  Autoregressive moving average (ARMA) models are frequently used to analyze
time series data. Despite the popularity of these models, likelihood-based
inference for ARMA models has subtleties that have been previously identified
but continue to cause difficulties in widely used data analysis strategies. We
provide a summary of parameter estimation via maximum likelihood and discuss
common pitfalls that may lead to sub-optimal parameter estimates. We propose a
random initialization algorithm for parameter estimation that frequently yields
higher likelihoods than traditional maximum likelihood estimation procedures.
We then investigate the parameter uncertainty of maximum likelihood estimates,
and propose the use of profile confidence intervals as a superior alternative
to intervals derived from the Fisher's information matrix. Through a series of
simulation studies, we demonstrate the efficacy of our proposed algorithm and
the improved nominal coverage of profile confidence intervals compared to the
normal approximation based on Fisher's Information.
