Stationarity
Stationarity is an important concept in the time series literature. When a process of a specific person is stationary, this implies that all the distributional characteristics (e.g., means, variances, covariances, lagged covariances, etc.) are invariant over time. Stationarity is one of the two fundamental requirements for ergodicity, which is necessary for being able to generalize from the level of the population to the individual and vice versa.
Thinking about stationarity is important when you want to determine how and when to measure a process. Moreover, many of the models that have been developed for ILD are based on the assumption that the process is stationary, and deviations from this require alternative modeling approaches.
In this article, you will find: 1) a discussion of how stationarity is related to various aspects of your study, in particular with respect to the way you measure a process; and 2) a description of various forms of non-stationarity, and how to handle this in the analysis of the data.
1 Stationarity in your study
When a time series is stationary, then it does not matter when you measure it, as you always obtain the same distribution (assuming we have a large enough number of time points). This implies that processes that are characterized by development—such as a smooth increasing or decreasing trend over time, or sudden transitions from one phase to the next—are by definition non-stationary.
Thinking about how stationarity—or a departure from this—may play a role in the process you are interested in, is important. There are various decisions about how to measure a process that will determine whether the measured process is (approximately) stationary or not. Below you can read how the time span of your study, the time frame of your measurements, and the temporal aggregation you use play key roles in this.
1.1 Stationarity and the time span of your study
Most processes that we consider in psychology are not stationarity for the entire life span. Often there is some kind of learning curve at the start, and perhaps also a form of decline towards the end. Such developmental changes may be relatively smooth, but may also be of a more step-wise nature.
Other forms of non-stationarity may be due to repetitive patterns such as seasonality, monthly patterns, day-of-the-week effects, and circadian rhythms. Such patterns may become the actual research focus, or it may be a nuisance that needs to be dealt with the analysis of the data.
Guong is interested in mood regulation on a day-to-day basis. When considering this process for a longer period of time, such as months, he expects that the process will contain some monthly recurrent pattern, and when zooming out even further, there may also be an annual pattern related to the seasons. Hence, when the time span increases, non-stationarity is more likely to become an issue, which needs to be dealt with in the analysis phase of the study, but when considering this process for a short period of two weeks, such long-term patterns are unlikely to emerge. However, Guong is concerned that there may be a day-of-the-week effect, and he therefore decides to also record which day of the week it is, such that this can be accounted for in the analysis phase.
When instead of looking at the entire lifespan, we zoom in and focus on a (much) shorter time span, we may find that many processes seem stationarity at least for some duration of time.
1.2 Stationarity and the time frame of your measurements
The time frame that you use when measuring a process, will also determine what fluctuations you get to see, and what fluctuations are being averaged over. The longer your time frame, the more you smooth over short-term variability, thereby obscuring possible non-stationarity that characterizes the process at shorter and faster timescales.
Jessica considers to set-up a daily diary study in which she will use end-of-day measurements asking participants how tired they were during that day. Such measurements ask participants to somehow average over all the fluctuations in their tiredness throughout the day. This implies that typical daily patterns in tiredness which can be understood as time-of-day effects—such as feeling tired when waking up, less tired after having had coffee, more tired after lunch, less tired in the early evening and more tired towards bedtime—will be smoothed over. The within-day nonstationarity of momentary tiredness thereby becomes invisible, while the stationary day-to-day process of tiredness is well captured due to the time frame of her measurements.
When thinking about how to measure a process, it is important to realize that using a different time frame may have large consequences for whether or not the measured process is stationarity.
1.3 Stationarity and the temporal aggregation you use
When you have fine grained measurements, you may decide to smooth over particular intervals, for instance because you are interested in the way a process operates at a higher timescale. Hence, instead of using a longer time frame in your original measurements (as described above), you create such longer time frames afterwards by taking averages across temporally more fine-grained measurements.
Artha studies friendships between adults. She is specifically interested in the degree to which friends share emotional content about themselves in their conversations. She wonders whether emotional disclosure between two long-term friends could be considered a stationary process. When looking at a series of meetings between the friends end considering the amount of emotional disclosure by each person in each of these meetings, she expects that this will fluctuate from one meeting to the next, but not that this is non-stationary.
But when thinking about what happens within each meeting, her expectations are quite different: She expects to see different phases within the duration of a conversation. The beginning of an interaction may be characterized by more rapid changes in taking turns while the content is light; after some time this changes to one of the friends talking for much longer stretches of time while sharing more emotional content when discussing struggles at work for instance; and towards the end she expects the friends to return to a phase where the exchanges are more rapid and contain less emotional content. Hence, she actually expects that non-stationarity shows up when zooming in and adopting a more granular temporal lens.
As with the change in the time frame of the measurements, smoothing over multiple measurements to get an aggregate that pertains to a different timescale also is likely to affect the kind of non-stationarity of the process.
2 Forms of non-stationarity
There are many ways in which a process can be non-stationary. One typical example is when there is a trend over time, for instance, an increasing or decreasing trajectory. It may also be the case that there is a repetitive trend. Examples of this are: time-of-day effects in experience sampling data, day-of-the-week effects in daily diary data, and seasonal effects in quarterly data.
A second commonly discussed form of non-stationarity in the time series literature is the presence of a unit root; the most canonical example of this is a [random walk] process. Another way in which a process can be non-stationary is when the variances changes as a function of time.
Given the various ways in which a process can be non-stationary, it follows that there are multiple [tests] to investigate whether an observed time series is stationary or not. Moreover, different forms of non-stationarity can be tackled using different strategies in the analysis step of your study, for instance: a) through the inclusion of a linear or cosinor trend in the model when there is evidence for such a deterministic pattern; b) by taking difference scores when there is evidence of a unit root process; or c) through choosing a model that allows for time-varying parameters such as in a regime-switching model, or a model that allows for more smoothly changing parameter values.
In general, non-stationarity should not be considered a problem, but rather an interesting characteristic that a process may have. In your study, it may be somewhat of a nuisance that you have to deal with when analyzing the data, but it can also be the actual focus of your study to determine what kind of non-stationary pattern characterizes a person’s process over time.
3 Takeaway
While it is likely that stationarity does not uphold for very long time spans (e.g., the entire life span), it may be present when we consider a shorter time span. But this should be taken to mean that stationarity is necessarily found at shorter timescales: It may be the case that if we zoom in further and use a more fine-grained way of observing the process, we actually also see non-stationarity at very short and fast timescales.
Whether a process is (approximately) stationary very much depends on the interplay between the time span of your study, the time frame that you use for your measurements (or to what degree you average over finer grained measurements to obtain a score that concerns a longer time frame), and the number of measurements that you have. Hence, stationarity or non-stationarity is something that is inherently related to how you (plan to) measure the process. Furthermore, different forms of non-stationarity require different analysis strategies when modeling the data, and whether a measured process is stationary or not can also be a research question in itself. It is therefore important to think about the consequences of measurement choices for the stationarity of the measured process, and to think about how to deal with this in the analysis phase of your study.
4 Further reading
We have collected various topics for you to read more about below.
- [Stationarity tests]
- Stationarity versus non-stationarity
Acknowledgments
This work was supported by the European Research Council (ERC) Consolidator Grant awarded to E. L. Hamaker (ERC-2019-COG-865468).
Citation
@article{hamaker2025,
author = {Hamaker, Ellen L.},
title = {Stationarity},
journal = {MATILDA},
number = {2025-05-23},
date = {2025-05-23},
url = {https://matilda.fss.uu.nl/articles/stationarity.html},
langid = {en}
}