Comprobaciones de mecanismos para seleccionar características de ETF
Notebook Machine Learning for Trading
Resumen
Este cuaderno amplía una evaluación de características de ETF, que abarcaba un único horizonte de retorno, para analizar diez características y tres horizontes, con corrección por pruebas múltiples. Después aplica diagnósticos basados en mecanismos al momentum de 12-1 con una ventana retrospectiva larga y a la reversión a corto plazo: desplazamientos temporales y persistencia de IC, un control de factor común mediante retornos de bonos del Tesoro y comparaciones entre regímenes de VIX. El análisis encuentra momentum con ventana larga entre los resultados significativos tras la corrección en el horizonte de cinco días, mientras que sus mayores coeficientes de información en un horizonte más largo no superan el umbral corregido. La característica de momentum seleccionada conserva información transversal, pero varía según las condiciones de volatilidad; la reversión no supera las comprobaciones de selección propuestas.
Las comprobaciones pretenden orientar la decisión de seguir investigando una característica, no identificar un efecto causal. Los placebos temporales pueden complicarse por la superposición de las entradas móviles, y la covariación con los bonos del Tesoro puede ser económicamente plausible para el momentum, en vez de constituir una refutación decisiva. Las divisiones por régimen revelan heterogeneidad, pero no permiten distinguir entre confusión y una verdadera modificación del efecto. Como cada prueba es bivariante, otras características del modelo aún pueden confundir la asociación observada; el análisis causal multivariante formal se deja para trabajos posteriores.
Ideas clave
- Analizar más horizontes puede revelar asociaciones entre características que no aparecen en una búsqueda de un solo horizonte, pero las pruebas múltiples requieren corrección.
- Los desplazamientos temporales, los resultados no relacionados y las divisiones por régimen exploran distintas formas en que una asociación de características puede ser frágil.
- La evidencia sobre el momentum con ventana retrospectiva larga varía según el horizonte y las condiciones de volatilidad del mercado.
- Las ventanas móviles de características pueden crear una persistencia aparente en las pruebas con etiquetas desplazadas debido a entradas compartidas.
- Estos diagnósticos bivariantes ayudan a seleccionar qué investigar, pero no demuestran causalidad ni descartan la confusión multivariante.
Etiquetas
Texto completo
# Mechanism Plausibility Checks for Feature Triage
# Mechanism Plausibility Checks for Feature Triage
**Docker image**: `ml4t`
**Chapter 7: Defining the Learning Task**
**Section Reference**: 7.5 - From Correlation to Causality
## Purpose
This notebook implements **lightweight falsification tests** for feature evaluation.
These tests complement the correlation-based IC analysis from Section 7.3 by checking
whether a feature-outcome association is *consistent* with a proposed mechanism.
**Important**: These checks do not identify causal effects. They are robustness
diagnostics guided by mechanism reasoning, testing one feature-outcome pair at
a time. They cannot detect multivariate confounding. Formal causal
identification is deferred to Chapter 15.
We begin with a **feature × horizon scan** that expands on the ETF results from
[`07_multiple_testing`](07_multiple_testing.ipynb), where no feature survived BH-FDR at a 5-day horizon.
Widening the search to 10 features × 3 horizons and correcting across all 30 tests,
the long-lookback momentum terms (252d and 12-1) are the only ones that survive, and
they survive at the 5-day horizon. Their largest ICs are at 21 days but do not clear
the corrected threshold there - a weaker result than the textbook finding
(Jegadeesh and Titman 1993, Asness et al. 2013) would lead you to expect from 92 ETFs
over 14 years, and the honest one. We then apply the three diagnostic checks to two
features selected from the scan:
- **12-1 Momentum** (12-month return skipping the most recent month): clears triage
with actionable caveats - it carries genuine cross-sectional information
(strongest at lag 0), does not predict Treasury returns, but concentrates in
low-volatility regimes.
- **Short-term reversal** (negated 1-day return): fails triage - no significant
IC at the 21-day horizon, and its one significant cell does not survive the
shifted-label check in Section 4.1.
## Learning Objectives
1. Run a multi-feature, multi-horizon IC scan to identify where signal exists
2. State a proposed mechanism using DAG vocabulary (confounder, mediator, collider)
3. Implement timing placebos (shift features, examine IC half-life)
4. Run shared-driver checks (unrelated outcomes should show no effect)
5. Assess regime heterogeneity (IC stability across VIX regime partitions)
6. Interpret results as triage decisions (proceed / revise / stop)
## Book Reference
Section 7.5 applies the three core checks to two features on the `etfs` universe
with a 21-day forward return label. This notebook produces the feature scan
heatmap and dual-feature diagnostics referenced in the worked example.
**Prerequisites**: Notebooks [`05_signal_evaluation`](05_signal_evaluation.ipynb) and [`06_ic_inference`](06_ic_inference.ipynb)
introduce IC analysis; [`07_multiple_testing`](07_multiple_testing.ipynb) motivates the horizon expansion.
```python
"""Mechanism Plausibility Checks for Feature Triage."""
from __future__ import annotations
from datetime import date
from pathlib import Path
import numpy as np
import plotly.graph_objects as go
import polars as pl
from IPython.display import display
from ml4t.diagnostic.metrics import compute_ic_hac_stats
from plotly.subplots import make_subplots
from scipy import stats
from statsmodels.stats.multitest import multipletests
from data import load_etfs, load_macro
from utils.paths import get_chapter_dir
from utils.reproducibility import set_global_seeds
from utils.style import ( # importing utils.style activates the ml4t Plotly template
COLORS,
show_plotly_with_alt,
)
```
```python
SEED = 42
START_DATE = "2010-01-01"
# A permutation test cannot resolve a p-value finer than 1/(B+1). At B=200 that
# floor is 0.005, which is coarser than the significance this notebook reports.
N_PERMUTATIONS = 1000
# Resolved from the chapter, not the working directory: the runner sets cwd to the chapter
# dir, so a repo-relative literal writes artifacts one level too deep.
OUTPUT_DIR = get_chapter_dir(7) / "output"
```
```python
set_global_seeds(SEED)
```
## Motivation: Why Expand the Search?
The multiple-testing scan in [`07_multiple_testing`](07_multiple_testing.ipynb) found that 0 out of 13
short-lookback features survived BH-FDR correction at a 5-day horizon on ETFs.
Two explanations are possible: the features are genuinely uninformative, or the
5-day horizon is too short for cross-sectional predictability in a 100-asset
universe. Cross-asset momentum is well-established at monthly+ horizons
(Asness, Moskowitz, and Pedersen 2013) but largely absent at weekly frequencies.
We resolve this by scanning 10 features across three horizons (5d, 21d, 63d).
This empirical scan determines which feature–horizon combinations carry signal,
and two are selected for the mechanism plausibility checks that follow.
## Load Data
We use the full ETF universe (~92 non-bond assets) and load VIX from FRED macro
data for regime conditioning. Treasury ETFs are excluded from the analysis
universe but IEF is retained for the shared-driver control.
```python
etfs = load_etfs()
END_DATE = date(2024, 1, 1)
etfs = etfs.filter(
(pl.col("timestamp") >= date.fromisoformat(START_DATE)) & (pl.col("timestamp") < END_DATE)
).sort(["symbol", "timestamp"])
print(f"ETF universe: {etfs['symbol'].n_unique()} symbols, {len(etfs):,} rows")
print(f"Date range: {etfs['timestamp'].min()} to {etfs['timestamp'].max()}")
```
```python
# Load VIX for regime conditioning
vix_raw = load_macro(series=["vixcls"])
vix = vix_raw.filter(
(pl.col("timestamp") >= date.fromisoformat(START_DATE)) & (pl.col("timestamp") < END_DATE)
).drop_nulls(subset=["vixcls"])
print(f"VIX: {len(vix):,} observations")
print(f"VIX range: {vix['vixcls'].min():.1f} to {vix['vixcls'].max():.1f}")
```
```python
# Compute Treasury 21d forward return (shared-driver control)
TREASURY_SYMBOL = "IEF" # 7-10 Year Treasury Bond ETF
LABEL_HORIZON = 21 # days
treasury_fwd = (
etfs.filter(pl.col("symbol") == TREASURY_SYMBOL)
.sort("timestamp")
.with_columns(
(pl.col("close").shift(-LABEL_HORIZON) / pl.col("close")).log().alias("treasury_fwd_return")
)
.select(["timestamp", "treasury_fwd_return"])
.drop_nulls()
)
# Fail loudly here rather than silently downstream; see the markdown above.
if treasury_fwd.is_empty():
raise ValueError(
f"No {TREASURY_SYMBOL} rows between {START_DATE} and {END_DATE}, so the "
f"shared-driver control cannot be built and the analysis frame would be empty. "
f"Universe contains: {sorted(etfs['symbol'].unique().to_list())}"
)
```
### Compute features and labels
The shared-driver control is joined with inner semantics, so an absent Treasury series
does not weaken the analysis - it empties it, and every statistic downstream is then
computed on nothing. The load below therefore fails loudly rather than letting a later
cell fail on whichever column happens to be missing first.
We compute 10 features spanning four families (momentum, reversal, trend,
volatility) and forward returns at three horizons (5d, 21d, 63d). All features
use data available at time $t$ (no lookahead). The 12-1 momentum follows the
academic convention of skipping the most recent month to avoid short-term
reversal contamination.
```python
# Exclude Treasury/Bond ETFs from the analysis universe
TREASURY_SYMBOLS = ["IEF", "TLT", "SHY", "AGG", "BND", "TIP", "GOVT", "BNDX", "VGSH"]
analysis = (
etfs.filter(~pl.col("symbol").is_in(TREASURY_SYMBOLS))
.with_columns(
# Daily return for volatility computation
pl.col("close").pct_change().over("symbol").alias("daily_ret"),
)
.with_columns(
# ── Momentum features ──
(pl.col("close") / pl.col("close").shift(21).over("symbol")).log().alias("mom_21d"),
(pl.col("close") / pl.col("close").shift(63).over("symbol")).log().alias("mom_63d"),
(pl.col("close") / pl.col("close").shift(126).over("symbol")).log().alias("mom_126d"),
(pl.col("close") / pl.col("close").shift(252).over("symbol")).log().alias("mom_252d"),
# 12-1: skip most recent month (Jegadeesh-Titman convention)
(pl.col("close").shift(21).over("symbol") / pl.col("close").shift(252).over("symbol"))
.log()
.alias("mom_12_1"),
# Trailing volatility (126d, annualized)
(pl.col("daily_ret").rolling_std(126).over("symbol") * np.sqrt(252)).alias("vol_126d"),
# ── Reversal features ──
(pl.col("close").shift(1).over("symbol") / pl.col("close")).log().alias("rev_1d"),
(pl.col("close").shift(5).over("symbol") / pl.col("close")).log().alias("rev_5d"),
# ── Trend ──
(
(pl.col("close") - pl.col("close").rolling_mean(200).over("symbol"))
/ pl.col("close").rolling_mean(200).over("symbol")
).alias("dist_200ma"),
# ── Volatility ──
(
pl.col("daily_ret").rolling_std(20).over("symbol")
/ pl.col("daily_ret").rolling_std(60).over("symbol")
).alias("rvol_ratio"),
# ── Forward returns (labels) ──
(pl.col("close").shift(-5).over("symbol") / pl.col("close")).log().alias("fwd_5d"),
(pl.col("close").shift(-21).over("symbol") / pl.col("close")).log().alias("fwd_21d"),
(pl.col("close").shift(-63).over("symbol") / pl.col("close")).log().alias("fwd_63d"),
)
)
```
```python
# Risk-adjusted features and label alias (requires vol_126d from previous step)
analysis = analysis.with_columns(
(pl.col("mom_126d") / pl.col("vol_126d")).alias("adj_mom_126d"),
pl.col("fwd_21d").alias("forward_return"),
)
```
```python
# Join VIX for regime conditioning
analysis = analysis.join(vix, on="timestamp", how="left").drop_nulls(subset=["vixcls"])
# Join Treasury forward return for shared-driver check
analysis = analysis.join(treasury_fwd, on="timestamp", how="left").drop_nulls(
subset=["treasury_fwd_return"]
)
n_symbols = analysis["symbol"].n_unique()
n_dates = analysis["timestamp"].n_unique()
print(f"Analysis panel: {len(analysis):,} rows, {n_symbols} symbols, {n_dates:,} dates")
if n_dates > 0:
print(f"Cross-section size: ~{len(analysis) // n_dates} assets per date")
else:
print("Cross-section size: N/A (no dates after join)")
```
The permutation helper below pre-computes and standardizes both sides' ranks once per
date. Ranks are invariant to relabeling, and a permutation changes neither a vector's
mean nor its standard deviation, so the Spearman IC of a permuted pair is the dot
product of the two standardized rank vectors divided by the count. That is identical to
the general correlation routine to floating-point noise and well over an order of
magnitude faster, which matters because the loop runs once per permutation per date.
## Cross-Sectional IC Function
We compute Spearman rank correlation between each feature and forward returns at
each date, producing a time series of ICs. This is the same IC framework from
Section 7.3.
```python
def _contiguous_groups(sorted_keys: np.ndarray) -> list[np.ndarray]:
"""Row indices of each run of equal keys, for an array already sorted by key."""
if len(sorted_keys) == 0:
return []
starts = np.flatnonzero(np.r_[True, sorted_keys[1:] != sorted_keys[:-1]])
return np.split(np.arange(len(sorted_keys)), starts[1:])
def block_permutation_null(
df: pl.DataFrame,
feature_col: str,
baseline_ic: float,
seed: int,
n_permutations: int,
) -> tuple[np.ndarray, float, float]:
"""Permutation null for a cross-sectional IC, blocked at the label horizon.
Shuffling assets independently on each date would imply the per-date ICs are
independent, so the null mean's spread would shrink like sigma/sqrt(n_dates).
The labels are LABEL_HORIZON-day forward returns sampled daily - consecutive
dates share all but one day of return - so both the returns and the per-date
ICs are strongly autocorrelated and the true spread is far larger. One asset
relabeling is therefore drawn per block of LABEL_HORIZON sessions and held
fixed across the block, exactly as ``05_signal_evaluation`` does.
Returns ``(null_ics, perm_p, perm_resolution)``. The p-value is
``(r + 1) / (B + 1)``: a permutation p-value can never be exactly zero, since
the observed assignment is itself one of the arrangements under the null, and
with B permutations the finest resolvable value is ``1 / (B + 1)``.
"""
perm_df = df.drop_nulls([feature_col, "forward_return"]).sort(["timestamp", "symbol"])
dates_arr = perm_df["timestamp"].to_numpy()
symbol_codes = perm_df["symbol"].cast(pl.Categorical).to_physical().to_numpy()
n_symbols = int(symbol_codes.max()) + 1
feat_arr_p = perm_df[feature_col].to_numpy()
ret_arr_p = perm_df["forward_return"].to_numpy()
# Ranks pre-computed and standardized once per date; see the markdown above.
def _z(v: np.ndarray) -> np.ndarray | None:
sd = v.std()
return (v - v.mean()) / sd if sd > 0 else None
date_groups = [idx for idx in _contiguous_groups(dates_arr) if len(idx) >= 20]
feat_ranks, ret_ranks, syms_by_date = [], [], []
for idx in date_groups:
fz = _z(stats.rankdata(feat_arr_p[idx]))
rz = _z(stats.rankdata(ret_arr_p[idx]))
if fz is None or rz is None:
continue
feat_ranks.append(fz)
ret_ranks.append(rz)
syms_by_date.append(symbol_codes[idx])
rng = np.random.default_rng(seed)
null_ics = []
for _ in range(n_permutations):
ic_per_date = []
block_keys = None
for i, f_ranks in enumerate(feat_ranks):
# New relabeling only when a block boundary is crossed
if i % LABEL_HORIZON == 0:
block_keys = rng.permutation(n_symbols)
# Same key vector across the block => same asset->asset map across the block
order = np.argsort(block_keys[syms_by_date[i]], kind="stable")
ic_per_date.append(float(f_ranks @ ret_ranks[i][order]) / len(f_ranks))
if ic_per_date:
null_ics.append(np.mean(ic_per_date))
null_ics = np.array(null_ics)
n_at_least = int(np.sum(np.abs(null_ics) >= abs(baseline_ic)))
return null_ics, (n_at_least + 1) / (len(null_ics) + 1), 1.0 / (len(null_ics) + 1)
def compute_cross_sectional_ic(
df: pl.DataFrame, feature_col: str, outcome_col: str, min_obs: int = 20
) -> tuple[float, float, list[float]]:
"""Compute cross-sectional IC (Spearman) at each timestamp.
Returns: (mean_ic, t_stat, ic_series)
"""
groups = df.select("timestamp", feature_col, outcome_col).partition_by(
"timestamp", as_dict=True
)
ic_values = []
for _key, group in groups.items():
if len(group) < min_obs:
continue
feature = group[feature_col].to_numpy()
outcome = group[outcome_col].to_numpy()
if np.std(feature) < 1e-10 or np.std(outcome) < 1e-10:
continue
ic, _ = stats.spearmanr(feature, outcome)
if not np.isnan(ic):
ic_values.append(ic)
if not ic_values:
return np.nan, np.nan, []
ic_array = np.array(ic_values)
mean_ic = np.mean(ic_array)
std_ic = np.std(ic_array, ddof=1)
t_stat = mean_ic / (std_ic / np.sqrt(len(ic_array))) if std_ic > 0 else np.nan
return mean_ic, t_stat, ic_values
```
## Feature × Horizon Scan
We scan 10 features across three forward-return horizons (5d, 21d, 63d) using
HAC-adjusted inference. The heatmap reveals a clear pattern: long-lookback
momentum features (126d+) carry significant cross-sectional information,
while short-term features remain noise regardless of horizon.
```python
SCAN_FEATURES = [
("mom_21d", "21d Momentum"),
("mom_63d", "63d Momentum"),
("mom_126d", "126d Momentum"),
("mom_252d", "252d Momentum"),
("mom_12_1", "12-1 Momentum"),
("adj_mom_126d", "Risk-Adj. Mom."),
("rev_1d", "1d Reversal"),
("rev_5d", "5d Reversal"),
("dist_200ma", "Dist. 200d MA"),
("rvol_ratio", "Vol Ratio 20/60"),
]
SCAN_HORIZONS = [("fwd_5d", "5d"), ("fwd_21d", "21d"), ("fwd_63d", "63d")]
```
```python
# Run scan: IC + HAC t-stat for each feature × horizon pair
scan_rows = []
for feat_col, feat_label in SCAN_FEATURES:
for hz_col, hz_label in SCAN_HORIZONS:
sub = analysis.drop_nulls(subset=[feat_col, hz_col])
_, _, ic_series = compute_cross_sectional_ic(sub, feat_col, hz_col)
if len(ic_series) < 50:
# Key carried even when empty, so the correction below fails on an empty
# scan rather than on a missing column.
scan_rows.append(
{
"feature_col": feat_col,
"feature": feat_label,
"horizon": hz_label,
"ic": np.nan,
"t_hac": np.nan,
"p_hac": None,
}
)
continue
hac = compute_ic_hac_stats(ic_series, label_horizon=int(hz_label[:-1]))
scan_rows.append(
{
"feature_col": feat_col,
"feature": feat_label,
"horizon": hz_label,
"ic": round(hac["mean_ic"], 4),
"t_hac": round(hac["t_stat"], 2),
"p_hac": hac["p_value"],
}
)
scan_df = pl.DataFrame(scan_rows, schema_overrides={"p_hac": pl.Float64})
```
### Correcting the scan for multiple testing
The scan above is not one test, it is one per cell of the grid. At a conventional
significance level a grid this size produces one or two false positives from noise
alone - the count is printed below - so reading `|t| > 2` off every cell is the error
[`07_multiple_testing`](07_multiple_testing.ipynb) exists to prevent. We control
the false discovery rate across the whole grid with Benjamini-Hochberg, and mark
significance with the corrected decision rather than the raw threshold.
```python
scan_valid = scan_df.drop_nulls("p_hac")
bh_reject, bh_qvalues = multipletests(scan_valid["p_hac"].to_numpy(), alpha=0.05, method="fdr_bh")[
:2
]
scan_df = scan_df.join(
scan_valid.select("feature", "horizon").with_columns(
q_bh=pl.Series(bh_qvalues), sig_bh=pl.Series(bh_reject)
),
on=["feature", "horizon"],
how="left",
)
n_raw = int((scan_df["p_hac"] < 0.05).sum())
n_bh = int(scan_df["sig_bh"].fill_null(False).sum())
print(f"Tests in the scan: {len(scan_valid)}")
print(f"Significant at raw p < 0.05: {n_raw}")
print(f"Significant after BH FDR control: {n_bh}")
print(f"Expected false positives if all 30 were null: {0.05 * len(scan_valid):.1f}")
```
```python
# Heatmap: HAC t-statistics by feature × horizon
feat_labels = [f[1] for f in SCAN_FEATURES]
hz_labels = [h[1] for h in SCAN_HORIZONS]
# Build matrices for the heatmap
z_matrix = []
text_matrix = []
for feat_label in feat_labels:
row_z = []
row_text = []
for hz_label in hz_labels:
match = scan_df.filter((pl.col("feature") == feat_label) & (pl.col("horizon") == hz_label))
t_val = match["t_hac"][0] if len(match) > 0 else np.nan
ic_val = match["ic"][0] if len(match) > 0 else np.nan
sig_bh = (
bool(match["sig_bh"][0]) if len(match) > 0 and match["sig_bh"][0] is not None else False
)
row_z.append(t_val if not np.isnan(t_val) else 0)
# The star is the BH-corrected decision across all 30 cells, not |t| > 2
sig = "*" if sig_bh else ""
row_text.append(f"IC={ic_val:.3f}<br>t={t_val:.1f}{sig}" if not np.isnan(t_val) else " - ")
z_matrix.append(row_z)
text_matrix.append(row_text)
```
```python
fig = go.Figure(
data=go.Heatmap(
z=z_matrix,
x=hz_labels,
y=feat_labels,
text=text_matrix,
texttemplate="%{text}",
textfont={"size": 11},
colorscale="RdBu",
zmid=0,
zmin=-5,
zmax=5,
colorbar=dict(title="HAC t-stat"),
)
)
fig.update_layout(
title="Cross-sectional IC and HAC t-statistic by feature and horizon",
xaxis_title="Forward Return Horizon",
height=450,
width=760,
# The feature names are long; without the left margin they render clipped
margin=dict(l=150),
yaxis=dict(autorange="reversed"),
)
show_plotly_with_alt(
fig,
alt=(
"A heatmap of ten candidate features against three forward-return horizons, each "
"cell annotated with its mean IC and HAC t-statistic and shaded on a "
"red-to-blue diverging scale centred on zero. Most of the grid is pale, meaning "
"t-statistics near zero. The darkest blue cells sit in the shortest-horizon "
"column, on the two longest momentum features, and a handful of mid-blue cells "
"appear elsewhere. Three cells carry an asterisk marking survival of the "
"false-discovery correction, and all three are in that shortest-horizon column: "
"the two long-lookback momentum features and the one-day reversal. Two cells at "
"the top of the grid are faintly red, marking small negative ICs."
),
)
```
**Findings from the scan.** Read the starred cells, not the $|t| > 2$ ones. Twice as
many cells clear a raw significance test as clear the grid-wide correction, and the
number that clear the correction is close to what this many null tests would throw up on
their own - both counts are printed with the corrected table above.
1. **Every cell that clears the correction sits at the shortest horizon**: the two
long-lookback momentum features and the one-day reversal. The shifted-label check
below disposes of the third, which leaves two to carry forward.
2. **The 21-day cells do not clear the correction**, even though the two largest ICs
anywhere in the grid are there. Against a grid this size, a large IC at a t in the
low twos is not enough. That is the honest reading and it is *weaker* than the
chapter's later worked example needs; the deep-dive below proceeds on 12-1 momentum
at 21 days anyway, with the caveat that its evidence is a large effect at borderline
significance rather than a clean one.
3. **At the longest horizon the signal fades.** Distance from the 200-day moving
average carries the largest remaining t and the momentum terms slip below it -
consistent with momentum being a monthly rather than a quarterly effect.
4. **Short-lookback and non-momentum features are noise throughout.** The short
momentum terms, risk-adjusted momentum, five-day reversal and the volatility ratio
never reach a t of two at any horizon.
Note what the correction changed. Reading $|t| > 2$ off the grid would have credited
twice as many discoveries, including two at the horizon the rest of the chapter uses.
The correction is not a formality here - it removes half of them.
The surviving pattern is consistent with well-established findings on cross-asset
momentum:
the effect requires long lookbacks (6–12 months) and manifests at monthly+
horizons.
### A Significant Cell That Does Not Hold Up Under a Shift of Its Label
One cell in the scan deserves a second look before we trust it: 1-day reversal at
the 5-day horizon. Write the two quantities out in logs, with $p_t = \log P_t$:
$$\text{rev1d}_t = p_{t-1} - p_t \qquad \text{fwd5d}_t = p_{t+5} - p_t$$
Both contain $-p_t$. Whatever noise sits in the close on day $t$ - a wide bid-ask
spread, a stale print, a bad tick - enters the feature and the label with the
*same* sign, so it induces positive covariance between them whether or not any
reversal exists. The feature and the label share an endpoint.
This is not a hypothetical. The check is to move the label one day forward, so it
spans $p_{t+6} - p_{t+1}$: the same five-day holding period and no shared endpoint.
**What this check can and cannot settle.** It is a *shift* of the return window,
not a removal of the shared price alone. Going from $p_{t+5} - p_t$ to
$p_{t+6} - p_{t+1}$ drops the $t \to t+1$ session **and adds a $t+5 \to t+6$
session**, on top of dropping the shared $p_t$. At least four things could
produce the decline we are about to see:
1. shared-endpoint noise inflated the original statistic;
2. a real effect concentrated in the $t \to t+1$ session, now excluded;
3. the added $t+5 \to t+6$ session contributes returns that offset the rest;
4. sampling variation, on a $t$ that was not large to begin with.
**This test distinguishes none of them.** One comparison with several
simultaneous changes cannot attribute its own result, and the only honest reading
is that the statistic moved when the window did.
Nor does the drop say anything about tradeability. A signal computed from the
day-$t$ close executes at the $t+1$ *open*, which is this repository's label
convention, so neither of these close-to-close labels is what a strategy would
earn. Separating the four needs measurements this notebook does not make: an
independent price for day $t$ - a quote midpoint rather than a last trade - to
isolate endpoint noise, one-session-at-a-time attribution across the window, and
an open-to-open label for the tradeable quantity. All belong with the execution
assumptions in Chapter 16.
What the check does establish is enough for the decision at hand: **the cell does
not survive a five-day label measured one session later.** A result that depends
on which five sessions are used is not one to build on, whichever of the four
explanations holds. That is a reason to withhold belief until it is re-measured -
not a demonstration that no reversal effect exists.
```python
shared_endpoint = analysis.with_columns(
# Same 5-day span, measured from t+1: shares no price with rev_1d
(pl.col("close").shift(-6).over("symbol") / pl.col("close").shift(-1).over("symbol"))
.log()
.alias("fwd_5d_from_t1"),
)
endpoint_rows = []
for label_col, label_desc in [
("fwd_5d", "p(t+5) - p(t) [shares p(t) with the feature]"),
("fwd_5d_from_t1", "p(t+6) - p(t+1) [no shared endpoint]"),
]:
sub = shared_endpoint.drop_nulls(subset=["rev_1d", label_col])
_, _, ic_series_ep = compute_cross_sectional_ic(sub, "rev_1d", label_col)
hac_ep = compute_ic_hac_stats(ic_series_ep, label_horizon=5)
endpoint_rows.append(
{
"label": label_desc,
"mean_ic": f"{hac_ep['mean_ic']:+.4f}",
"t_hac": f"{hac_ep['t_stat']:+.2f}",
"p_hac": f"{hac_ep['p_value']:.4f}",
"n_dates": len(ic_series_ep),
}
)
display(pl.DataFrame(endpoint_rows))
```
Removing the shared endpoint roughly halves the 1-day reversal t-statistic, printed in
the table above - from clearly significant to not - on the same panel and the same
five-day holding period, moved forward by one day.
So the scan's short-horizon reversal cell does not hold up under its own robustness
check, and the deep-dive below therefore treats 1-day reversal as a near-null
baseline rather than as a signal to explain. Note what that is *not*: it is not a
finding that one-day reversal is absent from this panel, and it is not a
tradeability result. Both would need the measurements named above.
**The general rule: a feature ending at $t$ and a label beginning at $t$ share a
price, and shared noise in that price induces covariance between them whether or
not any effect exists. That is a property of the construction, provable without
looking at data - which is exactly why it cannot be read backwards. Seeing a cell
fail this check does not establish that the mechanism fired; it establishes that
the cell has to be re-measured before it is believed.**
### Feature selection for diagnostic deep-dive
We select two features for the mechanism plausibility checks, chosen to
illustrate contrasting triage outcomes:
- **12-1 Momentum**, the largest IC in the scan at the 21-day horizon. Follows the
Jegadeesh-Titman convention of skipping the most recent month to separate momentum
from short-term reversal. The question: does it hold up under the mechanism checks,
or is the signal driven by a confound?
- **1-day Reversal**, whose IC at the same horizon is indistinguishable from zero and
which also flips sign across VIX regimes. Its one significant cell, at the five-day
horizon, is the one the shifted-label check above takes apart. We
expect the mechanism checks to return STOP - and the useful part is *which* check
catches it, since neither a timing placebo nor a shared-driver control was
designed to detect how a feature and its label are constructed.
```python
# Selected features for deep diagnostics
FEATURES = {
"mom_12_1": "12-1 Momentum",
"rev_1d": "1-day Reversal",
}
```
## Mechanism Hypotheses
Before running diagnostics, we state the assumed causal mechanism for each
feature. This structures the interpretation of the checks.
### Feature A: 12-1 Momentum
The 12-1 momentum signal (return from $t-252$ to $t-21$) is attributed to
behavioral underreaction: investors are slow to incorporate information, so past
risers keep rising. The alternative: momentum proxies a
risk-on/risk-off regime rather than encoding timely cross-sectional information.
```
Volatility Regime (VIX)
/ \
v v
12-1 Momentum -->? Forward Return
```
### Feature B: 1-day Reversal
Negated 1-day return. The hypothesized mechanism is microstructure-driven mean
reversion: temporary price dislocations reverse within days. At a 21-day
horizon, this mechanism should have no power - the effect decays too quickly.
```
Microstructure Friction
|
v
1-day Return -->? 21-day Forward Return
```
### Why run these checks?
These checks cost minutes; building a full model pipeline costs chapters.
Their primary value is **catching dead features early** (reversal correctly
gets STOP) and **generating actionable diagnostics** (momentum's regime
concentration informs Ch12+ modeling choices).
**REVISE is the expected outcome, not failure.** In efficient markets cross-sectional
ICs are small - the magnitudes the scan above produced are typical - and HAC-adjusted
significance is conservative. Few single features will cleanly pass all three bivariate
checks. The diagnostic information - *where* and *when* a feature works -
matters more than the triage label.
## Baseline IC (HAC-Adjusted)
The grid-wide BH decision for these same two cells is carried alongside the raw
p-value. Reporting a raw p here without it is what let this section call a cell
"significant" that the scan's own correction had already rejected.
The lookup joins on the feature column rather than the display label: the scan and the
deep-dive spell the same feature differently, so a label join silently returns nothing
for one of the two features - exactly the kind of quiet miss this table exists to
prevent.
We report the HAC-adjusted IC for both features at the 21-day horizon. This
establishes the baseline that the diagnostic checks will probe.
```python
baseline = {}
for feat_col, feat_label in FEATURES.items():
sub = analysis.drop_nulls(subset=[feat_col, "forward_return"])
ic, t, series = compute_cross_sectional_ic(sub, feat_col, "forward_return")
hac = (
compute_ic_hac_stats(series, label_horizon=LABEL_HORIZON)
if series
else {"t_stat": np.nan, "p_value": np.nan}
)
baseline[feat_col] = {"ic": ic, "t": t, "series": series, "hac": hac}
def _bh_for(feat_col: str) -> tuple[str, str]:
# Join on the feature column, not the display label; see the markdown above.
row = scan_df.filter(
(pl.col("feature_col") == feat_col) & (pl.col("horizon") == f"{LABEL_HORIZON}d")
)
if len(row) == 0:
raise KeyError(f"{feat_col} at {LABEL_HORIZON}d is not in the scan grid")
if row["q_bh"][0] is None:
return "n/a", "n/a"
return f"{row['q_bh'][0]:.4f}", "yes" if row["sig_bh"][0] else "no"
_bh = {f: _bh_for(f) for f in FEATURES}
print(
pl.DataFrame(
{
"feature": list(FEATURES.values()),
"mean_ic": [f"{baseline[f]['ic']:.4f}" for f in FEATURES],
"t_naive": [f"{baseline[f]['t']:.2f}" for f in FEATURES],
"t_hac": [f"{baseline[f]['hac']['t_stat']:.2f}" for f in FEATURES],
"p_hac_raw": [f"{baseline[f]['hac']['p_value']:.4f}" for f in FEATURES],
"q_bh_grid": [_bh[f][0] for f in FEATURES],
"survives_bh": [_bh[f][1] for f in FEATURES],
}
)
)
```
12-1 momentum has the largest IC in the grid and the smaller raw p-value of the two
features here, and it **does not clear the grid-wide BH correction** - the column
recording that decision says so, and the scan said so about the same cell. The raw $p$ is a
post-selection number: this feature and this horizon were chosen by looking at the scan,
so every comparison that produced the choice has to be paid for, and BH is the bill.
Reversal IC is indistinguishable from zero on any reading.
The naive t-statistic for momentum is higher than the HAC one because it ignores
autocorrelation in the IC series. That correction and the multiplicity correction are
separate, and both apply: HAC widens the interval for one test, BH sets the threshold
that test has to clear given all the others.
The deep-dive continues on 12-1 momentum regardless, because the chapter needs a
worked example and this is the strongest candidate the scan produced. What it is
an example of is a **large effect at borderline significance** - which is the
ordinary situation in cross-sectional equity work, and a more useful thing to
demonstrate the diagnostics on than an unambiguous result would be.
## Reusable Diagnostic Functions
We extract each falsification test into a reusable function so we can apply
the same checks to both features systematically.
```python
def run_timing_placebo(
df: pl.DataFrame,
feature_col: str,
outcome_col: str = "forward_return",
lags: list[int] | None = None,
) -> tuple[pl.DataFrame, str, str]:
"""Shift the feature backward by increasing lags and recompute IC.
Uses HAC-adjusted inference at each lag. Reports IC half-life as a
diagnostic rather than applying a hard decay threshold.
Returns: (lag_results_df, result_label, interpretation_msg)
"""
if lags is None:
lags = [0, 5, 21, 42, 63, 126, 252]
lag_rows = []
for lag in lags:
col_name = f"{feature_col}_lag{lag}"
lagged = df.with_columns(
pl.col(feature_col).shift(lag).over("symbol").alias(col_name)
).drop_nulls(subset=[col_name])
ic, _, ic_series = compute_cross_sectional_ic(lagged, col_name, outcome_col)
hac = (
compute_ic_hac_stats(ic_series, label_horizon=LABEL_HORIZON)
if ic_series
else {"t_stat": np.nan}
)
lag_rows.append({"lag": lag, "ic": ic, "t_hac": hac["t_stat"]})
lag_df = pl.DataFrame(lag_rows)
# IC half-life: smallest lag where |IC| drops to 50% of |IC at lag 0|
ic_0 = lag_df.filter(pl.col("lag") == 0)["ic"][0]
half_target = abs(ic_0) * 0.5
half_life = None
ics = lag_df["ic"].to_list()
lag_vals = lag_df["lag"].to_list()
for i in range(1, len(lag_vals)):
if abs(ics[i]) <= half_target and abs(ics[i - 1]) > half_target:
frac = (abs(ics[i - 1]) - half_target) / (abs(ics[i - 1]) - abs(ics[i]))
half_life = lag_vals[i - 1] + frac * (lag_vals[i] - lag_vals[i - 1])
break
# Assess: does IC at lag 0 have HAC significance AND meaningful decay?
t_0 = lag_df.filter(pl.col("lag") == 0)["t_hac"][0]
sig_0 = abs(t_0) > 2.0 if not np.isnan(t_0) else False
# Check for IC increasing at DISTANT lags (>= 42d, well beyond any lookback).
# Only flag if the increase is substantial - noise at nearby lags doesn't count.
distant = [(lag_vals[i], abs(ics[i])) for i in range(1, len(ics)) if lag_vals[i] >= 42]
ic_increases = any(abs_ic > abs(ic_0) * 1.5 for _, abs_ic in distant)
if sig_0 and half_life is not None:
result = "PASS"
msg = f"IC half-life ≈ {half_life:.0f}d (HAC t₀={t_0:.1f})"
elif not sig_0 and abs(ic_0) < 0.005:
result = "STOP"
msg = f"No timely signal (IC₀={ic_0:.4f}, HAC t₀={t_0:.1f})"
elif ic_increases:
result = "STOP"
msg = f"IC increases at distant lags (HAC t₀={t_0:.1f})"
elif half_life is not None:
result = "CAUTION"
msg = (
f"IC decays (half-life ≈ {half_life:.0f}d) but marginal significance (HAC t₀={t_0:.1f})"
)
else:
ic_range = [abs(ic) for ic in ics]
result = "CAUTION"
msg = (
f"IC persists without clear decay; |IC| range "
f"[{min(ic_range):.4f}, {max(ic_range):.4f}] (HAC t₀={t_0:.1f})"
)
return lag_df, result, msg
```
### Shared-Driver Check
The check below builds its IC series from a rolling rank correlation over a quarterly
window. Consecutive windows share all but one day of their data, so the effective sample
is much smaller than the number of windows. The HAC adjustment on the resulting series
accounts for part of that, and a borderline t-statistic should still be read with care.
Test whether a common exogenous factor (Treasury returns) or permutation
control explains the observed feature-outcome correlation.
```python
def run_shared_driver_check(
df: pl.DataFrame,
feature_col: str,
baseline_ic: float,
n_permutations: int = 200,
seed: int = 42,
) -> tuple[dict, str]:
"""Run shared-driver check (Treasury) and permutation control.
The Treasury check uses a cross-sectional mean approach with HAC inference:
at each date, compute the cross-sectional mean of the feature, then correlate
the daily mean-feature series with Treasury forward returns. This avoids
per-asset time-series regressions and produces a single IC series suitable
for HAC adjustment.
Returns: (metrics_dict, result_label)
"""
# Shared-driver check: cross-sectional mean of feature vs Treasury fwd return
daily_mean_feat = (
df.group_by("timestamp")
.agg(pl.col(feature_col).mean().alias("mean_feature"))
.sort("timestamp")
)
tsy_series = (
df.select("timestamp", "treasury_fwd_return").unique(subset=["timestamp"]).sort("timestamp")
)
shared = daily_mean_feat.join(tsy_series, on="timestamp", how="inner").drop_nulls()
# Rolling rank correlation as an IC series for HAC; overlap caveat in the markdown.
feat_arr = shared["mean_feature"].to_numpy()
tsy_arr = shared["treasury_fwd_return"].to_numpy()
WINDOW = 63
tsy_ic_series = []
for i in range(WINDOW, len(feat_arr)):
f_w = feat_arr[i - WINDOW : i]
t_w = tsy_arr[i - WINDOW : i]
if np.std(f_w) > 1e-10 and np.std(t_w) > 1e-10:
rho, _ = stats.spearmanr(f_w, t_w)
if not np.isnan(rho):
tsy_ic_series.append(rho)
if tsy_ic_series:
hac_tsy = compute_ic_hac_stats(tsy_ic_series, label_horizon=LABEL_HORIZON)
ic_tsy = hac_tsy["mean_ic"]
t_tsy = hac_tsy["t_stat"]
else:
ic_tsy, t_tsy = np.nan, np.nan
tsy_pass = abs(t_tsy) < 2.0 if not np.isnan(t_tsy) else True
# Permutation control: block permutation at the label horizon.
null_ics, perm_p, perm_resolution = block_permutation_null(
df, feature_col, baseline_ic, seed=seed, n_permutations=n_permutations
)
perm_signal = perm_p < 0.05
metrics = {
"ic_treasury": ic_tsy,
"t_treasury": t_tsy,
"tsy_pass": tsy_pass,
"perm_p": perm_p,
"perm_resolution": perm_resolution,
"perm_signal": perm_signal,
"null_std": float(np.std(null_ics)),
"n_rolling_windows": len(tsy_ic_series),
"n_permutations": len(null_ics),
}
# Only the Treasury arm decides this check; the permutation control is reported
# beside it and deliberately does not enter it. See the markdown above.
#
# So PASS here means "not explained by the Treasury driver", which for a
# feature with no signal is true and uninformative. Read it with the
# permutation p beside it; neither number means anything alone.
result = "PASS" if tsy_pass else "STOP"
return metrics, result
```
The outcome of the shared-driver check is about the shared driver, and only the Treasury
arm speaks to it. The permutation control answers a different question - *is there any
signal at all?* - so it is reported beside the outcome and does not enter it. An earlier
version of this function said exactly that and then gated on the permutation anyway,
which marked a feature as carrying a shared-driver confound when its real problem was
having no signal to confound.
### Regime Heterogeneity
Partition by VIX regime and test whether signal effectiveness varies
across market states.
```python
def run_regime_heterogeneity(
df: pl.DataFrame,
feature_col: str,
outcome_col: str = "forward_return",
vix_low: float = 15,
vix_high: float = 22,
) -> tuple[list[dict], str, str]:
"""Partition by VIX regime and assess IC heterogeneity.
This is a heterogeneity diagnostic, not a confounding test. It checks
whether the feature-outcome association varies across market states. It
cannot distinguish confounding from genuine effect modification.
Uses HAC-adjusted inference at each partition. Decision logic:
- PASS: IC maintains sign, magnitude varies < 2x
- CAUTION: sign stable but magnitude varies >2x; OR sign flips but
opposite-sign partition not significant (HAC |t| < 2)
- STOP: sign flips with significant opposite AND unconditional IC ≈ 0
Returns: (regime_results, result_label, interpretation_msg)
"""
partitions = [
("Low VIX", df.filter(pl.col("vixcls") < vix_low)),
("Mid VIX", df.filter((pl.col("vixcls") >= vix_low) & (pl.col("vixcls") <= vix_high))),
("High VIX", df.filter(pl.col("vixcls") > vix_high)),
]
# Unconditional IC with HAC
ic_unc, _, ic_series_unc = compute_cross_sectional_ic(df, feature_col, outcome_col)
hac_unc = (
compute_ic_hac_stats(ic_series_unc, label_horizon=LABEL_HORIZON)
if ic_series_unc
else {"t_stat": np.nan}
)
unc_sign = np.sign(ic_unc) if ic_unc != 0 else 0
unc_sig = abs(hac_unc["t_stat"]) > 2.0 if not np.isnan(hac_unc["t_stat"]) else False
regime_results = []
for name, partition_df in partitions:
ic, _, ic_series = compute_cross_sectional_ic(partition_df, feature_col, outcome_col)
hac = (
compute_ic_hac_stats(ic_series, label_horizon=LABEL_HORIZON)
if ic_series
else {"t_stat": np.nan}
)
regime_results.append(
{"regime": name, "ic": ic, "t_hac": hac["t_stat"], "n": len(partition_df)}
)
# Assess sign stability with HAC significance
valid = [r for r in regime_results if not np.isnan(r["ic"])]
has_significant_opposite = False
has_nonsig_opposite = False
for r in valid:
if r["ic"] != 0 and np.sign(r["ic"]) != unc_sign:
if not np.isnan(r["t_hac"]) and abs(r["t_hac"]) > 2.0:
has_significant_opposite = True
else:
has_nonsig_opposite = True
signs = [np.sign(r["ic"]) for r in valid if r["ic"] != 0]
all_same_sign = len(set(signs)) <= 1 if signs else False
if all_same_sign:
ic_vals = [abs(r["ic"]) for r in valid if r["ic"] != 0]
magnitude_ratio = (
max(ic_vals) / min(ic_vals) if ic_vals and min(ic_vals) > 1e-6 else float("inf")
)
if magnitude_ratio < 2.0:
result = "PASS"
msg = "IC maintains sign and magnitude across VIX regimes"
else:
result = "CAUTION"
msg = f"IC sign stable but magnitude varies {magnitude_ratio:.1f}x"
elif has_significant_opposite and not unc_sig:
result = "STOP"
msg = "IC flips sign (HAC sig.) and unconditional IC ≈ 0 - aggregation artifact"
elif has_significant_opposite and unc_sig:
result = "CAUTION"
msg = "IC flips sign (HAC sig.) but unconditional IC is itself significant"
else:
result = "CAUTION"
msg = "IC changes sign in one regime but not significantly (HAC)"
return regime_results, result, msg
```
## Run All Checks on Both Features
We apply each plausibility check to 12-1 momentum and 1-day reversal, collecting
results for the comparison scorecard. All t-statistics use Newey-West (HAC)
standard errors to account for serial dependence in overlapping IC series.
### Timing Placebo
For a rolling-window feature with lookback $L$, a $\Delta$-shifted version
shares roughly $(L - \Delta)/L$ of its inputs with the original. This creates
a mechanical floor on IC persistence. We therefore treat the IC-lag profile as
a **diagnostic** (how does IC behave as information ages?) rather than applying
a hard decay threshold. More weight should be placed on lags beyond the lookback
window.
For 12-1 momentum (lookback ~231 trading days), we extend the lag grid to
252d so the furthest lag reaches beyond the lookback window entirely.
```python
timing = {}
# 12-1 Momentum: extended lag grid for long lookback (~231d)
lag_df, result, msg = run_timing_placebo(
analysis.drop_nulls(subset=["mom_12_1", "forward_return"]),
"mom_12_1",
lags=[0, 5, 21, 42, 63, 126, 252],
)
timing["mom_12_1"] = {"lag_df": lag_df, "result": result, "msg": msg}
print(f"{FEATURES['mom_12_1']}: {result} - {msg}")
# 1-day Reversal: fine-grained grid for short lookback (1d)
lag_df, result, msg = run_timing_placebo(
analysis.drop_nulls(subset=["rev_1d", "forward_return"]),
"rev_1d",
lags=[0, 1, 2, 5, 10, 21],
)
timing["rev_1d"] = {"lag_df": lag_df, "result": result, "msg": msg}
print(f"{FEATURES['rev_1d']}: {result} - {msg}")
```
```python
# Side-by-side timing placebo visualization
fig = make_subplots(
rows=1,
cols=2,
subplot_titles=["(a) 12-1 Momentum", "(b) 1-day Reversal"],
shared_yaxes=True,
)
for i, feat_col in enumerate(FEATURES):
lag_df = timing[feat_col]["lag_df"]
fig.add_trace(
go.Bar(
x=[f"+{lag}d" for lag in lag_df["lag"].to_list()],
y=lag_df["ic"].to_list(),
marker_color=COLORS["blue"],
showlegend=False,
),
row=1,
col=i + 1,
)
fig.add_hline(y=0, line_dash="dash", line_color="gray", row=1, col=i + 1)
fig.update_layout(
height=350,
title="Mean IC against how stale the feature is, for two features",
)
fig.update_yaxes(title_text="Mean IC", row=1, col=1)
fig.update_xaxes(title_text="Feature Lag", row=1, col=1)
fig.update_xaxes(title_text="Feature Lag", row=1, col=2)
show_plotly_with_alt(
fig,
alt=(
"Two bar panels sharing a vertical IC axis, showing what happens to each "
"feature's mean IC as the feature is deliberately made stale. The left panel, "
"12-1 momentum, has tall bars at every lag from zero out to 252 days: the tallest "
"is at lag zero, and the rest sit between about two thirds and nine tenths of it "
"with no clear downward trend. The right panel, one-day reversal, has bars so "
"short they are barely distinguishable from the dashed zero line at every lag, "
"including lag zero."
),
)
```
12-1 momentum IC is strongest at lag 0 and decays only gradually out to the longest lag
tested, as the figure shows. The persistence is partly mechanical: with a lookback of
roughly a year, a shifted feature shares most of its input data with the original.
The key diagnostic is that IC peaks at lag 0 - the most recent version of the
feature is the most informative - and even the fully stale version (lag 252d,
beyond the lookback) retains some residual predictability. Reversal IC is near
zero at all lags - there is no timely information to decay.
### Shared-Driver Check Results
We test whether each feature's cross-sectional mean predicts Treasury (IEF)
21-day forward returns. For 12-1 momentum, Treasury co-movement through
risk-on/risk-off dynamics is plausible, so a non-zero Treasury IC would flag
a shared-driver concern rather than strictly falsify the mechanism.
```python
n_perms = N_PERMUTATIONS
nc = {}
for feat_col in FEATURES:
sub = analysis.drop_nulls(subset=[feat_col, "forward_return", "treasury_fwd_return"])
metrics, result = run_shared_driver_check(
sub, feat_col, baseline[feat_col]["ic"], n_permutations=n_perms
)
nc[feat_col] = {"metrics": metrics, "result": result}
m = metrics
print(
f"{FEATURES[feat_col]}: {result}\n"
f" Treasury IC={m['ic_treasury']:.4f} (HAC t={m['t_treasury']:.2f})\n"
f" Block-permutation p={m['perm_p']:.4f} "
f"(B={m['n_permutations']}, finest resolvable p={m['perm_resolution']:.4f}, "
f"null sd={m['null_std']:.4f})"
)
```
The reading for this row is the **Treasury** column, and only that column. Neither
feature's cross-sectional mean predicts Treasury forward returns, so neither shows the
shared-driver confound this check exists to detect.
The permutation column is reported beside it but does not decide the outcome, because it
answers a different question - *is there any signal here at all?* - and the timing check
already reports that. Gating the shared-driver reading on it would mark a feature as
having a confound when its actual problem is having nothing to confound.
Two things to read off the permutation column. First, the p-value is computed as
$(r+1)/(B+1)$ and printed beside the finest value $B$ permutations can resolve; a
permutation p-value is never exactly zero, because the observed assignment is
itself one of the arrangements under the null. Second, the null is **block**
permuted at the label horizon, so its spread reflects how persistent this data actually
is. An independent within-date shuffle would produce a far narrower null and would
reject nearly anything - `05_signal_evaluation` measures that gap on its own panel.
### Regime Heterogeneity (VIX Regimes)
A sign flip is only flagged as STOP if the opposite-sign partition has
HAC $|t| > 2$ **and** the unconditional IC is itself not significant. In
low-power subsamples, small ICs with the wrong sign may reflect noise rather
than genuine heterogeneity. This criterion is stated here as an a priori
design choice.
The regime cutoffs below are terciles. The reading is qualitatively the same under a
median split or quartiles, because the diagnostic is sign stability rather than the
exact boundary.
```python
VIX_LOW_THRESHOLD = 15
VIX_HIGH_THRESHOLD = 22
cond = {}
for feat_col in FEATURES:
sub = analysis.drop_nulls(subset=[feat_col, "forward_return"])
regime_results, result, msg = run_regime_heterogeneity(
sub, feat_col, vix_low=VIX_LOW_THRESHOLD, vix_high=VIX_HIGH_THRESHOLD
)
cond[feat_col] = {"regimes": regime_results, "result": result, "msg": msg}
print(f"\n{FEATURES[feat_col]}: {result} - {msg}")
for r in regime_results:
print(f" {r['regime']:10s}: IC={r['ic']:+.4f} (HAC t={r['t_hac']:+.2f}, n={r['n']:,})")
```
```python
# Side-by-side conditioning visualization
fig = make_subplots(
rows=1,
cols=2,
subplot_titles=["(a) 12-1 Momentum", "(b) 1-day Reversal"],
shared_yaxes=True,
)
for i, feat_col in enumerate(FEATURES):
regimes = cond[feat_col]["regimes"]
ic_unc = baseline[feat_col]["ic"]
x_labels = [r["regime"] for r in regimes] + ["Unconditional"]
y_values = [r["ic"] for r in regimes] + [ic_unc]
colors = [COLORS["blue"]] * len(regimes) + [COLORS["amber"]]
fig.add_trace(
go.Bar(x=x_labels, y=y_values, marker_color=colors, showlegend=False),
row=1,
col=i + 1,
)
fig.add_hline(y=0, line_dash="dash", line_color="gray", row=1, col=i + 1)
fig.update_layout(
height=350,
title="Mean IC by volatility regime, for two features",
)
fig.update_yaxes(title_text="Mean IC", row=1, col=1)
show_plotly_with_alt(
fig,
alt=(
"Two bar panels sharing a vertical IC axis, each showing a feature's mean IC in "
"the low, middle and high volatility terciles, with the unconditional IC beside "
"them in amber. For 12-1 momentum the bars fall steeply from low to high "
"volatility, the low-volatility bar standing about five times the "
"high-volatility one, and the unconditional bar sits between the low and middle "
"values. For one-day reversal the pattern runs the other way and crosses zero: "
"the low-volatility bar hangs below the axis, the middle one is slightly "
"positive, the high-volatility one is the tallest of the three, and the "
"unconditional bar is almost invisible."
),
)
```
The two panels fail the same check in opposite directions. Momentum's unconditional IC
is an average over regimes in which it behaves very differently, so the single number
understates it in calm markets and overstates it in turbulent ones. Reversal's
unconditional IC is near zero not because the feature does nothing but because its sign
flips across regimes and the halves cancel. In both cases the headline IC is an artifact
of mixing, and a model trained on the pooled sample is being asked to fit two different
relationships with one coefficient.
### Publication Figure Artifact
The book scorecard figure reads this compact artifact so formatting changes do
not rerun ETF/VIX preparation or permutation checks.
```python
FIGURE_7_10_LAGS = [0, 1, 5, 21, 63, 126, 252]
FIGURE_7_10_FEATURES = {"mom_12_1": "12-1 Momentum", "rev_1d": "1-day Reversal"}
def _figure_7_10_timing(df: pl.DataFrame, feature_col: str) -> np.ndarray:
# shift().over("symbol") is order-sensitive - sort chronologically within
# symbol first so the lagged value aligns with the correct earlier row.
df = df.sort(["symbol", "timestamp"])
lag_ics = []
for lag in FIGURE_7_10_LAGS:
shifted = df.with_columns(
pl.col(feature_col).shift(lag).over("symbol").alias("_shifted")
).drop_nulls(subset=["_shifted", "forward_return"])
lag_ics.append(compute_cross_sectional_ic(shifted, "_shifted", "forward_return")[0])
return np.array(lag_ics)
def _figure_7_10_regime(df: pl.DataFrame, feature_col: str) -> np.ndarray:
regimes = [
df.filter(pl.col("vixcls") < VIX_LOW_THRESHOLD),
df.filter(
(pl.col("vixcls") >= VIX_LOW_THRESHOLD) & (pl.col("vixcls") <= VIX_HIGH_THRESHOLD)
),
df.filter(pl.col("vixcls") > VIX_HIGH_THRESHOLD),
]
return np.array(
[compute_cross_sectional_ic(regime, feature_col, "forward_return")[0] for regime in regimes]
)
def _figure_7_10_permutation(
df: pl.DataFrame, feature_col: str, baseline_ic: float
) -> tuple[np.ndarray, float, float]:
# Same null as the shared-driver check above, for the reasons in the markdown.
return block_permutation_null(
df, feature_col, baseline_ic, seed=SEED, n_permutations=N_PERMUTATIONS
)
def write_figure_7_10_artifact() -> Path:
artifact_data: dict[str, np.ndarray | float] = {"common_lags": np.array(FIGURE_7_10_LAGS)}
for feature_col in FIGURE_7_10_FEATURES:
sub = analysis.drop_nulls(subset=[feature_col, "forward_return", "vixcls"])
baseline_ic = compute_cross_sectional_ic(sub, feature_col, "forward_return")[0]
perm_ics, perm_p, perm_resolution = _figure_7_10_permutation(sub, feature_col, baseline_ic)
artifact_data[f"baseline__{feature_col}"] = baseline_ic
artifact_data[f"timing__{feature_col}"] = _figure_7_10_timing(sub, feature_col)
artifact_data[f"regime__{feature_col}"] = _figure_7_10_regime(sub, feature_col)
artifact_data[f"perm__{feature_col}"] = perm_ics
artifact_data[f"perm_p__{feature_col}"] = perm_p
# The floor travels with the value: a figure caption quoting p without it
# cannot tell a measurement from the smallest number B can express.
artifact_data[f"perm_resolution__{feature_col}"] = perm_resolution
print(
f" {feature_col}: permutation p={perm_p:.4f} "
f"(B={N_PERMUTATIONS}, finest resolvable p={perm_resolution:.4f})"
)
OUTPUT_DIR.mkdir(parents=True, exist_ok=True)
artifact = OUTPUT_DIR / "figure_7_10_inputs.npz"
np.savez(artifact, **artifact_data)
return artifact
figure_7_10_artifact = write_figure_7_10_artifact()
print(f"Wrote publication figure artifact: {figure_7_10_artifact}")
```
12-1 momentum keeps a positive IC in all three VIX regimes, but the magnitude varies by
several times between the calmest and the most turbulent, and only the calm-regime
estimate is significant - the figures are printed above. This is consistent with the
well-documented "momentum crash": momentum strategies suffer in high-volatility
environments (Daniel and Moskowitz 2016). The sign is stable across regimes, so this
earns CAUTION for magnitude variation rather than STOP.
Reversal shows a significant sign flip: its IC is negative in the low-VIX regime and
positive in the high-VIX one, both at t-statistics past two in their respective
directions, and the two halves cancel into a near-zero unconditional IC. That is a
textbook aggregation artifact - the feature encodes opposite information depending on
the volatility state - and earns STOP.
**Important**: this check cannot distinguish confounding from genuine effect
modification. A feature whose IC varies by regime may be confounded *or* may
have a mechanism that operates differently across states. Chapter 15 provides
multivariate sensitivity analysis to separate these cases.
## Collider Bias: A Synthetic Demonstration
We place this simulation here - after the three main checks - because its
purpose is different: it illustrates a DAG concept from Section 2 rather
than diagnosing a specific feature. Readers who want to reinforce the DAG
vocabulary before running the checks can read this section first.
The text defines **collider bias**: conditioning on a common *effect* of feature
and label creates a spurious association between them. This short simulation
makes the concept concrete.
**Setup**: Momentum ($X$) and forward returns ($Y$) are independent. Fund flows
($Z$) are a function of both: $Z = \beta_1 X + \beta_2 Y + \varepsilon$. Because
$Z$ is a collider, conditioning on it opens a path between $X$ and $Y$.
```python
# Collider-bias simulation
rng_collider = np.random.default_rng(123)
n_obs = 5000
# X (momentum) and Y (forward return) are independent
X = rng_collider.standard_normal(n_obs)
Y = rng_collider.standard_normal(n_obs)
# Z (fund flows) is a common effect of both
Z = 0.5 * X + 0.5 * Y + rng_collider.standard_normal(n_obs) * 0.5
# Unconditional correlation: should be ~0
rho_uncond, _ = stats.spearmanr(X, Y)
# Conditional on Z > median (selecting funds with high flows): spurious correlation
high_Z = np.median(Z) < Z
rho_cond, _ = stats.spearmanr(X[high_Z], Y[high_Z])
print(
pl.DataFrame(
{
"condition": ["Unconditional", "Conditional on Z > median"],
"spearman_rho": [f"{rho_uncond:.4f}", f"{rho_cond:.4f}"],
"interpretation": [
"X and Y are independent (as generated)",
"Spurious negative correlation - collider bias",
],
}
)
)
```
Conditioning on high fund flows (the collider) induces a negative correlation
between momentum and returns: among high-flow funds, strong momentum implies
weaker returns and vice versa. This is an artifact of the conditioning, not a
real effect. The practical lesson: never condition on a variable that is
*caused by* both the feature and the label.
## PlausibiSe muestra íntegramente con atribución según la licencia de la fuente. Licencia: MIT
Este resumen lo redactó el agente de investigación de Stratmill a partir del original; no es una copia de la fuente.