A Least Squares Method for Detecting Multiple Change Points in a Univariate Time Series

Detecting and interpreting influential turning points in time series data is a routine research question in many disciplines of applied social science research. Here we propose a method for identifying important turning points in a univariate time series. The most rudimentary methods are inadequate when the researcher lacks preexisting expectations or hypotheses concerning where such turning points ought to exist. Other alternatives are computationally intensive and dependent on strict model assumptions. Our method is fused LASSO regression, a variant of regularized least squares method, providing a convenient alternative for estimation and inference of multiple change points under mild assumptions. We provide two examples to illustrate the method in social science applications. First, we assessed the validity of our method by reanalyzing the Greenback prices data used in (Willard et al. in Am Econ Rev 86:1001–1017, 1996). We next used the method to identify major change points in President Clinton’s approval ratings.

[1]  Stephen P. Boyd,et al.  1 Trend Filtering , 2009, SIAM Rev..

[2]  Michael Kagay.  Presidential Address: Public Opinion and Polling during Presidential Scandal and Impeachment , 1999 .

[3]  William Lasser,et al.  The Supreme Court in Periods of Critical Realignment , 1985, The Journal of Politics.

[4]  I. B. Cohen,et al.  Revolution in science , 1985 .

[5]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[6]  A. Rinaldo,et al.  Approximate Recovery in Changepoint Problems, from $\ell_2$ Estimation Error Rates , 2016, 1606.06746.

[7]  B. Western,et al.  A Bayesian Change Point Model for Historical Time Series Analysis , 2004, Political Analysis.

[8]  G. Wahba,et al.  A NOTE ON THE LASSO AND RELATED PROCEDURES IN MODEL SELECTION , 2006 .

[9]  R. Tibshirani,et al.  Sparsity and smoothness via the fused lasso , 2005 .

[10]  D. Donoho,et al.  Atomic Decomposition by Basis Pursuit , 2001 .

[11]  S. Chib.  Estimation and comparison of multiple change-point models , 1998 .

[12]  R. Tibshirani.  Adaptive piecewise polynomial estimation via trend filtering , 2013, 1304.2986.

[13]  S. Parker,et al.  TOWARDS AN UNDERSTANDING OF “RALLY” EFFECTS PUBLIC OPINION IN THE PERSIAN GULF WAR , 1995 .

[14]  Ryan J. Tibshirani,et al.  Efficient Implementations of the Generalized Lasso Dual Path Algorithm , 2014, ArXiv.

[15]  Steven Greene,et al.  The Role of Character Assessments in Presidential Approval , 2001 .

[16]  Wayne K. Talley,et al.  Diffusion indexes and a statistical test for predicting turning points in business cycles , 1989 .

[17]  David V. Hinkley,et al.  Inference about the change-point in a sequence of binomial variables , 1970 .

[18]  A. Rinaldo.  Properties and refinements of the fused lasso , 2008, 0805.0234.

[19]  Wenjiang J. Fu,et al.  Asymptotics for lasso-type estimators , 2000 .

[20]  Inchi Hu,et al.  Invited Discussion of "Sequential Analysis: Some Classical Problems and New Challenges , 2001 .

[21]  A. Zellner,et al.  Forecasting turning points in international output growth rates using Bayesian exponentially weighted autoregression, time-varying parameter, and pooling techniques , 1991 .

[22]  P. Green.  Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[23]  H. Rosen,et al.  Turning Points in the Civil War: Views from the Greenback Market , 1995 .

[24]  Adrian F. M. Smith,et al.  Hierarchical Bayesian Analysis of Changepoint Problems , 1992 .

[25]  Gary C. Jacobson.  Impeachment Politics in the 1998 Congressional Elections , 1999 .

[26]  W. Lance Bennett,et al.  Rethinking Media Politics and Public Opinion: Reactions to the Clinton‐Lewinsky Scandal , 2001 .

[27]  S. Geer,et al.  Locally adaptive regression splines , 1997 .

[28]  Holger Hoefling.  A Path Algorithm for the Fused Lasso Signal Approximator , 2009, 0910.0526.

[29]  Sungroh Yoon,et al.  High-Dimensional Fused Lasso Regression Using Majorization–Minimization and Parallel Processing , 2013, 1306.1970.

[30]  J. Stock,et al.  Recursive and Sequential Tests of the Unit Root and Trend Break Hypothesis: Theory and International Evidence , 1990 .

[31]  Larry W. Isaac,et al.  AHISTORICISM IN TIME-SERIES ANALYSES OF HISTORICAL PROCESS: CRITIQUE, REDIRECTION, AND ILLUSTRATIONS FROM U.S. LABOR HISTORY* , 1989 .

[32]  D. Siegmund.  Boundary Crossing Probabilities and Statistical Applications , 1986 .

[33]  Joong-Ho Won,et al.  Large-Scale Structured Sparsity via Parallel Fused Lasso on Multiple GPUs , 2017 .

[34]  John Zaller.  Monica Lewinsky's Contribution to Political Science , 1998 .

[35]  R. Tibshirani,et al.  The solution path of the generalized lasso , 2010, 1005.1971.

[36]  Larry Sabato,et al.  Feeding frenzy : how attack journalism has transformed American politics , 1991 .

[37]  V. O. Key,et al.  A Theory of Critical Elections , 1955, The Journal of Politics.

[38]  Robert J. Sampson,et al.  A Life-Course View of the Development of Crime , 2005 .

[39]  H. Zou.  The Adaptive Lasso and Its Oracle Properties , 2006 .

[40]  Lynn Kuo,et al.  Bayesian Binary Segmentation Procedure for a Poisson Process With Multiple Changepoints , 2001 .

[41]  John Brehm,et al.  History, heterogeneity, and presidential approval: a modified ARCH approach , 2002 .

[42]  Walter Dean Burnham,et al.  Critical Elections: And the Mainsprings of American Politics , 1970 .