# How are ergodicity and “weak dependence” related?

I understand that weak dependence is a broad concept, the definition I am referring to is the one Wooldridge (2013) uses as an assumption that has to be fulfilled (amongst other assumptions) so that the estimators in a time series linear regression model are asymptotically consistent. That is:

A process $\{X_t\}$ is weakly dependent if the correlation between $\{X_t\}$ and $\{X_{t+h}\}$ goes to zero relatively quickly as $h\to \infty$.

Instead of saying that stochastic processes have to be covariance stationary and weakly dependent, other authors say they have to be covariance stationary and ergodic.

How are ergodicity and weak dependence related? Are they interchangable in the context of time series OLS assumptions?

Thank you.

The concepts are not interchangeable. Ergodicity deals with studying the systems where different realizations of the process are not available. For instance, in coin toss we could reasonably argue that we could generate any number of realizations of the sequence of coin tosses. We'll toss 10 coins 1000 times, and it gives us the 1000 samples of the process. So, we could study the statistical properties of the 10 coin tosses. We could a few more 10 coin tosses and increase the sample, improve the estimates etc.

This is not always possible. In many cases we cannot generate many realizations of the process, but we can observe one realization for a long long time. So, the ergodicity suggests that we could replace many realization of the process with a very long observation over time. That over time we can obtain the same estimates of the parameters of the process as if we obtained many realizations.

Weak exogeneity (independence) deals with a one process, one time series. We're studying this process that is going on. We may need to forecast it in future etc. It's nice if the process has this property where the correlation doesn't stick for too long.

• But why is it, that sometimes authors state that stochastic processes have to be ergodic, while others state they "only" have to be weak dependent in the context of OLS regression? – shenflow Jan 28 '18 at 16:03
• Can you bring examples? Usually when you are interested in Time series you talk about exogeneity – Aksakal Jan 28 '18 at 16:06

I had the same question, and found these lecture notes. Page 8 states that a mixing process is ergodic (called Theorem 7) and that a mixing process is also called weakly dependent.

In other words, a weakly dependent process is ergodic.

It is my understanding that we require ergodicity to estimate the asymptotic covariance matrix of serially correlated series, such that assuming either ergodicity or weak dependence is sufficient.