How to interpret the constant for an ARMA model I'm trying to fit an ARMA(1,0) model for a timeseries that start at $10$ and drops slowly to $4$ in around $180$ steps. For this, I've tried to fit an ARMA model in python using the following:
# contrived dataset
data1 = data['beta_0'].tolist()

# fit model
model = ARMA(data1, order = (1,0))
model_fit = model.fit()

The results are as follows:

I'm having trouble understanding why the constant would be so big. Wouldn't this mean that if you say for instance that $y_0 = 10$ (as it is in my time series), then $y_1 = 6.8840 + 0.9916y_0 + \epsilon$. I would assume that then the values of this time series would blow up right?
Then how can it be that this is the output for a series that drops slowly from 10 to 4?
Any help would be much appreciated.
 A: Your equation is
$$[y(t)-6.8840][1-.9916B]= +ϵ(t)$$
or
$$y(t)= .0084\times6.8840 + .9916\cdot y(t-1)$$
$$y(t)= .0578 + .9916\cdot y(t-1)$$

What has you confused is for your stationary model the constant that is estimated is a Left-Hand side constant not the right-hand side that you were (normally !) expecting.
Your model in my opinion should be changed to a first difference model as your ar coefficient is not different from 1.0 .
EDITED AFTER COMMENTS BY @whuber:
I was intrigued by the interesting discussion and decided to roll out my trusty time series simulator   and generated a realization of the following model  . The nature of the model suggests that other realizations could be expected to have different patterns.
The series has its up's and down's much like what might be expected from a process where the expected value is the last value more or less.
The ACF and PACF are here

The estimated model is here  and here (with a non-significant BUT necessary constant ) 
Expressed as a pure right-hand side equation here 
The ACTUAL/FIT/FORECAST graph is here 
The history shows trends up and down for different segments .
Treated objectively the model is simply a random walk 
