# Where does the white noise come from in MA(q) model?

I'm having trouble understanding the intuition of the moving average model. How is summing up a bunch of white noises related to predicting your particular time series data?

Suppose I have a MA(q) model $$y_t = \mu + \epsilon_t + \theta_1 \epsilon_{t-1} + ... + \theta_q \epsilon_{t-q}$$, where do these $$\epsilon's$$ come from?

Are these $$\epsilon's$$ some residuals from some other models? If so, how does one estimate these $$\epsilon's$$?

Are these $$\epsilon's$$ just theoretical white noises? If so, why are they sequential?

• If you have a series generated by an MA(q) process, wouldn't it make sense to forecast it with an MA(q) model? Some real world series are roughly weighted cumulative sums of i.i.d. shocks, hence the MA(q). – Richard Hardy Dec 15 '18 at 10:48

While (I think) this answer will not provide the intuition behind, it hopefully will bring some insight.

One way to see where the white noise comes from in the $$MA(q)$$ representation is given by the Wold decomposition. The moving average $$MA(q)$$ and the autoregressive $$AR(p)$$ processes are specific cases of a general representation of stationary processes obtained by Wold.

Wold proved that any weakly stationary stochastic process, $$z_{t}$$, with finite mean, $$\mu$$, that does not contain deterministic components, can be written as a linear function of uncorrelated random variables, $$a_{t}$$, as:

$$\begin{array} \ z_{t} & = & \mu + a_{t} + \psi_{1} a_{t-1} + \psi_{2} a_{t-2} + \ldots \\ & = & \mu + \sum^{\infty}_{i = 0} \psi_{i} a_{t-i} & ; & \psi_{0} = 1 \end{array}$$

Where:

$$E(z_{t}) = \mu$$
$$E(a_{t}) = 0$$
$$Var(a_{t}) = \sigma^{2}$$
$$E(a_{t} a_{t-k}) = 0$$ for $$k>1$$

We can write, $$\tilde{z}_{t} = z_{t} - \mu$$ and using the lag operator, then we have:

$$\tilde{z}_{t} = \psi (B) a_{t} \tag{1}$$

With $$\psi(B) = 1 + \psi_{1} B + \psi_{2} B^{2} + \ldots$$

Equation $$(1)$$ is the general linear representation of a non-deterministic stationary process. This representation is important because it guarantees that any stationary process admits a linear representation. In general, the variables $$a_{t}$$ make up a white noise process, that is, they are uncorrelated with zero mean and constant variance.

Taken from Andrés M. Alonso Fernández slides. More here

How can answer provided by @IrishStat garnered a -2? I really don't understand. It fits better for the audience asking the question.

Anyway, let me try. The keyword here is "leftover shock". Take groceries sales as an example and assume it's stationary. Say there exists a mean of $$1000 sales (you know your sales are around$$1000). There will be white noise or "shocks" to the sales every day, so your sales won't be exactly \$1000 but around there. Sometimes shocks are introduced by poor weather, good weather, promotion, etc which affect the sales. Some shocks have instantaneous effect, some not, like coupon issue. Suppose we introduce shock to the system by issuing coupon. More people claim on the first day, then less people claim on the next day on the leftover coupons, and then less and less on the next day. In this case we see the shock today attributable to coupon issue is a proportion of the shock yesterday (θ1ϵt−1) and maybe, plus a proportion of the shock two days ago (θ2ϵt−2). If say, that proportion of the shock from 3 days ago (θ3ϵt−3) is insignificant, ie. θ3=0, then we have a MA(2) here. Note that we also have today's shock ϵt here as shocks can happen everyday and it can be attributable to that promotion, poor weather introduced today.

the "white noise" is the "leftover" after adjusting for previous "shocks"

Here MA (q) denotes a moving average model of order q. For a stationary time series,


the current observed value y(t) can be related to the current and all past unobservable shocks, a(t), a(t-1), …, a(t-q). This relationship is a moving average process. The MA (q) is a representation of this process. Here a shock is simply the one step ahead forecast error. For example, a(t) is the forecast error for period t. In practical modeling, q is always a finite number. The MA (q) model of order q follows:

y(t)= a(t) + 1a(t-1) + 2a(t-2) + … + pa(t-p)                    )

y(t) is an observed value, where, a(t), a(t-1) …, a(t-p), are unobservable shocks.


Because a(t), a(t-1) …, a(t-p), are unobservable, the OLS procedure can not be used to estimate the coefficients (). The unobserved a(t} series and s must be estimated iteratively with a nonlinear procedure. Under certain conditions, however, a MA model can be expressed as an AR model.

The term moving average for these types of models is somewhat misleading.


The term used here has nothing to do with the usual moving average calculations, which is simply an arithmetic average of a fixed number of terms in a sequence by moving the terms forward or backward.

In summary the errors are the result of an estimated set of coefficients.