Why the differenced at lag 12 time series of a SARIMA(0,0,0)(0,1,1)_12 model follow the MA(1) pattern with step 12?

I am trying to understand why the ACF of the seasonally differenced series reveals the AR of MA structure of the original series.

For example:

The following lines creates a SARIMA(0,0,0)(0,1,1)_12 time series

library(astsa)
ts.value = sarima.sim(D=1, sma=-0.6, S=12, n=200)


A typical plot of such a series is as follows:

From the ACF plot, we recognize the 12-period of the series. The PACF plot verifies further this remark.

Now, the following lines computes the first seasonal difference of the same series

ts.value.D1 <- diff(ts.value, lag = 12, differences = 1)


The corresponding plot of the differenced series is as follows.

From the ACF plot of the 12-differenced series a significant autocorrelation at lag 12 is appear. Since we know that MA(1) models has a single non-zero ACF value, we conclude that the original series is from an SARIMA(0,0,0)(0,1,1)_12 model.

This behavior seems reasonable, however it is not clear for me the theory behind this observation.

That is, why the differenced at lag 12 time series of a SARIMA(0,0,0)(0,1,1)_12 model should follows the MA(1) pattern with step 12?

Notes:

1. Analogous remarks can be easily reported for AR(1) models.
2. I found a post with analogous remarks but no theoretical explanation is given.

• Thank you for your time and effort. I am trying to understand it algebraically. So if $X_t$ ​~ SARIMA(0, 0, 0)(0, 1, 1)12 then $Y_t = X_t ​- X_{t-12} ​= w_t + Θw_{t-12}$. So, $Y_t - Y_{t-12} = w_t – w_{t-12} + Θ(w_{t-12} – w_{t-24}) = w’_t + Θw’_{t-12}$ and the last equation is the $ΜΑ(1)_{12}$ equation ($w’_t = w_t - w_{t-12} ~ Ν(0, 2σ^2)$ since $w_t$, iid N(0, $σ^2$)). However, I am not sure that this argument is correct. Is it possible to provide more details? Commented Apr 17 at 14:51