Breusch-Pagan Test for Heteroskedasticity, what is the correct form of the null hypothesis? For the Breusch-Pagan Test, under pg. 4 of this resource
It claims that the null is
$$
H_0: E(u\mid X) = \sigma^2
$$
But here it is stated that the null can be written as
$$
H_0: \delta_1 = \delta_2 = \ldots = \delta_k
$$
where the $\delta$'s come from regressing the error terms $\epsilon$ in an OLS regression against 
$$
\epsilon^2 =\delta_0 + \delta_1X_1 + \ldots + \delta_kX_k + e 
$$
where $e \sim N(0,1)$
Furthermore, I have seen that the original paper states
$$
H_0: \delta_1 = \delta_2 = \ldots = \delta_k = 0
$$
what is the difference between all these nulls? thanks.
 A: It should be the second. That is the null hypothesis given by:
$$
H_0: \delta_1 = \delta_2 = \ldots = \delta_k = 0
$$
Heteroskedasticity means that the variance is not constant across observations. If all $\delta$ are equal to each other then the variance would still depend on what the observations $X$ are (even if $\delta$ is constant $X$ is not).
The difference between them is that in first expression you just require all coefficients being exactly the same to each other whether in second expression you test whether they are all equal to each other and zero.
Also, according to the widely used Wooldridge introduction to econometrics, it’s the second equation. I found the same null (that is the second expression) also in Verbeeks guide to modern econometrics and also Pesaran time series and panel data econometrics - all highly cited textbooks from highly cited authors. Given this I think author of that pdf you shared just made a typo, I would hope that all those texts did not get it wrong.
