Transforming a time series

Question

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 1

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

Answer 2

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 $\varepsilon$.