Understanding Moving-Average model in time series

Question

I am not able to understand what the error/deviation/stochastic terms in moving average model stand for? What is the practical significance of the error term. Is the error term difference between the consecutive values in the series. Or is it the difference between the forecasted values and the observed values.

Answer 1

The error terms is neither the difference between the consecutive values nor the difference between the forecasted values and the observed values, though the latter is a somewhat close guess.

A moving-average model of order $q$, MA($q$), is $$ x_t=\varepsilon_t+\theta_1\varepsilon_{t-1}+\dots+\theta_q\varepsilon_{t-q}. $$ Its conditional mean, conditioning on information up to time $t-1$, $I_{t-1}$, is $$ \mathbb{E}(x_t|I_{t-1}) = \theta_1\varepsilon_{t-1}+\dots+\theta_q\varepsilon_{t-q} $$ and this is just $=x_t-\varepsilon_t$. Hence, $\varepsilon_t$ is the difference between $x_t$ and its conditional mean.

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The difference between two consecutive values is \begin{aligned} x_t-x_{t-1} &= (\varepsilon_t+\theta_1\varepsilon_{t-1}+\dots+\theta_q\varepsilon_{t-q}) - (\varepsilon_{t-1}+\theta_1\varepsilon_{t-2}+\dots+\theta_q\varepsilon_{t-q-1}) \\ &= (\varepsilon_t-\varepsilon_{t-1})+\theta_1(\varepsilon_{t-1}-\varepsilon_{t-2})+\dots+\theta_q(\varepsilon_{t-q}-\varepsilon_{t-q-1}) \\ &= \varepsilon_t+(\theta_1-1)\varepsilon_{t-1}+(\theta_2-\theta_1)\varepsilon_{t-2}+\dots+(\theta_q-\theta_{q-1})\varepsilon_{t-q}-\theta_{q}\varepsilon_{t-q-1} \\ &\neq\varepsilon_t. \end{aligned}

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The difference between a forecasted value and an observed value depends on the forecast. For example, the conditional mean is the optimal forecast under square loss. The conditional mean $\theta_1\varepsilon_{t-1}+\dots+\theta_q\varepsilon_{t-q}$ is unknown but can be estimated by $\hat\theta_1\hat\varepsilon_{t-1}+\dots+\hat\theta_q\hat\varepsilon_{t-q}$ where hats denote estimates of the true quantities. So when forecasting, the difference between an observed value $x_t$ and a forecasted value $\hat{x}_t$ is \begin{aligned} x_t-\hat{x}_t &= (\varepsilon_t+\theta_1\varepsilon_{t-1}+\dots+\theta_q\varepsilon_{t-q}) - (\hat\theta_1\hat\varepsilon_{t-1}+\dots+\hat\theta_q\hat\varepsilon_{t-q}) \\ &\neq\varepsilon_t. \end{aligned} If only the true values were known, the forecast error would coincide with the error term $\varepsilon_t$.