A note on maximum likelihood estimation in the first-order Gaussian moving average model

The likelihoood function of the Gaussian MA(1) zero-mean can be expressed in terms of the variance of the process and the first-order autocorrelation or alternatively in terms of the variance of the unobservable independent normal random variables and the moving average coefficient. The relations between the maximum likelihood estimates of these alternatives pairs are explored. It is noted that in a (finite) sample the maximizing value of the autocorrelation may not correspond to a real value of the moving average coefficient.