HANDOUT 7

Question 1

Constant variance CLRM assumption

Answer 1

V(€i I Xi) =sigma^2

NO i subscript = homoscedastic

Question 2

Variance if heteroscedastic

Answer 2

V(€i I Xi) =sigma i ^2

i subscript = varies across individuals = heteroscedastic.

Question 3

How does heteroscedasticity affect unbiasedness?

Answer 3

It does NOT

E(b1) = B1 only needs CLRM assumption 1 to hold: E(€i I X) = 0

Question 4

V(b1) for homoscedasticity

Answer 4

V(b1) = sigma^2 / sum[(xi - x bar)^2]

Question 5

V(b1) for heteroscedasticity

Answer 5

V(b1) = sum wi^2 sigma i^2
V(b1) = sum[(xi - x bar)^2 x sigma i ^2
/ [sum(xi - x bar)^2]^2

Question 6

So what is the problem with OLS when we have heteroscedasticity?

Answer 6

incorrect variance estimate = wrong SE = wrong t-ratios = CANNOT do hypothesis testing even tho coefficients are OK.

Question 7

Variance formula simplified White

Answer 7

V(b1) = sigma^2 v / [sum(xi - x bar)^2]^2
where vi = (xi - xbar)€i
and sigma^2 v = sum(xi - xbar)^2 €i^2

Question 8

How does heteroscedasticity affect the F statistic?

Answer 8

The F statistic ONLY holds for CONSTANT variance = we cannot use it now. Stata will give the correct F stat using a variance-covariance matrix, but we cannot calculate by hand using the RSS formula.

Question 9

Regression to detect heteroscedaticity

Answer 9

V(€i) = d0 + d1Z1i +…+ dpZpi
€i^2 = d0 + d1Z1i +…+ dpZpi + Ri
where Ri = well-behaved error term

Question 10

H0 and H1 for heteroscedasticity test

Answer 10

H0: d1 = d2 =…= dp = 0: V(€i) = do = CONSTANT
H1: any dj ≠ 0 - V(€i) = f(Zi) = NOT CONSTANT

Question 11

2 Problems with heteroscedasticity test and solutions

Answer 11

1. €i unobserved –> use residuals ei

2. Zi unknown –> find alternatives

Question 12

White’s alternatives for Zi

Answer 12

Z1i = X1i,…, Zki = Xki
Zk+1 i = X1i^2,…, Zpi = Xki^2
- use original explanatory variables and squares
- But NOT cross-products

Question 13

Breusch Pagan alternatives for Zi

Answer 13

Z1i = yi^

use fitted values

Question 14

ARCH alternatives for Zi

Answer 14

ARCH = autoregressive conditional heteroscedasticity
Z1i = ei-1 ^2 … Zpi = ei-1 ^2
- Today’s variance is a fucnction of yestedays
Good for time-series & particularly financial such as ER, stock prices

Question 15

Problem with heteroscedasticty test

Answer 15

We use proxies for Zi = LOW POWERED TEST
We often do not reject H0 when we should
We often find homoscedasticity when actually the variance is NOT constant.

Question 16

2 alternative statistics to do the heteroscedasticity test

Answer 16

1. The usual F test
2. Lagrange multiple for big sample sizes
   nR^2 - Chi-squared p

Question 17

Apparent heteroscedasticity can be caused by…

Answer 17

an omitted relevant variable
because the error term of the false model includes the omitted variable, hence the variance of the error term is not a constant. But this isn’t heteroscedasticity, just a symptom of misspecification.