What is a random walk and how does it relate to a unit root?

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AcadiFi · Accepted Answer

A random walk is a time series where today's value equals yesterday's value plus a random shock: x_t equals x_{t-1} plus epsilon_t. It is a special case of AR(1) with phi equal to 1.

A random walk is non-stationary because its variance grows linearly with time: Var(x_t) equals t times sigma squared. The series drifts without reverting to any mean. When phi equals 1 in an AR(1), the characteristic polynomial has a root on the unit circle — hence the term unit root. The process has infinite persistence: shocks are permanent, not transient.

A random walk with drift adds a constant: x_t equals mu plus x_{t-1} plus epsilon_t. The series drifts upward or downward on average, like a stock price with positive expected return.

Example: economist Varinder Chakraborty at Ganges Quantitative finds that monthly log-levels of USD-INR exchange rate fit a random walk with zero drift. Returns (first differences) are stationary around zero, consistent with efficient market hypothesis.

Why does this matter? Running standard OLS regression on non-stationary series produces spurious regressions — high R-squared and significant coefficients between unrelated variables purely due to their shared drifting nature. Engle and Granger's famous finding in the 1980s revolutionized macro-econometrics.

The solution: difference the series until it becomes stationary. A series requiring d differences to become stationary is called integrated of order d, written I(d). Most financial prices are I(1); returns are I(0).

What is a random walk and how does it relate to a unit root?

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