Interpretation differences between deterministic seasonality and deseasonalized data with X-13 SEATS

Question

I am running X-13 SEATS on r for monthly data in six years of observations and I think I got a (sufficiently) reasonable fit for the ARIMA model, but the output also shows me that my original series does not have significant seasonality, as it follows:

 Call:
seas(x = data_r[, 1], transform.function = "log", regression.aictest = NULL, 
    outlier = NULL, arima.model = "(0 1 1)(1 1 0)")

Coefficients:
                  Estimate Std. Error z value Pr(>|z|)    
AR-Seasonal-12     -0.6194     0.1110  -5.581 2.39e-08 ***
MA-Nonseasonal-01   0.6220     0.1093   5.690 1.27e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

SEATS adj.  ARIMA: (0 1 1)(1 1 0)  Obs.: 60  Transform: log
AICc: 773.4, BIC: 778.4  QS (no seasonality in final):    0  
Box-Ljung (no autocorr.): 20.04   Shapiro (normality): 0.9754
    > 
                qs p-val
    qsori        0     1
    qsorievadj   0     1
    qsrsd        0     1
    qssadj       0     1
    qssadjevadj  0     1
    qsirr        0     1
    qsirrevadj   0     1

(Still, there is also the fact that the irregular component seems to dominate the SI ratio for some specific months in some years. So maybe there is some dummy variable in the pre-adjustment that I am missing (right?))

But when I run a regression on Stata for yearly and monthly dummies on the original series -- assuming the seasonality is deterministic --, I cannot reject with an F test that they are all equal to zero. What does this show me? That my ARIMA fit is not correct?

Also, if someone could point me out the difference in interpretation that you should have when running a regression on seasonal dummies and deseasonalizing data with a X-13 SEATS, it would be also very helpful. Maybe that is what I am missing here.

Edit: is it by any chance a common practice, in some particular situations (when you are deseasonalizing a set of series), still deseasonalize a given series even if that series does not show significant seasonality?

Edit2: Adding the results of the automatic adjustment:

Coefficients:
                   Estimate Std. Error z value Pr(>|z|)    
Constant            59.1761    38.0551   1.555  0.11994    
Easter[15]        -903.6151   341.1891  -2.648  0.00809 ** 
MA-Nonseasonal-01    0.4974     0.1138   4.370 1.24e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

SEATS adj.  ARIMA: (0 1 1)  Obs.: 60  Transform: none
AICc: 925.6, BIC: 933.2  QS (no seasonality in final):    0  
Box-Ljung (no autocorr.):  21.9   Shapiro (normality): 0.9498 *

            qs p-val
qsori        0     1
qsorievadj   0     1
qsrsd        0     1
qssadj       0     1
qssadjevadj  0     1
qsirr        0     1
qsirrevadj   0     1 

I also, I get the following error for the monthplot function with the automatic adjustment:

Error in `[.default`(x$data, , "seasonal") : subscript out of bounds

Following this result from the automatic adjustment, the use of the dummy for easter, with the original specification, does not change that much the first output:

Coefficients:
                  Estimate Std. Error z value Pr(>|z|)    
Easter[15]        -0.08307    0.02690  -3.088  0.00202 ** 
AR-Seasonal-12    -0.63353    0.10816  -5.858  4.7e-09 ***
MA-Nonseasonal-01  0.50391    0.12075   4.173  3.0e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

SEATS adj.  ARIMA: (0 1 1)(1 1 0)  Obs.: 60  Transform: log
AICc: 767.9, BIC: 774.3  QS (no seasonality in final):    0  
Box-Ljung (no autocorr.): 29.37   Shapiro (normality): 0.9721  
            qs p-val
qsori        0     1
qsorievadj   0     1
qsrsd        0     1
qssadj       0     1
qssadjevadj  0     1
qsirr        0     1
qsirrevadj   0     1

Most recent observation: Now I Think I am fairly sure that there is no significant seasonality in this series, but I would be thankful if someone could show me other problems that I might not be considering. Still, I would like a possible canonical/scholarly answer on why I can reject the null hypothesis for the whole set of seasonal dummies being zero (though I had a small result for the F test with my data, ~4, but I still reject the null) and still get a reasonable ARIMA fit with which I cannot reject no seasonality in my original data. Does that have something to do with the difference of the adjustment with ARIMA models and deterministic seasonality? An intuitive answer on this difference would be of some help.

Answer 1

is it by any chance a common practice, in some particular situations (when you are deseasonalizing a set of series), still deseasonalize a given series even if that series does not show significant seasonality?

I don't see any reason to apply a seasonal adjustment procedure to a time series where seasonality cannot be identified. The common practice is to check for the presence of seasonality (by means of statistical tests, graphics) and then estimate and remove a seasonal component only for those series where seasonality is identifiable. Forcing to remove a seasonal component that is not present may distort the autocorrelation structure of the original series.

why I can reject the null hypothesis for the whole set of seasonal dummies being zero.

As mentioned in my comment above, it seems that you are testing for the significance of 17 variables (11 seasonal dummies plus other dummy variables, yearly dummies). It may be the latter set of variables what may lead to reject the null of zero coefficients for the whole set of 17 dummies.