auto.arima stationary parameter

Question

I fit data with auto.arima function in variant with exog. If I'm right, when exog is used, linear regression is used to fit data and residuals are fitted with arima, so coeffs of linear part must be always the same, but:

Var 1:

model = auto.arima(y,stationary=TRUE,stepwise=FALSE,parallel=TRUE,
        num.cores=cn,approximation=FALSE,xreg=train_x,x=y)

Output 1:

13
Series: y 
Regression with ARIMA(3,0,0)            errors 

Coefficients:
         ar1     ar2      ar3        T      R
      0.7887  0.5917  -0.8989  -0.0076  4e-04
s.e.  0.0507  0.0836   0.0506   0.0035  2e-04

---

Var 2:

model = auto.arima(y,stepwise=FALSE,parallel=TRUE,num.cores=cn,
        approximation=FALSE,xreg=train_x,x=y)

Output 2:

13
Series: y 
Regression with ARIMA(0,0,0)(2,1,0)[12] errors 

Coefficients:
        sar1     sar2    T    R
      1.2310  -0.3639    0    0
s.e.  0.1379   0.1565  NaN  NaN

Answer 1

The fitting is simultaneous, not stepwise, therefore the coefficients in the linear regression part vary with any variations in the ARIMA part.

Normally you would prefer simultaneous estimation because it is more efficient (statistically, not computationally), so the variation is a good thing. If not, then run a regression with lm, take the residuals and model them with arima or auto.arima. That will fix the regression estimation in the first stage and you can fiddle with the ARIMA part for the residuals independently.