A stochastic process X is homoskedastic if unconditional covariance matrices t Σ 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.
Financial markets experience random periods of high and low volatility. For this reason, conditionally heteroskedastic processes are often used in financial modeling.
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?
Explain in your own words the difference between covariance stationarity and homoskedasticity. Does covariance stationarity imply homoskedasticity? Does covariance stationarity imply conditional homoskedasticity?
Suppose x is a time series with –1x = 100.
- Calculate 0x if 0zsimple = 0.05.
- Calculate 0x if 0zlog = 0.05.
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.
For a strictly positive (tx > 0 for all t) univariate time series x, what is the range of values possible for:
- the simple returns tzsimple of x?
- the log returns tzlog of x?
Treating tzlog as a function of tx, construct a first-order Taylor polynomial for tzlog. Do so about the point t–1x. Simplifying the result, what do you obtain?