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05707f0 f784e93 05707f0 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | """Statistical and economic evaluation of out-of-sample forecasts."""
from __future__ import annotations
import numpy as np
import pandas as pd
from scipy import stats
def newey_west_t(x: np.ndarray, lags: int = 3) -> tuple[float, float]:
"""t-statistic and one-sided p-value (mean > 0) with Newey-West standard errors."""
x = np.asarray(x, dtype=float)
x = x[~np.isnan(x)]
n = len(x)
if n < 10:
return np.nan, np.nan
d = x - x.mean()
s = d @ d / n
for k in range(1, min(lags, n - 1) + 1):
w = 1 - k / (lags + 1)
s += 2 * w * (d[k:] @ d[:-k]) / n
se = np.sqrt(s / n)
t = x.mean() / se if se > 0 else np.nan
return t, 1 - stats.norm.cdf(t)
def r2_table(preds: pd.DataFrame, lags: int = 3) -> pd.DataFrame:
"""Pooled out-of-sample R^2 against two benchmarks.
R2_OS = 1 - sum (y - yhat)^2 / sum (y - bench)^2 (Campbell and Thompson 2008)
hist = the factor's own expanding mean (the standard benchmark)
shrink = that mean shrunk toward the regional mean of the same factor. A model
can beat `hist` merely by pooling noisy means across countries; only
beating `shrink` is evidence of time variation being forecast.
Significance: Clark-West (2007) adjusted MSPE difference, 2 (y - b)(yhat - b),
averaged across the cross-section each month; the monthly series is tested
with Newey-West errors, so correlation within a month does not inflate t.
"""
rows = []
for (model, scope), g in preds.groupby(["model", "scope"]):
y, p = g["target"].to_numpy(), g["pred"].to_numpy()
row = {"model": model, "scope": scope}
for tag, col in (("hist", "hist_mean"), ("shrink", "shrink_mean")):
b = g[col].to_numpy()
row[f"r2_vs_{tag}_pct"] = 100 * (1 - np.sum((y - p) ** 2) / np.sum((y - b) ** 2))
cw = 2 * (y - b) * (p - b)
monthly = pd.Series(cw).groupby(g["target_month"].to_numpy()).mean().to_numpy()
row[f"cw_t_{tag}"] = newey_west_t(monthly, lags)[0] if np.any(p != b) else np.nan
row["obs"], row["months"] = len(g), g["target_month"].nunique()
rows.append(row)
return pd.DataFrame(rows).sort_values("r2_vs_shrink_pct", ascending=False).reset_index(drop=True)
def r2_by_country(preds: pd.DataFrame, model: str, scope: str) -> pd.DataFrame:
g = preds[(preds["model"] == model) & (preds["scope"] == scope)]
rows = []
for c, h in g.groupby("country"):
y, p = h["target"].to_numpy(), h["pred"].to_numpy()
r2 = {f"r2_vs_{t}_pct": 100 * (1 - np.sum((y - p) ** 2) / np.sum((y - h[c2].to_numpy()) ** 2))
for t, c2 in (("hist", "hist_mean"), ("shrink", "shrink_mean"))}
rows.append({"country": c, **r2, "obs": len(h)})
return pd.DataFrame(rows).sort_values("r2_vs_shrink_pct", ascending=False).reset_index(drop=True)
def timing_weights(g: pd.DataFrame, how: str) -> np.ndarray:
"""Weights across factors within one country-month; gross exposure sums to 1."""
n = len(g)
if how == "static":
return np.full(n, 1.0 / n)
if how == "sign":
s = np.sign(g["pred"].to_numpy())
return s / max(np.abs(s).sum(), 1)
if how == "long_only":
p = np.maximum(g["pred"].to_numpy(), 0.0)
tot = p.sum()
return p / tot if tot > 0 else np.full(n, 1.0 / n)
p = g["pred"].to_numpy()
tot = np.abs(p).sum()
return p / tot if tot > 0 else np.full(n, 1.0 / n)
def portfolio_returns(preds: pd.DataFrame, cost_bps: float = 0.0) -> pd.DataFrame:
"""Monthly returns of timing portfolios, averaged equally across countries.
For each model/scope the country portfolio holds each factor in proportion to
its forecast (gross exposure 1). The static benchmark holds every factor
equally. Turnover is sum |w_t - w_{t-1}| within a country; the cost applies
only to these timing trades -- the factor returns themselves are gross of the
costs of their own monthly rebalancing.
"""
out = []
static_done = False
for (model, scope), g in preds.groupby(["model", "scope"]):
variants = [("proportional", "prop")]
if model == "fmom":
variants = [("sign", "sign")]
if not static_done:
variants.append(("static", "static"))
for label, how in variants:
recs = []
for (country, tm), h in g.groupby(["country", "target_month"], sort=True):
w = timing_weights(h, how)
recs.append((country, tm, h["factor"].to_numpy(), w, float(w @ h["target"].to_numpy())))
df = pd.DataFrame(recs, columns=["country", "target_month", "factors", "w", "gross"])
df["turnover"] = 0.0
for country, h in df.groupby("country"):
prev = None
for i, row in h.iterrows():
cur = pd.Series(row["w"], index=row["factors"])
if prev is not None:
df.at[i, "turnover"] = cur.sub(prev, fill_value=0).abs().sum()
prev = cur
df["net"] = df["gross"] - cost_bps / 1e4 * df["turnover"]
monthly = df.groupby("target_month")[["gross", "net", "turnover"]].mean().reset_index()
name = "static_ew" if how == "static" else f"{model}:{scope}"
monthly["strategy"] = name
out.append(monthly)
if how == "static":
static_done = True
return pd.concat(out, ignore_index=True)
def portfolio_table(monthly: pd.DataFrame) -> pd.DataFrame:
rows = []
for s, g in monthly.groupby("strategy"):
gr, nt = g["gross"].to_numpy(), g["net"].to_numpy()
rows.append(
{
"strategy": s,
"mean_ann_pct": 1200 * gr.mean(),
"vol_ann_pct": 100 * np.sqrt(12) * gr.std(ddof=1),
"sharpe_gross": np.sqrt(12) * gr.mean() / gr.std(ddof=1),
"sharpe_net": np.sqrt(12) * nt.mean() / nt.std(ddof=1),
"turnover_monthly": g["turnover"].mean(),
"months": len(g),
}
)
return pd.DataFrame(rows).sort_values("sharpe_net", ascending=False).reset_index(drop=True)
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