I have a time series $x_t$ which may go through different phases of volatility. One example might be some stock that has high variance from 9 AM to 11 AM, low variance from 11 AM to 2 PM, and then high variance again afterwards. Is there a way to identify these different periods of variance?
I am thinking of taking a sliding window of length $L$, computing the variance in that window and running change point detection on that, but I think this requires me to know the distribution of the estimated variance. Another idea I had was to take the variance of the windows and try to fit a Markov chain to it, but I don't know beforehand how many states there should be.
Sorry, the motivating example is not just in finance. I would like to have a way to model "risk" in a time series, which might be risk associated with a financial asset. Another example is in wind energy production – energy produced from wind turbines is highly volatile and unpredictable, but it would be nice to have a learning algorithm that could identify what days of the month is variance of wind energy production high and when is it low
Best Answer
There is a lot of literature for testing the change in mean. If it is known that mean does not change, and you need to test the variance, you can convert the problem of testing for change in variance to the one of testing for change in mean with simple transformation.
Suppose your initial data is $X_i$, then define $Y_i=(X_i-\mu)^2$, where $\mu$ is the mean. Then the change in mean of $Y_i$, $EY_i=E(X_i-\mu)^2=Var(X_i)$ will be a change of variance in $X_i$.