ARIMA model : How does the ARIMA model forecast?

Question

The aim of my request is to know how the forecast function of arima model works.

here I am trying to make predictions on my monthly time serie. I know that it is stationary. I am performing a simple ARIMA(1,0,0) with the pmdarima package like following:

model_auto = pm.auto_arima(df_train.values, start_p=1, start_q=0,max_p=1, max_q=0,d=0,trace=True)

Of course I forced it to return an ARIMA(1,0,0) in order to see how it calculates the predictions. By hand, the calculation would be : X_t = a*X_{t-1} + intercept, with a and intercept given by the model. The length of my test df is 10 so I want to forecast 10 values. The first value I want to forecast is the value at 01/01/2021. So the X_{t-1} is the value at 01/12/2020, which is 27779546.0.

The arima model brings me the parameters : a = 0.700, intercept = 8.204e+06.

The forecasted value with the forecast function for 01/01/2021 is 27658114.17.

The calculation by hand is X_t = aX_{t-1} + intercept = X_{01/01/2021} = aX_{01/12/2020} + intercept = 27779546.0 * 0.700 + 8.204e+06 = 27649682.2.

So my question is : why I get two different values with the two methods? (forecasted : 27658114.17, calculate by hand : 27649682.2)

I would appreciate your help 😊 Thank you!