# Auto.arima coefficients with exogeneous variables

I have the following output from the auto.arima function with specified xreg:

Series: Individuals
Regression with ARIMA(2,1,1) errors

Coefficients:
ar1      ar2     ma1  inflation  GDP growth  EURIBOR
-0.8366  -0.3928  0.4008    -0.0013      -8e-04   0.0139
s.e.   0.2155   0.0863  0.2333     0.0023       1e-03   0.0036

sigma^2 estimated as 0.0004495:  log likelihood=453.32
AIC=-892.63   AICc=-892   BIC=-870.09


Does it hold that

$$y_t = -0.0013*inflation_t -8*10^{-4}*GDP_t + 0.0139*EURIBOR_t + n_t$$

$$n_t = -0.8366*n_{t-1} - 0.3928*n_{t-2} + e_t + 0.4008*e_{t-1}$$

or does the differencing change something?

I get the residuals by the following code

nt <- residuals(fitInd, type = "regression")
et <- residuals(fitInd, type = "innovation")


When I check with the residuals "nt" it does match, for example

nt=0.2621466


and per the first equation:

$$y_4 - (-0.0013*inflation_4 -8*10^{-4}*GDP_4 + 0.0139*EURIBOR_4)= n_4 = 0.2621466$$

However, it does not match for the innovation errors "et" per the second equation which should be equal to nt:

-0.8366*nt - 0.3928*nt + et + 0.4008*et = -0.3318558


Could someone clarify what I am doing wrong. Thanks in advance.

• Did you test for and possibly consider lags of any of the three predictors ? Were there any untreated anomalies in the residuals ? Was the error variance of the residuals constant over time ? These are items that need to be considered in determining a useful model. Oct 7, 2019 at 14:32

Your equation would have been right if you used $$\Delta n_t$$ instead of $$n_t$$. It's the differenced model, like ARMA of $$\Delta n_t$$:
$$\Delta n_t = -0.8366*\Delta n_{t-1} - 0.3928*\Delta n_{t-2} + e_t + 0.4008*e_{t-1}$$