MA(1) Process Model

Definition

The first-order moving average process: $Z_t=a_t-\theta a_{t-1}$ where $\{a_t\}$ is white noise.

Autocorrelation Bounds

For an MA(1) process, the lag-1 autocorrelation is: ${\rho }_1=\frac{-\theta }{1+{\theta }^2}$ This value is bounded: $-0.5\le {\rho }_1\le 0.5$. The maximum value $0.5$ is reached when $\theta =-1$, and the minimum $-0.5$ when $\theta =1$.

Model Non-Uniqueness

The MA(1) model is not uniquely determined by its autocorrelation function. Replacing $\theta$ with $1/\theta$ results in the same ${\rho }_1$. To ensure uniqueness, we impose the invertibility condition ($|\theta |<1$).

Invertibility Condition

An MA(1) process is invertible if it can be represented as an infinite-order AR process. This requires: $|\theta |<1$ Invertibility allows us to express current shocks $a_t$ as a function of current and past observations $Z_t$.