ARMA model selection involves selecting a number of autoregressive lags and moving average lags. Meaningfully estimating the coefficients (i.e. without overfitting) requires the length of the series to be far greater than the number of parameters. If the series is quite short, say 10 time points, then a model that considers all 10 lags to predict the next observation is out of the question (let's ignore Bayesian approaches for the moment).
However, suppose that instead of a single series I have 1000 series, and suppose that each of these is generated independently but from the same underlying random process. If all I care about are the AR coefficients, I could fit a linear regression of the final observation on all of the previous ones -- each series is a single row of a linear model design matrix -- right? Is there a standard algorithm/software that similarly extends the fitting process for the MA terms to incorporate multiple series?