Generate prediction intervals for a simple moving average model

Question

I'm using a simple moving average to generate a forecast. Say I have $t$ observations. Then the forecast for time $t+1$ is given by

\begin{equation} \hat{Y}_{t+1}= \frac{Y_t+Y_{t-1}+\dots Y_{t-m+1}}{m} \end{equation}

If the forecasting horizon is longer than one period, then the forecasts are given by $\hat{Y}_{t+h} = \hat{Y}_{t+1}$. This obviously means that the forecasts are flat.

How can I generate a prediction interval for these forecasts? (say 95%). I've seen that Hyndman et al. (2021) give a formula for some simple methods, and I've been wondering if there is something like that for the simple moving average.

Answer 1

Inspired by one of the comments made by @mlofton, I used the following solution. It might not be the best, but for now it works for my particular problem (generating prediction intervals for a window average)

Consider a moving average model of order $q$, denoted MA(q). Then

$X_t = \mu + \epsilon_t +\theta_1 \epsilon_{t-1} + \dots + \theta_q \epsilon_{t-q}$

If we pick $q=0$, then

$X_t = \mu + \epsilon_t$

Hence the value for $X_t$ is simply the mean plus an error.

So let's consider an ARIMA(p,q,d) model. Using $p=q=d=0$, the ARIMA model is equal to the historic average. The prediction intervals for this particular ARIMA model can be easily computed using the forecast R package. The key here is only using the data in the window that we're considering. The point forecast of the ARIMA model will be equal to the point forecast of the Window Average. And we can use the prediction intervals of that ARIMA model as prediction intervals of the Window Average.