Terminology: unconditional heteroskedasticity

Question

I have seen mentions of both unconditional and conditional heteroskedasticity. The latter is fine with me but I am struggling to uderstand the former. It appears I am not the only one to question unconditional heteroskedasticity. A recent comment by @whuber reads:

Heteroscedasticity literally means "different amounts of scatter." It applies purely descriptively to data, for instance, where one subgroup of data may have a strikingly different spread than another (disjoint) subgroup. It may apply to time series (where it is a particular kind of failure of weak second order stationarity). But the general sense of it is that scatter varies, meaning it depends on something, and in that respect the adjective "unconditional" looks like an oxymoron.

In some contexts the contrast between unconditional and conditional heteroskedasticity seems to lie in the size of the conditioning set. E.g. in a time series context, unconditional heteroskedasticity seems to be heteroskedasticity that is not conditioned on any variable in the model but rather the time index (so it actually appears to be conditional despite the name). Meanwhile, conditional heteroskedasticity conditions on some variable in the model.

Could someone who feels comfortable with the term of unconditional heteroskedasticity explain it?

Answer 1

You can treat it as a shorthand for "heteroskedasticity that does not depend on variables explicitly recognized by the model".

In a more practical sense, it is "heteroskedasticity that does not depend on variables on which we have data".

So "conditional heteroskedasticity" would be "heteroskedasticity that does depend on..."

It is always a good thing to enquire about the philosophical meaning of a term, but sometimes, utilitarian considerations as regards the use of labels to communicate take over.

---

RESPONSE TO COMMENTS
Suppose that we are examining the output of a specific industry at a specific period of time and geographical area, having available a cross-sectional sample that contains data related to production (output, inputs). We are pretty sure that we have data on all inputs $\mathbf x$ and we consider the model

$$Q_i = F(\mathbf x_i) + \varepsilon_i$$

What does $\varepsilon_i$ represents? Many things, but (structural reasoning) based on our knowledge of the industry and the specific market, we are certain of this: customer preferences and demand for the product of this industry depends on whether the customer lives in cities or in the country. In particular, city dwellers have much more volatile preferences, chasing the latest fashion etc, while country folks have much more stable preferences. Each firm has a mixed customer base, both city and country. This means that we expect (structural reasoning) the error term to differ depending on the customer mix of each firm (observation).

A way to model this knowledge is to postulate a classical zero-mean, independent, homoskedastic etc, disturbance $u_i$ that represents the error term when the customer base is city-only, then consider a variable $m_i \in [0,1]$ that reflects the proportion of city-customers for each firm, and set

$$\varepsilon_i = m_iu_i.$$

The variable $m_i$ is at least party deterministic, in the sense that it is directly influenced by the marketing/management decisions of each firm. Assume also that whatever part of $m_i$ is stochastic, is not statistically associated with the inputs.

SITUATION A : Unconditional Heteroskedasticity (effectively)

Suppose that your sample does not contain data on $m_i$. In this case, the regression error is conditionally heteroskedastic given $m_i$, but you cannot do anything about it. Moreover, because $m_i$ is deterministic, you end up with an unconditionally heteroskedastic error term.

SITUATION B : Conditional heteroskedasticity (implementable)

Suppose now that you have a data series on the $m_i$ variable. Then you can effectively condition on $m_i$. For example, in an OLS estimation with a linear regression function setting we will get

$${\rm Var}(\hat \beta \mid X, m) = \sigma^2_u(X'X)^{-1}[X'{\rm diag}\{m_i^2\}X](X'X)^{-1}$$

SITUATION C: Clustered Standard Errors (heir of "group-wise heteroskedasticity")

Suppose now that you do not have data on $m_i$, but instead you have a classification variable that meta-classifies the observations in three groups according to customer mix: "more country-oriented", "more city-oriented", "balanced". Then you can do something with this variable and implement a "clustered standard errors" estimation based on it, to at least capture to a degree the variability that comes from the customer mix.