Solved – Weak stationarity vs strict stationarity

Question

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?

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Answer 1

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.