The Dickey-Fuller Test

Now we know what integration is and why it might be a problem. The next question that we can ask ourselves is somehow evident: “How do we know if a series is actually integrated?” The answer: The Dickey-Fuller test!

The standard Dickey-Fuller test

Remember our definition of random walk:

In the above equation,$\epsilon_t$is Gaussian random noise. We have added a constant in front of $y_{t-1}$, $\beta$. The reason why we didn’t see it there before is that, in a random walk, $\beta$ is simply 1. Now, however, it’s a very good time to introduce it, as it will help us find out whether or not $y_t$ is an $I(1)$ process.

The idea behind the standard Dickey-Fuller (DF) test is pretty simple: Is $y_t$ a random walk? To answer the question, we can estimate the model above and formulate a hypothesis test for the value of $\beta$ ...