Is it true that values of parameters do not vary from sample to sample? This is something I was told.  
However, it seems to me that parameters of a population can be functions of time, in certain situations, in which case the values of parameters could vary from sample to sample, if the parameter varies with time, unless the samples are all taken at the same time.
Or, am I missing something?  Are populations with time-varying parameters ill-defined, in some way?  If so, what are the criteria that makes a population 'well-defined'?
Thanks in advance.
 A: There are two possible sources of variation here: variation between the population and the sample, and variation of the population itself.
The former is due to randomness.
The prevailing frequentist assumption is typically that the relationship describing the latter is invariant, even if the actual values change over time. Somewhere in there, there's a "true" parameter, or at the very least a "true" function that produces parameter values. This is what the person was probably trying to tell you.
This distinction can get a little murky in a time series. You might assume first that there is a true, underlying function that describes how the time series evolves over time. So if you had infinitely-closely-spaced observations, you could figure out the function. But your observations have gaps between them -- that is, you only have a sample of the infinite population of time points -- so there will also be variation due to the fact that your sample is random, and therefore your estimates could vary owing to that randomness.
This is why we test for structural breaks. Did the underlying relationship between GDP and inflation change, or did we just get a funky spin from the great roulette wheel in the sky that dictates what values we observe?
But really this is really just a modeling choice; obviously you can never know if a true underlying value exists if you can't observe it. In Bayesian statistics, for instance, you typically assume that there is no true underlying parameter, and that instead it itself is a random variable. This really just ends up passing the buck to other assumptions down the line -- you eventually have to assume something to learn anything. But note that here the source of variation in the population is in fact due to randomness, rather than a deterministic (if unknowable) function. That is an important but subtle distinction, at least on a conceptual level.
