What is lag in a time series? I am curious about what a lagging time series is.
On investopedia, I saw an article that said that:
"Autocorrelation is degree of similarity between time series and a lagged version of itself over successive intervals." Someone please explain to me what "lagged" means, and why autocorrelation matters in relation to time series analysis. Does autocorrelation mean that the time series will perform like the past?
Thanks!
Edit: Thanks for everyone's answers, especially the 2 thumbs-up answer earlier. That was very helpful.
Now I am wondering why autocorrelation even matters. Sure a function may correlate with a shifted version of itself, but who says that that function will perform like that? Is it just through correlation? Why does this matter in context of autoregressive models, and how did we develop this autocorrelation, then ARM/ARIMA kinda thing to model time series in the first place. Who developed time series?
 A: Lag is essentially delay. Just as correlation shows how much two timeseries are similar, autocorrelation describes how similar the time series is with itself.
Consider a discrete sequence of values, for lag 1, you compare your time series with a lagged time series, in other words you shift the time series by 1 before comparing it with itself. Proceed doing this for the entire length of time series by shifting it by 1 every time. You now have autocorrelation function.
From the values of autocorrelation function, you can see how much it correlates with itself. For any time series you will have perfect correlation at lag/delay = 0, since you're comparing same values with each other. As you shift your time series you begin to see the correlation values decreasing. Note that if timeseries comprises of completely random values, you will only have correlation at lag=0, and no correlation everywhere else. In most of the datasets/time series this is not the case, as values tend to decrease over time, thus having some correlation at low lag values.
Now, consider a long periodic time series, for example outdoor temperature over a few years, sampled hourly. Your time series will correlate with itself on daily basis (day/night temperature drop) as well as yearly (summer/winter temperatures). Lets say your first datapoint is at 1 pm in mid summer. Lag=1 represents one hour. The autocorrelation function at lag=1 will experience a slight decrease in correlation. At lag=12 you will have the lowest correlation of the day, after what it will begin to increase. Move forward 6 month to 1 pm. Your time series is still somewhat correlated. Move lag to 6 months and 1 am. This might be your lowest correlation point in the time series. At lag of 12 months your timeseries is again close to the peak value.
You might have noticed from the previous example that autocorrelation function reveals frequency components of a time series. Indeed, it is closely tied to frequency domain, and is just fourier transform from becoming a power spectra.
For a random time series, autocorrelation function will show you how quickly it becomes unsimilar with itself, while periodic time series will show at what delay/lag values time series is similar with itself.
Hope this isn't as confusing as it seems.
