# Do coefficients in moving average process add to 1?

I'm studying weakly stationary stochastic processes, and I'm confused by the title of the "moving average" representation of such a process.

Suppose that $$y_t$$ is a weakly stationary stochastic process, so that it can be written in the following form: $$y_t=\epsilon_t+c_1\epsilon_{t-1}+c_2\epsilon_{t-2}+...$$

where the epsilons are following a white noise distribution. This is supposedly referred to as the MA($$\infty$$) representation of a weakly stochastic process, but because it is referred to as a moving average, I am inclined to interpret the $$c$$ coefficients as being weights, in which case I believe they should add up to one. Is this correct?

• No, they do not add to one. The term is simply misleading. – Richard Hardy May 10 at 16:13
• Because the (common) variance $\sigma^2$ of the $\epsilon_t$ is unspecified, the sum of the coefficients (including the initial $1$) has no meaning. Without any loss of generality, then, the sum could be constrained to be in the set $\{0,1,-1\}$ through a suitable choice of $\sigma.$ This slightly generalizes the sum-to-unity ("average") case. Thus, maybe the terminology is not that misleading. – whuber May 10 at 16:40