj) under SLR

Answer 4

Look @ notes

Question 5

Var(Bj) under MLR

Answer 5

Look @ notes

Question 6

Reasons to use WLS over OLS

Answer 6

If variance is correctly specified, WLS is more efficient than OLS.

Question 7

WLS formula: The form of Hetero is known

Answer 7

Var(u|x) = σ^2h(x)

Question 8

Transform model with heteroskedastic errors

Answer 8

notes

Question 9

What do we divide by root hi?

Answer 9

To transform the heteroskedastic errors into a constant homoskedastic result

Question 10

GLS, why do we use it?

Answer 10

Technique when correlation is suspected between the residuals.

Question 11

GLS: Interpret B1 after transformation

Answer 11

β1 is the change in yi/ √ hi given a one-unit change in (xi1/ √ hi), ceteris paribus.

Question 12

When do use GLS/WLS?

Answer 12

• 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

Question 13

Feasible GLS (FGLS) estimator

Answer 13

When the form of heteroskedasticity is unknown, weight residuals by hi^

Question 14

FGLS

Answer 14

Var(u│x)=σ^2 exp⁡ (δ0+ δ1 x1+⋯+δk xk)v

Question 15

Why do we use an exponential function in FLGS?

Answer 15

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

Question 16

Why do we use an exponential function in FLGS?

Answer 16

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

Question 17

What does the v mean in FGLS?

Answer 17

Mean unity, conditional on x

Question 18

Assume v is independent of x, rewrite the equation using log form to linearise model

Answer 18

log (u^2) = … look @ notes

Question 19

What does the Breusch-Pagan test assume about U^2

Answer 19

Test assumes u^2 is a linear function of the independent variables.

Question 20

Write out the steps for the BP test

Answer 20

Find in notes

Question 21

White test assumption

Answer 21

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.

Question 22

Weakness of white test?

Answer 22

Uses too many degrees of freedom, fix this by using the modified white test.

Question 23

Special Case white test

Answer 23

Uses residuals and fitted values, WILL ALWAYS HAVE TWO restrictions.

Question 24

If you reject the null, what does this mean?

Answer 24

There is evidence of heteroskedasticity within the model

Question 25

What does the LPM always violate?

Answer 25

Always violates homoskedasticity assumption, unless all slope parameters are 0

Question 26

Why does the LPM always violate MLR5? How to overcome this

Answer 26

Due to Var(y|x) = Var (u|x) = p(x)[1-p(x)]
Overcome this by estimating the LMP using FGLS

Question 27

For LPM, need to weight each observation i by 1/h^i, what does h^i entail?

Answer 27

h^i = y^i(1-y^i), NEED TO ENSURE 0 < y^i < 1

Question 28

How do we ensure 0 < y^i < 1 in LPM?

Answer 28

• Throw away observations such that fitted values 0 < y^i < 1
• Use OLS with hetero robust standard errors