I was reading about unit root test, when I started to get slightly confused about the setting for the Null hypothesis vs Alternative hypothesis, and so I thought of asking the experts opinion.
In the augmented Dickey-Fuller test, the null hypothesis is that there IS a unit root. My confusion comes from the fact that I think the null hypothesis should be that there is NO unit root. Allow me to explain:
The reason why I think so (and I know that I am probably wrong but I am hoping someone might point out my error), is more philosophical rather than mathematical. By this I mean: Accepting the null hypothesis, implies that what it says MIGHT be true (statistically), and rejecting the null hypothesis based on the observed data, means that there is a a very little (tiny) chance (p-value) that the null hypothesis is true, given the data, but very very unlikely (and hence we reject the null hypothesis).
But if we accept the null hypothesis, and we transform the data (by differencing it) to get rid of the unit root, then we have acted on what MIGHT be true, and as result we would be modeling a different time series.
IF (and that is a big if) the null hypothesis was that there is no unit root, then after running (my hypothetical) unit root test, I would only transform the data, if there is a very little chance that the magnitude of the root is less than 1.
Thanks for correcting my wrong thoughts in advance.
Best Answer
The null hypothesis is "the differences, $y_{t+1} - y_t$, are stationary". You're suggesting switching it to the opposite, but one won't be able to carry out such a test, as very-close-to-stationary will look just like stationary.
But what you are really saying is that one should only take differences and act as if they're stationary if there is good evidence that they are stationary.
You might satisfy this concern by being less stringent about the conclusion of non-stationarity of the differences.