# Log return of sp500. Stationary vs strictly stationary

By first glance of this time series; will you say it is stationary?

I can easily see some "seasonality" which means that this is not strictly stationary since the distribution will not be the same; higher variance around 1987 and 2008. But is it weakly stationary?

The expected value at any timepoint will be zero, so I will say yes it is weakly stationary. Am I right?

• I don't think "seasonality" is the right term for what I think you're referring to (the patches of low and high variance). Also, how do get the expected value being zero (which implies a return of 1 on the original scale, which I presume means no actual return from holding those shares, which seems unlikely) - just by eyeballing it? For any of these things I think you need additional analysis not just this chart (although it is a start). – Peter Ellis Jan 28 '18 at 18:35
• Constant mean and constant variance (i.e. mean and variance that do not vary with time) are the requirements to define a series as weakly stationary. However, as @Peter points out, you need to run some statistical tests that determine whether or not this series is, in fact, weakly stationary. One critical assumption of weakly stationary series is that their ACF's should rapidly tend towards zero. – rmrouse88 Jan 28 '18 at 18:52
• One way to start with checking the assumption would be a unit root test (like Dickey Fuller), whose null is that the data are not weakly stationary and assesses the fit of the data against random walk models (since random walk models assume unit root). If you can reject the D-F null, then you may be able to apply some AR/MA/ARMA models to the data. You can use appropriate identification techniques for these model orders that depend on your model of choice (PACF for AR, ACF for MA, EACF for ARMA). – rmrouse88 Jan 28 '18 at 19:00
• Also, constant mean across time does not have to mean zero mean across time; although it could sometimes be the case. t-tests of the sample mean are not going to help you validate stationarity, because they are not telling you whether the mean is changing with time, just the expected value of the data across all time. – rmrouse88 Jan 28 '18 at 19:17

The commenters are correct in saying that various tests allow you to check for stationarity or not when in doubt.

For anyone looking into this question, there's an enlightening discussion around your cross-post in the Quantitative Finance section.

And I'll repeat my QF answer here too:

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They probably can be modelled using a weakly stationary process.

To quote Section 1.2.1 from these lecture notes:

[Asset] returns [...] typically fluctuate around a constant level, suggesting a constant mean over time. [...] In fact most asset returns can be modeled as a stochastic process with at least time-invariant first two moments.

Mathematically, a time series $\{ Y_t \}$ is weakly stationary if, for all time indices $k,s,t$

1. $\text{E}[Y_s] = E[Y_t]$, i.e. the first moment (the mean) is constant
2. $\text{Cov}[Y_t, Y_{t+k}] = \text{Cov}[Y_s,Y_{s+k}]$, i.e. the second moment is constant

From a visual inspection of your series of asset returns,

1. the mean/first moment does indeed appear to be constant
2. the series clearly exhibits the phenomenon of volatility clustering, implying that it has a non-constant "conditional volatility" / exhibits heteroskedasticity - however we cannot make any visual judgements as to the behaviour of the unconditional volatility.