Time-Series Models for Financial Data › Moving Average (MA) Models

While AR models express current values as functions of past values, moving average models take a fundamentally different approach by expressing current values as functions of past shocks. The terminology can be confusing because we are not computing averages in the usual sense. Instead, the current observation is modeled as a weighted combination of recent random innovations. This perspective views the time series as arising from the accumulation of shocks, with each shock having both an immediate impact and lingering effects over subsequent periods.

The MA(1) model expresses the current value as the mean plus the current shock and a fraction of the previous shock:

$$Y_t = \mu + \varepsilon_t + \theta \varepsilon_{t-1}$$

where:

• $Y_t$: value of the series at time $t$
• $\mu$: mean of the process
• $\varepsilon_t$: white noise shock at time $t$
• $\theta$: moving average coefficient
• $\varepsilon_{t-1}$: white noise shock at time $t-1$

Unlike AR models, MA models are always stationary regardless of parameter values. This happens because the model is defined as a finite linear combination of independent random variables, which always has well-defined finite moments. However, we typically impose invertibility conditions for meaningful interpretation and to ensure unique parameter identification, as we will discuss later.

The mean of the MA(1) process is straightforward to compute by taking expectations:

$$\begin{aligned} E[Y_t] &= E[\mu + \varepsilon_t + \theta \varepsilon_{t-1}] \\ &= \mu + E[\varepsilon_t] + \theta E[\varepsilon_{t-1}] && \text{(linearity of expectation)} \\ &= \mu + 0 + 0 && \text{(zero mean noise)} \\ &= \mu \end{aligned}$$

where:

• $E[Y_t]$: expected value of the series
• $\mu$: mean parameter
• $\varepsilon_t$: white noise shock at time $t$
• $\theta$: moving average coefficient

The variance calculation uses the independence of the white noise terms at different times. Since $\varepsilon_t$ and $\varepsilon_{t-1}$ are uncorrelated, their contributions to the variance simply add:

$$\begin{aligned} \text{Var}(Y_t) &= \text{Var}(\mu + \varepsilon_t + \theta \varepsilon_{t-1}) \\ &= \text{Var}(\varepsilon_t) + \theta^2 \text{Var}(\varepsilon_{t-1}) && \text{(independence of shocks)} \\ &= \sigma^2 + \theta^2 \sigma^2 && \text{(constant variance)} \\ &= \sigma^2(1 + \theta^2) && \text{(factor out } \sigma^2 \text{)} \end{aligned}$$

where:

• $\text{Var}(Y_t)$: variance of the series
• $\sigma^2$: variance of the white noise shock $\varepsilon_t$
• $\theta$: moving average coefficient

For autocovariances, the MA(1) model exhibits a striking feature: only lag 1 is nonzero. To see why, we expand the covariance between adjacent observations:

$$\begin{aligned} \gamma_1 &= E[(Y_t - \mu)(Y_{t-1} - \mu)] \\ &= E[(\varepsilon_t + \theta \varepsilon_{t-1})(\varepsilon_{t-1} + \theta \varepsilon_{t-2})] && \text{(substitute definition)} \\ &= E[\varepsilon_t \varepsilon_{t-1} + \theta \varepsilon_{t-1}^2 + \theta \varepsilon_t \varepsilon_{t-2} + \theta^2 \varepsilon_{t-1} \varepsilon_{t-2}] && \text{(expand product)} \\ &= 0 + \theta E[\varepsilon_{t-1}^2] + 0 + 0 && \text{(cross-terms are zero)} \\ &= \theta \sigma^2 \end{aligned}$$

where:

• $\gamma_1$: autocovariance at lag 1
• $\theta$: moving average coefficient
• $\sigma^2$: variance of the white noise shock
• $Y_t$: value of the series at time $t$
• $\mu$: mean of the process
• $\varepsilon_t, \varepsilon_{t-1}, \varepsilon_{t-2}$: white noise shocks

The cross-terms vanish because white noise shocks at different times are uncorrelated by assumption. The only surviving term is $\theta E[\varepsilon_{t-1}^2] = \theta \sigma^2$, which arises from the single shock $\varepsilon_{t-1}$ that appears in both $Y_t$ and $Y_{t-1}$. For $k \geq 2$, there is no overlap in the $\varepsilon$ terms entering $Y_t$ and $Y_{t-k}$, so $\gamma_k = 0$.

The ACF is therefore:

$$\rho_k = \begin{cases} \frac{\theta}{1+\theta^2} & \text{if } k = 1 \\ 0 & \text{if } k > 1 \end{cases}$$

where:

• $\rho_k$: autocorrelation at lag $k$
• $\theta$: moving average coefficient

This cutoff property of the ACF is the diagnostic signature of MA models. An MA(q) process has $\rho_k = 0$ for all $k > q$, meaning the autocorrelation function drops abruptly to zero after lag $q$. This sharp cutoff contrasts with the gradual decay of AR models and provides the basis for identifying MA order from sample ACF plots.

The MA(q) model extends to include $q$ lagged shock terms:

$$Y_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2} + \cdots + \theta_q \varepsilon_{t-q}$$

where:

• $Y_t$: value of the series at time $t$
• $\mu$: constant mean
• $\varepsilon_{t-k}$: white noise shock at lag $k$
• $\theta_k$: moving average coefficient at lag $k$

Using the lag operator:

$$Y_t = \mu + \Theta(L) \varepsilon_t$$

where:

• $Y_t$: value of the series at time $t$
• $\Theta(L) = 1 + \theta_1 L + \theta_2 L^2 + \cdots + \theta_q L^q$: moving average polynomial
• $\mu$: constant mean
• $\varepsilon_t$: white noise series

The moving average polynomial $\Theta(L)$ captures the distributed impact of shocks over time. The ACF cuts off after lag $q$, while the PACF decays gradually. This is the opposite pattern from AR models. When we observe sample ACF with a sharp cutoff and PACF with gradual decay, this strongly suggests an MA process.

An MA model is invertible if it can be expressed as an AR($\infty$) process, meaning we can write current shocks in terms of past observations. For MA(1), invertibility requires $|\theta| < 1$. The importance of invertibility goes beyond mathematical elegance. Non-invertible MA models create identification problems: an MA(1) with parameter $\theta$ and another with parameter $1/\theta$ produce identical autocorrelation structures, making the models observationally equivalent. By convention, we restrict attention to invertible representations to ensure unique parameter estimates.