# What is the difference between an AR process and autocorrelation?

Or is it maybe the same thing? I see that autocorrelation is when Yt is correlated with its lag Yt-1. But isn't that essentially what an AR process (say AR(1)) is? We are assuming that there IS correlation between its previous time period since we might see a directional trend from its initial data, right?

• An autocorrelation is a function of a pair of times. An AR process is a stochastic process: thus, they aren't even remotely the same kinds of mathematical objects. On the face of it, then, your question makes no mathematical sense. Could you clarify what you're trying to ask? – whuber May 12 '19 at 19:53

Think of it this way:

If the index $$t$$ denotes time, then a stochastic process is simply a collection of random variables indexed by time.

There are stochastic processes (e.g., AR(1)) for which we can explicitly indicate how the value of $$Y$$ at time t (that is, $$Y_t$$) depends on values of Y at previous times. For example, an AR(1) process with mean zero stipulates that $$Y_t = \phi*Y_{t-1}+ \epsilon_t$$, where $$\epsilon_t$$ is a white noise process with zero mean and constant variance $$\sigma^2$$.

The autocorrelation function provides a measure of similarity between the values of $$Y$$ at times $$t$$ and $$s$$ by computing the correlation between $$Y_t$$ and $$Y_s$$, namely $$Cor(Y_t, Y_s)$$. (For a weekly stationary stochastic process, this correlation only depends on how apart in time $$t$$ and $$s$$ are from each other, it does not depend on the actual values of $$t$$ and $$s$$.)

For the above AR(1) process, the correlation between $$Y_t$$ and $$Y_{t-1}$$ is given by:

$$Cor(Y_t, Y_{t-1}) = Cor( \phi*Y_{t-1}+ \epsilon_t, Y_{t-1}) = \phi$$

$$Cor(Y_t, Y_{t-2}) = \phi^2$$

and, more generally,

$$Cor(Y_t, Y_{t-k}) = \phi^k$$, where $$k > 2$$ is an integer. (See https://www.mathstat.dal.ca/~stat5390/Section_3_ACF.pdf.)

• Your opening characterization of stochastic processes seems quite at odds with standard conceptions of them. Did you perhaps mean to describe AR models? – whuber May 12 '19 at 21:17
• Thanks, @whuber! I made some edits to the original answer to reflect your concerns - let me know if they look fine to you. I want to keep things simple. – Isabella Ghement May 12 '19 at 21:24