📂Statistical Analysis

Moving Average Process

Model ¹

The process defined as follows for white noise $\left\{ e_{t} \right\}_{t \in \mathbb{N}}$ and according to $Y_{t} := e_{t} - \theta_{1} e_{t-1} - \theta_{2} e_{t-2} - \cdots - \theta_{q} e_{t-q}$, is called the $q$th order moving average process $MA(q)$.

• (1): $MA(1) : Y_{t} = e_{t} - \theta e_{t-1}$
• (2): $MA(2) : Y_{t} = e_{t} - \theta_{1} e_{t-1} - \theta_{2} e_{t-2}$
• (q): $MA(q) : Y_{t} = e_{t} - \theta_{1} e_{t-1} - \theta_{2} e_{t-2} - \cdots - \theta_{q} e_{t-q}$
• (∞): $MA( \infty ) : Y_{t} = e_{t} - \theta_{1} e_{t-1} - \theta_{2} e_{t-2} - \cdots$

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• $\mathbb{N}$ denotes the set of natural numbers $\left\{ 1, 2, 3 , \cdots \right\}$.

Explanation

The green, red, orange, and purple lines in the following figure are referred to as moving average lines.

Moving average lines are widely used in graphs, especially in the stock market, as they allow observing the overall trend rather than the extreme changes day by day. However, it might be difficult to understand why $MA(q)$ is called a “moving average process” just by looking at the formula. As a simple example, consider $MA(2) : Y_{t} = e_{t} - \theta_{1} e_{t-1} - \theta_{2} e_{t-2}$, which becomes $\displaystyle Y_{t} = {{ e_{t-1} + e_{t-2} } \over {2}} + e_{t}$ when $\displaystyle \theta_{1} = \theta_{2} = - {{1} \over {2}}$.

If a variable is growing or declining smoothly, that is, with minimal local fluctuations, then looking at the moving averages would be meaningless. Similarly, moving averages merely smooth out changes without altering the scale itself. Similarly, $MA(q)$ focuses on identifying patterns over shorter periods than $q$, not on the specific values or overall trends. In other words, it can only be used with data that has stationarity.