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Given a unique time series $TS = (t_{0},t_{1},.....,t_{n})$ and a unique triplet $(p,q,d)$, is the corresponding ARIMA$(p,q,d)$ model unique or are there multiple possible ARIMA models?

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No, it isn't unique, but the differences are usually minor.

First of all, ARIMA models are sometimes defined as having an intercept parameter, sometimes not.

And even if we are clear about whether there is an intercept or not, there are different valid ways of estimating the parameters. You can run full maximum likelihood (and still need to maximize numerically, so in ill-posed cases, you can end up with different final parameter estimates), or minimize conditional sums of squares (CSS), or use heuristics to estimate the starting values and then run CSS, and so forth. Here is an earlier answer on this.

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