Political Analysis Advance Access originally published online on February 24, 2007
Political Analysis 2007 15(2):97-100; doi:10.1093/pan/mpm001
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From Statistical Nuisances to Serious Modeling: Changing How We Think About the Analysis of Time-Series–Cross-Section Data
Department of Politics, New York University, New York, NY 10012-1119
e-mail: nathaniel.beck{at}nyu.edu
This special issue of Political Analysis hopefully marks a watershed in how we as a discipline think about modeling time-series–cross-section (TSCS) data. By that I mean moving beyond thinking of TSCS data as only presenting a series of technical estimation problems that are seen as violations of the Gauss-Markov assumptions, problems that can be treated by better estimation methods. The articles in this special issue point to ways of moving forward in thinking of TSCS data as providing opportunities to model intrinsic and important features of the data (while also paying attention to the various statistical issues).
The use of TSCS data in political science, and particularly comparative politics and international relations, has become increasingly common. Adolph, Butler, and Wilson (2005) estimate that about 5% of all political science articles (on JSTOR) published during the last decade used TSCS or related designs. Figure 1, reproduced from their paper, graphically shows the linear increase in the use of such designs over the last three decades.
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TSCS data became popular, particularly in political economy, because the initial complicated regressions on 15 or 20 observations were bound to be uninformative. These regressions were very sensitive to inclusion or exclusion of one particular country, or other seemingly arbitrary choices. Political economy scholars naturally gravitated toward TSCS designs that seemed to make it possible to move from only 15 or 20 data points to 20 or 30 times more than that. Fortunately, at the same time, the seminal piece by Stimson (1985) made clear the complications of such data, and the need to take those complications seriously.
Unfortunately most users of TSCS data worried mostly about violations of the Gauss-Markov assumptions (nuisances), rather than interesting features of the data that cried out for modeling. When Katz and I (Beck and Katz 1995, 1996) first published our work, we were reacting to some fixes for violations of the Gauss-Markov assumptions that had quite poor properties for the types of data being analyzed. We attempted to provide a simple methodology that would allow for some technical issues to be easily handled. One reason we wanted a simple method was so that political scientists could pay attention to the important issues of model specification that required the insights of political scientists and not statisticians or econometricians. This is not to say that the technical issues dealt with by statisticians and econometricians are unimportant, but they should not be dealt with at the expense of ignoring more substantive issues raised by TSCS designs.
Alas, things did not work out quite as simply as Katz and I might have hoped. The first article in this issue, by Wilson and Butler, makes this point clearly. They analyze a large number of published studies and find that many analysts thought that the P in PCSE stood for Panacea and not Panel, i.e., that PCSEs solved all TSCS problems, rather than simply providing correct SEs. Wilson and Butler then show how important it is to take into account various critical features of TSCS data, such as unit heterogeneity and complicated dynamics. Their conclusion is not some new method, but an injunction to take the various modeling issues presented by TSCS data seriously. Thus, I take it that Wilson and Butler have written a coda to current practice, and suggest we enter a new era where we deal with TSCS data as presenting interesting modeling decisions, based on substantive political science, not statistical nuisances that have a statistical "fix."
The second article, by Plümper and Troeger, starts with an "old" issue and then takes on a new one. It is well known that the use of fixed effects does not allow for time-invariant covariates, such as geography. They present a method for estimating coefficients and SEs for such variables. But they then take on the more interesting case of covariates that vary, but do not vary much. Slowly changing covariates, such as political institutions, are of great interest to political scientists.
It is, of course, easy to run OLS (with unit dummies) on a specification with slowly changing covariates without running afoul of the Gauss-Markov assumptions. However, the use of such a procedure typically leads to a finding that slowly changing variables have no impact. The correct interpretation is that it has no impact over time, but it still may have a huge cross-sectional impact, and it is often the latter that is of interest. But omitting the fixed effects leads to biased estimates. Plümper and Troeger show that one can deal with this issue by allowing for an estimator that is biased (and inconsistent), but often has better mean squared error properties than the unbiased OLS fixed effects estimator. They investigate conditions under which one might prefer their new estimator, and show that analysts must think about how the data were generated before choosing either the traditional OLS with fixed effects or their new estimator.
The last three papers deal more directly with issues of specification that only arise in TSCS (and related) data. The usual (implicit) assumption is that the various units (most often countries) are unrelated to each other. But there is a whole world of spatial econometrics built on the idea that "nearby" units are related in some way. This is particularly relevant in comparative political economy, where outcomes in open economies are, by definition, heavily related to outcomes in their trading partners. While this is hardly a startling observation, few have modeled this interrelationship explicitly. Franzese and Hays' contribution is to do exactly this. Through a variety of examples they show how important spatial interdependence can be, and discuss various ways of both modeling it directly and also dealing with econometric issues induced by spatial interdependence in TSCS data.
The last two papers deal with the hierarchical (or multilevel) nature of TSCS data, where we can think of data at one level (states) observed over a number of time points (years). Shor, Bafumi, Keele, and Parks use this insight to note that analysts can allow for unit heterogeneity by thinking of data as a Bayesian multilevel analyst would; this allows for a coherent discussion of modeling unit heterogeneity by the use of random effects, i.e., the units differ from each other, but only randomly. The article then shows that these Bayesian multilevel models can be estimated by Markov chain Monte Carlo methods, and that the Bayesian estimator performs well compared to a variety of classical estimators.
My article with Katz also treats TSCS data as hierarchical, although estimation is done classically (via maximum likelihood). We show that the classical model, which is usually known as the random coefficients model, has very nice theoretical and statistical properties. This model allows for all model parameters to vary randomly over units (the Shor et al. article only considers random variation of the intercepts) and thus allows for the process that translates the covariates into the dependent variable to vary randomly across units, but not to vary in a completely arbitrary way (i.e., there is some, but not complete, uniformity in the models that describe the various units). We show that the maximum likelihood estimator for this model performs well, in that it accurately estimates variability (and also does not find variability when none exists).
One particularly nice feature of the various hierarchical models is that they can allow for parameters to vary across units as a function of institutional variables that are stable within a unit. The models discussed by Franzese and Hays are also concerned with how institutions mediate between external shocks and internal outcomes. The Plümper and Troeger paper also deals with estimating the impact of institutions that often vary only very slowly over the course of observations on a country. Thus, all the final four papers can be seen as using different methods to assess the impact of differing institutions on how economic and political inputs are translated into outputs.
While I have stressed the role of substantive modeling of the features of TSCS data, this does not mean we should ignore technical issues. There is no sense producing estimates that are so inefficient that they tell us nothing, or SEs that give wildly inaccurate assessments of parameter variability. All the articles in this issue do a very nice job of discussing the estimation of models using TSCS data that are technically competent and allow for modeling the key features of TSCS data. The challenge is now to build on these endeavors, and for applied researchers to take advantage of the interesting modeling possibilities advocated by all the authors in this special issue.
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Adolph Christopher, Butler Daniel M., Wilson Sven E. (2005) Which time-series cross-section estimator should I use now? Guidance from Monte Carlo experiments. (Paper presented at the 2005 annual meeting of the American Political Science Association, Washington, DC).
Beck Nathaniel and Katz Jonathan N. (1995) What to do (and not to do) with time-series cross-section data. American Political Science Review 89:634–47.[CrossRef][Web of Science]
———. (1996) Nuisance vs. substance: Specifying and estimating time-series–cross-section models. Political Analysis 6:1–36.[Medline]
Stimson James. (1985) Regression in space and time: A statistical essay. American Journal of Political Science 29:914–47.
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