Moving Average Process

Definition

Let $\{e_t\}$ : White noise

A moving average process is defined as:

$$Y_t=\frac{e_t+e_{t-1}}{2}$$

Properties

$$\begin{array}{rl}E[Y_t]& =E\left[\frac{e_t+e_{t-1}}{2}\right]\\ & =\frac{E[e_t]+E[e_{t-1}]}{2}\\ & =0\\ & \\ \text{Var}(Y_t)& =\text{Var}\left(\frac{e_t+e_{t-1}}{2}\right)\\ & =\frac{1}{4}[\text{Var}(e_t)+\text{Var}(e_{t-1})]\\ & =\frac{1}{4}\cdot 2{\sigma }_e^2\\ & =\frac{1}{2}{\sigma }_e^2\end{array}$$

For covariance,

$$\begin{array}{rl}\text{Cov}(Y_t,Y_{t-k})& =\text{Cov}\left(\frac{e_t+e_{t-1}}{2},\frac{e_{t-k}+e_{t-k-1}}{2}\right)\\ & =\frac{1}{4}\text{Cov}(e_t+e_{t-1},e_{t-k}+e_{t-k-1})\\ & =\frac{1}{4}[\text{Cov}(e_t,e_{t-k})+\text{Cov}(e_t,e_{t-k-1})+\text{Cov}(e_{t-1},e_{t-k})+\text{Cov}(e_{t-1},e_{t-k-1})]\end{array}$$

Notice that $\text{Cov}(e_s,e_s)=\text{Var}(e_s)={\sigma }_e^2,\forall s$

Consider all cases for $k$:

If $k=0$, then

$$\begin{array}{rl}\text{Cov}(Y_t,Y_{t-k})& =\frac{1}{4}[{\sigma }_e^2+0+0+{\sigma }_e^2]\\ \text{Var}(Y_t)& =\frac{1}{2}{\sigma }_e^2\end{array}$$

If $k=1$, then

$$\text{Cov}(Y_t,Y_{t-k})=\frac{1}{4}[0+0+{\sigma }_e^2+0]=\frac{1}{4}{\sigma }_e^2$$

Otherwise, if $k\ge 2$

$$\text{Cov}(Y_t,Y_{t-k})=\frac{1}{4}[0+0+0+0]=0$$