HACとブロックブートストラップによるシグナル情報係数の推論

# 06_ic_inference.py


```py
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# %% [markdown]
# # IC Inference: HAC Adjustment and Block Bootstrap
#
# **Docker image**: `ml4t`
#
# **Chapter 7: Defining the Learning Task**
# **Section Reference**: 7.3 - Feature and Label Evaluation as Triage
#
# ## Purpose
#
# This notebook addresses **statistical inference for IC**: how confident should we
# be that a signal's IC is not just noise? We cover HAC adjustment for autocorrelated
# IC series and block bootstrap for robust confidence intervals.
#
# ## Learning Objectives
#
# 1. Understand why naive t-statistics fail for IC inference
# 2. Apply HAC (Newey-West) adjustment for proper standard errors
# 3. Implement block bootstrap for distribution-free confidence intervals
# 4. Distinguish practical vs statistical significance
# 5. Plan track records using HAC-adjusted effective sample size
#
# ## Data Policy
#
# Examples use **real ETF data** from the case study store.
#
# ## Prerequisites
#
# - `05_signal_evaluation` - produces the IC time series whose inference we
#   formalize here.
# - Familiarity with autocorrelation, the Newey-West HAC estimator, and
#   block bootstrap.

# %%
"""IC Inference - statistical testing and confidence intervals for information coefficients."""

from __future__ import annotations

import json
import logging
import warnings

import numpy as np
import plotly.graph_objects as go
import polars as pl
from arch.bootstrap import StationaryBootstrap
from IPython.display import display
from ml4t.diagnostic.evaluation.autocorrelation import analyze_autocorrelation
from ml4t.diagnostic.metrics import compute_ic_hac_stats
from ml4t.diagnostic.signal import analyze_signal
from plotly.subplots import make_subplots
from scipy import stats

from data import load_etfs
from utils.reproducibility import set_global_seeds
from utils.style import (  # importing utils.style activates the ml4t Plotly template
    COLORS,
    show_plotly_with_alt,
)

# Quiet ml4t library INFO logging. Child loggers set their own level, so raising the
# parent alone does not silence them - set every already-created ml4t logger explicitly.
for _lg in list(logging.root.manager.loggerDict):
    if _lg.startswith("ml4t"):
        logging.getLogger(_lg).setLevel(logging.WARNING)

# %% tags=["parameters"]
SEED = 42
START_DATE = ""
N_BOOT = 2000
# Forward-return horizon, in trading days. Every dependence-aware choice in this
# notebook is derived from it: the momentum lookback, the IC evaluation period,
# the HAC lag truncation, and the bootstrap block length.
LABEL_HORIZON = 21

# %%
set_global_seeds(SEED)


# %% [markdown]
# ## The Autocorrelation Problem
#
# IC time series exhibit **autocorrelation** due to:
#
# 1. **Overlapping forward returns** at longer horizons
# 2. **Persistent signal values** (momentum doesn't flip daily)
# 3. **Regime persistence** (markets stay in trends/ranges)
#
# Naive t-statistics assume iid observations and **underestimate standard errors**,
# leading to inflated significance.

# %%
# Load real ETF data and compute momentum signal
etfs = load_etfs()

if START_DATE:
    etfs = etfs.filter(pl.col("timestamp") >= pl.lit(START_DATE).str.to_date())

# Compute momentum over the label horizon
factor_df = (
    etfs.sort(["symbol", "timestamp"])
    .with_columns(
        [
            (pl.col("close") / pl.col("close").shift(LABEL_HORIZON).over("symbol") - 1).alias(
                "factor"
            )
        ]
    )
    .filter(pl.col("factor").is_not_null())
    .select(["timestamp", "symbol", "factor"])
)

prices_df = etfs.select(["timestamp", "symbol", "close"]).rename({"close": "price"})

# Run signal analysis to get IC series
result = analyze_signal(
    factor_df,
    prices_df,
    periods=(LABEL_HORIZON,),
    quantiles=5,
    ic_method="spearman",
    date_col="timestamp",
    asset_col="symbol",
)

ic_series = np.array(result.ic_series.get(f"{LABEL_HORIZON}D", []))
print(f"IC series length: {len(ic_series)}")
print(f"Mean IC: {np.mean(ic_series):.4f}")

# %% [markdown]
# ### Autocorrelation in IC Series
#
# Let's examine the autocorrelation structure directly.


# %%
# Compute autocorrelation function
def compute_acf(series, nlags=20):
    """Compute autocorrelation function."""
    n = len(series)
    series = series - np.mean(series)
    acf = []
    for lag in range(nlags + 1):
        if lag == 0:
            acf.append(1.0)
        else:
            acf.append(np.corrcoef(series[lag:], series[:-lag])[0, 1])
    return np.array(acf)


n_acf_lags = LABEL_HORIZON - 1
acf_values = compute_acf(ic_series, nlags=n_acf_lags)

# Significance bounds for white noise
n = len(ic_series)
sig_bound = 1.96 / np.sqrt(n)

fig = go.Figure()

n_significant_lags = sum(abs(acf_values[1:]) > sig_bound)

fig.add_trace(
    go.Bar(
        x=list(range(1, n_acf_lags + 1)),
        y=acf_values[1:],
        name="ACF",
        marker_color=[
            COLORS["amber"] if abs(v) > sig_bound else COLORS["blue"] for v in acf_values[1:]
        ],
    )
)

fig.add_hline(
    y=sig_bound, line_dash="dash", line_color=COLORS["negative"], annotation_text="95% band"
)
fig.add_hline(y=-sig_bound, line_dash="dash", line_color=COLORS["negative"])
fig.add_hline(y=0, line_color=COLORS["neutral"])

fig.update_layout(
    title=f"Autocorrelation of the daily IC series, lags 1 to {LABEL_HORIZON - 1}",
    xaxis_title="Lag (days)",
    yaxis_title="Autocorrelation",
    template="ml4t",
    height=350,
)
show_plotly_with_alt(
    fig,
    alt=(
        "A bar chart of the autocorrelation of the daily IC series against lag, from one "
        "day out to one day short of the label horizon, with dashed red lines marking the "
        "white-noise band either side of zero. The bars start just above 0.8 at lag one "
        "and fall away in a smooth, almost linear decline, reaching the band around lag "
        "sixteen. Every bar up to that point is amber, marking it outside the band; only "
        "the last few lags are inside it and drawn in navy."
    ),
)
print(f"\nSignificant autocorrelation lags: {n_significant_lags}/{n_acf_lags}")

# %% [markdown]
# The decline is smooth and reaches the white-noise band near the label horizon, which is
# consistent with overlap driving it: two IC values computed $k$ days apart share $h - k$
# days of return, so the shared window shrinks as $k$ grows. It is consistent with
# persistent signal ranks as well, and the two are not separated here. Nor can this figure
# say what happens *past* the horizon - it stops at $h-1$, and a series of rank
# correlations can stay dependent beyond the overlap through regimes or a slow-moving
# signal. What the figure does establish is enough for the section's purpose: a naive
# t-statistic treats these values as independent draws, and they are not.

# %% [markdown]
# ### Library Autocorrelation Analysis
#
# The library version includes PACF and the **Ljung-Box portmanteau test**,
# which tests the joint null that all autocorrelations up to lag $L$ are zero.

# %%
acf_analysis = analyze_autocorrelation(ic_series, max_lags=LABEL_HORIZON - 1, alpha=0.05)

print("=== ml4t-diagnostic Autocorrelation Analysis ===\n")
print(f"Significant ACF lags:  {acf_analysis.significant_acf_lags}")
print(f"Significant PACF lags: {acf_analysis.significant_pacf_lags}")
print(f"Is white noise:        {acf_analysis.is_white_noise}")
print(f"Suggested ARIMA order: {acf_analysis.suggested_arima_order}")

# %% [markdown]
# ## HAC (Newey-West) Adjustment
#
# HAC (Heteroskedasticity and Autocorrelation Consistent) standard errors account
# for serial correlation in the IC series.
#
# The Newey-West estimator uses a weighted sum of autocovariances:
#
# $$\hat{\sigma}^2_{HAC} = \hat{\gamma}_0 + 2\sum_{j=1}^{L} w_j \hat{\gamma}_j$$
#
# Where $w_j = 1 - j/(L+1)$ (Bartlett kernel) and $L$ is the lag truncation.
#
# **$L$ should not be left to the sample size here.** The usual automatic rule
# $L = \lfloor 4(T/100)^{2/9} \rfloor$ reads only $T$, and on this series it
# returns 9. The labels are $h$-day *overlapping* forward returns, so date $t$ and
# date $t+j$ share part of their outcome window for every $j < h$ - and the signal
# is 21-day momentum, whose cross-sectional ranks change slowly. Overlapping
# outcomes measured against persistent ranks give ICs that move together over
# roughly the label horizon.
#
# **Both halves of that sentence are doing work, and the overlap alone is not
# enough.** The IC is a rank correlation, not a return: if the signal ranks were
# redrawn independently every day, the ICs could be serially uncorrelated no matter
# how much the return windows overlap. So $h-1$ is not a theorem about overlapping
# labels - it is a **conservative horizon-aware bandwidth for a persistent signal**,
# and the evidence that it is the right order of magnitude is the measured ACF in
# the figure above, which stays outside the white-noise band until about lag 16.
#
# What is not defensible is $L = 9$, chosen by a rule that never looked at the
# labels or the signal. Passing `label_horizon` makes the library take
# $L = \max(h-1, \text{auto rule})$; the ACF is what justifies that choice, and the
# next section measures what it is worth.

# %% [markdown]
# ### What the Bandwidth Is Worth
#
# Worth measuring rather than asserting, so the cell below runs the estimator
# both ways on the same IC series. The block bootstrap below resamples contiguous blocks
# of length $h$, so it already respects the overlap. If HAC is given a bandwidth that does
# not, the two will not agree - and the gap is then a property of the bandwidth, not of the
# estimator families, which is what the side-by-side is there to establish. The bootstrap
# section shows the two agreeing once both are horizon-aware.
#
# Calling the estimator without `label_horizon` raises a `UserWarning` saying the automatic
# bandwidth may be anti-conservative for overlapping labels. That warning is the library
# doing its job, and the call below triggers it on purpose; it is silenced for that one
# call so it does not read as a defect in the run.

# %%
# Compute HAC-adjusted statistics
hac_result = compute_ic_hac_stats(ic_series, label_horizon=LABEL_HORIZON)

# The same estimator with the bandwidth left to the sample-size rule. Omitting
# label_horizon is deliberate here and the library warns about it, correctly; the warning
# is silenced for this one call, by category and message, so it does not read as a defect.
with warnings.catch_warnings():
    warnings.filterwarnings(
        "ignore",
        category=UserWarning,
        message="label_horizon was not provided.*",
    )
    hac_auto = compute_ic_hac_stats(ic_series)
print("=== Bandwidth: sample-size rule vs label-horizon aware ===\n")
print(
    f"  auto rule            L={hac_auto['effective_lags']:>3}  "
    f"SE={hac_auto['hac_se']:.6f}  t={hac_auto['t_stat']:.4f}  p={hac_auto['p_value']:.4f}"
)
print(
    f"  label_horizon={LABEL_HORIZON:<8} L={hac_result['effective_lags']:>3}  "
    f"SE={hac_result['hac_se']:.6f}  t={hac_result['t_stat']:.4f}  p={hac_result['p_value']:.4f}"
)
print(f"\n  SE ratio (horizon-aware / auto): {hac_result['hac_se'] / hac_auto['hac_se']:.3f}x")

# Compare naive vs HAC
naive_se = np.std(ic_series, ddof=1) / np.sqrt(len(ic_series))
naive_t = np.mean(ic_series) / naive_se

print("=== Naive vs HAC Inference ===\n")
print(
    f"HAC lag truncation: {hac_result['effective_lags']} "
    f"(horizon-aware choice: label horizon {LABEL_HORIZON} -> L = {LABEL_HORIZON - 1}, "
    f"consistent with the measured ACF above)"
)
print(f"Mean IC: {hac_result['mean_ic']:.4f}")
print("\nStandard Errors:")
print(f"  Naive SE:    {naive_se:.4f}")
print(f"  HAC SE:      {hac_result['hac_se']:.4f}")
print(f"  Inflation:   {hac_result['hac_se'] / naive_se:.2f}x")

print("\nt-Statistics:")
print(f"  Naive t:     {naive_t:.2f}")
print(f"  HAC t:       {hac_result['t_stat']:.2f}")

print("\np-Values:")
print(f"  Naive p:     {2 * stats.t.sf(abs(naive_t), len(ic_series) - 1):.4f}")
print(f"  HAC p:       {hac_result['p_value']:.4f}")

# Significance conclusion
alpha = 0.05
naive_sig = "Yes" if abs(naive_t) > stats.t.ppf(1 - alpha / 2, len(ic_series) - 1) else "No"
hac_sig = "Yes" if hac_result["p_value"] < alpha else "No"

print("\nSignificant at α=0.05?")
print(f"  Naive:       {naive_sig}")
print(f"  HAC:         {hac_sig}")

if naive_sig == "Yes" and hac_sig == "No":
    print("\n  WARNING: Naive test finds significance that HAC adjustment removes!")

# %% [markdown]
# ### Effective Sample Size
#
# The HAC SE inflation factor tells us how much the autocorrelation reduces our
# effective sample size:
#
# $$T_{eff} = T \times \left(\frac{\sigma_{naive}}{\sigma_{HAC}}\right)^2$$

# %%
# Compute effective sample size
inflation_factor = hac_result["hac_se"] / naive_se
effective_n = len(ic_series) / (inflation_factor**2)

print("=== Effective Sample Size ===\n")
print(f"Actual observations: {len(ic_series)}")
print(f"HAC inflation factor: {inflation_factor:.2f}x")
print(f"Effective sample size: {effective_n:.0f}")
print(f"Efficiency loss: {(1 - effective_n / len(ic_series)) * 100:.0f}%")

# %% [markdown]
# ## Block Bootstrap for Robust CI
#
# When IC series has autocorrelation, **iid bootstrap is invalid**. We use
# **block bootstrap** which preserves the dependence structure by resampling
# contiguous blocks of observations.
#
# Two common approaches:
# 1. **Moving Block Bootstrap (MBB)**: Fixed block length
# 2. **Stationary Bootstrap**: Random block lengths (geometric distribution)
#
# We use the `arch` library's `StationaryBootstrap` for robust inference.


# %%
# Block bootstrap implementation
def block_bootstrap_ic_ci(
    ic_series: np.ndarray,
    n_bootstrap: int = 1000,
    block_length: int = 21,
    alpha: float = 0.05,
    random_state: int = 42,
) -> dict:
    """
    Compute block bootstrap confidence interval for mean IC.

    Uses the Politis-Romano stationary bootstrap from `arch.bootstrap`:
    random block lengths drawn from a geometric distribution with
    expected length equal to ``block_length``.

    Parameters
    ----------
    ic_series : array
        Time series of IC values
    n_bootstrap : int
        Number of bootstrap replications
    block_length : int
        Expected block length (mean of the geometric distribution)
    alpha : float
        Significance level (default 0.05 for 95% CI)
    random_state : int
        Random seed for reproducibility

    Returns
    -------
    dict with mean, CI bounds, and bootstrap distribution
    """
    ic_series = np.array(ic_series)

    bs = StationaryBootstrap(block_length, ic_series, seed=random_state)
    bootstrap_means = np.array([np.mean(data[0][0]) for data in bs.bootstrap(n_bootstrap)])

    # Percentile confidence interval
    ci_lower = np.percentile(bootstrap_means, 100 * alpha / 2)
    ci_upper = np.percentile(bootstrap_means, 100 * (1 - alpha / 2))

    return {
        "mean": np.mean(ic_series),
        "ci_lower": ci_lower,
        "ci_upper": ci_upper,
        "bootstrap_std": np.std(bootstrap_means),
        "bootstrap_means": bootstrap_means,
        "method": "Stationary Bootstrap",
        "block_length": block_length,
    }


# %%
# Run block bootstrap
n_boot = N_BOOT
block_result = block_bootstrap_ic_ci(
    ic_series, n_bootstrap=n_boot, block_length=LABEL_HORIZON, random_state=SEED
)

# Compare with HAC CI
hac_ci_lower = hac_result["mean_ic"] - 1.96 * hac_result["hac_se"]
hac_ci_upper = hac_result["mean_ic"] + 1.96 * hac_result["hac_se"]

print("=== Block Bootstrap vs HAC Confidence Intervals ===\n")
print(f"Method: {block_result['method']} (block length={block_result['block_length']})")
print(f"Bootstrap replications: {n_boot}")
print(f"\nMean IC: {block_result['mean']:.4f}")
print("\n95% Confidence Intervals:")
print(f"  Block Bootstrap: [{block_result['ci_lower']:.4f}, {block_result['ci_upper']:.4f}]")
print(f"  HAC (Newey-West): [{hac_ci_lower:.4f}, {hac_ci_upper:.4f}]")
print("\nStandard Errors:")
print(f"  Block Bootstrap: {block_result['bootstrap_std']:.4f}")
print(f"  HAC:             {hac_result['hac_se']:.4f}")

# Does CI include zero?
ci_includes_zero = block_result["ci_lower"] <= 0 <= block_result["ci_upper"]
print(f"\nBlock Bootstrap CI includes zero: {'Yes' if ci_includes_zero else 'No'}")

# %% [markdown]
# ### Library vs Manual Bootstrap: When to Use Each
#
# The `ml4t-diagnostic` library provides `stationary_bootstrap_ic()` for
# **cross-sectional** bootstrap: given arrays of predictions and returns for a
# single date, it resamples asset pairs and recomputes Spearman IC. This is useful
# when you want a confidence interval for a single cross-section's IC.
#
# For **time-series** inference on the IC series itself - "is the mean IC over
# many dates significantly different from zero?" - the correct tool is the block bootstrap
# used above, which preserves temporal dependence.
#
# ```python
# # Cross-sectional bootstrap (library) - CI for one date's IC
# from ml4t.diagnostic.evaluation.stats.hac_standard_errors import stationary_bootstrap_ic
# result = stationary_bootstrap_ic(signals_t, returns_t, n_samples=1000)
#
# # Time-series bootstrap (manual) - CI for mean IC across dates
# block_bootstrap_ic_ci(ic_series, n_bootstrap=2000, block_length=21)
# ```
#
# Throughout this notebook we use the manual block bootstrap because we are
# testing whether the *time-averaged* IC is significantly positive.

# %%
print("=== Block Bootstrap Summary (recommended for IC time series) ===\n")
print(f"  IC: {block_result['mean']:.4f}")
print(f"  Bootstrap SE: {block_result['bootstrap_std']:.4f}")
print(f"  95% CI: [{block_result['ci_lower']:.4f}, {block_result['ci_upper']:.4f}]")
print(f"  HAC CI: [{hac_ci_lower:.4f}, {hac_ci_upper:.4f}]")

# %%
# Visualize bootstrap distribution
fig = make_subplots(
    rows=1,
    cols=2,
    subplot_titles=["Bootstrap distribution of mean IC", "95% CI: block bootstrap vs HAC"],
)

# Bootstrap distribution
fig.add_trace(
    go.Histogram(
        x=block_result["bootstrap_means"],
        nbinsx=50,
        name="Bootstrap means",
        marker_color=COLORS["blue"],
        opacity=0.7,
    ),
    row=1,
    col=1,
)

# Mark CIs
fig.add_vline(
    x=block_result["ci_lower"],
    line_dash="dash",
    line_color=COLORS["negative"],
    annotation_text="2.5%",
    row=1,
    col=1,
)
fig.add_vline(
    x=block_result["ci_upper"],
    line_dash="dash",
    line_color=COLORS["negative"],
    annotation_text="97.5%",
    row=1,
    col=1,
)
fig.add_vline(
    x=0, line_dash="dot", line_color=COLORS["neutral"], annotation_text="Null", row=1, col=1
)

# CI comparison
methods = ["Block Bootstrap", "HAC"]
ci_lowers = [block_result["ci_lower"], hac_ci_lower]
ci_uppers = [block_result["ci_upper"], hac_ci_upper]
means = [block_result["mean"], hac_result["mean_ic"]]

for i, method in enumerate(methods):
    # CI as error bars
    fig.add_trace(
        go.Scatter(
            x=[method],
            y=[means[i]],
            error_y=dict(
                type="data",
                symmetric=False,
                array=[ci_uppers[i] - means[i]],
                arrayminus=[means[i] - ci_lowers[i]],
            ),
            mode="markers",
            marker=dict(size=12, color=[COLORS["blue"], COLORS["amber"]][i]),
            name=method,
        ),
        row=1,
        col=2,
    )

fig.add_hline(y=0, line_dash="dot", line_color=COLORS["neutral"], row=1, col=2)

fig.update_layout(
    title="Mean IC: block-bootstrap distribution and the two interval estimates",
    height=400,
    template="ml4t",
    showlegend=False,
)
fig.update_xaxes(title_text="Mean IC", row=1, col=1)
fig.update_yaxes(title_text="Frequency", row=1, col=1)
fig.update_yaxes(title_text="Mean IC", row=1, col=2)

show_plotly_with_alt(
    fig,
    alt=(
        "Two panels. The left panel is a histogram of the block-bootstrap distribution of "
        "the mean IC, a symmetric bell centred on zero and running out to roughly plus and "
        "minus 0.045, with dashed red lines marking the 2.5th and 97.5th percentiles and a "
        "dotted line at the null, which falls on the mode. "
        "The right panel plots the two interval estimates side by side as a point with "
        "error bars: the block bootstrap in navy and HAC in amber. Both point estimates "
        "sit on the dotted zero line and both intervals span it, reaching roughly the "
        "same distance above and below, so the two methods are hard to tell apart."
    ),
)

# %% [markdown]
# The two intervals are close enough to overlay, which is the result the bandwidth section
# predicted: once HAC is given a lag truncation matched to the label horizon, it agrees
# with a bootstrap that resamples blocks of that same length. They agree about the answer
# as well - both intervals contain zero, so the mean IC is not distinguishable from no
# skill at this sample size, whichever dependence-aware method is used.

# %% [markdown]
# ## Practical vs Statistical Significance
#
# Statistical significance of an IC time-series mean is a separate question from whether
# the signal is economically tradeable.
#
# There is no fixed IC magnitude that counts as detectable. Detectability is set by the
# standard error, and this notebook has already estimated the one that applies here - the
# HAC standard error on this series, at this sample size, with this label horizon. The
# cell below uses it rather than a table of conventions: it reports the smallest mean IC
# this series could distinguish from zero at the stated confidence, and then sweeps a
# range of candidate magnitudes through the same test.
#
# Net P&L after costs is a separate calculation, handled by the break-even analysis in
# `05_signal_evaluation` and by the case-study cost models in Chapters 16 to 18. A factor
# can clear the test below and still lose money.

# %% tags=["results"]
IC_CANDIDATES = (0.01, 0.02, 0.03, 0.05, 0.10)
CONFIDENCE = 0.95

z_crit = stats.norm.ppf(0.5 + CONFIDENCE / 2)
hac_se = hac_result["hac_se"]
min_detectable_ic = z_crit * hac_se

print(f"Sample: {len(ic_series):,} daily IC values, label horizon {LABEL_HORIZON}")
print(f"HAC standard error: {hac_se:.4f}  (lag truncation {hac_result['effective_lags']})")
print(f"Smallest |mean IC| separable from zero at {CONFIDENCE:.0%}: {min_detectable_ic:.4f}")
print()
print(f"{'candidate IC':>13}{'t vs 0':>9}{'separable':>11}")
print("-" * 33)
for candidate in IC_CANDIDATES:
    t_stat = candidate / hac_se
    print(f"{candidate:>13.2f}{t_stat:>9.2f}{'yes' if t_stat > z_crit else 'no':>11}")
print()
print(f"This series' observed mean IC: {hac_result['mean_ic']:.4f} (t={hac_result['t_stat']:.2f})")

# %%
# Practical significance analysis
print("=== Practical Significance Analysis ===\n")

observed_ic = np.mean(ic_series)
print(f"Observed IC: {observed_ic:.4f}")
print(f"HAC t-stat: {hac_result['t_stat']:.2f}")
print(f"HAC p-value: {hac_result['p_value']:.4f}")

# Cost-adjusted interpretation
# Simplified model: Net IC ≈ IC - cost_factor × turnover
# Assuming ~50% monthly turnover and 10bps round-trip cost
turnover_assumption = 0.5  # 50% monthly turnover
cost_per_trade = 0.001  # 10bps
cost_impact = turnover_assumption * cost_per_trade * 12  # Annualized

print("\nCost assumptions:")
print(f"  Monthly turnover: {turnover_assumption:.0%}")
print(f"  Round-trip cost: {cost_per_trade * 10000:.0f} bps")
print(f"  Annual cost drag: {cost_impact:.2%}")

# Rough IR approximation: IR ≈ IC × √(252) × some multiplier
# This is a simplified Grinold approximation
ir_approx = observed_ic * np.sqrt(252)
print(f"\nGrinold IR approximation (raw): {ir_approx:.2f}")
print("  (This is a raw, pre-cost upper bound - net IR depends on turnover and costs;")
print("   the break-even cost analysis in `05_signal_evaluation` evaluates feasibility.)")

# %% [markdown]
# ## Track Record Planning
#
# How many observations do we need to be confident in our IC estimate?
#
# Using HAC-adjusted effective sample size:
#
# $$T_{required} = \left(\frac{z_{\alpha/2} \times \sigma_{IC}}{IC_{target}}\right)^2 \times \text{HAC inflation}^2$$


# %%
def min_track_record_ic(
    target_ic: float,
    ic_std: float,
    confidence: float = 0.95,
    hac_inflation: float = 1.5,
) -> int:
    """
    Minimum track record to confirm IC with given confidence.

    Uses HAC-adjusted effective sample size.

    Parameters
    ----------
    target_ic : float
        Minimum IC we want to detect
    ic_std : float
        Standard deviation of IC series
    confidence : float
        Required confidence level
    hac_inflation : float
        HAC SE inflation factor, measured as ``hac_se / naive_se`` on the
        IC series in hand rather than assumed

    Returns
    -------
    int : minimum periods needed (actual, not effective)
    """
    z = stats.norm.ppf((1 + confidence) / 2)

    # Base formula without HAC
    base_n = (z * ic_std / target_ic) ** 2

    # Adjust for autocorrelation
    actual_n = base_n * (hac_inflation**2)

    return int(np.ceil(actual_n))


# %%
# Track record requirements
ic_std_observed = np.std(ic_series)
hac_inflation = hac_result["hac_se"] / naive_se

print("=== Minimum Track Record for IC Detection ===\n")
print(f"Observed IC std: {ic_std_observed:.4f}")
print(f"HAC inflation factor: {hac_inflation:.2f}x")
print()

target_ics = [0.01, 0.02, 0.03, 0.05]
track_record_rows = []
for target in target_ics:
    min_90 = min_track_record_ic(target, ic_std_observed, 0.90, hac_inflation)
    min_95 = min_track_record_ic(target, ic_std_observed, 0.95, hac_inflation)
    min_99 = min_track_record_ic(target, ic_std_observed, 0.99, hac_inflation)
    track_record_rows.append(
        {
            "target_ic": target,
            "days_90pct": min_90,
            "years_90pct": round(min_90 / 252, 1),
            "days_95pct": min_95,
            "years_95pct": round(min_95 / 252, 1),
            "days_99pct": min_99,
            "years_99pct": round(min_99 / 252, 1),
        }
    )

track_record_df = pl.DataFrame(track_record_rows)
print("Days of cross-sectional IC observations needed at each confidence level:")
display(track_record_df)

# %% [markdown]
# ### Interpretation
#
# The track record requirements are substantial because:
#
# 1. **IC is noisy**: daily cross-sectional IC has high variance.
# 2. **Autocorrelation reduces the effective sample**: overlapping forward returns inflate
#    the standard error by the factor printed above.
# 3. **Small effects need large samples**: the smallest separable mean IC printed earlier
#    is the bar this series sets, and it moves with the sample and the dependence rather
#    than sitting at a conventional number.
#
# **Implication**: a "predictive" factor claimed from one or two years of data deserves
# skepticism. The test to apply is its own standard error, not a threshold read off a
# table - which is what the cell above demonstrates on this series.

# %% [markdown]
# ## IC Inference Report
#
# Export a structured report for downstream use.

# %%
# Build IC inference report
inference_report = {
    "signal_name": f"momentum_{LABEL_HORIZON}d",
    "n_observations": len(ic_series),
    "mean_ic": round(float(np.mean(ic_series)), 4),
    "ic_std": round(float(np.std(ic_series)), 4),
    "inference": {
        "naive": {
            "se": round(float(naive_se), 4),
            "t_stat": round(float(naive_t), 2),
            "p_value": round(float(2 * stats.t.sf(abs(naive_t), len(ic_series) - 1)), 4),
        },
        "hac": {
            "se": round(float(hac_result["hac_se"]), 4),
            "t_stat": round(float(hac_result["t_stat"]), 2),
            "p_value": round(float(hac_result["p_value"]), 4),
            "inflation_factor": round(float(hac_inflation), 2),
        },
        "block_bootstrap": {
            "method": block_result["method"],
            "block_length": block_result["block_length"],
            "n_replications": n_boot,
            "se": round(float(block_result["bootstrap_std"]), 4),
            "ci_95_lower": round(float(block_result["ci_lower"]), 4),
            "ci_95_upper": round(float(block_result["ci_upper"]), 4),
        },
    },
    "effective_sample_size": round(float(effective_n), 0),
    "ci_includes_zero": bool(ci_includes_zero),
    "autocorrelation_lags_significant": int(n_significant_lags),
}

print("=== IC Inference Report ===\n")
print(json.dumps(inference_report, indent=2))

# %% [markdown]
# ## Summary
#
# ### Key Concepts
#
# | Concept | Description |
# |---------|-------------|
# | **HAC SE** | Newey-West adjustment for autocorrelated IC |
# | **Block Bootstrap** | Preserves dependence structure via block resampling |
# | **Effective N** | True sample size after accounting for autocorrelation |
# | **Track Record** | Minimum observations to detect a given IC |
#
# ### Best Practices
#
# 1. **Always use HAC-adjusted t-statistics** - naive t-stats overstate significance
# 2. **Report block bootstrap CIs** - distribution-free and robust
# 3. **Consider practical significance** - IC must exceed cost drag
# 4. **Plan for adequate track records** - 1-2 years rarely sufficient for small IC
# 5. **Check effective sample size** - may be much smaller than actual observations
#
# ### API Reference
#
# ```python
# from ml4t.diagnostic.metrics import compute_ic_hac_stats
#
# # HAC-adjusted statistics
# hac = compute_ic_hac_stats(ic_series, label_horizon=21)
# print(f"HAC t-stat: {hac['t_stat']:.2f}")
# print(f"HAC p-value: {hac['p_value']:.4f}")
#
# # Block bootstrap (requires arch library)
# from arch.bootstrap import StationaryBootstrap
# bs = StationaryBootstrap(21, ic_series)  # 21-day blocks
# ci = bs.conf_int(np.mean, 1000, method='percentile')
# ```
#
# ### Next Notebooks
#
# - [`07_multiple_testing`](07_multiple_testing.ipynb) - FDR control when evaluating many factors
# - [`08_causal_sanity_checks`](08_causal_sanity_checks.ipynb) - Causal falsification tests

```