stat.MESep 28, 2026
SaveForecast-Necessary Causal Discovery for Nonlinear Political Panel Data: Feedback, Functional Form, and the Dynamics of Democratization
Organizations: Department of Political Science, University of Notre Dame, Notre Dame, IN, USA · Kellogg Institute, University of Notre Dame, Notre Dame, IN, USA · Lucy Family Institute for Data & Society, University of Notre Dame, Notre Dame, IN, USA
Abstract
A non-significant coefficient in a dynamic panel model need not imply the absence of a relationship. It may instead reflect heterogeneous effects averaged toward zero, reciprocal dynamics overlooked by a recursive specification, or relationships masked by the omission of correlated covariates. Standard linear estimators cannot distinguish among these possibilities. We develop an inferential workflow for political panel data that resolves this ambiguity by combining flexible autoregressive estimation, forecast-necessity testing, functional characterization, and same-data linear benchmarking. The workflow first identifies relationships required for out-of-sample prediction, then characterizes their functional form across political contexts, and finally, distinguishes differences arising from estimator flexibility from those due to model specification. Applied to the causal sequence model of democratization on the V-Dem panel of 113 countries, the workflow reproduces the model's central finding - the protective belt of civil society, the rule of law, and institutionalized parties - while recovering reciprocal relationships from democracy to its institutional supports that a linear model cannot detect. Most importantly, three weak published direct effects, of which two are null, and one is marginally significant, receive three different diagnoses: one dissolves under the full specification, one reflects heterogeneous effects averaged toward zero, and one was masked by the reduced variable set. The workflow corrects the published record in both directions, removing one relationship and recovering two. More broadly, the workflow provides a framework for evaluating dynamic political theories under a model class capable of representing nonlinear and reciprocal mechanisms while preserving relationship-level interpretation and explicit inferential standards.
Figures & tables
Figure 1: The workflow applied to the causal sequence model. The questions at left state what each stage determines, independent of the application; the quantities at right report each stage’s output for the present one. Each stage’s output for the present application appears at right: the exogeneity screen admits five structural variables as causes only; flexible estimation yields a causal score matrix over 396 candidate relationships; forecast-necessity testing retains 160; the automatic threshold retains 84; and functional characterization assigns each retained relationship a sign, a functional form, and a stability measure across democratic contexts. A linear benchmark, estimated on identical data and variables, enters at the final stage: the diagnosis of each relationship follows from the comparison between the two model classes.
| Baseline ( Coppedge et al., 2022 ) | This article | |
|---|---|---|
| Data | V-Dem panel, 113 countries, 1900–2001 | Same, standardized |
| Outcome | Electoral Democracy Index | Same |
| Variables | 10 of 23 candidates | All 23 consistently observed |
| Estimator | Linear recursive path model | Neural additive autoregression |
| Functional form | Constant coefficients | Flexible per relationship |
| Reciprocal effects | Lagged cross-paths; few significant | Estimated jointly; tested for necessity |
Table 1: Design of the baseline model and of the present re-estimation. The data, outcome, and conceptual organization of the variables are held constant; the estimator, variable set, lag depth, and retention criterion change.
| Quantity | Value |
|---|---|
| Candidate relationships (23 candidate causes 18 dynamic outcomes, self-effects excluded) | 396 |
| Forecast-necessary relationships (Diebold–Mariano test) | 160 |
| Dominant-necessary relationships (automatic threshold, ): the estimated graph | 84 |
| Positively / negatively signed net effects | 47 / 37 |
| Relationships with a stable sign across democratic contexts | 84 of 84 |
| Relationships nonlinear in at least one democratic regime | 73 of 84 |
Table 2: The estimated system at a glance. All quantities are from the seed-42 reference run; ensemble ranges appear in I , and the complete forecast comparison in H .
Figure 2: Signed net effects and functional forms across the estimated system. Cell color reports the direction and magnitude of each source variable’s net effect on an outcome, aggregated from the regime-specific response curves: blue denotes positive effects, red negative, and grey cells are uncharacterized. Cell text reports the consensus functional form across democratic contexts (LIN linear, THR threshold, SAT saturating, SIG sign-changing, OTH other); where forms differ across regimes, the modal form is shown, with the per-regime record in Table 3 and in G . Two system-level patterns are visible at a glance: the uniformly positive row for polyarchy, summarizing the reciprocal dynamics of Section 5.2 , and the uniformly negative row for agricultural income, summarizing the indirect suppression established in Section 5.3 .
Figure 3: Three weak direct effects, three diagnoses. Each panel shows the estimated response of polyarchy to one variable, computed separately within low, middle, and high democratic contexts. The direct effect of agricultural income (left) is flat throughout: no direct relationship survives once intermediate channels enter the model, and the baseline’s small negative estimate dissolves. The effect of GDP per capita (center) reverses direction across the range of development within every context, the pattern that a constant coefficient averages toward zero. The effect of literacy (right) is positive throughout and was masked in the baseline by the reduced variable set: once the specification expands, even a linear model detects it. In the published model these three direct effects appear as marginally significant, null, and null; the response curves show one effect dissolving and two emerging, a correction running in both directions.
Figure 4: The dominant-necessary causal network (84 edges, seed 42). Nodes are arranged in the causal-sequence layout of the baseline model’s path diagram: structural and distal causes at the left, then intermediate and proximate/episodic variables, the protective belt, and the outcome (polyarchy) at the right. Structural variables appear as sources only; edge width is proportional to the NAVAR causal score; blue arrows denote positive ICE-derived effects, red negative, and grey mixed or sign-changing. Edge inclusion is read directly from the GMM-filtered adjacency matrix ( w_active_gmm.csv ).
| Relationship | Causal score (sign) | Retention | Baseline coefficient | Functional form | Diagnosis |
| Protective belt polyarchy | |||||
| Civil society (CSO) | 0.073 | 5/5 | 0.0515*** | linear sat. | Replicated |
| Rule of law | 0.039 | 5/5 | 0.0483*** | linear sat. | Replicated |
| Party institutionalization | 0.033 | 5/5 | 0.097 (n.s.) | linear sat. | Strengthened |
| Modernization variables polyarchy | |||||
| Literacy rate | 0.040 | 4/5 | 0.0002 (n.s.) | linear sat. | Masked by specification |
Table 3: Relationship-level comparison: flexible workflow vs. linear baseline. Causal scores are unsigned contribution magnitudes from the dominant-necessary graph; signs and functional forms are derived from the functional characterization. Baseline coefficients from Coppedge et al. (2022) , Table 8.1 and Figure 8.2. The two columns are not on a common scale; the comparison is between verdicts, not magnitudes.
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
| Variable | Code | Source | Category |
|---|---|---|---|
| Electoral democracy (polyarchy) | v2x_polyarchy | V-Dem v9 | Outcome |
| CSO participatory environment | v2csprtcpt | V-Dem v9 | Protective belt |
| Rule of law / state capacity | v2clrspct | V-Dem v9 | Protective belt |
| Institutionalized parties | v2xps_party | V-Dem v9 | Protective belt |
| Majoritarian electoral system | elecsys | V-Dem v9 | Intermediate |
| Anti-system movement | v2csantimv | V-Dem v9 | Proximate |
Table 4: The 23 variables of the final model. Categories follow Coppedge et al. (2022) .
Figure 5: The structural exogeneity screen. Bars show the intraclass correlation coefficient of each model variable, the fraction of its variance lying between countries rather than within them. The five variables above the 0.88 threshold (red) have essentially no within-country movement over the observation period and are designated structurally exogenous: they enter the analysis as candidate causes but are never modeled as outcomes. The threshold falls within a wide empty interval, between 0.856 (party institutionalization) and 0.987 (Protestant population share), so the classification does not depend on its exact placement. Slow-moving institutional stocks (party institutionalization, rule of law, literacy) sit high among the dynamic variables; episodic variables (campaigns, growth) sit lowest.
Figure 6: Estimated causal score matrix. Each cell reports how actively a source variable (rows) moves the prediction of an outcome variable (columns), before necessity testing. Shaded columns mark the five structurally exogenous variables, which enter as candidate causes only. The concentration of scores in a small share of cells reflects the sparsity that the penalty induces; the columns for polyarchy and its institutional supports carry the strongest signal. The complete matrix is reported numerically in Table 5 .
Figure 7: Forecast-necessity testing across the 396 candidate relationships. Left: one-sided Diebold–Mariano -values for the exact ablation of each source (rows) from each target equation (columns); green cells are significant at , blank cells are excluded self-effects and masked structural targets, and -values above 0.2 share the terminal color. Right: each candidate’s DM -value against its causal score magnitude, with the dashed line at and -values floored at for display; blue points are the 160 forecast-necessary relationships. Necessity is not a monotone function of score: candidates with some of the largest scores fall below the line, and candidates with small scores clear it.
Figure 8: The automatic dominance criterion. Left: a two-component mixture fitted to the logarithm of mean loss differentials across the 160 forecast-necessary relationships separates a near-zero cluster from a substantive one; the dashed line marks their intersection, , which serves as the retention threshold. Right: the same 160 relationships ranked by loss differential, with 84 above the threshold. No researcher-specified parameter enters the decision.
Figure 9: ICE response curves for the nine edges in Table 1 of the main text, computed from the pipeline’s ICE output. Each panel overlays the low (solid blue), mid (dashed grey), and high (dotted red) democratic regimes, defined as terciles of the lag-1 standardized polyarchy level of the prediction window. Horizontal axis: source value on the normalized 81-point grid; vertical axis: mean change in the predicted target. Curves are plotted as estimated, with no smoothing beyond the model itself.
| Model | Masked val. MSE |
|---|---|
| OLS (23 variables, ) | 0.2289 |
| Ridge ( , grid-selected) | 0.2270 |
| NAVAR (seed 42) | 0.2032 |
Table 6: Masked validation MSE, dynamic targets, terminal 10% split.
| Source Polyarchy | (classical ) | (HC1 Wald) | Lag-sum |
| Civil society (CSO) | 0.0001 | 0.0040 | |
| Nonviolent campaign | 0.0001 | 0.0001 | |
| Literacy rate | 0.0001 | 0.0021 | |
| Party institutionalisation | 0.0003 | 0.0717 | |
| Strict presidentialism | 0.0006 | 0.1283 | |
| Rule of law | 0.0021 | 0.0538 |
Table 7: Linear edge verdicts. Block test that all eight lag coefficients of the source are zero in the target equation; lag-sum is the sum of the eight coefficients.
| Edge | Retention | Score mean (sd) | Score range |
|---|---|---|---|
| Civil society (CSO) Polyarchy | 5/5 | 0.072 (0.001) | [0.070, 0.073] |
| Rule of law Polyarchy | 5/5 | 0.041 (0.001) | [0.039, 0.042] |
| Party institutionalization Polyarchy | 5/5 | 0.031 (0.001) | [0.030, 0.033] |
| Literacy rate Polyarchy | 4/5 | 0.039 (0.001) | [0.038, 0.041] |
| Log GDP per capita Polyarchy | 4/5 | 0.034 (0.002) | [0.033, 0.037] |
| Nonviolent campaign Polyarchy | 3/5 | 0.031 (0.001) | [0.030, 0.031] |
Table 8: Seed-ensemble robustness of headline edges (5 seeds). Score statistics are computed over seeds in which the edge is retained.