Back to the Future: Modeling Time Dependence in Binary Data

1. Back to the Future: Modeling Time Dependence in Binary Data David B. Carter & Curtis Signorino — University of Rochester Abstract Since BKT, the use of time dummies or splines has become the standard method to model temporal dependence in binary data. We show that there are problems with each of these approaches, especially in the case of time dummies. We propose a simpler alternative: using t, t2, and t3, which serves as a Taylor series approximation to the hazard. This cubic polynomial is trivial to implement and avoids problems with time dummies such as complete separation and problems with spline such as interpretation or knot selection. It also accommodates non-proportional hazards in a much simpler way than either time dummies or splines. We show via monte carlo analysis that our method performs as well as splines and better than time dummies. We also demonstrate this method with reanalyses of numerous empirical studies such as Oneal and Russett (1997) and Crowley and Skocpol (2001). • Solution: t, t2, & t3 • Taylor Series Approximation to hazard. • Simple to implement in both estimation and interpretation of hazard. • Easy to account for non-proportional hazards • How does it perform? • Empirical Examples • Oneal and Russett (1997) find that trade has significant negative effect on conflict. • BKT show that this finding weakens considerably when time taken seriously. • t, t2, & t3 produces same finding. • Crowley and Skocpol (2001) study of associational formation, 1860-1930. • Use logit with time dummies. • Find effects of the Civil War have greatest impact, not modernization variables. • Replicate their results with t, t2, & t3. • In both cases, Vuong test does not discriminate between methods, Clarke test shows t, t2, & t3 does better. • Non-Proportional Hazards • Relatively easy to deal with. • Interact regressor of interest with t, t2, & t3. • Test with Likelihood Ratio Test • Pension $ Per Pensioner highly non-proportional, as we expected. • Time Dummies • A dummy variable for each individual duration. • Maximum Duration of 40 =>40 time dummies • Perfect and quasi-perfect separation issues. • Drop “perfectly predicting” time dummies and some observations. • Monte Carlo results show severity of problem under different hazard shapes. • Monte Carlo Analysis • Assume data-generating process of logit with time dummies: • Pr(y=1)=1/1+exp(-(Xß+kt)) • Decreasing, Increasing, and Non-Monotonic Hazards Decreasing Hazard Increasing Hazard Non-Monotonic Hazard • Introduction • How do political phenomena change over time? • Democracies more stable the longer established? • Nations less likely to go to war the longer at peace? • Existing Approach • Scholars increasingly analyzing binary version of event history data: does event occur in some discrete slice of time. • Beck, Katz, and Tucker (1998) advocate logit with either time dummies or splines. • Researchers have taken BKT seriously, but have not necessarily taken time seriously. • Additionally, existing methods have problems, especially time dummies. • Splines • Splines generally not well understood. • Many different kinds of splines: quadratic spline, cubic spline, natural spline, b-spline. • Issue of knot selection. • Have to know shape of hazard to intelligently select knots. • However, do not know shape of hazard until post-estimation, which relies on knot selection. • Default: Use BKT’s 1,4, & 7. • Conclusions • Never use time dummies. • Knot selection not innocuous (1,4, & 7 used). • t, t2, & t3 performs very well in both Monte Carlos and empirical examples. • On average, t, t2, & t3 recovers true hazard in Monte Carlos. • Does as well or better in empirical examples. • Take Time Seriously, but do it with t, t2, & t3.