Conditional homoskedasticity vs heteroskedasticity From Econometrics, by Fumio Hayashi (Chpt 1):
Unconditional Homoskedasticity:


*

*The second moment of the error terms E(εᵢ²) is constant across the observations

*The functional form E(εᵢ²|xi) is constant across the observations


Conditional Homoskedasticity:


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*The restriction that the second moment of the error terms E(εᵢ²) is constant across the observations is lifted

*

*Thus the conditional second moment E(εᵢ²|xi) can differ across the observations through possible dependence on xᵢ.



So then, my question:
How does Conditional Homoskedasticity differ from Heteroskedasticity?
My understanding is that there is heteroskedasticity when the second moment differs across observations (xᵢ). 
 A: I will begin by just quoting from Hayashi to help anybody else who would like to comment.  I have tried to preserve formatting and original equation numbers.
Begin quote from Hayashi page 126, section 2.6:
Conditional versus Unconditional Homoskedasticity
The conditional homoskedasticity assumption is:
Assumption 2.7 (conditional homoskedasticity):
\begin{align*}
\tag{2.6.1} E(\epsilon_i^2|\mathbf{x}_i)=\sigma^2>0.
\end{align*}
This assumption implies that the unconditional second moment $E(\epsilon_i^2)$ equals $\sigma^2$ by the Law of Total Expectations.  To be clear about the distinction between unconditional and conditional homoskedasticity, consider the following example [Example 2.6 (unconditionally homoskedastic but conditionally heteroskedastic errors)...]
End quote.
Some relevant equations from Hayashi pages 11-14 (Section 1.1):
\begin{align*}
\tag{1.1.12} E(\epsilon_i^2|\mathbf{X})=\sigma^2>0 \quad (i=1,2,\ldots,n) \\\
\tag{1.1.17} E(\epsilon_i^2|\mathbf x_i)=\sigma^2>0 \quad (i=1,2,.\ldots,n).
\end{align*}
The subsection "The Classical Regression Model for Random Samples" on page 12 discusses the implications of a sample being iid.  Quoting from Hayashi pages 12-13: "The implication of the identical distribution aspect of a random sample is that the joint distribution of $(\epsilon_i,\mathbf x_i)$ does not depend on $i$.  So the unconditional second moment $E(\epsilon_i^2)$ is constant across $i$ (this is referred to as unconditional homoskedasticity) and the functional form of the conditional second moment $E(\epsilon_i^2|\mathbf x_i)$ is the same across $i$.  However Assumption 1.4---that the value of the conditional second moment is the same across $i$---does not follow.  Therefore, Assumption 1.4 remains restrictive for the case of a random sample; without it, the conditional second moment $E(\epsilon_i^2|\mathbf x_i)$ can differ across $i$ through its possible dependence on $\mathbf x_i$.  To emphasize the distinction, the restrictions on the conditional second moments, (1.1.12) and (1.1.17), are referred to as conditional homoskedasticity."
[No further quotes from Hayashi, just my understanding after this point.]
I assume the original question was about the above discussion on pages 12-13.  In that case, I think the first bullet under "Conditional Homoskedasticity" isn't technically correct (though I understand what you mean): Hayashi says (1.1.17) is "conditional homoskedasticity," and if $E(\epsilon_i^2|\mathbf x_i)=\sigma^2$, then $E(\epsilon_i^2)=E[E(\epsilon_i^2|\mathbf x_i)]=E[\sigma^2]=\sigma^2$, as Hayashi notes on page 126 (that conditional homoskedasticity implies unconditional homoskedasticity by the Law of Total Expectations).
So I think part of the issue may be the interpretation of Hayashi's statements.  Conditional homoskedasticity says (1.1.17) even for different $\mathbf x_i$, the variance of $\epsilon_i$ is the same constant $\sigma^2$.  Unconditional homoskedasticity is a weaker statement, in that you could have $E(\epsilon_i^2)=\sigma^2$ but $E(\epsilon_i^2|\mathbf x_i)\ne\sigma^2$; Examples 2.6 (page 127) illustrates this.  It also perhaps answers the question of the overlap between homo- and heteroskedasticity: it gives an example where there is unconditional homoskedasticity as well as conditional heteroskedasticity.
These are confusing concepts, especially without a lot of experience with conditional expectations/distributions, but hopefully this adds some clarity (and source material for any future discussions).
