How Does the Volatility of the Markets Change Through Time?

Using Autocorrelation Functions to Describe Non-Stationarity in Volatility with Self-Similarity

Photo by Dan Cristian Pădureț on Unsplash

In my previous article I exhibited how, year-by-year, the variance of market returns doesn’t seem to take on constant values. I showed that the observed sequences of variance ratios for the returns of the S&P 500 Index are incompatible with the idea that they represent sequential, but noisy and therefore different, measurements of the same quantity! That article is linked to below.

Let’s Talk About Heteroskedasticity

It’s true that the tests presented, which examine how one year’s estimates compare to the next one from 1928 to date, are somewhat arbitrary. Why one year, not two, or four, or six months? Why start on the 2nd. of January and end on the 31st. of December, perhaps some other choice of groupings would deliver a different result.

These are valid criticisms but, I don’t think, undermine the key result. If the variance were stationary it would not fail any of these tests, no matter how the time boundaries are arranged. These tests are sufficient to reject that hypothesis, even if they are inefficient with regards to pinning down the actual manner in which the variance changes through time. But, these tests are definitely not the best way to learn about the nature of the true hetereoskedasticity of returns. That requires a sharper pencil than such a crude “grouping” analytic.

How Can we Measure Changing Variance?

At the risk of being overly wordy, what does variance actually mean? The variance is the expected value of the squared deviation from the population mean. That is

for some random variable, x. The original definition can be replaced with a simpler alternative, that is easier to compute:

Measuring Variance

We know from the Law of Large Numbers, that the arithmetic mean of a quantity is a sample estimator for…