Generating random AR() and MA() Toeplitz correlation matrices

Methods are proposed for generating random ( p + 1 ) × ( p + 1 ) Toeplitz correlation matrices that are consistent with a causal AR( p ) Gaussian time series model. The main idea is to first specify distributions for the partial autocorrelations that are algebraically independent and take values in ( − 1 , 1 ) , and then map to the Toeplitz matrix. Similarly, starting with pseudo-partial autocorrelations, methods are proposed for generating ( q + 1 ) × ( q + 1 ) Toeplitz correlation matrices that are consistent with an invertible MA( q ) Gaussian time series model. The density can be uniform or non-uniform over the space of autocorrelations up to lag p or q , or over the space of autoregressive or moving average coefficients, by making appropriate choices for the densities of the (pseudo)-partial autocorrelations. Important intermediate steps are the derivations of the Jacobians of the mappings between the (pseudo)-partial autocorrelations, autocorrelations and autoregressive/moving average coefficients. The random generating methods are useful for models with a structured Toeplitz matrix as a parameter.