Using Fractional Differencing to Preserve Memory in Stationary Series
Summary
The discussion presents fractional differencing as a way to make a time series more stationary while retaining more of its long-term memory than ordinary differencing. The differencing order d is fractional, with the answer describing values between zero and one and noting that intermediate values may be useful. It illustrates the method with S&P 500 closing prices and compares augmented Dickey–Fuller test results across transformations.
In the example, ordinary daily returns pass the test but are said to lose memory. Fractional orders of 0.35 and 0.40 do not pass at conventional critical values, while 0.45 produces a statistic beyond the listed one-percent threshold; the author says this transformed series still resembles the original. These are example-specific results, not a general guarantee. The response does not provide implementation steps, discuss alternative stationarity tests, or resolve the separate issue of time-varying volatility. A second answer recommends checking whether the data follow an autoregressive model and examining residual stationarity.
Key ideas
- Fractional differencing uses a non-integer order to seek stationarity while retaining long-term memory.
- The example tests several differencing orders on S&P 500 closing-price data.
- In that example, the reported order of 0.45 passes the ADF test at the stated critical values.
- The results are illustrative and do not guarantee stationarity or solve every form of changing volatility.
- Checking for autoregressive structure and testing residuals is another suggested first step.
Tags
Full text
# Transforming a time series
# Transforming a time series
I have a time series that displays time varying volatility how would I take this time series an turn it into a more stationary process
this is what the time series looks like , if one can provide r code to help me our that would be really helpful as well
## Answer by Sergei Rodionov (score 4, accepted)
https://quant.stackexchange.com/a/61390
Fractional differentiation (or differencing) is a technique that transforms an input series to a stationary series while retaining "long-term" memory.
Consider the following example based on S&P 500 closing prices.
The daily returns pass the ADF test however the memory is now lost:
```
t-stat: -13.77
p-value: 0.00
CV 1%: -3.43
CV 5%: -2.86
CV 10%: -2.57
```
The question is, are there transformations that produce stationary series but retain most of the features of the underlying series? One of the solutions is applying differentiation with a factor that is not an integer, but a fraction. This parameter is often called `d` and is typically constrained to `[0, 1]` range, and often produces reasonable results in the `[0.25, 0.50]` range.
- FracDiff `d=0.35` ` t-stat: -1.84 p-value: 0.36 1%: -3.43 5%: -2.86 10%: -2.57 `
```
t-stat: -1.84
p-value: 0.36
1%: -3.43
5%: -2.86
10%: -2.57
```
- FracDiff `d=0.40` ` t-stat: -2.61 p-value: 0.09 1%: -3.43 5%: -2.86 10%: -2.57 `
```
t-stat: -2.61
p-value: 0.09
1%: -3.43
5%: -2.86
10%: -2.57
```
- FracDiff `d=0.45` ` t-stat: -3.50 p-value: 0.01 1%: -3.43 5%: -2.86 10%: -2.57 `
```
t-stat: -3.50
p-value: 0.01
1%: -3.43
5%: -2.86
10%: -2.57
```
At `d=0.45` we have a series that passes the ADF test and yet resembles the underlying to a significant extent.
I used a Python package for these examples, but there should be a similar implementation on r
## Answer by NicholasLP (score 1)
https://quant.stackexchange.com/a/58326
As a first step, I would check whether this time series is autoregressive, that is, of the form
$$ y_t = c + \phi_1 y_{t-1} + \ldots + \phi_p y_{t-p} + \varepsilon_t. $$
If this is the only feature of your data, then you should have stationary residuals $\varepsilon$.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.