# Weak stationarity vs strict stationarity

I was wondering why in practical terms (aka when modeling and so on), why it's more useful to work a weakly stationary time series than a strict stationary one?

Cheers

• It is less restrictive but it may not be the most appropriate depending on your series. – Michael R. Chernick Apr 25 '17 at 15:32

With the stochastic process model for a time series, there is usually just one sample path or realization of the process to work with, and a weakly stationary model is much easier to fit than a a strictly stationary model. Remember that all we need to fit a weakly stationary model is the value of the (constant) mean, and this is easily estimated as $$E[X_i] \approx\mu = \frac 1n \sum_{k=0}^{n-1} x_k$$ from the single available sample path $x_0, x_1, \cdots, x_{n-1}$, Similarly, the autocorrelation function $R_X(\ell) = E[X_iX_{i+\ell}]$ can be estimated as $$R_X(\ell) = E[X_iX_{i+\ell}] \approx \frac 1n \sum_{k=0}^{n-1-\ell} x_kx_{k+\ell}, ~ \ell = 0, 1, 2, \ldots$$ with the caveat that the estimate is likely to be suspect for values of $\ell$ close to $n-1$. In contrast, the fitting of a strictly stationary model requires estimation of the distribution of the $X_i$, the estimation of the joint distribution of $X_i$ and $X_{i+\ell}$ for each $\ell$, the joint distribution of $X_i$, $X_{i+\ell}$ and $X_{i+\ell + m}$, and so on, all of which estimations should be viewed with a great deal of skepticism.
• @CagdasOzgenc With a single sample path, we can estimate the distribution by, for example, ooking at a histogram of $x_0, x_1, \ldots$ but this very act presupposes that the process is stationary. If we had several sample paths $\mathbf x^{(i)}$, we could look at the histogram of $x_i^{(1)}, x_i^{(2)}, x_i^{(3)},\ldots$ to get an empirical distribution for $X_i$ and compare it to the empirical distribution for $X_k$ to see whether the assumption of stationarity is even reasonably defendable, and if so, we can proceed. With a single sample path, this is not possible. – Dilip Sarwate Apr 25 '17 at 18:25