Testing the Random-walk Hypothesis with a Dickey-Fuller Test07 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.
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”.
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,
The variance is given by
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.”
This was my first technical post! I’ll follow up with example code and more time-series techniques.
Dicky and Fuller https://www.jstor.org/stable/2286348?&seq=1
Granger and Newbold http://wolfweb.unr.edu/~zal/STAT758/Granger_Newbold_1974.pdf