# 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

• what is it a time series of Sep 28, 2020 at 6:37

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


• FracDiff d=0.40

  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


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

• Would you happen to know how to get the inverse of the now differentiated series? Mar 21, 2022 at 23:22

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 $$\varepsilon$$.

• I already know this time series is not stationary it has time varying volatility the question I have is how do I make it more stationary . Oct 4, 2020 at 12:40
• Normally, you do not make a time series stationary, but you look forward to making the residuals as such. As a first step, I would look into autoregression and see what the residuals look like. Oct 4, 2020 at 18:25
• Just to expound on what NicholasLP said about the residuals. Assuming that one starts off with the log(closing prices) as the series, then if one takes the first difference of that series, he-she is then calculating the returns so, in this manner, analyzing returns is somewhat similar to analyzing residuals. Feb 27, 2021 at 2:53