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

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
qsrsd        0     1
qsirr        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?

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
qsrsd        0     1
qsirr        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
qsrsd        0     1
qsirr        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.

• Which model is chosen if you use automatic model identification? If you can post the data you may get further feedback. – javlacalle May 12 '15 at 6:54
• I cannot share the data, sadly. But this series does not seem to have seasonality in the original series, right? The F test for the whole set of seasonal dummies (17 in total) says that those dummies cannot be jointly accepted as zero. But the total result of the F test is also not very strong (~4). So I do believe a large portion of those dummies might actually be insignificant if I get to test them accurately (F tests for subsets of theseseasonal dummies). – John Doe May 12 '15 at 17:19
• Later, at night, I'll add the results from the automatic adjustment. – John Doe May 12 '15 at 17:58
• I'm sorry for not answering earlier. I had some personal matters to attend, but now I've added the results from the automatic adjustment in the question above, plus some new considerations. I hope this helps. – John Doe May 17 '15 at 0:32
• Is it correct when you say that you have 17 seasonal dummies? there are 12 seasons in your data, so will have 11 seasonal dummies plus the intercept in the regression model. You also mention that yearly and seasonal dummies are included in the model. Is your F-test based on a regression with seasonal dummies or there are other dummies involved in the test. – javlacalle May 17 '15 at 16:04