Is the ar1 coefficient from an AR(1) model the "first order correlation of the noise" of a time series?
I'm using R's
aws package and one of the arguments of the
aws function allows you to "specify a first order correlation of the noise". Instinctively, I did
arima (foo, c(1, 0, 0))
and used the ar1 coefficient as this correlation value. Plugging it into
aws makes the results look more like what I'd expect, so I was quite happy…
Until it struck me that AR coefficients can fall outside of the range [-1, 1], while the correlation cannot. So maybe I just got lucky. Then it struck me that if the ar1 coefficient of an AR(1) model falls outside of [-1, 1], the model's not stationary, so perhaps a well-fitting AR(1) model's ar1 coefficient is essentially restricted to [-1, 1].
But then I thought: just because they share the same range doesn't mean they are the same. I did some searching and didn't find anything saying this is a correlation. I did some simulations and it appears that it is close to a correlation.
Anyone have an authoritative statement on the matter?
I've also looked at
arima (foo, c(1, 0, 0), xreg=theDate) to account for a trend. I've thought that if the AR coefficient is a correlation: what if an AR(2) model is a better fit than an AR(1) model -- is the ar1 coefficient still what I'm looking for or do I have to combine it with the ar2 coefficient, or do I still go with the AR(1) model?