I'm trying to teach myself a bit of time series analysis. I came across this equation that looks like it may be solvable, but I can't be sure. I've just gotten through the first few chapters of Hamiltons Time-Series Analysis, so I apologize if this question is a bit premature.
So its the standard $Ax = b$ equation. We know $x$ and $b$ vectors, and all but the first row of matrix $A$. The rest of matrix $A$ looks like the identity matrix shifted down a row. So if $A$ was $3\times 3$, row $1 = [x,~y,~z]$; row $2 = [1,~0,~0]$; row $3 = [0,~1,~0]$.
Since $xx^T$ wouldn't have an inverse, I can't move it to the other side. So I'm not sure if I would be able to find a solution to this, it just looks so close. I think in the book they are assuming these $x,y,z$ (phi's) are known at this point. But it would be much more useful to find a solution/approximation for these.
If anyone is familiar with time series or ARMA/ARIMA models, do you know how these phi values are approximated? I know phi can be approximated as $1 - \frac{d}{2}$ ($d = $ Durbin Watson statistic) for a first order model. But how are phi values approximated for higher order models?