Cross Validated Asked by wasif on January 2, 2021

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} + xregtimes {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 an old post, but since there is no accepted answer, I still want to provide an explanation for future readers. You have the following R output:

```
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
```

(1) y = intercept + xreg * x series name + n_t(error term)

(2) n_t(error term) = ar1 * n(t-1) + ar2 * n(t-2) + white noise epsilon + ma1 * epsilon(t-1)

(3) white noise epsilon ~ NID(0,0.08929)

To explain it further:

(1) This formula focuses on xreg coefficient. The R outputs gives you a coefficient for xreg -- the effect of the exogenous variable on Y at it's current value.

(2) This formula focuses on unpacking the error term left unexplained by formula (1). Formula 1 and 2 together is also called regression with ARIMA error (ARMA error in this case). This error term itself is an ARIMA (2,0,1) process, according to your output. This is what ar1, ar2, ma2 in the R output corresponds to.

(3) This formula is describing what is left after formula 1 and 2. sigma^2 describes the variance of the white noise series -- epsilon in most Econometric textbooks.

Answered by RachelSunny on January 2, 2021

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