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

Methods are proposed for generating random (p+1)×(p+1)(p+1)×(p+1) Toeplitz correlation matrices that are consistent with a causal AR(pp) 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)(−1,1), and then map to the Toeplitz matrix. Similarly, starting with pseudo-partial autocorrelations, methods are proposed for generating (q+1)×(q+1)(q+1)×(q+1) Toeplitz correlation matrices that are consistent with an invertible MA(qq) Gaussian time series model. The density can be uniform or non-uniform over the space of autocorrelations up to lag pp or qq, 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.