{"blocks":[12,24,60],"default_block":12,"diffs":[{"band":{"12":{"hi":[3.818911151e-06,0.1406012814,0.3028496468,0.636375738,0.8586005363,1.003888318],"lo":[-2.470462588e-06,-0.1537943235,-0.06741402617,0.09172653772,0.239361224,0.3420487978]},"24":{"hi":[3.996027166e-06,0.1557514387,0.312550878,0.639153686,0.8664699187,1.030165308],"lo":[-2.529188626e-06,-0.1674612638,-0.08751410377,0.06367403772,0.2022409828,0.2985952614]},"60":{"hi":[3.92358772e-06,0.1610689774,0.3353132193,0.6600966413,0.8801597166,1.035129677],"lo":[-1.842419089e-06,-0.2127797893,-0.1477617678,0.005620256396,0.1495613551,0.2407653817]}},"diff":[4.473535007e-07,-0.007080329591,0.1230298962,0.3816806804,0.5766205669,0.6965657853],"id":"gram_minus_euclid","label":"Gram minus Euclidean fit","label_zh":"\u683c\u62c9\u59c6\u62df\u5408\u51cf\u6b27\u6c0f\u62df\u5408","printed":["0.00","\u22120.01","0.12","0.38","0.58","0.70"],"printed_ci":[["0.00","0.00"],["\u22120.15","0.14"],["\u22120.07","0.30"],["0.09","0.64"],["0.24","0.86"],["0.34","1.00"]]},{"band":{"12":{"hi":[0.2614737695,0.6669007305,0.8696635618,1.13621344,1.316766606,1.430122029],"lo":[-0.4449760175,-0.05059254792,0.207926406,0.4376384978,0.5496811174,0.6139606441]},"24":{"hi":[0.2258915058,0.6964494026,0.9028961038,1.177184224,1.364602498,1.489393157],"lo":[-0.3885165081,-0.009424660436,0.2238862489,0.4069383976,0.4960843438,0.5350235482]},"60":{"hi":[0.184935474,0.6727215476,0.8962214791,1.222283348,1.440711536,1.562575496],"lo":[-0.4073443826,-0.08311099011,0.1522465452,0.3210813791,0.3921940376,0.4254980597]}},"diff":[-0.08250357388,0.2965684924,0.5312253631,0.802120493,0.9546176228,1.032621432],"id":"rppca_minus_euclid","label":"RP-PCA minus Euclidean fit","label_zh":"RP-PCA \u51cf\u6b27\u6c0f\u62df\u5408","printed":["\u22120.08","0.30","0.53","0.80","0.95","1.03"],"printed_ci":[["\u22120.44","0.26"],["\u22120.05","0.67"],["0.21","0.87"],["0.44","1.14"],["0.55","1.32"],["0.61","1.43"]]}],"headline":{"diff_ref":"0.38","diff_ref_ci":["0.09","0.64"],"euclid_ref":"0.69","gram_ref":"1.07","locator":"Table I, p. 19; Table II, p. 24; Figure 1, p. 4","unit_consistent_ref":"0.69","unpenalised":"1.62","unpenalised_gram":"1.62","weights_one_set_median":"1.0 \u00d7 10\u207b\u00b9\u00b9","weights_own_units_median":"0.17"},"lambda_labels":["0","0.001","0.003","0.01","0.03","0.1"],"lambdas":[0.0,0.001,0.003,0.01,0.03,0.1],"meta":{"built_at":"2026-10-16T15:54:45Z","exhibit":"Figure 1 (p. 4); Table I, Panel A (p. 19); Table II, Panels A-B (p. 24); Figure 3 (p. 26)","locators":{"diff":"Table II, Panel B, p. 24","families":"Figure 3, p. 26","figure":"Figure 1, p. 4","identity":"Proposition 1, p. 12; Section IV.B, p. 22","levels":"Table I, Panel A, p. 19","penalty_removal":"Table II, Panel C, p. 24; Section IV.C, pp. 23-24","weights":"Table II, Panel A, p. 24"},"notes":{"bands":"Pointwise 95% intervals from a moving-block bootstrap (12-month blocks in the paper's figures) that holds the fitted models fixed.","gross":"Gross returns at unit gross exposure, before trading costs; 594 months, June 1975 to November 2024.","lambda_axis":"lambda = 0 is a categorical tick on a logarithmic axis (Figure 1, p. 4).","one_set":"Written in one set of units, the Gram fit with the penalty in Euclidean units holds the Euclidean fit's portfolio; the paper computes this row at lambda = 0.01 only and, by Proposition 1, it equals the Euclidean row at every lambda (Table I caption, p. 19).","same_forecasts":"The two fits' stock forecasts differ by a median relative gap of 4.8 x 10^-6 (Table II, Panel A, p. 24)."},"pipeline_version":"1.0.0","sample":{"first":"1975-06","label":"594 months, June 1975 to November 2024","last":"2024-11","months":594},"source_files":["10_sharpe_by_lambda_594.csv","10_contrasts_594.csv","13_ladder_594.csv","13_holdings_594.json"],"source_release":"Same Forecasts, Different Portfolios: Factor Normalization and Regularization (October 2026), replication data","statistic":"annualised gross Sharpe ratio at unit gross exposure"},"penalty_removal":[{"diff":0.9328656837,"hi":1.252376924,"lo":0.6050927945,"mean_diff_pp":2.607394188,"mean_hi":4.009688453,"mean_lo":1.411944024,"printed":{"diff":"0.93","hi":"1.25","lo":"0.61","mean_diff_pp":"2.61","mean_hi":"4.01","mean_lo":"1.41"},"series":"euclid"},{"diff":0.5511854507,"hi":0.7745032404,"lo":0.3474514648,"mean_diff_pp":1.864793301,"mean_hi":3.094752429,"mean_lo":0.799538479,"printed":{"diff":"0.55","hi":"0.77","lo":"0.35","mean_diff_pp":"1.86","mean_hi":"3.09","mean_lo":"0.80"},"series":"gram"},{"diff":0.04824161682,"hi":0.3012759051,"lo":-0.1988441231,"mean_diff_pp":-0.3657289202,"mean_hi":0.4481896311,"mean_lo":-1.167765568,"printed":{"diff":"0.05","hi":"0.30","lo":"\u22120.20","mean_diff_pp":"\u22120.37","mean_hi":"0.45","mean_lo":"\u22121.17"},"series":"rppca"},{"diff":0.1721993734,"hi":0.3884787939,"lo":-0.03251490881,"mean_diff_pp":1.396097495,"mean_hi":2.547921218,"mean_lo":0.3210929741,"printed":{"diff":"0.17","hi":"0.39","lo":"\u22120.03","mean_diff_pp":"1.40","mean_hi":"2.55","mean_lo":"0.32"},"series":"ff5"}],"reference_lambda":0.01,"series":[{"ann_mean_pct":[6.63604503,5.83080593,5.088437553,4.028650842,3.106900404,2.351652351],"ann_vol_pct":[4.099739319,4.445457089,4.878361585,5.874511721,6.961429525,7.724456301],"band":{"12":{"hi":[2.061625488,1.771139287,1.511844828,1.194624834,0.9622820958,0.810833236],"lo":[1.246973003,0.9198673005,0.6388102753,0.2736442992,0.06891164076,-0.05680245387]},"24":{"hi":[2.133368375,1.852486696,1.588240361,1.272923701,1.044346093,0.9076678783],"lo":[1.200235844,0.8607487598,0.5784325025,0.2175874669,0.01117159619,-0.1155026337]},"60":{"hi":[2.357698198,2.064050657,1.797128885,1.458698204,1.21983465,1.086620949],"lo":[1.07233238,0.7533528629,0.4772202466,0.1286288095,-0.08033552017,-0.2146112325]}},"id":"euclid","label":"IPCA, Euclidean units (plain ruler)","label_zh":"IPCA\uff0c\u6b27\u6c0f\u5355\u4f4d\uff08\u666e\u901a\u5c3a\u5b50\uff09","palette_key":"ipca","printed":["1.62","1.31","1.04","0.69","0.45","0.30"],"sharpe":[1.618650483,1.311632485,1.043062812,0.6857847993,0.4463020696,0.3044424435]},{"ann_mean_pct":[6.63604902,5.543650193,5.087909134,4.771255719,4.62574672,4.538573488],"ann_vol_pct":[4.099740652,4.249466125,4.363211517,4.469704931,4.522088529,4.534002176],"band":{"12":{"hi":[2.061624429,1.744925755,1.585075819,1.455884028,1.402271869,1.376379272],"lo":[1.246972645,0.9343911832,0.8124505216,0.729284103,0.6923622712,0.6771849801]},"24":{"hi":[2.133367795,1.812581888,1.641453003,1.50583536,1.44485879,1.419790519],"lo":[1.200238892,0.8771466253,0.7612074297,0.6859974047,0.652609746,0.638787949]},"60":{"hi":[2.357697034,1.988556858,1.780328594,1.626431339,1.559457713,1.534891669],"lo":[1.072335259,0.8114562532,0.7048738662,0.6326118007,0.5987773562,0.5865442963]}},"id":"gram","label":"IPCA, Gram units (characteristic ruler)","label_zh":"IPCA\uff0c\u683c\u62c9\u59c6\u5355\u4f4d\uff08\u7279\u5f81\u5c3a\u5b50\uff09","palette_key":"qz_ipca","printed":["1.62","1.30","1.17","1.07","1.02","1.00"],"sharpe":[1.61865093,1.304552156,1.166092708,1.06746548,1.022922636,1.001008229]},{"ann_mean_pct":[5.43427747,5.940447569,5.944519566,5.800006391,5.666316437,5.59808533],"ann_vol_pct":[3.537602711,3.693846511,3.776004711,3.898101862,4.044711819,4.186849583],"band":{"12":{"hi":[1.99961711,2.087256087,2.039632412,1.949767526,1.867372081,1.804243734],"lo":[1.11386414,1.166142731,1.139634657,1.06970391,0.9871818309,0.9148820058]},"24":{"hi":[2.095732694,2.187952465,2.133370478,2.023783127,1.939302841,1.890965902],"lo":[1.056441961,1.111924483,1.089405602,1.009433995,0.9245308315,0.8573371531]},"60":{"hi":[2.284806383,2.381849891,2.319675433,2.20240532,2.101456735,2.042900831],"lo":[0.9261084918,0.9655153926,0.9392339378,0.8842980074,0.8150522754,0.7658410285]}},"id":"rppca","label":"RP-PCA","label_zh":"\u98ce\u9669\u6ea2\u4ef7\u4e3b\u6210\u5206\u5206\u6790\uff08RP-PCA\uff09","palette_key":"rp_pca","printed":["1.54","1.61","1.57","1.49","1.40","1.34"],"sharpe":[1.536146909,1.608200978,1.574288175,1.487905292,1.400919692,1.337063876]},{"ann_mean_pct":[5.303565339,4.420083617,4.115385238,3.907467844,3.817369516,3.77972078],"ann_vol_pct":[5.729592199,5.30768757,5.22297998,5.186134806,5.177549575,5.175488621],"band":{"12":{"hi":[1.298687211,1.22132216,1.162248994,1.12235613,1.102865647,1.094005564],"lo":[0.6094278145,0.4993703393,0.4577650625,0.4264859832,0.4113269435,0.4046724149]},"24":{"hi":[1.35020169,1.264222953,1.206755693,1.163809121,1.142397337,1.133541579],"lo":[0.5758600959,0.4305143261,0.3862709304,0.356709338,0.3433236273,0.3375600386]},"60":{"hi":[1.349573774,1.280066765,1.227676437,1.184718275,1.165121741,1.154379801],"lo":[0.5679907525,0.4284153306,0.379794544,0.3451064232,0.3287560927,0.3211984628]}},"id":"ff5","label":"FF5 (characteristic-instrumented)","label_zh":"FF5\uff08\u4ee5\u7279\u5f81\u4e3a\u5de5\u5177\uff09","palette_key":"ff5","printed":["0.93","0.83","0.79","0.75","0.74","0.73"],"sharpe":[0.9256444707,0.8327701204,0.7879381605,0.7534450973,0.7372927019,0.7303118715]}],"units":{"default":"own","one_set":{"check":{"abs_diff_from_curve":3.26e-07,"fit":"gram","lambda":0.01,"printed":"0.69","sharpe":0.6857844737,"source":"Table I, Panel A, row 'IPCA, unit-consistent penalty' (p. 19)"},"curve":"euclid","label":"One set of units (Euclidean)","label_zh":"\u7edf\u4e00\u5355\u4f4d\uff08\u6b27\u6c0f\uff09","printed":["1.62","1.31","1.04","0.69","0.45","0.30"],"sharpe":[1.618650483,1.311632485,1.043062812,0.6857847993,0.4463020696,0.3044424435],"status":["computed","by_identity","by_identity","computed","by_identity","by_identity"]},"one_set_gram":{"check":{"abs_diff_from_curve":1.06e-06,"fit":"euclid","lambda":0.01,"printed":"1.07","sharpe":1.067466535,"source":"Table I, Panel A, row 'IPCA, unit-consistent penalty (variant F): table note only' (p. 19)"},"curve":"gram","label":"One set of units (Gram)","label_zh":"\u7edf\u4e00\u5355\u4f4d\uff08\u683c\u62c9\u59c6\uff09","printed":["1.62","1.30","1.17","1.07","1.02","1.00"],"sharpe":[1.61865093,1.304552156,1.166092708,1.06746548,1.022922636,1.001008229],"status":["computed","by_identity","by_identity","computed","by_identity","by_identity"]},"own":{"curves":["euclid","gram"],"label":"Each normalisation's own units","label_zh":"\u5404\u81ea\u7684\u5355\u4f4d"}},"weights":{"forecast_gap":{"max":0.0001360969311771,"median":4.848806729060372e-06,"printed":{"max":"1.4 \u00d7 10\u207b\u2074","median":"4.8 \u00d7 10\u207b\u2076"}},"one_set_euclidean":{"l1_distance":{"max":5.1948743318854234e-05,"median":4.54752605072379e-06,"printed":{"max":"5.2 \u00d7 10\u207b\u2075","median":"4.5 \u00d7 10\u207b\u2076"}},"label":"Gram fit, penalty in Euclidean units","one_minus_cosine":{"max":1.3825272038303638e-09,"median":1.013683581518876e-11,"printed":{"max":"1.4 \u00d7 10\u207b\u2079","median":"1.0 \u00d7 10\u207b\u00b9\u00b9"}}},"one_set_gram":{"l1_distance":{"max":6.943387952536864e-05,"median":3.0618707621222687e-06,"printed":{"max":"6.9 \u00d7 10\u207b\u2075","median":"3.1 \u00d7 10\u207b\u2076"}},"label":"Euclidean fit, penalty in Gram units","one_minus_cosine":{"max":2.5653943414027935e-09,"median":4.8386850082238206e-12,"printed":{"max":"2.6 \u00d7 10\u207b\u2079","median":"4.8 \u00d7 10\u207b\u00b9\u00b2"}}},"own_units":{"l1_distance":{"max":1.03780040321276,"median":0.5769175061845865,"printed":{"max":"1.04","median":"0.58"}},"label":"Reference penalty, own units","one_minus_cosine":{"max":0.5460934736314783,"median":0.17124294771809814,"printed":{"max":"0.55","median":"0.17"}}},"unpenalised":{"l1_distance":{"max":9.361225643945861e-05,"median":3.4923461540858205e-06,"printed":{"max":"9.4 \u00d7 10\u207b\u2075","median":"3.5 \u00d7 10\u207b\u2076"}},"label":"Both unpenalised (lambda = 0)","one_minus_cosine":{"max":4.632016437078334e-09,"median":6.208311642552644e-12,"printed":{"max":"4.6 \u00d7 10\u207b\u2079","median":"6.2 \u00d7 10\u207b\u00b9\u00b2"}}}}}
