Autocovariance, Autocorrelation and Autocorrelation coefficient

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.

• 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. Commented Jan 29, 2017 at 22:47
• Yea, that was a typo! Thanks for pointing it out. I just added the self-study tag! Commented Jan 30, 2017 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.

• 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.
– whuber
Commented Jan 30, 2017 at 15:28
• I'll try to improve it. Commented Jan 30, 2017 at 16:21
• 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.
– whuber
Commented Jan 30, 2017 at 16:43
• That's correct. I'll work it in. Commented Jan 30, 2017 at 16:56
• alright i added the appropriate caveat and linked wikipedia Commented Jan 31, 2017 at 9:25

Disclaimer: This is meant to be an intuitive explanation without going into mathematical details, given that the original question seemed fairly elementary.

To understand the difference between auto-covariance and auto-correlation, it helps to understand the difference between covariance and correlation.

Covariance is a measure of how much two paired variables v1 and v2 vary in the same way/direction. It is positive when v1 is above its mean at the same time that v2 is above its mean and/or v1 is below its mean at the same time that v2 is below its mean. It is negative when the opposite happens, i.e. whenever v1 is above its mean, v2 is below its mean and vice versa.

It is important to note that covariance only gives you an idea of the direction of the relationship but its hard to interpret the magnitude of the relation, since it is very dependent on the units that are used. (For instance, if your variables are in cm and then you transform it into inches, the absolute value of the covariance will be very different.

Correlation addresses this by scaling the covariance, and putting it into the interval between -1 and 1. Correlation of 1 means your two variables are perfectly positively correlated, -1 means they are perfectly negatively correlated (whenever v1 goes up, v2 goes down), 0 means that there is no correlation at all.

Now, for time-series, the "auto-" prefix indicates that you calculate the covariance and correlation between one variable at time t1 and the the same variable at a later time t1+k. E.g. for k=1, you calculate the covariance and correlation between one time and the next time. The k indicates the difference or lag between the time points, e.g. if you had monthly data, with k=12 you would inspect the relationship between the variables in the first year and the following year.

The auto-correlation coefficients then give you the auto-correlation for each lag k. Comparing the coefficients for different lags can tell you if there is seasonality in the data - e.g. temperatures tend to change with seasons, so you would expect a higher autocorrelation coefficient at around k=12 for monthly data.