Is the order important when applying unit root tests? I'm using KPSS unit root test with a vector of 700 observations.
The test return a p-value of 0.1 (so it seems stationary), but if I reverse the vector the KPSS test tells me the vector is not stationary. With "reverse" I mean 1, 2, 3, 4 becomes 4, 3, 2, 1. (Obviously those are not the numbers I have.)
So the questions is: Is the order important?
 A: It is a time series data that you are using for unit root test. Therefore, order is usually important.
There are some stochastic processes though for which the property of "time reversibility "  also holds. 
According to Wikipedia on Random Walk
, "A random walk on a graph is a very special case of a Markov chain. Unlike a general Markov chain, random walk on a graph enjoys a property called time symmetry or reversibility. Roughly speaking, this property, also called the principle of detailed balance, means that the probabilities to traverse a given path in one direction or in the other have a very simple connection between them (if the graph is regular, they are just equal). This property has important consequences." 
While testing for unit root though, there can be a drift term or a deterministic trend as well. Then it seems that reversibility may or may not hold. 
In general though, it seems to be the case that you need to know whether the nature of the series is suited to allow for the reversibility to hold. There are some statistical tests for that. 
