I am taking a course about time series this year, and this is a brand new topic for me. I just got my first assignment and I am confused by a couple of terminologies.

Q1. What is the difference between autocovariance, autocorrelation and autocorrelation coefficient? I tried to google it, but most of them don't really make sense to me.

Q2. The question asks to estimate lag-1 autocorrelation coefficient, but what is lag? Is it a variable?

Any help would be really appreciated!
We don't lecture slides or any additional materials for this course, so if you have any textbooks, papers or slides that are helpful, feel free to comment below.

  • $\begingroup$ Is Utocorrelation coefficient a typo? Should it be Autocorrelation coefficient? I can edit the pot for you? You also need to add the self study tag. $\endgroup$ Jan 29 '17 at 22:47
  • $\begingroup$ Yea, that was a typo! Thanks for pointing it out. I just added the self-study tag! $\endgroup$
    – user311290
    Jan 30 '17 at 14:26

Q1: Say you have observations over time on a variable $\{x_t\}, t=\{1,...,T\}$. If they are generated from a second-order stationary stochastic process (Click) you may apply the following techniques to find the first autocovariance and the first autocorrelation coefficient.

Calculate the covariance of observations $x_t, \forall t>1$ and $x_{t-1}$, this gives the first autocovariance. This generalizes: $Cov(x_t,x_{t-n})$ is the n-th autocovariance. If you divide the autocovariance by the variance of the $x_t$ you obtain the autocorrelation coefficient: $\rho_1=\frac{Cov(x_t,x_{t-1})}{Var(x_t)}$.

Autocorrelation is the property of the variable $x_t$ indicated by the autocorrelation coefficient, which tells you how much the realization $x_t$ depends on the last realization $x_{t-1}$. This naturally leads into the idea of an autoregressive process. For example an autoregressive process of order 1 is given by $x_t=\rho_1 x_{t-1}+\epsilon_t$, where $\epsilon_t$ is a standard-normal random variable. So, $x_{t-1}$ will contribute to $x_t$ according to the parameter $\rho_1$. Click

Q2: You can piece it together from the answer to Q1. The lag-operator shifts the period on a variable like $x_t$ to a previous period. For example, $L(x_t)=x_{t-1}$, where $L()$ is the one-period lag-operator. You can find the estimate of $\rho_1$ by running a linear regression of present values of your variable on their realizations one period back.

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    $\begingroup$ I find much of this discussion confusing, for several reasons. One is that you seem not to distinguish between data, random variables, and random variables at different times. Another is that $\rho_1 \ne 1$ for a random walk. $\endgroup$
    – whuber
    Jan 30 '17 at 15:28
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    $\begingroup$ I'll try to improve it. $\endgroup$ Jan 30 '17 at 16:21
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    $\begingroup$ Thank you. Let me be a little more specific about some concerns. At the outset you now distinguish between a (random?) "variable $x_t$" and time-series observations. (You don't seem to maintain this distinction later, though.) This already is potentially confusing, because $x_t$ is really a set of variables indexed by time $t$. Thus, when you get to describing autocovariance, you have implicitly assumed the series is (second order) stationary, entailing (by definition) $\operatorname{Cov}(x_t,x_{t-n})=\operatorname{Cov}(x_s,x_{s-n})$ for all $t,s,n$ where these expressions make sense. $\endgroup$
    – whuber
    Jan 30 '17 at 16:43
  • $\begingroup$ That's correct. I'll work it in. $\endgroup$ Jan 30 '17 at 16:56
  • $\begingroup$ alright i added the appropriate caveat and linked wikipedia $\endgroup$ Jan 31 '17 at 9:25

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