# Time series analysis: since volatility depends on time, why are returns stationary?

I run Dickey Fuller test in order to know if stock returns are stationary.
I get that no matter which stock I take, his return is stationary.
I don’t know why I get this result since it is clear that volatility depends on time (hence, returns are not stationary since their variance depends on time).
I’d like to get both a mathematical and intuitive answer.

I think your problem is that you confuse the UNconditional variance and the conditional variance. Indeed, you can have a time-varying conditional volatility but a constant unconditional variance.

First, I illustrate what Dickey-Fuller does and why it is a very specific test. Second, I explain why you can have a time-varying conditional volatility but a constant unconditional variance.

Firstly, consider the framework :

$y_{t}=\rho y_{t-1}+\epsilon_{t}$ where $\epsilon_t\sim_{iid}\mathcal{N}(0,\sigma^2)$ for $t\in[1,T]$.

If you compute the expectation and (unconditional) variance of $y_t$, you get

$\mathbb{E}[y_t]=\rho^{t-1}y_{1}$ and $\mathbb{V}[y_t]=\sigma^2\sum_{l=0}^{t-1}\rho^{2l}$

Dickey-Fuller test performs $H0:”\rho=1″$ vs $H1:”\rho<1″$.

If $\rho=1$, then $\mathbb{V}[y_t]=t\sigma^2$, what means the unconditional variance increases linearly with time.

If it is inferior to 1, then the unconditional variance tends to be constant with time due to the geometric series of its expression. If $\rho<1$ and $t\rightarrow\infty$,$\mathbb{V}[y_t]\rightarrow\frac{\sigma^2}{1-\rho^2}<+\infty$ what implies it is covariance-stationary.

That is why, if DF test rejects H0, you cannot accept that the unconditional variance increases linearly with time when compared with the covariance-stationnary hypothesis, but it just concerns a specific form of nonstationarity.

Second, consider the following process (ARCH(1)):

$y_{t}=\sigma_t\epsilon_t$
with $\sigma_t^2=\alpha+\beta y_{t-1}^2$

where $\alpha>0$ and $0<\beta< 1$, $\epsilon_t\sim_{iid}\mathcal{N}(0,1)$, $\sigma_t$ being independent of $\epsilon_t$.

Here, you can see the volatility parameter $\sigma_t$ depends on time. However, this parameter is the variance of $y_t$ conditionally to the information we get at time $t$. Actually, the UNconditional variance of $y_t$ is:

$\mathbb{V}[y_t]=\mathbb{E}[y_t^{2}]=\mathbb{E}[\sigma_t^2]=\alpha+\beta \mathbb{E}[y_{t-1}^2]$

If $y_t$ is covariance-stationary, $\mathbb{V}[y_t]=\mathbb{E}[y_t^{2}]=\mathbb{E}[y_{t-1}^{2}]$ what implies : $\mathbb{V}[y_t]=\frac{\alpha}{1-\beta}<+\infty$

So, $y_t$ can be covariance-stationary while displaying locally some clusters of volatility.

To think further, you can go to see this paper proposing a framework to test if the UNconditional variance is constant or not: Sansó, A., Aragó, V. and Carrion-i-Silvestre, J. Ll. (2004): “Testing for Changes in the Unconditional Variance of Financial Time Series”.