# 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.

## 2 Answers

This problem is encountered in quality control/statistical process control settings. There's a large literature, as you have hinted, because different parameters as estimated in various ways from different forms of sampling different distributions can be expected to vary in different ways. The purpose is to detect that variation on-line as soon as possible after it occurs without triggering too many false detections along the way. Consider using a control chart (1, 2). In your concrete situation a good choice is a combined Shewhart-CUSUM control chart.

• I should note that my problem is not 'online': I have all the data at my disposal, and do not need to e.g. identify the first time my process goes haywire. – shabbychef Nov 19 '10 at 1:36

This is a pretty general problem in time series analysis. I'd probably start by looking at some descriptive statistics like the cross-correlation to see if the samples are roughly independent over time. You could also test whether the correlation between successive samples is significant.

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.