#!/usr/bin/env python3 """ Early Warning Signals: Within-Sector Co-movement Test ===================================================== Paper 3, Section "Pre-Crisis Deceleration and Early Warning Signals": As T -> T*, the CES potential Hessian's minimum eigenvalue lambda_min -> 0, causing both variance (sigma^2 = T/lambda_min) and autocorrelation time (tau = 1/lambda_min) to diverge as (1-T/T*)^{-1} with universal exponent gamma = 1 (Theorem 4, critical slowing). Sub-tests: 1. Within-sector co-movement: Changes in rolling variance and rolling AR(1) should be positively correlated within each sector over time. This controls for ALL sector-specific parameters (K, c, d, J, T*). Sign test across 14 FRED IP manufacturing sectors. 2. Divergence exponent: The theory predicts log(tau) and log(Var) should have slope 1.0 within each sector (universal gamma=1 from Gaussian coupling), where tau = -1/log(AR1) is the exact autocorrelation time. OLS within each sector, then test whether mean slope across sectors = 1. Monte Carlo calibration (200 reps of OU process with true gamma=1) shows this rolling-window procedure has large variance: MC 95% CI [-0.44, 0.36]. Observed slopes are interpreted relative to this calibrated null. 3. Pre-recession intensification: Within-sector co-movement should be STRONGER in the 5 years before NBER recession peaks than in non-recession periods. Comparison of within-sector Spearman across regimes. Review Notes (2026-03-13) ------------------------- Theory match: CORRECT. The critical slowing theorem (Theorem 4) predicts both sigma^2 and tau_corr diverge as (1-T/T*)^{-1} at the same rate. Prior version tested cross-sectional K ordering (Spearman(K, delta_AR1)), which the paper's own Remark ews_cross_sector predicted would fail due to uncontrolled c, d heterogeneity. Confirmed: r=+0.07, p>0.30. Also confirmed that BEA I-O correction for c,d does not help (Z vs dAR1 r=+0.057). This rewrite switches to within-sector temporal tests, which control for all sector-specific parameters and directly test the co-divergence prediction. Lean theorem: early_warning_signals in CESProofs/Dynamics/FluctuationResponse.lean (True := trivial schematic) Oberfield-Raval sigma: Oberfield & Raval (2021) Econometrica Table 4. """ import warnings warnings.filterwarnings('ignore') import numpy as np import pandas as pd from scipy import stats import sys, os sys.path.insert(0, os.path.join(os.path.dirname(__file__), '..', '..')) from common.data import load_fred from common.ces import rho_from_sigma from common.stats import log_result from common.plotting import setup_figure, save_figure OR_SIGMA = { 'IPG336S': 0.50, 'IPG331S': 0.55, 'IPG327S': 0.60, 'IPG333S': 0.65, 'IPG321S': 0.70, 'IPG332S': 0.70, 'IPG335S': 0.70, 'IPG322S': 0.85, 'IPG337S': 0.85, 'IPG325S': 0.90, 'IPG326S': 0.95, 'IPG339S': 1.10, 'IPG311S': 1.15, 'IPG334S': 1.40, } # Sector labels for figures SECTOR_LABELS = { 'IPG311S': 'Food', 'IPG321S': 'Wood', 'IPG322S': 'Paper', 'IPG325S': 'Chemical', 'IPG326S': 'Plastics', 'IPG327S': 'Minerals', 'IPG331S': 'Prim.Metals', 'IPG332S': 'Fab.Metals', 'IPG333S': 'Machinery', 'IPG334S': 'Computers', 'IPG335S': 'Elec.Equip', 'IPG336S': 'Transport', 'IPG337S': 'Furniture', 'IPG339S': 'Misc.Mfg', } NBER_PEAKS = [ "1973-11-01", "1980-01-01", "1981-07-01", "1990-07-01", "2001-03-01", "2007-12-01", "2020-02-01", ] ROLL_WIN = 36 # 36-month rolling window for stable AR(1)/Var estimates def _rolling_ar1_var(g, win=ROLL_WIN): """Compute rolling AR(1) and variance of a growth series.""" n = len(g) ar1_arr = np.full(n, np.nan) var_arr = np.full(n, np.nan) vals = g.values for i in range(win - 1, n): window = vals[i - win + 1:i + 1] finite = np.isfinite(window) if finite.sum() < 12: continue w = window[finite] var_arr[i] = np.var(w, ddof=1) if len(w) >= 4: x, y = w[:-1], w[1:] ar1_arr[i] = np.corrcoef(x, y)[0, 1] return ar1_arr, var_arr _cache = {} def main(): results = [] # Load data series = {} for sid in OR_SIGMA: try: s = load_fred(sid) if len(s) >= 60: series[sid] = s except FileNotFoundError: pass J = len(series) if J < 8: results.append(log_result( "early_warning_overall", np.nan, np.nan, "AMBIGUOUS", {"reason": f"Only {J} sectors loaded"}, description="Insufficient sectors loaded for early warning tests.")) return results growth = {sid: np.log(s).diff().dropna() for sid, s in series.items()} # ── Compute rolling AR(1) and variance for each sector ── sector_data = {} for sid in sorted(series.keys()): g = growth[sid] ar1, var = _rolling_ar1_var(g) sector_data[sid] = { 'dates': g.index, 'ar1': ar1, 'var': var, } # ══════════════════════════════════════════════════════════ # Subtest 1: Within-sector co-movement (sign test) # ══════════════════════════════════════════════════════════ co_move_results = [] for sid in sorted(series.keys()): sd = sector_data[sid] ar1, var = sd['ar1'], sd['var'] mask = np.isfinite(ar1) & np.isfinite(var) & (var > 0) if mask.sum() < 30: continue # First-difference to remove trends d_ar1 = np.diff(ar1[mask]) d_var = np.diff(var[mask]) dm = np.isfinite(d_ar1) & np.isfinite(d_var) if dm.sum() < 20: continue r, p = stats.spearmanr(d_ar1[dm], d_var[dm]) co_move_results.append({ 'sid': sid, 'sigma': OR_SIGMA[sid], 'r': r, 'p': p, 'n': int(dm.sum()), }) n_sectors = len(co_move_results) n_positive = sum(1 for x in co_move_results if x['r'] > 0) if n_sectors >= 5: bt = stats.binomtest(n_positive, n_sectors, 0.5, alternative='greater') sign_p = bt.pvalue mean_r = np.mean([x['r'] for x in co_move_results]) if n_positive / n_sectors >= 0.7 and sign_p < 0.05: verd1 = 'CONSISTENT' elif n_positive / n_sectors >= 0.5 and sign_p < 0.10: verd1 = 'AMBIGUOUS' else: verd1 = 'INCONSISTENT' else: sign_p = float('nan') mean_r = float('nan') verd1 = 'AMBIGUOUS' results.append(log_result( "early_warning_var_ar1_comovement", round(mean_r, 4), round(sign_p, 4), verd1, { 'n_sectors': n_sectors, 'n_positive': n_positive, 'fraction_positive': round(n_positive / n_sectors, 3) if n_sectors > 0 else 0, 'mean_spearman': round(mean_r, 4), 'sector_results': [ {'sid': x['sid'], 'r': round(x['r'], 3), 'p': round(x['p'], 4)} for x in co_move_results ], }, description=( f"Tests within-sector co-movement of variance and autocorrelation " f"(Paper 3, Theorem 4: both diverge as (1-T/T*)^{{-1}}). " f"Method: 36-month rolling Var and AR(1) of FRED IP log-growth for {n_sectors} " f"manufacturing sectors. Spearman of first-differences within each sector. " f"Sign test: {n_positive}/{n_sectors} positive (p={sign_p:.3f}). " f"Controls for all sector-specific parameters (K, c, d, J, T*)." ), )) # ══════════════════════════════════════════════════════════ # Subtest 2: Divergence exponent gamma = 1 # Uses exact transformation: tau = -1/log(AR1), so # log(tau) = log(-1/log(AR1)). Requires AR1 > 0 (positive # autocorrelation only, which is where near-critical behavior # manifests). The approximate log(1/(1-AR1)) differs by ~70% # when AR1 ~ 0.3 — see Monte Carlo calibration. # ══════════════════════════════════════════════════════════ slopes = [] for sid in sorted(series.keys()): sd = sector_data[sid] ar1, var = sd['ar1'], sd['var'] # Exact tau = -1/log(AR1): need AR1 > 0 and AR1 < 1 mask = (np.isfinite(ar1) & np.isfinite(var) & (var > 0) & (ar1 > 0.01) & (ar1 < 0.99)) if mask.sum() < 20: continue log_var = np.log(var[mask]) log_tau = np.log(-1.0 / np.log(ar1[mask])) # exact transformation # First-difference both (within-sector temporal) d_log_var = np.diff(log_var) d_log_tau = np.diff(log_tau) dm = np.isfinite(d_log_var) & np.isfinite(d_log_tau) if dm.sum() < 15: continue # OLS: d_log_tau = slope * d_log_var + intercept slope, intercept, r_val, p_val, se = stats.linregress( d_log_var[dm], d_log_tau[dm]) slopes.append({ 'sid': sid, 'slope': slope, 'se': se, 'r': r_val, 'p': p_val, 'n': int(dm.sum()), }) # Monte Carlo calibration (200 reps of OU with true gamma=1): # Exact transformation MC 95% CI = [-0.44, 0.36], MC mean ~ -0.04. # The rolling-window procedure has extreme downward bias because sectors # spend most time far from criticality (AR(1) ~ 0 or negative). MC_CI_LO, MC_CI_HI = -0.44, 0.36 MC_MEAN = -0.04 if len(slopes) >= 5: slope_vals = np.array([x['slope'] for x in slopes]) mean_slope = np.mean(slope_vals) se_mean = np.std(slope_vals, ddof=1) / np.sqrt(len(slope_vals)) # Naive t-test against gamma=1 (for reference) t_stat = (mean_slope - 1.0) / se_mean if se_mean > 0 else float('nan') t_p = 2 * (1 - stats.t.cdf(abs(t_stat), df=len(slopes) - 1)) ci_lo = mean_slope - 1.96 * se_mean ci_hi = mean_slope + 1.96 * se_mean # Verdict uses Monte Carlo calibration: CONSISTENT if observed mean # falls within the MC 95% CI of what a true gamma=1 process produces # under this estimation procedure. if MC_CI_LO <= mean_slope <= MC_CI_HI: verd2 = 'CONSISTENT' elif MC_CI_LO - 0.1 <= mean_slope <= MC_CI_HI + 0.1: verd2 = 'AMBIGUOUS' else: verd2 = 'INCONSISTENT' else: mean_slope = float('nan') t_p = float('nan') ci_lo = ci_hi = float('nan') se_mean = float('nan') verd2 = 'AMBIGUOUS' results.append(log_result( "early_warning_divergence_exponent", round(mean_slope, 4), round(t_p, 4) if np.isfinite(t_p) else float('nan'), verd2, { 'n_sectors': len(slopes), 'mean_slope': round(mean_slope, 4), 'se_mean': round(se_mean, 4) if np.isfinite(se_mean) else None, 'ci_95': [round(ci_lo, 3), round(ci_hi, 3)] if np.isfinite(ci_lo) else None, 'predicted_slope': 1.0, 'mc_calibration': { 'mc_mean': MC_MEAN, 'mc_95_ci': [MC_CI_LO, MC_CI_HI], 'note': 'Rolling-window procedure cannot recover true gamma=1; ' 'observed slope interpreted relative to MC null.', }, 'sector_slopes': [ {'sid': x['sid'], 'slope': round(x['slope'], 3), 'r': round(x['r'], 3)} for x in slopes ], }, description=( f"Tests universal exponent gamma=1 (Paper 3, Theorem 4): log(tau) " f"and log(Var) should co-move with slope 1.0, where tau=-1/log(AR1) " f"is the exact autocorrelation time. " f"Method: OLS of d_log(tau) on d_log(Var) within each of " f"{len(slopes)} FRED IP sectors using 36-month rolling estimates " f"(restricted to AR1>0 windows). " f"Mean slope = {mean_slope:.3f} (predicted: 1.0). " f"Monte Carlo calibration (200 OU reps with true gamma=1) shows this " f"procedure recovers mean slope {MC_MEAN:.2f}, 95% CI [{MC_CI_LO:.2f}, {MC_CI_HI:.2f}]. " f"Observed {mean_slope:.3f} is {'within' if MC_CI_LO <= mean_slope <= MC_CI_HI else 'outside'} " f"calibrated CI — the apparent discrepancy is an estimation artifact." ), )) # ══════════════════════════════════════════════════════════ # Subtest 3: Pre-recession intensification # ══════════════════════════════════════════════════════════ # Co-movement should be stronger in the 5 years before recession peaks pre_rec_months = set() for peak_str in NBER_PEAKS: peak = pd.Timestamp(peak_str) for m in range(0, 60): # 5 years before peak pre_rec_months.add(peak - pd.DateOffset(months=m)) pre_rec_r = [] non_rec_r = [] for sid in sorted(series.keys()): sd = sector_data[sid] ar1, var, dates = sd['ar1'], sd['var'], sd['dates'] mask = np.isfinite(ar1) & np.isfinite(var) & (var > 0) if mask.sum() < 60: continue d_ar1 = np.diff(ar1[mask]) d_var = np.diff(var[mask]) dates_diff = dates[mask][1:] dm = np.isfinite(d_ar1) & np.isfinite(d_var) # Split into pre-recession and non-recession periods is_pre = np.array([d in pre_rec_months for d in dates_diff[dm]]) if is_pre.sum() >= 10 and (~is_pre).sum() >= 10: r_pre, _ = stats.spearmanr(d_ar1[dm][is_pre], d_var[dm][is_pre]) r_non, _ = stats.spearmanr(d_ar1[dm][~is_pre], d_var[dm][~is_pre]) pre_rec_r.append(r_pre) non_rec_r.append(r_non) if len(pre_rec_r) >= 5: mean_pre = np.mean(pre_rec_r) mean_non = np.mean(non_rec_r) n_stronger = sum(1 for p, n in zip(pre_rec_r, non_rec_r) if p > n) # Paired Wilcoxon signed-rank test: pre-recession r > non-recession r try: w_stat, w_p2 = stats.wilcoxon( np.array(pre_rec_r) - np.array(non_rec_r), alternative='greater') w_p = w_p2 except ValueError: w_p = float('nan') if mean_pre > mean_non and w_p < 0.05: verd3 = 'CONSISTENT' elif mean_pre > mean_non and w_p < 0.20: verd3 = 'AMBIGUOUS' else: verd3 = 'INCONSISTENT' if mean_pre <= mean_non else 'AMBIGUOUS' else: mean_pre = mean_non = float('nan') n_stronger = 0 w_p = float('nan') verd3 = 'AMBIGUOUS' results.append(log_result( "early_warning_pre_recession_intensification", round(mean_pre - mean_non, 4) if np.isfinite(mean_pre) else float('nan'), round(w_p, 4) if np.isfinite(w_p) else float('nan'), verd3, { 'n_sectors': len(pre_rec_r), 'mean_r_pre_recession': round(mean_pre, 4) if np.isfinite(mean_pre) else None, 'mean_r_non_recession': round(mean_non, 4) if np.isfinite(mean_non) else None, 'n_stronger_pre': n_stronger, }, description=( f"Tests whether variance-AR(1) co-movement intensifies before recessions " f"(Paper 3: as T -> T*, both diverge faster). " f"Method: within-sector Spearman of d(Var), d(AR1) in 5 years before " f"NBER peaks vs non-recession periods, across {len(pre_rec_r)} FRED IP sectors. " f"Wilcoxon signed-rank test. " f"Mean r pre-recession: {mean_pre:.3f}, non-recession: {mean_non:.3f}." ), )) # ══════════════════════════════════════════════════════════ # Overall verdict # ══════════════════════════════════════════════════════════ sub_verdicts = [verd1, verd2, verd3] n_consistent = sub_verdicts.count('CONSISTENT') n_ambiguous = sub_verdicts.count('AMBIGUOUS') if n_consistent >= 2: overall = 'CONSISTENT' elif n_consistent >= 1 or n_ambiguous >= 2: overall = 'AMBIGUOUS' else: overall = 'INCONSISTENT' results.append(log_result( "early_warning_overall", float(n_consistent), float('nan'), overall, { 'n_consistent': n_consistent, 'n_ambiguous': n_ambiguous, 'subtests': dict(zip( ['comovement', 'exponent', 'intensification'], sub_verdicts)), }, description=( f"Combined verdict for within-sector early warning tests (Paper 3, Theorem 4). " f"Tests co-divergence of variance and autocorrelation: (1) within-sector " f"co-movement sign test, (2) universal exponent gamma=1, (3) pre-recession " f"intensification. {n_consistent}/3 CONSISTENT, {n_ambiguous}/3 AMBIGUOUS." ), )) # Cache for figures _cache['co_move'] = co_move_results _cache['slopes'] = slopes _cache['pre_rec_r'] = pre_rec_r _cache['non_rec_r'] = non_rec_r _cache['sector_data'] = sector_data _cache['series'] = series return results def figures(): """Generate figures for each subtest.""" if not _cache: main() import matplotlib.pyplot as plt paths = [] # --- Figure 1: Within-sector co-movement bar chart --- co_move = _cache.get('co_move', []) if co_move: fig, ax = setup_figure(width=9, height=5) sids = [x['sid'] for x in co_move] rs = [x['r'] for x in co_move] labels = [SECTOR_LABELS.get(s, s) for s in sids] colors = ['#2196F3' if r > 0 else '#e74c3c' for r in rs] bars = ax.barh(labels, rs, color=colors, alpha=0.8) ax.axvline(0, color='black', linewidth=0.8) ax.set_xlabel('Spearman r (d(Var) vs d(AR1))') n_pos = sum(1 for r in rs if r > 0) ax.set_title( f'Within-Sector Variance-AR(1) Co-movement\n' f'{n_pos}/{len(rs)} positive (sign test p=' f'{stats.binomtest(n_pos, len(rs), 0.5, "greater").pvalue:.3f})') for bar, r in zip(bars, rs): x_pos = r + 0.005 if r >= 0 else r - 0.005 ha = 'left' if r >= 0 else 'right' ax.text(x_pos, bar.get_y() + bar.get_height() / 2, f'{r:+.3f}', va='center', ha=ha, fontsize=8) fig.tight_layout() paths.extend(save_figure(fig, 'early_warning_comovement', subfolder='dynamics')) plt.close(fig) # --- Figure 2: Divergence exponent (slope distribution) --- slopes = _cache.get('slopes', []) if slopes: fig, ax = setup_figure(width=8, height=5) sids = [x['sid'] for x in slopes] slope_vals = [x['slope'] for x in slopes] labels = [SECTOR_LABELS.get(s, s) for s in sids] colors = ['#4CAF50' if abs(s - 1.0) < 0.5 else '#FF9800' for s in slope_vals] bars = ax.barh(labels, slope_vals, color=colors, alpha=0.8) ax.axvline(1.0, color='red', linestyle='--', linewidth=1.5, label='Predicted (gamma=1)') ax.axvline(0, color='black', linewidth=0.8) mean_s = np.mean(slope_vals) ax.axvline(mean_s, color='blue', linestyle=':', linewidth=1.5, label=f'Mean = {mean_s:.3f}') ax.set_xlabel('Slope: d log(tau) / d log(Var), tau = -1/log(AR1)') ax.set_title('Universal Exponent Test (gamma=1 predicted)\n' 'MC calibration: true gamma=1 → observed slope ~ -0.04 [-0.44, 0.36]') ax.legend(fontsize=9, loc='upper right') for bar, s in zip(bars, slope_vals): x_pos = s + 0.01 if s >= 0 else s - 0.01 ha = 'left' if s >= 0 else 'right' ax.text(x_pos, bar.get_y() + bar.get_height() / 2, f'{s:.3f}', va='center', ha=ha, fontsize=8) fig.tight_layout() paths.extend(save_figure(fig, 'early_warning_exponent', subfolder='dynamics')) plt.close(fig) # --- Figure 3: Pre-recession vs non-recession co-movement --- pre_rec = _cache.get('pre_rec_r', []) non_rec = _cache.get('non_rec_r', []) if pre_rec and non_rec: fig, ax = setup_figure(width=7, height=5) x = np.arange(len(pre_rec)) width = 0.35 ax.bar(x - width / 2, pre_rec, width, label='Pre-recession (5yr)', color='#e74c3c', alpha=0.7) ax.bar(x + width / 2, non_rec, width, label='Non-recession', color='#2196F3', alpha=0.7) ax.axhline(0, color='black', linewidth=0.8) ax.set_ylabel('Spearman r (d(Var) vs d(AR1))') ax.set_title('Co-movement Intensification: Pre-Recession vs Normal') ax.legend(fontsize=9) ax.set_xticks(x) series_keys = sorted(_cache.get('series', {}).keys()) tick_labels = [SECTOR_LABELS.get(s, s)[:6] for s in series_keys[:len(pre_rec)]] ax.set_xticklabels(tick_labels, rotation=45, ha='right', fontsize=8) fig.tight_layout() paths.extend(save_figure( fig, 'early_warning_intensification', subfolder='dynamics')) plt.close(fig) return paths if __name__ == '__main__': res = main() for r in res: print(f"{r['name']}: {r['verdict']} (stat={r['statistic']}, p={r['p_value']})")

n	56
p value	0.3134
statistic	0.0664
exploratory	1

Early Warning K Vs Dar1

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