Time-series Auto-Covariance vs. Stochastic Process Auto-Covariance

Question

My background is more on the Stochastic processes side, and I am new to Time series analysis. I would like to ask about estimating a time-series auto-covariance:

$$ \lambda(u):=\frac{1}{T-u}\sum_{t=1}^{T-u}(Y_{t+u}-\bar{Y})(Y_{t}-\bar{Y}) $$

When I think of the covariance of Standard Brownian motion $W(t)$ with itself, i.e. $Cov(W_s,W_t)=min(s,t)$, the way I interpret the covariance is as follows: Since $\mathbb{E}[W_s|W_0]=\mathbb{E}[W_t|W_0]=0$, the Covariance is a measure of how "often" one would "expect" a specific Brownian motion path at time $s$ to be on the same side of the x-axis as as the same Brownian motion path at time t.

It's perhaps easier to think of correlation rather than covariance, since $Corr(W_s,W_t)=\frac{min(s,t)}{\sqrt(s) \sqrt(t)}$: with the correlation, one can see that the closer $s$ and $t$ are together, the closer the Corr should get to 1, as indeed one would expect intuitively.

The main point here is that at each time $s$ and $t$, the Brownian motion will have a distribution of paths: so if I were to "estimate" the covariance from sampling, I'd want to simulate many paths (or observe many paths), and then I would fix $t$ and $s=t-h$ ($h$ can be negative), and I would compute:

$$ \lambda(s,t):=\frac{1}{N}\sum_{i=1}^N(W_{i,t}-\bar{W_i})(W_{i,t-h}-\bar{W_i}) $$

For each Brownian path $i$.

With the time-series approach, it seems to be the case that we "generate" just one path (or observe just one path) and then estimate the auto-covariance from just that one path by shifting throught time.

Hopefully I am making my point clear: my question is on the intuitive interpretation of the estimation methods.

Answer 1

First, let us be strict about what we are discussing. Auto covariance is defined here. Let me assume $\bar{Y}_t \equiv 0$ for all $t$, just to make life easier.

It looks like you are missing an implicit assumption in the calculation of the auto covariance. For the observed time series, stationarity is implicitly assumed. Otherwise, writing $\lambda(u)$ does not make much sense - it has to depend on $t$!!! Under the assumption of stationarity, the quantity $\lambda(u) = \mathbb{E}[Y_{t+u} Y_t]$ is well defined and does not depend on $t$. Hence, one can estimate it by averaging lags at different time indices $t$ as you have suggested (again, Wikipedia helps here).

BM (contditioned on $W_0 = 0$) is a completely different story. It is not stationary and cannot be made statinary! Thus, an autocovariance has to be calculated by averaging over different realizations.

Answer 2

Having given this some thought (and since no one has answered the question yet), I offer the following intuitive explanation (but I look forward to more answers and comments here):

(i) Time-Series: it would appear that we use time series predominantly for phenomena that only ever offer us "one observation path": whether that be an earthquake, historical rain-falls, or annual milk-consumption in a specific region, etc.

(ii) Brownian Motion: it would appear that we use Brownian motion (and other similar stochastic processes) to simulate predominantly processes that can be performed in a lab in a controlled experiment: and therefore we assume that these can be repeated many times with "the same" conditions.

Therefore the auto-covariance and auto-correlation functions have a slightly different interpretation.

In a time-series model, it would appear that the auto-covariance and auto-correlation tell us how "often" the various observations at different points in time lie on the same side of the series "mean" as other, previous observations with a specific lag. The full population in this case is a time-series stretching to infinity over time.

In a Brownian-motion model, as alluded to in my question, the population is the same experiment or phenomena performed infinitely many times under the same conditions. The auto-covariance and auto-correlation then tell us how "often" we would expect the various observations at specific points in time to lie on the same side of the "mean" as other observations at other specific points in time.

Ps: I do look forward to more comments, answers and general discussion that might appear here. For example, the fascinating thing is that we use both time-series as well as Brownian motion, to simulate stock-prices.