What is the effect of performing first-order differencing on a random walk time series?

Performing first-order differencing on a random walk time series transforms the series into a white noise process. A random walk is characterized by the property that its future values are based on its past values plus a stochastic (random) term. Mathematically, this means that each value is dependent on the one before it, leading to non-stationarity and a trend over time.

When first-order differencing is applied to a random walk, it involves subtracting the previous value from the current value at each time point. This operation effectively eliminates the unit root characteristic of the random walk, which contributes to its non-stationary nature. The result is a series of differences that are statistically independent and identically distributed, which is the defining property of white noise.

White noise, in this context, consists of a sequence of random variables that have a mean of zero and constant variance, with no correlation between successive observations. Thus, differencing a random walk yields a series with these properties, signifying its transformation to white noise.

This understanding clarifies why the assertion that first-order differencing results in a white noise series is accurate, highlighting the impact of differencing on the characteristics of the original series.