Moving Average, Exponential Smoothing, and Random Walk for Forecasting I would like to confirm my understanding. Is it true that a (simple) exponential smoothing model with alpha (smoothing constant) = 1 is the same as MA(1), which is in turn the same as a random walk model? (i.e. using only the most recent observation as the forecast for all future periods)?
 A: 
Is it true that a (simple) exponential smoothing model with alpha (smoothing constant) = 1 is the same as MA(1), which is in turn the same as a random walk model? (i.e. using only the most recent observation as the forecast for all future periods)?

No, it is not. Here are the forecasts by the three models:


*

*Simple exponential smoothing (SES; see section 7.1 of Hyndman & Athanasopoulos "Forecasting: Principles and Practice"):
$$
\hat x_{t+1|t} = \alpha x_t + \alpha(1-\alpha)x_{t-1} + \alpha(1-\alpha)^2 x_{t-2} + \alpha(1-\alpha)^3 x_{t-3} + \dots
$$

*Moving average of order 1 (MA(1); see section 8.4 of the same textbook):
$$
\hat x_{t+1|t} = \mu + \alpha\hat\varepsilon_t
$$
where $\varepsilon_t = x_t - \hat x_{t-1}$. By iterated substitution,
$$
\hat x_{t+1|t} = \mu(1-\alpha+\alpha^2-\alpha^3+\dots) + \alpha x_{t-1} - \alpha^2 x_{t-2} + \alpha^3 x_{t-3} - \dots
$$

*Random walk (RW):
$$
\hat x_{t+1|t} = x_t.
$$
As you can see, they are all quite different. It is only RW that uses the most recent observation as the forecast for all future periods. Meanwhile, both SES and MA(1) (implicitly) use a linear combination of all past observations to forecast the future.
When $\alpha=1$ for SES (but not for MA(1)), you get 


*

*SES: $\hat x_{t+1|t} = x_t$.


Hence, there SES coincides with RW but MA(1) is different. Also, even if we take $\alpha=1$ for MA(1), it still does not coincide with SES or RW. However, if you replaced MA(1) with AR(1), then your conjecture would be correct.
Edit: If by MA(1) you mean moving average of one element rather than the moving average model of order 1 (which it is the standard notation for), then indeed that forecast will coincide with the RW and SES forecasts under $\alpha=1$.
A: A moving average can have a choice of a large set of decreasing weights into the past. One such is an exponential function, and exponential functions are memoryless in the sense that they have a constant rate of decreased weightings looking retrospectively. Thus, although exponential functions find use for moving averages, power functions and many other functions do as well.
{Random walk](https://en.wikipedia.org/wiki/Random_walk) is something entirely different. Morn gives this example in Wikipedia

As you can see, these many random walks have increasing noise the further out they are extrapolated. Moving averages are smooths and have less noise than any single datum beyond the first. Moving averages are not extrapolations, they are estimates of location.
