# Conditional homoskedasticity vs heteroskedasticity

From Econometrics, by Fumio Hyashi (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:

• 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).

-
Maybe this will help: www2.econ.iastate.edu/classes/econ674/falk/… –  whuber Apr 7 '11 at 18:48
There a slight issue in that the lecture says "Therefore, conditional homoskedasticity implies unconditional homoskedasticity" in contradiction to the Econometrics book. They seem to be conditioning on different things. –  Henry Apr 7 '11 at 23:02
@Henry It is difficult to tell from the present question which definitions are accurate and which ones not--some of them do not seem to make sense out of the context of the textbook. Some clarification would be welcome. –  whuber Apr 8 '11 at 0:25

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 *un*conditional 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).

-