Improve identification of a SARIMA model Does it make sense to build a $SARIMA(1,0,0)(2,0,0)$ when $\Phi_1$ is not significant and $\phi_1$ and $\Phi_2$ are? I have this case, here is the plot:

Here are the data:
data<-ts(c(5.8,5.4,5.3,5.4,5.7,5.7,5.6,5.6,6.0, 
      5.2,4.9,4.9,5.2,5.0,4.9,4.4,4.5,3.7,4.0,4.3,5.0,5.3,5.7,5.3, 
      5.2,4.9,4.8,4.4,5.1,4.4,4.0,4.8,4.7,4.8,5.1,4.9,5.7,5.0,5.3, 
      5.4,6.0,5.4,5.6,5.6,5.8,5.8,5.6,5.7,6.3,6.1,6.1,5.6,6.0,5.4), start=c(2004,1), end=c(2017,2), frequency=4)

I'm assuming that the Data Generating Process has a high AR component (so I'm refusing to treat this with a unit root inside).
Here are the ACF and PACF:

The first model I choose is an $AR(1)$, but then I would reject the Ljung-Box test of on the residuals. It seems to be a seasonal component like $SARIMA(1,0,0)(1,0,0)$ or $(1,0,0)(2,0,0)$. 
I try, but the first gives the seasonal coefficient not significant, and the second, as I mentioned before, gives only the second seasonal coefficient signficant (plus $\phi_1$).
Then residuals are ok, but I'm wondering if this identification is good and could someone suggest to me how to improve it in another way.
This question originated  from this nother question here.
 A: I don't think it does make sense in this case to have a seasonal AR component at lag of 2 periods but not 1.  The interpretation would be that somehow the unemployment this quarter is impacted on unemployment in the same quarter two years ago, but not by unemployment last year.  This seems unlikely.
I suspect you've got as far as you can with this dataset as a univariate time series.  My preference, based on KPSS tests and AIC, would be an ARIMA(1,1,0) model (this is what Hyndman's auto.arima comes up with) ie differenced, but no seasonality.  However, the forecasts from ARIMA(1,1,0), ARIMA(1,0,0) and SARIMA(1,0,0)(2,0,0) all look very similar, with big chunks of uncertainty about the predictions.  I would say that this data:


*

*is difficult to model well or predict with any precision 

*shows mild evidence of non-stationarity (ie against your assumption, noted in the previous question, that unemployment is just hovering around the non-accelerating inflation rate of unemployment) and appears to be secularly increasing; but more evidence needed to be really sure

*doesn't show evidence of seasonality (this is a bit surprising - are you sure you're not working with figures that have already been seasonally adjusted?  Maybe link to the original).


An interesting feature of this data is that it shows signs of being less volatile as it gets bigger; basically because of the down and up during the global financial crisis.  Some kind of ARCH model might be useful, depending on your purpose.
A: I've never had much luck trying to choose the parameters using ACF or PACF plots for seasonal arima models.  One option is to use the methodology in the forecast package in R.  The other is to implement a cross-validation approach.
I ran your data through the latter and it identified the following model as having the best predictive accuracy SARIMA (0,1,1) (0,0,1)4 with constant.  I included a couple sample iterations to show how the simulations behave.  Hope that helps.

