A Fast Algorithm for the Exact Likelihood of Stationary and Partially Nonstationary Vector Autoregressive-Moving Average Processes
提出一种数值高效的算法，用于计算平稳和部分非平稳向量自回归移动平均过程的精确似然函数，避免因过度差分导致的非可逆性和参数可识别性问题。
An expression for the likelihood function of a stationary vector autoregressive-moving average process is developed. The expression is very efficient numerically and applies to any stationary but not necessarily invertible model. In particular, when the multivariate process is autoregressive, the exact likelihood can be evaluated with a small number of operations depending on the order of the autoregressive operator and the process dimension, but not on the size of the observed series. The expression also provides an efficient method for the evaluation of the exact likelihood of a partially nonstationary vector autoregressive-moving average process, for which the determinant of the autoregressive operator has at least one unit root and the remaining roots are outside the unit circle. This method does not require differencing the series, so that complications caused by over-differencing the series, such as noninvertibility and parameter identifiability problems, are avoided. The results for autoregressive models are also applied to testing the stationarity and invertibility of any autoregressive-moving average model with given parameter values.