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Vector autoregressions (VARs) have an associated order $p$; conditional onobservations at the preceding $p$ time points, the variable at time $t$ isconditionally independent of all the earlier history. Learning the order of themodel is therefore vital for its characterisation and subsequent use inforecasting. It is common to assume that a VAR is stationary. This prevents thepredictive variance of the process from increasing without bound as theforecast horizon increases and facilitates interpretation of the relationshipsbetween variables. A VAR is stable if and only if the roots of itscharacteristic equation lie outside the unit circle, constraining theautoregressive coefficient matrices to lie in the stationary region.Unfortunately, the geometry of the stationary region is very complicated whichimpedes specification of a prior. In this work, the autoregressive coefficientsare mapped to a set of transformed partial autocorrelation matrices which areunconstrained, allowing for straightforward prior specification, routinecomputational inference, and meaningful interpretation of the magnitude of theelements in the matrix. The multiplicative gamma process is used to build aprior for the unconstrained matrices, which encourages increasing shrinkage ofthe partial autocorrelation parameters as the lag increases. Identifying thelag beyond which the partial autocorrelations become equal to zero thendetermines the order of the process. Posterior inference is performed usingHamiltonian Monte Carlo via Stan. A truncation criterion is used to determinewhether a partial autocorrelation matrix has been effectively shrunk to zero.The value of the truncation threshold is motivated by classical theory on thesampling distribution of the partial autocorrelation function. The work isapplied to neural activity data in order to investigate ultradian rhythms inthe brain.

Bayesian inference on the order of stationary vector autoregressions