# How to test for parameter stationarity?

Suppose I have time series observations from distributions drawn from some population. That is, I observe $X_{t,i}$ for $t=1,2,...,T,$ and $i=1,2,...,n$, where I believe that $X_{t,i}$ have pdf $f(\theta_i)$. (I have some idea about the distribution of the $\theta_i$, but that may not be important here.) I have some sample statistic which is a good estimator of $\theta_i$ given some observations.

However, there is the suspicion that, in fact, the $\theta_i$ are not stationary, rather the observations come from $f(\theta_{t,i})$, where the $\theta_{t,i}$ are changing slowly over time. How can I test this, either by a formal hypothesis test or an 'eyeball' test? The amount of data available in the time domain is not so great (i.e. $T$ is not so large), thus partitioning the time domain and computing the sample estimate on each partition would only be advisable for a small number (say 5) of partitions (because otherwise the standard error of the estimate is too great). However, the number of series, $n$, is largeish, say 10,000.

I realize there are a number of gaps in this question, e.g. how the $\theta_{t,i}$ might be varying with time, the standard error of the parameter estimator, etc. However, any hints would be appreciated.

To be concrete, one could think of the $X_{t,i}$ as being normally distributed with mean $\theta$ and standard deviation $1$, and the sample statistic is the sample mean.

Or you could go the model-fitting route in which case one simple thing to do is to fit an auto-regressive model with some order k and then do model comparison versus the static model. If you assume that $\theta$ just follows a Gaussian random walk, then model you're describing is exactly a Kalman filter. So that might be another thing to look at.