HANDOUT 7 Flashcards
Constant variance CLRM assumption
V(€i I Xi) =sigma^2
NO i subscript = homoscedastic
Variance if heteroscedastic
V(€i I Xi) =sigma i ^2
i subscript = varies across individuals = heteroscedastic.
How does heteroscedasticity affect unbiasedness?
It does NOT
E(b1) = B1 only needs CLRM assumption 1 to hold: E(€i I X) = 0
V(b1) for homoscedasticity
V(b1) = sigma^2 / sum[(xi - x bar)^2]
V(b1) for heteroscedasticity
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
So what is the problem with OLS when we have heteroscedasticity?
incorrect variance estimate = wrong SE = wrong t-ratios = CANNOT do hypothesis testing even tho coefficients are OK.
Variance formula simplified White
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
How does heteroscedasticity affect the F statistic?
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.
Regression to detect heteroscedaticity
V(€i) = d0 + d1Z1i +…+ dpZpi
€i^2 = d0 + d1Z1i +…+ dpZpi + Ri
where Ri = well-behaved error term
H0 and H1 for heteroscedasticity test
H0: d1 = d2 =…= dp = 0: V(€i) = do = CONSTANT
H1: any dj ≠ 0 - V(€i) = f(Zi) = NOT CONSTANT
2 Problems with heteroscedasticity test and solutions
- €i unobserved –> use residuals ei
2. Zi unknown –> find alternatives
White’s alternatives for Zi
Z1i = X1i,…, Zki = Xki
Zk+1 i = X1i^2,…, Zpi = Xki^2
- use original explanatory variables and squares
- But NOT cross-products
Breusch Pagan alternatives for Zi
Z1i = yi^
use fitted values
ARCH alternatives for Zi
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
Problem with heteroscedasticty test
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.
2 alternative statistics to do the heteroscedasticity test
- The usual F test
- Lagrange multiple for big sample sizes
nR^2 - Chi-squared p
Apparent heteroscedasticity can be caused by…
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.
