I have time-series data generated by a machine over two disjoint periods of time - roughly one month in 2016 and another month in 2018.
It is hypothesized by domain experts that at each time step $t$, an observed variable $Y^t$ can be explained by another set of observed variables, $X_1^t, \ldots, X_d^t$.
How can I test whether this process has changed over time? Note that I am not trying to test if the distribution for the variable $Y$ has changed over time. I want to test if the relationship between the $X_i$s and $Y$ has changed over time.
Suppose I fit a time-series model (e.g., a Gaussian Process) on the data from 2016 to predict $Y^t$ given $X_1^t, \ldots, X_d^t$ as a way to model the underlying process that generated $Y^t$.
The domain experts have suggested that maybe we can try to use this model to predict the variables $Y^t$s given the $X^t$s from 2018 and use the residuals somehow to infer that the model (representing the process in 2016) is or is no longer the same in 2018. I am uncertain how to continue after this point.
What I'm considering
Should I test if the residues from 2016 and 2018 are generated from the same distribution, or perform a goodness of fit test using something like Kolmogorov-Smirnov test? My concern with this approach is that the out-of-sample data from 2018 is likely to have larger errors than the in-sample training data from 2016, so this test will likely give rise to false positives. Is there any way to adjust/account for this effect?
Should I fit two models, one for 2016 and another for 2018, and use some way to test that these two models are "same" or "different"? For e.g., one possibility is to compute the KL divergence between the 2 Gaussian Processes fitted respectively on the 2016 and 2018 data. Are there any other suggestions or problem with this approach?
I saw some posts on cointegration. But I do not fully understand this concept. Is this relevant?
In general, how might one approach this type of problem? I've tried searching for this online, but maybe due to a lack of precision of my query (I'm not familiar in this area), I'm not getting many relevant results. I appreciate even simple hints/comments on the topic(s)/keywords to search, or books/papers to look through.
Kindly note that I am looking for principled (preferably statistical) approaches and not methods based on heuristics. Good examples are the answers suggesting the Chow test and its variants below.