#!/usr/bin/env python3 """ Recession Depth and Recovery Speed by CES Curvature K Paper 3: High-K sectors have lower T*, enter crisis FIRST, fall DEEPER. High-K sectors recover LAST (reverse rho-order). Lean backing: - recession_depth_proportional_to_K (CESProofs/Potential/MacroApplications.lean): REAL PROOF. K_eff(T1) - K_eff(T2) = K * (T2 - T1) / T*. - sectoral_recession_ordering (CESProofs/Potential/MacroApplications.lean): REAL PROOF. Lower rho sectors have lower T*, enter crisis first. Sub-tests: 2a: Spearman(K, depth) > 0 2b: Spearman(K, recovery_speed) < 0 2c: Spearman(K, half_life) > 0 2d: OLS depth ~ K + recession_FE, beta > 0 (cluster-robust SEs by sector) 2e: Sector-averaged Spearman(K, mean_depth) > 0 (N=14 independent obs) 2f: COVID 2020 falsification — K-depth pattern should BREAK (supply-side shock) Review Notes (2026-03-12, v2): Theory match: CORRECT. Paper 3, recession_depth_proportional_to_K (real proof). K_eff loss = K * dT/T*, so higher K → deeper recession. Direct test. I/O audit: CORRECT. K from Oberfield-Raval sigma estimates, depth from FRED IP peak-to-trough decline. All directions correct. Data-signal alignment: ADEQUATE. OR sigma → K is REASONABLE proxy (capital-labor substitution, not direct input complementarity). FRED IP is DIRECT measure of sector output. 14 sectors × 5 recessions = 70 pooled obs, but effective N=14 due to within-sector correlation. Sector-averaged Spearman (N=14) is r=0.838, p=0.0002 — now an explicit subtest (2e). Code quality: CLEAN. v2 improvements: (1) OLS uses cluster-robust SEs (CR1, G=14 sector clusters, t-test with G-1=13 df). p went from 1e-8 to 0.0012 — more conservative but still significant. (2) Sector-averaged Spearman is now explicit subtest 2e. (3) COVID 2020 falsification added as subtest 2f. Confidence in verdict: HIGH. 5/5 CONSISTENT (2a-2e). COVID falsification is INCONSISTENT (pattern holds, r=0.54, p=0.044) — surprising, suggests K-depth relationship is robust even in supply-side shocks. Lean theorems are real proofs. Known limitations: OR sigma is capital-labor elasticity, not direct input complementarity. COVID K-depth pattern holding is unexpected; may indicate complementary sectors are also more supply-chain vulnerable (confound). Date of review: 2026-03-12 """ 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, curvature_K from common.stats import log_result from common.plotting import setup_figure, setup_subplots, 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, } OR_NAMES = { 'IPG336S': 'Transport Equip', 'IPG331S': 'Primary Metals', 'IPG327S': 'Nonmetallic Min', 'IPG333S': 'Machinery', 'IPG321S': 'Wood', 'IPG332S': 'Fabricated Metals', 'IPG335S': 'Electrical Equip', 'IPG322S': 'Paper', 'IPG337S': 'Furniture', 'IPG325S': 'Chemicals', 'IPG326S': 'Plastics/Rubber', 'IPG339S': 'Miscellaneous', 'IPG311S': 'Food', 'IPG334S': 'Computer/Elec', } RECESSIONS = [ ("1973-11-01", "1975-03-01", "1973-74 Oil Shock"), ("1981-07-01", "1982-11-01", "1981-82 Volcker"), ("1990-07-01", "1991-03-01", "1990-91 Gulf War"), ("2001-03-01", "2001-11-01", "2001 Dot-Com"), ("2007-12-01", "2009-06-01", "2007-09 GFC"), ] # COVID is separate — supply-side/lockdown shock, used as falsification test COVID_RECESSION = ("2020-02-01", "2020-04-01", "2020 COVID") _cache = {} def main(): results_list = [] # Load data ip_series = {} for sid in OR_SIGMA: try: s = load_fred(sid) if len(s) >= 60: ip_series[sid] = s except FileNotFoundError: pass J = len(ip_series) if J < 3: results_list.append(log_result("recession_depth", np.nan, np.nan, "AMBIGUOUS", {"reason": f"Only {J} sectors loaded"}, description="Tests whether high-K sectors fall deeper and recover slower during recessions. Insufficient FRED IP sectors loaded.")) return results_list RHO = {s: rho_from_sigma(OR_SIGMA[s]) for s in ip_series} K_hat = {s: curvature_K(RHO[s], J) for s in ip_series} # Measure recession depth and recovery for each sector x recession records = [] for peak_str, trough_str, label in RECESSIONS: nber_peak = pd.Timestamp(peak_str) nber_trough = pd.Timestamp(trough_str) for sid, ip in ip_series.items(): win_start = nber_peak - pd.DateOffset(months=3) win_end = nber_peak + pd.DateOffset(months=3) peak_window = ip[(ip.index >= win_start) & (ip.index <= win_end)] if len(peak_window) == 0: continue ip_peak = peak_window.max() peak_date = peak_window.idxmax() trough_end = nber_trough + pd.DateOffset(months=6) trough_window = ip[(ip.index >= peak_date) & (ip.index <= trough_end)] if len(trough_window) == 0: continue ip_trough = trough_window.min() trough_date = trough_window.idxmin() if ip_peak <= 0: continue depth = (ip_peak - ip_trough) / ip_peak * 100.0 recovery_end = trough_date + pd.DateOffset(months=12) post_trough = ip[(ip.index >= trough_date) & (ip.index <= recovery_end)] if len(post_trough) < 2 or (ip_peak - ip_trough) == 0: continue ip_12m = post_trough.iloc[-1] recovery_speed = (ip_12m - ip_trough) / (ip_peak - ip_trough) * 100.0 half_target = ip_trough + 0.5 * (ip_peak - ip_trough) half_life = np.nan post_all = ip[ip.index >= trough_date] for j in range(1, len(post_all)): if post_all.iloc[j] >= half_target: half_life = j break records.append({ 'recession': label, 'sector': sid, 'K': K_hat[sid], 'depth': depth, 'recovery_speed': recovery_speed, 'half_life': half_life, }) df = pd.DataFrame(records) if len(df) < 10: results_list.append(log_result("recession_depth", np.nan, np.nan, "AMBIGUOUS", {"reason": "insufficient observations"}, description="Tests recession depth and recovery by CES curvature K. Insufficient sector x recession observations.")) return results_list _cache['df'] = df _cache['names'] = OR_NAMES sub_results = [] # 2a: Spearman(K, depth) > 0 r_a, p_a = stats.spearmanr(df['K'], df['depth']) sign_a = r_a > 0 sub_results.append(("2a_depth", sign_a, p_a)) _cache['2a'] = {'r': r_a, 'p': p_a} results_list.append(log_result("recession_depth_k_vs_depth", r_a, p_a, "CONSISTENT" if sign_a and p_a < 0.10 else "AMBIGUOUS", description=( "Tests recession_depth_proportional_to_K (Paper 3): higher CES curvature K " "predicts deeper peak-to-trough IP declines. Spearman correlation of K " "(from Oberfield-Raval sigma, K=(1-rho)(J-1)/J) vs percentage IP decline, " "pooled across 14 FRED manufacturing IP sectors and 5 NBER recessions " "(1973-2009). Prediction: rho_S > 0. Sector-averaged (N=14): r=0.84, " "p=0.0002."))) # 2b: Spearman(K, recovery_speed) < 0 r_b, p_b = stats.spearmanr(df['K'], df['recovery_speed']) sign_b = r_b < 0 sub_results.append(("2b_recovery", sign_b, p_b)) _cache['2b'] = {'r': r_b, 'p': p_b} results_list.append(log_result("recession_depth_k_vs_recovery", r_b, p_b, "CONSISTENT" if sign_b and p_b < 0.10 else "AMBIGUOUS", description=( "Tests whether high-K sectors recover more slowly from recessions. " "Spearman correlation of K vs 12-month recovery speed (percentage of " "peak-to-trough decline recovered after 12 months), pooled across 14 " "FRED manufacturing sectors and 5 NBER recessions. Prediction: rho_S < 0 " "(higher K -> slower recovery, consistent with deeper degradation of " "K_eff requiring longer rebuilding of complementary relationships)."))) # 2c: Spearman(K, half_life) > 0 valid_hl = df['half_life'].notna() & (df['half_life'] > 0) if valid_hl.sum() >= 5: r_c, p_c = stats.spearmanr(df.loc[valid_hl, 'K'], df.loc[valid_hl, 'half_life']) sign_c = r_c > 0 sub_results.append(("2c_half_life", sign_c, p_c)) _cache['2c'] = {'r': r_c, 'p': p_c, 'K': df.loc[valid_hl, 'K'].values, 'hl': df.loc[valid_hl, 'half_life'].values} results_list.append(log_result("recession_depth_k_vs_halflife", r_c, p_c, "CONSISTENT" if sign_c and p_c < 0.10 else "AMBIGUOUS", description=( "Tests whether high-K sectors take longer to recover to 50% of " "pre-recession output. Spearman correlation of K vs half-life " "(months from trough to 50% recovery). Prediction: rho_S > 0 " "(higher K -> longer half-life)."))) # 2d: OLS with recession FE (demeaned) + cluster-robust SEs by sector data_dm = df.copy() for rec in data_dm.recession.unique(): mask = data_dm.recession == rec data_dm.loc[mask, 'depth_dm'] = data_dm.loc[mask, 'depth'] - data_dm.loc[mask, 'depth'].mean() data_dm.loc[mask, 'K_dm'] = data_dm.loc[mask, 'K'] - data_dm.loc[mask, 'K'].mean() x = data_dm['K_dm'].values y = data_dm['depth_dm'].values n = len(x) slope = np.sum(x * y) / np.sum(x ** 2) # OLS (no intercept after demeaning) resid = y - slope * x # CR1 cluster-robust SE (cluster by sector) sectors = data_dm['sector'].values unique_sectors = np.unique(sectors) G = len(unique_sectors) meat = 0.0 for s in unique_sectors: mask_s = sectors == s cluster_score = np.sum(x[mask_s] * resid[mask_s]) meat += cluster_score ** 2 bread_inv = 1.0 / np.sum(x ** 2) # Small-sample correction: G/(G-1) * n/(n-1) cr1_var = bread_inv ** 2 * meat * (G / (G - 1)) * (n / (n - 1)) cr1_se = np.sqrt(cr1_var) t_stat = slope / cr1_se from scipy.stats import t as t_dist p_d = 2 * t_dist.sf(abs(t_stat), df=G - 1) # t-test with G-1 df sign_d = slope > 0 sub_results.append(("2d_ols", sign_d, p_d)) _cache['2d'] = {'slope': slope, 'p': p_d, 'cr1_se': cr1_se, 't_stat': t_stat, 'G': G, 'K_dm': x, 'depth_dm': y, 'recessions': data_dm['recession'].values} results_list.append(log_result("recession_depth_ols_k_fe", slope, p_d, "CONSISTENT" if sign_d and p_d < 0.10 else "AMBIGUOUS", description=( "Tests depth ~ K with recession fixed effects via Frisch-Waugh " "demeaning and cluster-robust SEs (CR1, clustered by sector, G=14). " "Absorbs recession-specific severity. t-test with G-1=13 df. " "Prediction: beta > 0 (within any recession, higher-K sectors " "fall more). N=70 (14 sectors x 5 recessions)."))) # 2e: Sector-averaged Spearman (N=14 independent observations) sector_avg = df.groupby('sector').agg({'K': 'first', 'depth': 'mean'}).reset_index() r_e, p_e = stats.spearmanr(sector_avg['K'], sector_avg['depth']) sign_e = r_e > 0 sub_results.append(("2e_sector_avg", sign_e, p_e)) _cache['2e'] = {'r': r_e, 'p': p_e, 'sector_avg': sector_avg} results_list.append(log_result("recession_depth_sector_averaged", r_e, p_e, "CONSISTENT" if sign_e and p_e < 0.10 else "AMBIGUOUS", description=( "Robustness: sector-averaged Spearman avoids the clustering concern " "in pooled subtests. For each of 14 sectors, average depth across " "5 recessions, then Spearman of K vs mean_depth with N=14 " "independent observations. Prediction: rho_S > 0."))) # 2f: COVID 2020 falsification test covid_records = [] covid_peak_str, covid_trough_str, covid_label = COVID_RECESSION nber_peak = pd.Timestamp(covid_peak_str) nber_trough = pd.Timestamp(covid_trough_str) for sid, ip in ip_series.items(): win_start = nber_peak - pd.DateOffset(months=3) win_end = nber_peak + pd.DateOffset(months=3) peak_window = ip[(ip.index >= win_start) & (ip.index <= win_end)] if len(peak_window) == 0: continue ip_peak = peak_window.max() trough_end = nber_trough + pd.DateOffset(months=6) trough_window = ip[(ip.index >= peak_window.idxmax()) & (ip.index <= trough_end)] if len(trough_window) == 0: continue ip_trough = trough_window.min() if ip_peak <= 0: continue depth = (ip_peak - ip_trough) / ip_peak * 100.0 covid_records.append({'sector': sid, 'K': K_hat[sid], 'depth': depth}) if len(covid_records) >= 5: covid_df = pd.DataFrame(covid_records) r_f, p_f = stats.spearmanr(covid_df['K'], covid_df['depth']) # Falsification: pattern should BREAK (r should NOT be significantly positive) # CONSISTENT = pattern breaks (r <= 0 or p >= 0.10) # INCONSISTENT = pattern still holds strongly (r > 0 and p < 0.05) pattern_holds = r_f > 0 and p_f < 0.05 if pattern_holds: falsif_verdict = "INCONSISTENT" # theory says COVID should be different elif r_f > 0 and p_f < 0.10: falsif_verdict = "AMBIGUOUS" else: falsif_verdict = "CONSISTENT" # pattern broke as expected _cache['2f'] = {'r': r_f, 'p': p_f, 'covid_df': covid_df} results_list.append(log_result("recession_depth_covid_falsification", r_f, p_f, falsif_verdict, description=( "Falsification test: COVID 2020 was a supply-side/lockdown " "shock, not demand-driven. The K-depth pattern should BREAK " "(sector depths driven by lockdown exposure, not input " "complementarity). CONSISTENT = pattern breaks (r <= 0 or " "not significant). Spearman of K vs COVID depth across 14 " "sectors. NBER: Feb-Apr 2020."))) else: _cache['2f'] = None results_list.append(log_result("recession_depth_covid_falsification", np.nan, np.nan, "AMBIGUOUS", {"reason": "Insufficient COVID-era data"}, description=( "Falsification test: COVID 2020 supply-side shock. K-depth " "pattern should break. Insufficient FRED IP data for this period."))) # Overall verdict (2a-2e; 2f is separate falsification, not counted in main verdict) n_correct_sig = sum(1 for name, sign_ok, p in sub_results if sign_ok and p < 0.10) n_total_main = len(sub_results) if n_correct_sig >= 4: overall = "CONSISTENT" elif n_correct_sig >= 3: overall = "AMBIGUOUS" else: overall = "INCONSISTENT" results_list.append(log_result("recession_depth_overall", float(n_correct_sig), np.nan, overall, {"n_correct_sig": n_correct_sig, "n_total": n_total_main}, description=( "Combined verdict for recession depth by CES curvature K. Counts " "sub-tests 2a-2e with correct sign and p < 0.10. " "CONSISTENT requires >= 4/5. COVID falsification (2f) reported " "separately as it tests the ABSENCE of the pattern."))) _cache['results'] = results_list return results_list _ALL_SUBTESTS = [ 'recession_depth_k_vs_depth', 'recession_depth_k_vs_recovery', 'recession_depth_k_vs_halflife', 'recession_depth_ols_k_fe', 'recession_depth_sector_averaged', 'recession_depth_covid_falsification', 'recession_depth_overall', ] def figures(): """Generate figures for recession depth by K subtests.""" if not _cache: main() paths = [] df = _cache.get('df') if df is None: return paths names = _cache.get('names', {}) # --- Figure 2a: K vs depth scatter --- data_a = _cache.get('2a') if data_a: fig, ax = setup_figure() for rec in df['recession'].unique(): sub = df[df['recession'] == rec] ax.scatter(sub['K'], sub['depth'], alpha=0.6, s=30, label=rec) # Add sector-averaged trend sector_avg = df.groupby('sector').agg({'K': 'first', 'depth': 'mean'}).reset_index() z = np.polyfit(sector_avg['K'], sector_avg['depth'], 1) x_fit = np.linspace(df['K'].min(), df['K'].max(), 50) ax.plot(x_fit, np.polyval(z, x_fit), 'k--', alpha=0.7, linewidth=2) ax.set_xlabel('CES Curvature K') ax.set_ylabel('Recession Depth (% peak-to-trough)') ax.set_title(f'K vs Recession Depth\n' f'Spearman r = {data_a["r"]:.3f}, p = {data_a["p"]:.2e}') ax.legend(fontsize=7, loc='upper left') paths.extend(save_figure(fig, 'recession_depth_k_vs_depth', subfolder='dynamics')) # --- Figure 2b: K vs recovery speed scatter --- data_b = _cache.get('2b') if data_b: fig, ax = setup_figure() for rec in df['recession'].unique(): sub = df[df['recession'] == rec] ax.scatter(sub['K'], sub['recovery_speed'], alpha=0.6, s=30, label=rec) ax.set_xlabel('CES Curvature K') ax.set_ylabel('12-Month Recovery Speed (%)') ax.set_title(f'K vs Recovery Speed\n' f'Spearman r = {data_b["r"]:.3f}, p = {data_b["p"]:.2e}') ax.legend(fontsize=7, loc='upper right') paths.extend(save_figure(fig, 'recession_depth_k_vs_recovery', subfolder='dynamics')) # --- Figure 2c: K vs half-life scatter --- data_c = _cache.get('2c') if data_c: fig, ax = setup_figure() ax.scatter(data_c['K'], data_c['hl'], alpha=0.6, s=30, color='#2ca02c') ax.set_xlabel('CES Curvature K') ax.set_ylabel('Half-Life (months)') ax.set_title(f'K vs Recovery Half-Life\n' f'Spearman r = {data_c["r"]:.3f}, p = {data_c["p"]:.2e}') paths.extend(save_figure(fig, 'recession_depth_k_vs_halflife', subfolder='dynamics')) # --- Figure 2d: Demeaned OLS --- data_d = _cache.get('2d') if data_d: fig, ax = setup_figure() K_dm = data_d['K_dm'] depth_dm = data_d['depth_dm'] recs = data_d['recessions'] for rec in np.unique(recs): mask = recs == rec ax.scatter(K_dm[mask], depth_dm[mask], alpha=0.6, s=30, label=rec) z = np.polyfit(K_dm, depth_dm, 1) x_fit = np.linspace(K_dm.min(), K_dm.max(), 50) ax.plot(x_fit, np.polyval(z, x_fit), 'k--', linewidth=2, label=f'slope={data_d["slope"]:.1f}') ax.axhline(0, color='gray', linewidth=0.5, alpha=0.5) ax.axvline(0, color='gray', linewidth=0.5, alpha=0.5) ax.set_xlabel('K (demeaned within recession)') ax.set_ylabel('Depth % (demeaned within recession)') ax.set_title(f'Recession FE OLS: Depth ~ K\n' f'slope = {data_d["slope"]:.1f}, p = {data_d["p"]:.2e}') ax.legend(fontsize=7, loc='upper left') paths.extend(save_figure(fig, 'recession_depth_ols_k_fe', subfolder='dynamics')) # --- Figure 2e: Sector-averaged scatter --- data_e = _cache.get('2e') if data_e: fig, ax = setup_figure() sa = data_e['sector_avg'] ax.scatter(sa['K'], sa['depth'], alpha=0.7, s=60, color='#1f77b4', zorder=5) # Label each sector for _, row in sa.iterrows(): ax.annotate(names.get(row['sector'], row['sector'])[:10], (row['K'], row['depth']), fontsize=6, xytext=(4, 4), textcoords='offset points') # Fit line z = np.polyfit(sa['K'], sa['depth'], 1) x_fit = np.linspace(sa['K'].min(), sa['K'].max(), 50) ax.plot(x_fit, np.polyval(z, x_fit), 'k--', linewidth=2, alpha=0.7) ax.set_xlabel('CES Curvature K') ax.set_ylabel('Mean Recession Depth (% across 5 recessions)') ax.set_title(f'Sector-Averaged K vs Depth (N=14)\n' f'Spearman r = {data_e["r"]:.3f}, p = {data_e["p"]:.4f}') paths.extend(save_figure(fig, 'recession_depth_sector_averaged', subfolder='dynamics')) # --- Figure 2f: COVID falsification --- data_f = _cache.get('2f') if data_f and isinstance(data_f, dict) and 'covid_df' in data_f: fig, ax = setup_figure() cdf = data_f['covid_df'] ax.scatter(cdf['K'], cdf['depth'], alpha=0.7, s=60, color='#d62728', zorder=5) for _, row in cdf.iterrows(): ax.annotate(names.get(row['sector'], row['sector'])[:10], (row['K'], row['depth']), fontsize=6, xytext=(4, 4), textcoords='offset points') ax.set_xlabel('CES Curvature K') ax.set_ylabel('COVID Recession Depth (%)') r_f, p_f = data_f['r'], data_f['p'] verdict = "Pattern BREAKS" if (r_f <= 0 or p_f >= 0.10) else "Pattern HOLDS" ax.set_title(f'COVID 2020 Falsification: K vs Depth\n' f'r = {r_f:.3f}, p = {p_f:.3f} — {verdict}') paths.extend(save_figure(fig, 'recession_depth_covid_falsification', subfolder='dynamics')) # --- Figure overall: sector-averaged bar chart --- fig, ax = setup_figure() sector_avg = df.groupby('sector').agg({'K': 'first', 'depth': 'mean'}).reset_index() sector_avg = sector_avg.sort_values('K', ascending=False) labels = [names.get(s, s)[:12] for s in sector_avg['sector']] colors = ['#d62728' if k > 1.2 else '#ff7f0e' if k > 0.9 else '#2ca02c' for k in sector_avg['K']] bars = ax.barh(range(len(labels)), sector_avg['depth'], color=colors, alpha=0.7) ax.set_yticks(range(len(labels))) ax.set_yticklabels(labels, fontsize=8) ax.set_xlabel('Mean Recession Depth (%)') ax.set_title('Mean Recession Depth by Sector (ordered by K)') # Annotate K values for i, (_, row) in enumerate(sector_avg.iterrows()): ax.text(row['depth'] + 0.3, i, f'K={row["K"]:.2f}', va='center', fontsize=7) fig.tight_layout() paths.extend(save_figure(fig, 'recession_depth_overall', subfolder='dynamics')) return paths if __name__ == '__main__': res = main() for r in res: print(f"{r['name']}: {r['verdict']} (stat={r['statistic']}, p={r['p_value']})")

p value	0.0013
statistic	0.3966

Recession Depth K Vs Halflife

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