Heteroskedasticity - Value-at-Risk: Theory and Practice

4.6.6 Heteroskedasticity

A stochastic process X is homoskedastic if unconditional covariance matrices ^tΣ of terms ^tX are constant. It is heteroskedastic if they are not constant. A process is conditionally homoskedastic if conditional covariance matrices ^t|t–1Σ for terms ^tX are constant. It is 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.

Exhibit 4.8: 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

4.11

Consider the process Y, which we described earlier—all terms ^tY are equal and are unconditionally U(0,1); two realizations are indicated in Exhibit 4.7.

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

Solution

4.12

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

4.13

Suppose x is a time series with ^–1x = 100.

Calculate ⁰x if ⁰z_simple = 0.05.
Calculate ⁰x if ⁰z_log = 0.05.

Solution

4.14

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.

Exhibit 4.9: Values are indicated for a time series x. The table is to be completed for Exercise 4.14.

Solution

4.15

For a strictly positive (^tx > 0 for all t) univariate time series x, what is the range of values possible for:

the simple returns ^tz_simple of x?
the log returns ^tz_log of x?

Solution

4.16

Treating ^tz_log as a function of ^tx, construct a first-order Taylor polynomial for ^tz_log. Do so about the point ^t^–1x. Simplifying the result, what do you obtain?
Solution