# What data generation process corresponds to exponential moving average prediction?

For each ARMA process formulation, there is an optimal prediction. E.g.:

• When you believe that $y_{t+1}=\alpha y_t + \varepsilon_{t+1}$, where $\varepsilon_{t+1}$ are IID, you predict $\hat{y}_{t+1}=\alpha y_t$.

• When you believe that $y_{t+1}=\beta \varepsilon_{t} + \varepsilon_{t+1}$, where $\varepsilon_{t+1}$ are IID, you predict $\hat{y}_{t+1}=\beta y_t - \beta^2 y_{t-1} + \beta^3 y_{t-2}-\beta^4 y_{t-3}+...$.

The question is:

For which data generating process (except the obvious AR($\infty$)), the exponentially-moving-average forecast $\hat{y}_{t+1}=\alpha\sum_{i=0}^{\infty}(1-\alpha)^iy_{t-i}$ is optimal?

First, you mean $$\hat{y}_{t+1} = \alpha \sum_{i=0}^\infty (1-\alpha)^i y_{t-i}.$$ That is the optimal prediction for several models, as explained in Chatfield et al (JRSSD 2001). The best known of these models are the ARIMA(0,1,1) process where the moving average parameter $\theta=1-\alpha$ and the ETS(A,N,N) and ETS(M,N,N) models. See http://otexts.org/fpp2/sec-7-ETS.html for details.