AR(1) coefficient is correlation? 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?
 A: For a second-order stationary series it is the correlation coefficient between the dependent value and its lag.
Specify
$$y_{t+1} = \beta y_t + u_{t+1}\qquad u_{t+1}= \text{white noise}$$
The correlation coefficient between $y_{t+1}$ and $y_{t}$ is defined as usual
$$\rho_{(1)} = \frac{\text{Cov}(y_{t+1},y_{t})}{\sigma(y_{t+1})\sigma(y_t)}$$
Now 
$$\text{Cov}(y_{t+1},y_{t}) = E(y_{t+1}y_{t}) - E(y_{t+1})E(y_{t})$$
$$ = E\Big((\beta y_t+u_{t+1})y_{t}\Big) - E(y_{t+1})E(y_{t}) = E\Big(\beta y_t^2+u_{t+1}y_{t}\Big) - E(y_{t+1})E(y_{t})$$
Now $u_{t+1}, y_{t}$ are independent. Also, the expected value of the $y$-series is zero, given the specification. Using these facts we end up with
$$\text{Cov}(y_{t+1},y_{t}) =\beta E(y_t^2) = \beta\text{Var}(y_t) $$
Since we assume 2nd-order stationarity, $\text{Var}(y_t) = \text{Var}(y_{t+1}) = \text{Var}(y)$
Inserting all this back to the correlation coefficient
$$\rho_{(1)} = \frac{\beta\text{Var}(y)}{\sigma(y)\sigma(y)} = \frac{\beta\text{Var}(y)}{\text{Var}(y)} = \beta. $$
