EMA Residuals, Fractional Differencing, and Variance Smoothing

Summary

The document asks how subtracting a five-second exponential moving average from a process, then summing the residuals, relates to the original process and its unknown variance. The author suggests that the residual resembles fractional differencing and that accumulation may retain correlation with the original series while smoothing its variance. These are posed as hypotheses and questions rather than established results.

No derivation, data, or empirical evidence is provided, so the proposed equivalence and variance behavior remain unverified. The effect depends on details such as the EMA weighting, sampling interval, process dynamics, and how the residuals are accumulated. The text points toward studying time-series transforms, but does not identify a specific reference or establish a trading strategy.

Key ideas

• Subtracting an exponential moving average creates a filtered residual series.
• The author asks whether this residual is equivalent to fractional differencing.
• Accumulating the residuals is hypothesized to preserve correlation with the original process while smoothing variance.
• The document provides questions and conjectures rather than a proof or empirical test.

Tags

# Process Transforms (Fractional Difference)


# Process Transforms (Fractional Difference)












Let's say I have a process $X_t$ with unknown variance process $V_t$. Then, I write $\mathrm{EMA}[X_t]$ to be the 5 sec exponential moving average of $X_t$.

Consider the transformation $$\sum (X_t-\mathrm{EMA}[X_t]).$$

What can we say about the transformation in terms of the original $X_t$ and $V_t$? (Doesn't have to be proof, but I do like proofs)

As far as I know, a process minus it's exponential moving average is equivalent to fractional differencing. Summing the differential produces a process that is highly correlated to X, and has a more normalized (smoothed) variance process.

If you could point me in the right direction to study something like this I would be very appreciative.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.