Predicting a value from the previous value

In a time series without trend, positive autocorrelation means that values tend to be followed by others of similar size.

We can try to predict yt from its previous value by fitting a straight line by least squares to this scatterplot,

yt  =  b0  +  b1 yt−1

This is a type of auto-regressive model, and is usually called an AR(1) model.

Forecasting further into the future

The model only directly predicts one time period into the future,

t  =  b0  +  b1 yt − 1

but we can predict further into the future using:

t + k  =  b0  +  b1 t + k −1