12.1 Moving Average Processes

The moving average process of order $q$, MA($q$), is defined as

$$X_t = \beta_1 \varepsilon_{t-1} + \ldots + \beta_q \varepsilon_{t-q} + \varepsilon_t$$ (12.1)

with white noise $\varepsilon_t$. With the Lag-Operator $L$ (see Definition 10.13) instead of (11.1) we can write

$$X_t = \beta(L) \varepsilon_t$$ (12.2)

with $\beta(L) = 1+ \beta_1 L + \ldots + \beta_q L^q.$ The MA($q$) process is stationary, since it is formed as the linear combination of a stationary process. The mean function is simply $\mathop{\text{\rm\sf E}}(X_t)=0$. Let $\beta_0=1$, then the covariance structure is

$$\gamma_\tau = \mathop{\text{\rm Cov}}(X_t,X_{t+\tau})$$

$$= \mathop{\text{\rm Cov}}(\sum_{i=0}^q \beta_i \varepsilon_{t-i}, \sum_{j=0}^q \beta_i \varepsilon_{t+\tau-j})$$

$$= \sum_{i=0}^q \sum_{j=0}^q \beta_i \beta_j \mathop{\text{\rm Cov}}(\varepsilon_{t-i},\varepsilon_{t+\tau-j})$$

$$= \sum_{i=0}^{q-\vert\tau\vert}\beta_i \beta_{i+\vert\tau\vert} \sigma^2, \:\: \vert\tau\vert \le q.$$

For the ACF we have for $\vert\tau\vert \le q$

$$\rho_\tau = \frac{\sum_{i=0}^{q-\vert\tau\vert} \beta_i \beta_{i+\vert\tau\vert}} {\sum_{i=0}^q \beta_i^2},$$ (12.3)

and $\rho_\tau=0$ for $\vert\tau\vert > q$, i.e., the ACF breaks off after $q$ lags.

As an example consider the MA(1) process

$$X_t = \beta \varepsilon_{t-1} + \varepsilon_t,$$

which according to (11.3) holds that $\rho_1 = \beta/(1+\beta^2)$ and $\rho_\tau=0$ for $\tau>1$. Figure 11.1 shows the correleogram of a MA(1) process.

Fig.: ACF of a MA(1) process with $\beta=0.5$ (left) and $\beta=-0.5$ (right). SFEacfma1.xpl

Obviously the process

$$X_t = 1/\beta \varepsilon_{t-1} + \varepsilon_t$$

has the same ACF, and it holds that

$$\rho_1 = \frac{1/\beta}{1+(1/\beta)^2} = \frac{\beta}{1+\beta^2}.$$

In other words the process with the parameter $\beta$ has the same stochastic properties as the process with the parameter $1/\beta$. This identification problem can be countered by requiring that the solutions of the characteristic equation

$$1+\beta_1 z + \ldots + \beta_q z^q = 0$$ (12.4)

lie outside of the complex unit circle. In this case the linear filter $\beta(L)$ is invertible, i.e., there exists a polynomial $\beta^{-1}(L)$ so that $\beta(L)\beta^{-1}(L) = 1$ and

$\beta^{-1}(L) = b_0 + b_1 L + b_2 L^2 + \ldots.$ Figure 11.2 displays the correlogram of a MA(2) process $X_t = \beta_1 \varepsilon_{t-1} + \beta_2 \varepsilon_{t-2} + \varepsilon_t$ for some collections of parameters.

Fig.: ACF of a MA(2) process with $(\beta_1=0.5,\beta_2=0.4)$ (top left), $(\beta_1=0.5,\beta_2=-0.4)$ (top right), $(\beta_1=-0.5,\beta_2=0.4)$ (bottom left) and $(\beta_1=-0.5,\beta_2=-0.4)$ (bottom right). SFEacfma2.xpl