In the online forecasting book of Hyndman (https://otexts.com/fpp2/MA.html) firstly the use of differencing is explained. After that he shows the formula for a moving average model:

$$y_t = c + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2} + \ldots + \theta_q \varepsilon_{t-q}$$

I don't see why to add the constant denoted by 'c' in the upper formula. Is there any difference between introducing 'c' and using first differencing before trying to fit a MA model? Furthermore, introducing 'c' would imply that it actually ís allowed to have a nonstationary time series. It seems like I'm making a mistake in my thinking...


The two are not equivalent.

If you difference an invertible MA and fit another MA, the differenced population model is not invertible (this is overdifferencing).

The first most obvious issue is if you try to fit an MA of the same order you started with you get a different model. Consider an MA(1):

$$y_t = c + \epsilon_t + \theta_1 \epsilon_{t-1}$$

and its first difference:

\begin{eqnarray} y_t - y_{t-1} &=& c-c + \epsilon_t + \theta_1 \epsilon_{t-1} - \epsilon_{t-1} - \theta_1 \epsilon_{t-2}\\ &=& \epsilon_t - (1-\theta_1) \epsilon_{t-1} - \theta_1 \epsilon_{t-2} \end{eqnarray}

Well it got rid of the $c$ but it's no longer an MA(1). So we immediately see the differenced model is not equivalent to the model with a constant.

It looks like an MA(2) with the two parameters related. But it's no longer invertible; it has a root on the unit circle. In this case the algebra to see it is straightforward, but I'm going to show you the observed effect on the inverse characteristic roots for a simulated series (since that's something you could do with real data that might help you notice you have overdifferenced).

First I generated 109 observations from an MA(1) with $\theta_1= -0.5$ (in your parameterization) and $c=13.1$, and $\sigma=1$ by hand (this is two short lines in R).

Then using Rob Hyndman's code here I generated a plot of the inverse roots for an MA(1) fitted to the y, and an MA(1) fitted to the differenced y (since that's what we'd do if we naively assumed we'd only got rid of the constant without changing anything else), and then an MA(2) fitted to the differenced y (which is what we'd do if we did the above algebra and noticed it looked like an MA(2)).

inverse characteristic roots for MA(1), MA(1) on differenced series and MA(2) on differenced series. Points are at 0.5, at 1 and at 0.5 & 1 respectively

We can see in both plots for the differenced series - whether we fitted an MA(1) or an MA(2) that there's a root on (or extremely close to) the unit circle. This makes it clear that we fundamentally changed the characteristics of the model by differencing an already stationary-and-invertible series.

Having a constant $c$ doesn't make the series nonstationary. If you start with a stationary series, you don't want to difference it.

The R code I used (after running the code to define the functions on Rob's page at the link above):

 eps <- rnorm(110)
 y <- 13.1 + eps[2:110] -.5*eps[1:109]
 # don't forget to set the mfrow parameter back 

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