If I have the autocovariance function $\gamma_\tau$ (numerically over a given set of lags $\tau = 0 \ldots n - 1$) of a stationary linear stochastic process, is there an efficient way to determine the coefficients $(\phi, \theta)$ of a matching ARMA process for fixed order $(p,q)$?

I know that there are several ways to estimate ARMA coefficients from data without first estimating the ACF. However, in my application the ACF is already given, and I cannot go back to the underlying data.

I'm aware that one can calculate the ACF for given ARMA coefficients, e.g. following Brockwell and Davis (2002, Chapter 3). I could then combine this forward model with a distance function and let a numeric optimizer do the inversion. That approach works, but is too slow for my application.

So I'm wondering whether there is a more direct way to fit an ARMA model to a given ACF, similar to Yule–Walker equations for AR models?


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