ARMA Estimation Efficiency of the Mean Usually in books related to ARMA Time Series, it is assumed that the series is 0 mean.
If not, the recommended standard procedure is to simply subtract the sample mean from the series and continue business as usual.
My question is whether there is a difference in estimation efficiency of ARMA parameters and the mean parameter when estimated altogether with MLE procedure vs first subtracting the sample mean and then estimating only ARMA parameters with MLE procedure. 
 A: Provided that it is a not a rule to subtract to sample mean from the series before fitting a model (de-meaning is possible but not a gold standard in all applications), then we could say the following. 
Theoretically, you are perfectly right: all the parameters of MLE should be estimated altogether. So, in this case, also the sample mean, which is indeed the sample estimate of the unconditional mean. However, the real problem for the efficiency of the MLE comes when you make the WRONG assumption about the true value of the parameters that you have pre-estimated. In this case, if you have enough data, the estimation of the sample mean can be considered reliable as a proxy for the population mean. If this assumption is correct and you have correctly specified the model, then you should have no relevant loss of efficiency. Instead it is difficult to pre-estimate reliably other information (think about the shape of the distribution of innovations for example). That is a different pair of shoes, very “Error-prone”.
