This is the final post I have planned in the autocorrelation series. I’ll introduce a new concept called partial autocorrelation and show a couple examples of the concept in action. We’ll learn some important things about the stock market.

Without getting into the math, here’s a couple definitions you should commit to memory:

**Autocorrelation (“ACF”)**measures the correlation between observations in a time series and observations a fixed unit of time away.**Partial autocorrelation (“PACF”)**measures the correlation between observations in a time series and observations a fixed unit of time away while controlling for indirect effects.

For example, here’s both ACF and PACF for monthly U.S. unemployment.

This is a common pattern you’ll see in the field often: a slow decay of ACF, and a spike in PACF. Starting at a lag of two months, PACF is basically zero.

This means that January’s unemployment influences March’s unemployment, but only because January influences February and February influences March. If unemployment were high in January and low in February, we would predict March’s unemployment to also be low.

In the Box-Jenkins methodology, ACF and PACF plots like this one are used to determine the correct model. Applying this step to a preexisting model will address seasonal patterns in residuals.

A few notes: Continue reading →