###### 4.6.6 Heteroskedasticity

A stochastic process ** X** is

**homoskedastic**if unconditional covariance matrices

^{t }**Σ**of terms

*are constant. It is*

^{t}**X****heteroskedastic**if they are not constant. A process is

**conditionally homoskedastic**if conditional covariance matrices

^{t|t–1}

**Σ**for terms

*are constant. It is*

^{t}**X****conditionally heteroskedastic**if they are not.

These distinctions are easy to grasp intuitively with a picture. Exhibit 4.8 depicts realizations for two processes. The realization on the left exhibits constant conditional standard deviations consistent with homoskedasticity and conditional homoskedasticity. The one on the right exhibits nonconstant standard deviations consistent with heteroskedasticity or conditional heteroskedasticity.

Financial markets experience random periods of high and low volatility. For this reason, conditionally heteroskedastic processes are often used in financial modeling.

###### Exercises

Consider the process ** Y**, which we described earlier—all terms

*are equal and are unconditionally*

^{t}Y*U*(0,1); two realizations are indicated in Exhibit 4.7.

- Is
stationary?*Y* - Is it unconditionally homoskedastic?
- Is it conditionally homoskedastic?
- What is the unconditional standard deviation
^{1}σ? - What is the conditional standard deviation
^{1|0}σ?

Explain in your own words the difference between covariance stationarity and homoskedasticity. Does covariance stationarity imply homoskedasticity? Does covariance stationarity imply conditional homoskedasticity?

Solution

Suppose ** x** is a time series with

^{–1}

*x*= 100.

- Calculate
^{0}*x*if^{0}*z*= 0.05._{simple} - Calculate
^{0}*x*if^{0}*z*= 0.05._{log}

Exhibit 4.9 indicates values for a time series ** x**. Complete the table by calculating the corresponding differences, simple returns, and log returns for the time series.

*. The table is to be completed for Exercise 4.14.*

**x**For a strictly positive (* ^{t}x* > 0 for all

*t*) univariate time series

**, what is the range of values possible for:**

*x*- the simple returns
of^{t}z_{simple}?*x* - the log returns
of^{t}z_{log}?*x*

Treating * ^{t}z_{log}* as a function of

*, construct a first-order Taylor polynomial for*

^{t}x*. Do so about the point*

^{t}z_{log}

^{t}^{–1}

*x*. Simplifying the result, what do you obtain?

Solution