[R-sig-ME] multilevel time series?

[Malcolm Fairbrother]

Dear all,

In macro-social science, it's become fairly conventional to analyse repeated cross-sectional survey data using three-level models. Individual survey espondents (level-1) are nested in state-years (level-2), which are in turn nested within states (level-3). One big pay-off is the ability to examine how time-constant or time-varying state-level variables affect level-1 outcomes.

A co-author and I recently had a reviewer question whether this approach is adequate, however. He/she suggested that this approach could generate very misleading results, if the data are nonstationary. (We just included a linear time effect in our models.) So I'm thinking about how to proceed (and I'm not particularly knowledgeable about time series analysis). Any advice would be much appreciated. We used lme4 to fit the models in our paper, and we have several tens of thousands of respondents nested in 48 states, each observed about 15 or 16 times over about a 30-year period.

(1) Is the reviewer's query? Is he/she right to question this approach?

(2) How might we test for nonstationarity? The reviewer mentioned differencing the outcome variable, but in a multilevel context I'm not sure how to do that... Perhaps we could calculate an *aggregate* value for every state-year, and check the aggregated data for autocorrelation? My understanding is that autocorrelation across multiple lags is a strong indicator of nonstationarity (while, conversely, the absence of multiple-lag autocorrelation is almost a guarantee of stationarity). I believe this can be done with nlme, as a two-level model, with state-years nested within states.

(3) However, that approach would seem to throw away a lot of level-1 information (about individual respondents), and I'm not sure about the implications for any significance tests. An alternative approach would seem to be "multilevel time series", where autocorrelation at the *group* rather than individual/first level is specifically allowed for in the model. However, I can't find any references to R packages (or other software) that allow for the specification of, for example, AR1 processes at anything other than level-1 in multilevel models.

In short, I'd be curious to hear what people think... (especially if anyone out there happens to be a whiz at both multilevel and time series analysis). I hope I've been clear about the problem, but I'm happy to elaborate. Thanks in advance for any help.

Cheers,
Malcolm


Dr Malcolm Fairbrother
Lecturer
School of Geographical Sciences
University of Bristol