In recent years, political science has seen a boom in the use of sophisticated methodological tools for time series analysis. One such tool is the general error correction model (GECM), originally introduced to political scientists in the pages of this journal over 20 years ago (Durr 1992; Ostrom and Smith 1992) and re-introduced by De Boef and Keele (2008), who advocate its use for a wider set of time series data than previously considered appropriate. Their article has proven quite influential, with numerous papers justifying their methodological choices with reference to De Boef and Keele's contribution.

Grant and Lebo (2016) take issue with the increasing use of the GECM in political science and argue that the methodology is widely misused and abused by practitioners. Given the recent surge of research conducted using error correction methods, there is every reason to take their suggestions seriously and provide a fuller discussion of the points they raise in their paper. The present symposium serves such a role. It consists of Grant and Lebo's critique, a detailed response by Keele, Linn, and Webb (2016b), and shorter comments by Esarey (2016), Freeman (2016), and Helgason (2016). Finally, Lebo and Grant (2016) and Keele, Linn, and Webb (2016a) reflect on the contributions made in the symposium, as well as discuss outstanding issues.

While traditionally considered for non-stationary and cointegrated data, DeBoef and Keele suggest applying a General Error Correction Model (GECM) to stationary data with or without cointegration. The GECM has since become extremely popular in political science but practitioners have confused essential points. For one, the model is treated as perfectly flexible when, in fact, the opposite is true. Time series of various orders of integration–stationary, non-stationary, explosive, near- and fractionally integrated–should not be analyzed together but researchers consistently make this mistake. That is, without equation balance the model is misspecified and hypothesis tests and long-run-multipliers are unreliable. Another problem is that the error correction term's sampling distribution moves dramatically depending upon the order of integration, sample size, number of covariates, and the boundedness of Yt. This means that practitioners are likely to overstate evidence of error correction, especially when using a traditional t-test. We evaluate common GECM practices with six types of data, 746 simulations, and five paper replications.

In this article, we highlight three points. First, we counter Grant and Lebo's claim that the error correction model (ECM) cannot be applied to stationary data. We maintain that when data are properly stationary, the ECM is an entirely appropriate model. We clarify that for a model to be properly stationary, it must be balanced. Second, we contend that while fractional integration techniques can be useful, they also have important weaknesses, especially when applied to many time series typical in political science. We also highlight two related but often ignored complications in time series: low power and overfitting. We argue that the statistical tests used in time-series analyses have little power to detect differences in many of the sample sizes typical in political science. Moreover, given the small sample sizes, many analysts overfit their time-series models. Overfitting occurs when a statical model describes random error or noise instead of the underlying relationship. We argue that the results in the Grant and Lebo replications could easily be a function of overfitting.

Two contributions in this issue, Grant and Lebo and Keele, Linn, and Webb, recommend using an ARFIMA model to diagnose the presence of and estimate the degree of fractional integration, then either (i) fractionally differencing the data before analysis or, (ii) for cointegrated variables, estimating a fractional error correction model. But Keele, Linn, and Webb also present evidence that ARFIMA models yield misleading indicators of the presence and degree of fractional integration in a series with fewer than 1000 observations. In a simulation study, I find evidence that the simple autodistributed lag model (ADL) or equivalent error correction model (ECM) can, without first testing or correcting for fractional integration, provide a useful estimate of the immediate and long-run effects of weakly exogenous variables in fractionally integrated (but stationary) data.

Cointegration was introduced to our discipline by Renée Smith and Charles Ostrom Jr. and by Robert Durr more than two decades ago at political methodology meetings in Washington University�St. Louis and Florida State University. Their articles, along with comments by Neal Beck and John T. Williams, were published in a symposium like this one in the fourth volume of Political Analysis. Keele, Lin, and Webb (2016; hereafter KLW) and Grant and Lebo (2016; hereafter GL) show how, in the years that followed, cointegration was further evaluated by political scientists, and the related idea of error correction subsequently was applied.

Have the last twenty-plus years witnessed significant progress in modeling nonstationary political time series? In some respects, the answer is yes. The present symposium represents progress in understanding equation balance, analyzing bounded variables, and decomposing short- and longterm causal effects. In these respects KLW's and GL's articles deserve wide dissemination. But KLW and GL leave important methodological issues unresolved. They do not address some critical methodological challenges. From a historical perspective, the present symposium shows that we have made relatively little progress in modeling nonstationary political time series.

Grant and Lebo (2016) and Keele, Linn, and Webb (2016) provide diverging recommendations to analysts working with short time series that are potentially fractionally integrated. While Grant and Lebo are quite positive about the prospects of fractionally differencing such data, Keele, Linn, and Webb argue that estimates of fractional integration will be highly uncertain in short time series. In this study, I simulate fractionally integrated data and compare estimates from the general error correction model (GECM), which disregards fractional integration, to models using fractional integration methods over thirty-two simulation conditions. I find that estimates of short-run effects are similar across the two models, but that models using fractionally differenced data produce superior predictions of long-run effects for all sample sizes when there are no short-run dynamics included. When short-run dynamics are included, the GECM outperforms the alternative model, but only in time series that consist of under 250 observations.

The papers in this symposium agree on several points. In this article, we sort through some remaining areas of disagreement and discuss some of the practical issues of time series modeling we think deserve further explanation. In particular, we have five points: (1) clarifying our stance on the general error correction model in light of the comments in this issue; (2) clarifying equation balance and discussing how bounded series affects our thinking about stationarity, balance, and modeling choices; (3) answering lingering questions about our Monte Carlo simulations and exploring potential problems in the inferences drawn from long-run multipliers; (4) reviewing and defending fractional integration methods in light of the questions raised in this symposium and elsewhere; and (5) providing a short practical guide to estimating a multivariate autoregressive fractionally integrated moving average model with or without an error correction term.

This issue began as an exchange between Grant and Lebo (2016) and ourselves (Keele, Linn, and Webb 2016) about the utility of the general error correction model (GECM) in political science. The exchange evolved into a debate about Grant and Lebo's proposed alternative to the GECM and the utility of fractional integration methods (FIM). Esarey (2016) and Helgason (2016) weigh in on this part of the debate. Freeman (2016) offers his views on the exchange as well. In the end, the issue leaves readers with a lot to consider. In his comment, Freeman (2016) argues that the exchange has produced little significant progress because of the contributors' failures to consider a wide array of topics not directly related to the GECM or FIM. We are less pessimistic. In what follows, we distill what we believe are the most important elements of the exchange–the importance of balance, the costs and benefits of FIM, and the vagaries of pre-testing.

The most commonly used statistical models of civil war onset fail to correctly predict most occurrences of this rare event in out-of-sample data. Statistical methods for the analysis of binary data, such as logistic regression, even in their rare event and regularized forms, perform poorly at prediction. We compare the performance of Random Forests with three versions of logistic regression (classic logistic regression, Firth rare events logistic regression, and L1-regularized logistic regression), and find that the algorithmic approach provides significantly more accurate predictions of civil war onset in out-of-sample data than any of the logistic regression models. The article discusses these results and the ways in which algorithmic statistical methods like Random Forests can be useful to more accurately predict rare events in conflict data.

Sensitivity diagnostics has recently been put high on the agenda of methodological research into Qualitative Comparative Analysis (QCA). Existing studies in this area rely on the technique of exhaustive enumeration, and the majority of works examine the reactivity of QCA either only to alterations in discretionary parameter values or only to data quality. In this article, we introduce the technique of combinatorial computation for evaluating the interaction effects between two problems afflicting data quality and two discretionary parameters on the stability of QCA reference solutions. In this connection, we challenge a hitherto unstated assumption intrinsic to exhaustive enumeration, show that combinatorial computation permits the formulation of general laws of sensitivity in QCA, and demonstrate that our technique is most efficient.

Much of the data used in social science is aggregated into spatial units, even if the analysis itself does not explicitly incorporate that information. A key concern with such aggregation, however, is that changes in the units of aggregation themselves cause difficulty in comparing data gathered on the old boundaries and the new boundaries. Such changes present serious concerns to researchers who may exclude observations or cases due to a lack of comparable units or omit certain key variables. While geographers have long examined this problem and created methods of projecting data from one spatial unit into another, known as areal interpolation, it is telling that a recent article notes the difficulty for researchers in implementing even the most basic solutions without relying heavily on programming skills or proprietary software.