# Interpretation of TSA::arimax output model is presented in R

I have a relatively simple problem, but yet taking some time to solve it. I am using the arimax() function from the TSA package. (Note: not arima() from the stats package.) This is the model:

out <- arimax(sub_s_t_series, order=c(2,0,1), xreg=sub_r_t_series, method=c("ML"))


and these are my coefficients:

Call
arimax(x = sub_s_t_series, order = c(2, 0, 1), xreg = sub_r_t_series, method = c("ML"))

Coefficients:
ar1      ar2      ma1  intercept     xreg
1.4825  -0.6613  -0.8516  52745.107  -1.0132
s.e.  0.0295   0.0294   0.0064     40.828   0.0012

sigma^2 estimated as 0.08929:  log likelihood = -105.98,  aic = 221.97


All I am trying to do is to interpret the results. According to my understanding and the help given in the TSA package, the above ARIMAX(2,0,1) model is represented as follows: $${\rm sub\_s\_t\_series\_hat[k]} = {\rm intercept} + xreg\times {\rm sub\_r\_t\_series[k]} + \frac{a_{t[k]}+ma1*a_{t[k-1]}}{a_{t[k]}-ar1*a_{t[k-1]}-ar2*a_{t[k-2]}} \tag{1}$$ where $a_t$ are the residuals. When I use e_t = fitted(out)-sub_s_t_series_hat to measure the error / residuals myself, e_t matches exactly to the values obtained by out[["residuals"]].

But when I use (1) as follows: e_t_hat = sub_s_t_series_hat - sub_s_t_series, e_t_hat does not match with out[["residuals"]], in fact the results deviate by a magnitude of almost 4.

My questions is: did an ARIMAX(2,0,1) fit would result in (1) or am I missing something?

• This is a question about how to interpret output & understand the ARIMAX model. Although it mentions R, it should be considered on topic here, IMO. – gung Jul 29 '15 at 16:54
• Have you seen this useful blog post by Rob J. Hyndman? – Richard Hardy Aug 3 '15 at 16:45
• As per @RichardHardy's comment, you may want to switch to stats::arima() or forecast::auto.arima() , both of which can also model covariates via an xreg parameter, since you don't seem to use the transfer function feature that TSA::arima() was written to include after all. The difference is that we know that stats::arima() and forecast::auto.arima() fit regressions with ARIMA errors - easy to interpret -, while we don't know what exactly TSA::arima() does without going into the code. Probably a "real" ARIMAX model - hard to interpret. – Stephan Kolassa Jul 14 '16 at 6:30