I am reviewing the fundamentals of time series modelling and came across the following question regarding the concept of stationarity:
A time series is strictly stationary if its unconditional distribution (all moments) is invariant in time.
Consider the following stationary time series $\{{X_t}\}_{t\in{Z}}$ and let us focus on the first two moments of the distribution, namely:
- $E[X_t]= constant$
- $var[X_t]= constant$
The time invariant unconditional moments does not imply that the conditional moments ($E[X_t|X_{t-1}]$ and $var[X_t|X_{t-1}]$) are time invariant as well. I understand that the conditional expectation is random, due the random nature of conditioning part of $X_{t-1}$. But, I am not sure about $var[X_t|X_{t-1}]$ .
If we show a typical path of time series generated by an $AR(1)$ model, we observe a time-varying conditional distribution. Notice that the One realized path has a time varying conditional mean.
Given that a time series is strictly stationary, what can we say about the conditional distribution?
How is it possible to determine that a time series is stationary based on a single realization path?
EDIT
I believe the conceptual issue I am facing relates to conditioning. Moreover, I think that the question may be answered by:
Law of total expectation: $E(x)=E[E(x|y)]$
Law of total variance: $var(x)=E[var(x|y)] + var[E(x|y)]$