Testing the Random-walk Hypothesis with a Dickey-Fuller Test

07 Oct 2015

I came across this excellent article on using NIST tests designed for random number generators to investigate the random-walk hypothesis in the stock market.

It reminded me of other ways to test for random walks from Econometrics.

The Dickey-Fuller Test

Suppose you model a time series with an autoregressive model.

\[y_t = \beta y_{-t} + u_t\]

where $y_t $ is the value of $y$ at time $t$, $\beta$ is a constant and $u_t$ is the random error term. Then the processes is called a random-walk if $\beta=1$, because then it’s next move only depends on the random error term. This hypothesis test is conducted under the null-hypothesis that $\beta=1$. If we fail to reject the null hypothesis, it is said that there is a “unit autoregressive root” or just “unit root”.

This is a very simplistic test. Some other tests for a unit root are the Augmented Dickey-Fuller and the Phillips–Perron tests.

Estimation Problems

A unit root can cause estimation problems because it implies that the underlying distribution is non-stationary, since the variance increases over time. If $\beta = 1$ in the following model,

\[y_t = \beta y_{-t} + u_t = y_{t-1} + u_t\]

then

\[y_t = y_0 \sum_{j=1}^t u_t\]

The variance is given by

\[\operatorname{Var}(y_t) = \sum_{j=1}^t \sigma^2=t \sigma^2\]

The variance diverges to infinite with time. When the stochastic process is non-stationary, OLS can produce invalid results despite high t-statistics and $R^2$. Granger and Newbold called such results “spurrious regressions.”