Dynamic Models for Compositional Data

Time Series, Panel Data & the “Dynamic PIE” Framework

Whitten · Lipsmeyer · Philips · Rutherford & Collaborators

2026-07-31

What Is Compositional Data?

Multivariate vectors of proportions that sum to a constant (1 or 100%). Each observation lives on the (J−1)-simplex.

Core properties

  • Constant-sum constraint — any increase in one component must be offset by decreases in others
  • Relative information — only ratios between parts carry meaning; absolute scale is irrelevant
  • Zero-sum trade-offs — modeling must respect the mutually exclusive, competitive structure of the parts

Examples

  • Party vote shares (Labour / Conservative / Liberal Democrat)
  • Government budget allocations across spending categories
  • Income quintile shares
  • Public opinion composition on Most Important Problem

Why Standard Methods Fall Short

  1. Redundant degree of freedom — with J components summing to 1, J−1 determine the last; including all J creates perfect collinearity
  2. Constraint violations — OLS predictions are unconstrained and can exceed 1 or go negative
  3. Ignored trade-off structure — modeling each category against an omnibus “rest” misrepresents competition across all categories
  4. Static-only priors — earlier compositional methods (Katz & King 1999; Tomz, Tucker & Wittenberg 2002) had no dynamic structure, no lagged effects, no error-correction

Core Methodology: The Dynamic PIE Framework

Step 1 — Additive log-ratio (ALR) transformation (Aitchison 1982)

Express J categories as J−1 log ratios relative to a baseline category J:

s_{tj} = \ln\!\left(\frac{y_{tj}}{y_{tJ}}\right), \quad j = 1, \ldots, J-1

Places the composition in unrestricted real space.

Step 2 — Error-correction model within seemingly unrelated regression (SUR)

\Delta s_{tj} = \alpha\, s_{t-1,j} + \gamma\, \Delta x_t + \delta\, x_{t-1} + \varepsilon_{tj}

Estimated as a joint system via SUR to capture cross-equation error correlation.

Step 3 — Simulation-based inference

Draw 1,000 coefficient vectors from the multivariate normal posterior (CLARIFY; Tomz, Wittenberg & King 2003), apply counterfactual shocks, back-transform to proportions with confidence intervals.

Step 4 — Back-transformation

Inverse ALR returns all predictions to the simplex (positive, summing to 1). Displayed as change-from-baseline or predicted-proportion trajectories.

Key Papers: Time-Series Foundations (2015–2016)

Philips, Rutherford & Whitten (2015)
Electoral Studies 39: 264–274
First application in political science. Models monthly UK party vote shares using ECM-SUR.

Philips, Rutherford & Whitten (2016a)
American Journal of Political Science 60(1): 268–283
The definitive methods paper. Addresses two prior gaps: non-dynamic compositional models and graphical tools that broke down for J > 3. Illustrates with UK party support (2004–2010) and US federal budget categories (1947–2009).

Philips, Rutherford & Whitten (2016b)
Stata Journal 16(3): 662–677
Introduces dynsimpie, wrapping ECM-SUR estimation and CLARIFY simulation into a single command with built-in graphics.

Key Papers: Extension to Panel / TSCS Data (2017–2019)

Lipsmeyer, Philips & Whitten (2017)
Journal of European Public Policy 24(6): 912–930
First compositional panel application. Shows how immigration stocks alter welfare state spending trade-offs across European countries.

Lipsmeyer, Philips, Rutherford & Whitten (2019)
Political Science Research & Methods 7(3): 523–540
Extends Dynamic PIE from single time series to time-series cross-sectional (TSCS) data. Innovations:

  • Unit fixed effects for stable cross-sectional differences
  • Contextual variables for cross-unit variation in composition levels
  • Spatial weighting matrix for cross-unit spillovers

Illustrated with U.S. state budget data (50 states, developmental / allocational / redistributive / educational spending).

Recent Advances (2020–2024)

Jung, Souza, Philips, Rutherford & Whitten (2020)
Stata Journal 20(3): 584–603
Updates dynsimpie with cfbplot, effectsplot, and dynsimpiecoef postestimation commands; expanded model specification options.

Philips, Souza & Whitten (2020)
Political Science Research & Methods 8(3): 509–525
Applies the compositional panel model to cross-national income quintile share data; traces globalization effects on distributional trade-offs.

Kagalwala, Philips & Whitten (2021)
Social Science Quarterly 102(4): 1534–1552
Demonstrates the dynamic compositional approach on U.S. income share data; gains over piecemeal regression of inequality indices.

Lipsmeyer, Philips, Rutherford & Whitten (2024)
Social Science Quarterly 105(4): 934–947
Economic shocks and partisan composition of state budgets.

Jokinsky, Lipsmeyer, Philips, Williams & Whitten (2024)
Social Science Quarterly 105(4): 913–933
Compositional modeling of Most Important Problem public opinion.

Eastman, Junqueira, Kagalwala, Philips & Whitten (2024)
Social Science Quarterly 105(4): 965–979
Introduces compositional volatility as a new estimand — modeling variance, not just mean, of simplex proportions over time.

Computing Implementation: dynsimpie

Installation

* Prerequisite
net install clarify, from(https://gking.harvard.edu/clarify)

* From SSC
ssc install dynsimpie

* Latest version
net install dynsimpie, ///
  from(https://raw.githubusercontent.com/andyphilips/dynsimpie/master/)

Basic usage

tsset time

dynsimpie dv1 dv2 dv3,   ///
  ivs(x1 x2 x3)          ///  independent variables
  shockvar(x1)            ///  variable to shock
  shockval(1)             ///  shock size (1 s.d.)
  time(20) sims(1000)

cfbplot         // change-from-baseline plot
effectsplot     // predicted proportions over time
dynsimpiecoef   // SUR coefficient plot

What the command handles automatically

  • ALR transformation of raw proportions or percents
  • Construction of lags and first differences
  • Joint SUR estimation via sureg
  • CLARIFY simulation and back-transformation to the simplex
  • 90% and 95% confidence intervals

Source: https://github.com/andyphilips/dynsimpie · GPL v3

Applications

Electoral politics

  • UK monthly party vote shares (Labour / Conservative / Liberal Democrat), 2004–2010
  • Multi-party electoral competition across European democracies
  • Effects of economic evaluations and leader approval on vote-share composition

Government budgeting

  • US federal budget: defense, welfare, social security, interest, other (1947–2009)
  • US state budgets: developmental / allocational / redistributive / educational spending (50 states)
  • Effects of partisanship, ideology, and economic shocks on budget trade-offs

Comparative policy

  • European welfare state spending under immigration and integration pressure
  • Globalization effects on cross-national income quintile shares
  • Local government budgets by gender of chief executive (Funk & Philips 2019)

Public opinion

  • Most Important Problem composition (economy / social / foreign policy)
  • Nativism and its consequences for multi-party support compositions

Three Illustrative Cases

Case 1 — UK party vote shares (time series)
Dependent variable: monthly proportions, Labour / Conservative / Liberal Democrat
Shock: +1 s.d. increase in Liberal Democrat leader approval
Finding: Immediate significant rise in Lib Dem support (~25%); symmetric declines in Labour and Conservative; long-run plateau significantly above baseline
Source: Philips, Rutherford & Whitten (2016a)

Case 2 — US federal budget (time series)
Dependent variable: annual shares — defense, welfare, social security, interest, other, 1947–2009
Shock: −1 s.d. decrease in policy mood liberalism
Finding: Conservative mood reduces defense share (counterintuitive); significant or borderline-significant changes across all categories, demonstrating full interdependence
Source: Philips, Rutherford & Whitten (2016a)

Case 3 — US state budgets (panel)
Dependent variable: developmental, allocational, redistributive, educational spending shares, 50 states
Shock: shift in partisan composition of state government
Finding: Partisanship significantly drives trade-offs across all four categories; spatial spillovers between neighboring states; unemployment and per capita income matter as contextual variables
Source: Lipsmeyer, Philips, Rutherford & Whitten (2019)

Takeaways & Future Directions

What the programme delivers

  • A complete toolkit: theory, Stata software, time-series and TSCS estimators, and extensive applied examples — all with open replication data
  • ALR transformation + ECM-SUR + CLARIFY simulation is the canonical pipeline
  • Results are interpretable as simplex-constrained predicted trajectories following a counterfactual shock

Scope beyond political science

The framework applies wherever proportions compete over time: economics (portfolio and budget shares), sociology (population composition), ecology (species abundance), public health (resource allocation).

Open research frontiers

  • Bayesian estimation of compositional time series (cf. DARMA — Zheng et al.)
  • Cointegration testing for log-ratio series
  • Machine-learning extensions for high-dimensional compositions
  • Integration with large-N panel data and non-linear dynamics