Fractional Differencing Derivation Walkthrough (FD Part 2)
Just a quick warning before I start, this post is going to be math heavy. Those who are not brave enough to traverse these waters, be forewarned! Let's get right to it: To recap, last time I talked about a few basic statistical concepts regarding time series. Stationarity, Memory and reconciling them both using an idea called fractional differencing. This post walks through how we do this mathematically and gets down to the brass tacks'. Before starting, I need to explain a few intermediate steps to avoid confusion. 1. The Taylor series expansion for the function f(x) = (1+x)^d for any complex number d, is the binomial series. Don't worry too much about the complex number part. Essentially, what this is saying is the following: We have a function (1+x)^d that's a bit messy to deal with so we use a mathematical technique called Taylor-series approximation to get a bit of insight into how we construct this function as a sum of polynomials. We'll come...