Autocorrelation and lags definition

Question

I'm looking for a way to define lags in autocorrelation graphs / corellograms. I understand how to interpret such a graph but can't define a lag in a clear concise manner. Does a definition for this concept already exist? If not, what would be a good way to construct it?

Answer 1

You have a time series variable, $X$, with realizations at discrete points in time indexed by $t$, so the time series data is $x_1,x_2,\ldots,x_T$.

An autocorrelation asks to what degree realizations with a certain time difference - a lag - co-move. For example, the autocorrelation with a lag of 1 would compute the correlation of the following pairs: $(x_1,x_2)$; $(x_2,x_3)$; $(x_3,x_4)$, and so on. Formally, the lag $l$ defines $Corr(x_t,x_{t+l})$. Informally, the lag defines the time-distance for which you compare realizations of the time series.

An autocorrelation graph usually plots these correlations for different values of $l$, starting from $1$ to, say, $10$ (depends on context).

For example, $x_t$ could denote the price of Microsoft at the end of day $t$, and the autocorrelation with lag $l=1$ tells you how much the stock price co-moves from day to day. The lag $l=365$ tells you how much the stock price today co-moves with the stock price next year.