Why EMA Residual Volatility Depends on the Time Series
Summary
The document asks whether the standard deviation of a time series minus its exponential moving average can be determined from the series’ standard deviation and the EMA smoothing parameter alone. The author observes that the residual spread tends toward the original series’ spread for a smoothing parameter near zero and toward zero near one. Plots across several series suggest a common curve for longer samples, while short four-point examples look much less regular.
The key issue is that standard deviation alone does not specify serial dependence or the sample’s realized pattern, both of which affect how closely an EMA tracks observations. The post asks whether an additional statistic could produce a deterministic formula or whether the residual spread must be estimated, but it presents no derivation or final answer. Its visual observations are suggestive rather than formal evidence, and finite-sample behavior and the EMA initialization convention are not addressed.
Key ideas
- The standard deviation of the difference between a series and its EMA is the quantity under investigation.
- The author observes residual spread approaching the series spread at low smoothing parameter and zero at the high end.
- Sample length appears to affect how regular the plotted relationship looks.
- The post does not establish a formula, and the series’ dependence structure may matter beyond its standard deviation.
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Full text
# Standard deviation of the difference between a time series and its EMA?
# Standard deviation of the difference between a time series and its EMA?
I have a time series $Q={\{q_t\}}$ of known standard deviation $\sigma$, and its EMA of parameter $\alpha$ : $\{EMA_t(\alpha)\}$.
My question is : I'm looking for a formula that would give the standard deviation $\sigma_{\alpha}$ of the time series made by the difference between $Q$ and its EMA for any $\alpha$, depending of the known Q standard deviation $\sigma$ : $$ \sigma_{\alpha} = f_\sigma(\alpha) \space\space? $$
### Here is what I did so far
Intuitively, one could think that, the shorter is the EMA (= the closer is $\alpha$ to 1), the more it will fit the shape of the time series, so the less the stddev of the difference between the TS and its EMA, $\sigma_\alpha$, will be. Here is a time series with $\sigma = 2.82$ and their EMAs with different values for alpha to illustrate this intuition :
To get a further intuition of what that function could be, I plotted $\sigma_\alpha$ as a function of $\alpha$ :
As expected, when $\alpha$ is close to 0, $\sigma_\alpha$ approches $\sigma$ ; and when $\alpha$ is close to 1, the $EMA = Q$ : the standard deviation of the difference is naturally 0.
When I plot that with many different $Q$, this curve looks like pretty much the same, so it appears there is some deterministic function f, but I'm not able to find its expression. I plotted the derivative of $f$ for many different Q, and unfortunalty, it has a different shape depending on Q. Here are some plot of $(\sigma_\alpha)'$ :
In order to try to understand better the problem, I plotted $\alpha_\sigma$ and its derivative for a Q containing a very small number of points (4). The shape of f and its derivative looks now much more random... Here are $\alpha_\sigma$ and $(\alpha_\sigma)'$ for different Q containing 4 points :
So finally, it looks like the more the time series has points, the more $f$ will tend to be something deterministic. Though, I can't see intuitively what can be the missing information different than Q standard deviation $\sigma$ that make f stochastic, especially with a little number of points.
Am I missing some constant that I could compute from $Q$, whose depend an existing deterministic $f$ (appart from $\sigma$) ? Or is $f$ eventually stochastic and I can only have an estimation of $\sigma_\alpha$ by doing a regression ?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.