ARMA(1,1) Unique Solution Assume that we have a white noise process. When you try to fit an ARMA(1,1) model on it (clearly wrong model but bear with me):
$y_t=ay_{t-1}+b\epsilon_{t-1}+\epsilon_t$
you will end up with a variety of different $(a,b)$ where $a+b=0$ depending on where the optimizer will start.
Of course I rather get $a=b=0$ to slap me in the face to show me this is not the right model.
Is there any way to constrain the optimization so that I get the $(0,0)$ solution?
Thanks
 A: Since, as you write, you obtain estimates $(\hat a, \hat b)\; : \;\hat b = -\hat a$, your estimator is telling you that
$$y_t=ay_{t-1}-a\epsilon_{t-1}+\epsilon_t = a(y_{t-1}-\epsilon_{t-1})+\epsilon_t$$
But 
$$y_{t-1}-\epsilon_{t-1}= ay_{t-2}-a\epsilon_{t-2} = a(y_{t-2}-\epsilon_{t-2})$$
and so 
$$y_t - \epsilon_t=   a^2(y_{t-2}-\epsilon_{t-2})  $$
Setting $z_t = y_t - \epsilon_t$ we have 
$$z_t = a^nz_{t-n}$$
So if the $\hat a$'s you obtain are $< |1|$, the estimator tells you that $z_t=y_t - \epsilon_t \rightarrow 0$ (oscillating perhaps) and so, essentially, that $ y_t = \epsilon_t$, i.e. it's ...telling the truth.
If $\hat a=1$ then the estimator tells us that
$$y_t -\epsilon_t =  y_{t-1} -\epsilon_{t-1} =... = y_{t-n} -\epsilon_{t-n} \Rightarrow \sum y_t = \sum\epsilon_t\Rightarrow y_t = \epsilon_t $$
...again telling the truth.
Finally, if  $\hat a> |1|$, the estimator is telling us that the process, explodes, either oscillating or exponentially, which clearly is against the visual you have (since the process is actually a white noise). So again the estimator tells us the truth by loudly signalling "what a useless model did you make me estimate".
I really don't see any need to complicate matters by constraining  the optimization procedure.
