Sample Partial Autocorrelation Function of a Multivariate Time Series

The choice of a matrix square root in order to define a correlation coefficient is crucial for the notion of partial autocorrelation function (PACF) for a multivariate time series. Here this topic is revisited and, introducing a new matrix link coefficient between two random vectors, a general framework for estimating the PACF is given. This leads to new autoregressive estimation methods based on sample estimators of the partial autocorrelation coefficients. Moreover, some generalizations of the scalar Burg's technique fit in this framework making the comparison of all these methods easier.