Persistence and stationarity together in volatility analysis

Question

I am trying to analyse a time series. I want to get only quantitative results (so, I'm excluding things like "looking at this plot we can note..." or "as you can see in the chart ...").

In my job, I analyse stationarity and persistence. First, I run ADF test and get "stationary" or "non-stationary" as results. Then, I need to work on persistence. To do so, I use ACF.

My question is: suppose I got "non-stationary" time series. Is it right to run ACF on it (without differencing)? I would like to comment upon stationarity and persistency without having to differentiate (so, just run tests on the original data and getting "answers" like "strong positive persistence", "weak negative persistence", ...).

I am writing here my question since I am working on close prices, returns and volatility. The time series are about them and I am trying to see if there are "better" time series than others (looking at general time series characteristics as non-stationarity...).

Thanks to who will even just read my question.

Answer 1

ADF tests for a unit root. Autocorrelation function of a unit root process does not make sense. For example let

$$y_{t+1}=y_t+\epsilon_{t+1}$$

Here $\epsilon_t$ is i.i.d white noise. Then the one period autocovariance is

$$Cov(y_{t+1},y_{t})=Cov(y_t+\epsilon_{t+1},y_{t})=Var(y_t)$$

For a unit root process $Var(y_t) \rightarrow \infty$ as $t\rightarrow \infty$. This autocovariance is hence not well defined and the sample autocorrelations grow as the sample length increases . If your data features a unit root you should not look at autocorrelations.

Note that prices are rather non-stationary because of a time trend. Here it would be equally wrong to look at autocorrelation functions. But you can solve most of these issues by taking differences. But all in all: only look at autocorrelations if your variable is stationary.