Partial Autocorrelation

The partial autocorrelation coefficient

The partial autocorrelation coefficient $\lambda_j$ of a time series $y_t$ measures the linear relationship between observations that are $j$ periods apart, once the effect of all observations in between has been accounted for. In other words, $\lambda_j$ tells us how strong the impact that $y_{t-j}$ has on $y_t$ after subtracting the impact that observations $y_{t-j+1}, y_{t-j+2}, ..., y_{t-1}$ might also have on $y_t$.

The partial autocorrelation coefficient $\lambda_j$ can be extracted from a linear model of $y_t$ with $j$ lagged values. For example, imagine that we want to get the partial autocorrelation coefficient of order 3, $\lambda_3$. The following model would determine the coefficient: