IV-RVスプレッドと株式リターンの二重機械学習

# 12_causal_dml.py


```py
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# %% [markdown]
# # S&P 500 Equity+Option - IV-RV Spread: Causal DML Sensitivity Analysis
#
# Does the negative association between the implied-realized volatility spread
# and future equity returns remain after adjustment for observed confounders?
# This notebook applies DML to `ivrv_spread` across a retrospective panel drawn
# from the current-constituent roster, with option-derived features at daily
# frequency.
#
# **Treatment rationale**: The IV-RV spread captures option mispricing - when
# implied volatility exceeds realized, options are "expensive." Realized
# volatility (20d), equity momentum (21d), and risk-reversal skew (25-delta,
# 30d) confound this relationship because all three co-move with investor fear
# and positioning.
#
# The estimand, the analysis population, the cross-fitting geometry and the
# refutation design are resolved by the shared causal request rather than
# assembled here. The request verifies that the treatment and every confounder
# exist, excludes outcomes whose horizon reaches the holdout, keeps complete
# decision-time panels, cross-fits with an embargo, and records the placebo
# block design in the causal identity. This notebook inspects what it resolved,
# runs it, and reports the result.
#
# **Learning Objectives**:
# - Keep the holdout outside an exploratory causal analysis
# - Read the estimand and its temporal controls off the resolved specification
# - Compare the adjusted estimate with naive OLS and a block-permutation null
#
# **Book Reference**: Chapter 15, Section 15.7 (Case study causal evidence)
#
# **Prerequisites**: `03_financial_features.py`, `04_model_based_features.py`

# %% [markdown]
# ## Identifying Assumptions
#
# DML identifies a causal effect only if three assumptions hold:
# 1. **Conditional ignorability**: No unobserved confounders - all backdoor paths
#    between treatment and outcome are blocked by the observed confounders.
# 2. **Overlap (positivity)**: Every unit has a nonzero probability of receiving
#    any treatment level, conditional on confounders.
# 3. **SUTVA**: One unit's treatment doesn't affect another's outcome.
#
# These are untestable. The refutation test below provides indirect evidence
# but cannot prove the assumptions hold.

# %%
"""Estimate and register the configured equity-option causal DML specification."""

import matplotlib.pyplot as plt
import numpy as np
import polars as pl
import yaml

from case_studies.research import ExecutionTier, causal_supersedes, open_study
from utils.paths import get_case_study_dir
from utils.reproducibility import set_global_seeds
from utils.style import COLORS, FIGSIZE, add_message_title

# %% tags=["parameters"]
CASE_STUDY_ID = "sp500_equity_option_analytics"
PRIMARY_LABEL = ""
MAX_SYMBOLS = 0
CV_FOLDS = 0
MAX_SAMPLES = 0
N_PLACEBO = 0
FORCE_RETRAIN = False
# The tier is declared, not inferred from whether a reduction happens to be set: a reduced run
# that still opened the case study's own artifacts in place would be writing down the
# production path. WORKSPACE is the other half - a preview has nowhere else to write.
EXECUTION_TIER = "canonical"
WORKSPACE: str | None = None
# `b47bd0ec208a` is not the hash named here. That is the capped fit of 2026-08-26 - 37,240
# rows, 9.9% of the panel this request resolves - written under a spec with no
# `identity_version`, so `current_causal_identities` does not return it and no reader resolves
# it. It is stranded rather than superseded, and naming it would fail the write for declaring a
# predecessor that is not current. `18f777683c40` is the full-panel row that followed it and is
# current, so that is the one this run retires.
#
# The value has to be resolved against the registry rather than supplied at run time:
# `run-production-notebook.sh` executes with no parameter overrides, so a hash given only as an
# override could never be stamped. `causal_supersedes` withholds the declaration against a
# reader's clone, which holds no causal rows at all, so one committed value is right for both.
# Get it wrong and registration refuses the write after the fit and all 100 placebo refits have
# been paid for.
# Retired by this run: the block-permutation refutation now compares the HAC t-statistic
# rather than the raw effect, so CAUSAL_RUNNER_VERSION moved and every causal identity with
# it. The rows named here hold a p-value computed on the shrunken placebo effects; this run
# supersedes them rather than correcting them, because the statistic is different, not the
# arithmetic. Read out of each registry's current canonical identity per label, 2026-09-10.
SUPERSEDES_CAUSAL: str = "18f777683c40"

# %% [markdown]
# ## Resolve the estimand and analysis population
#
# Nonzero fold, sample, symbol, or placebo limits declare a preview. They are recorded in
# identity and excluded from canonical evidence, so the published estimate cannot silently be a
# reduced one.

# %%
setup = yaml.safe_load((get_case_study_dir(CASE_STUDY_ID) / "config" / "setup.yaml").read_text())
label = PRIMARY_LABEL or setup["labels"]["primary"]

if FORCE_RETRAIN:
    raise ValueError("an identical complete causal request is reused; change the request to refit")

REDUCTION_PARAMETERS = {
    "max_symbols": MAX_SYMBOLS or None,
    "n_folds": CV_FOLDS or None,
    "max_samples": MAX_SAMPLES or None,
    "n_placebo": N_PLACEBO or None,
}
reductions = {key: value for key, value in REDUCTION_PARAMETERS.items() if value is not None}
tier = ExecutionTier(EXECUTION_TIER)
if tier is ExecutionTier.PREVIEW and not reductions:
    raise ValueError("preview execution must declare at least one reduction")
if tier is ExecutionTier.CANONICAL and reductions:
    raise ValueError(f"canonical execution cannot carry reductions: {sorted(reductions)}")

study = open_study(CASE_STUDY_ID, execution_tier=tier, workspace=WORKSPACE or None)
request = study.causal(
    method="dml",
    label=label,
    config_name="dml",
    execution_tier=tier,
    preview_reductions=reductions,
    overrides={},
    supersedes=causal_supersedes(
        study, SUPERSEDES_CAUSAL, label, labels=[label], execution_tier=tier.value
    ),
)
resolved = request.resolve()
computation = resolved.spec["computation"]
estimand = computation["estimand"]
# Seeded from the identity rather than from a notebook parameter. The seed the estimate depends
# on is `dml.yaml`'s, and it is in the resolved specification and therefore in the hash; a
# parameter beside it would be a dial a reader can turn that changes nothing, and worse, turning
# it would leave the identity untouched and reload the cached result under a seed it was not fit
# with. The nuisance estimator and the placebo permutation carry this seed explicitly; the global
# call covers whatever else in the stack reads a process-wide one.
set_global_seeds(int(resolved.spec["seed"]))

print(f"Case study: {CASE_STUDY_ID}")
print(f"Treatment: {estimand['treatment']}")
print(f"Outcome: {estimand['outcome']} (horizon {estimand['outcome_horizon']})")
print(f"Confounders: {', '.join(estimand['confounders'])}")
print(f"Execution tier: {tier.value} | seed {resolved.spec['seed']}")
print(f"Last admissible outcome endpoint: {estimand['holdout_endpoint_cutoff']}")

# %% [markdown]
# > **Scope**: This is a development-period sensitivity analysis, not a holdout
# > evaluation. The request excludes every observation whose outcome window could
# > reach past `holdout_endpoint_cutoff`, and it keeps complete cross-sections, so
# > neither folds nor embargoes cut through a decision time.
#
# > **Population scope**: The source universe is a current-constituent roster,
# > not point-in-time S&P 500 membership. Firms removed from the index before
# > the roster date are absent. The estimate therefore describes this
# > retrospective roster during the development period; it does not generalize
# > to the historical index-membership process or a prospective S&P 500
# > population.

# %% [markdown]
# ## 1. Inspect the temporal controls
#
# Three quantities decide how this estimate is measured, and they answer different questions.
#
# The **embargo** separates a fold's training window from its test window, so a label measured
# in training cannot still be running when testing starts. `labels.buffer` sets it, and that
# buffer is declared deliberately longer than the outcome it protects, so it is not a statement
# about the outcome. The **outcome horizon** is how long one outcome stays open, and it is what
# the Driscoll-Kraay bandwidth is sized by: a bandwidth shorter than the overlap between
# successive outcomes understates the standard error. The **permutation block size** is the
# scale of the dependence the placebo has to preserve: shuffling in blocks shorter than that
# scale pulls dependent observations apart and the placebo degrades towards an independent draw
# - the permutation that is too easy to pass and therefore proves nothing.
#
# **Two scales qualify for the block size, and it has to cover both.** One is the outcome
# horizon. The other is the treatment's own persistence: `ivrv_spread` subtracts a 20-session
# rolling realized volatility from implied, so consecutive values share most of their input and
# stay dependent over that rolling window whatever the label does. It is the longer of the two
# here, and `causal.treatment_window` in `config/setup.yaml` declares it beside the derivation
# that produced it, because no rule can read a construction window off a column name.
#
# The table prints the block the run used beside both scales it has to span, read from the
# resolved specification rather than from `setup.yaml`, so it shows what the run used rather
# than what the file asks for.

# %%
pl.DataFrame(
    {
        "label": [resolved.spec["label"]],
        "cross_fitting_folds": [computation["cv"]["n_folds"]],
        "embargo_periods": [computation["cv"]["embargo_periods"]],
        "fold_unit": [computation["cv"]["fold_unit"]],
        "label_buffer_steps": [computation["refutation"]["label_buffer_steps"]],
        "label_horizon_steps": [computation["refutation"]["label_horizon_steps"]],
        "treatment_window_steps": [computation["refutation"]["treatment_window_steps"]],
        "placebo_method": [computation["refutation"]["method"]],
        "placebo_block_size": [computation["refutation"]["block_size"]],
        "block_size_basis": [computation["refutation"]["block_size_basis"]],
        "placebo_draws": [computation["refutation"]["n_placebo"]],
        "analysis_rows": [computation["analysis_population"]["n_rows"]],
        "analysis_timestamps": [computation["analysis_population"]["n_timestamps"]],
        "analysis_key_digest": [computation["analysis_population"]["key_digest"]],
    }
)

# %% [markdown]
# ## 2. Estimate or reload the causal result
#
# Registration happens only after the fit returns a finite effect and HAC standard error. A
# missing treatment, confounder, or valid analysis population fails before any causal row is
# written.

# %% tags=["results"]
result = resolved.run()
if not result.complete:
    raise RuntimeError(f"the causal result for {label} is incomplete")
# Compare the identity-bearing computation, not the whole specification. `provenance` records
# the git commit of the run that registered the result, so a full-spec comparison fails on any
# re-run made after any commit - it would assert that nothing had been committed since, which is
# not a property of the causal estimate.
if result.spec["computation"] != computation:
    raise RuntimeError(f"the registered causal computation for {label} differs")
reloaded = request.resolve().run()
if reloaded.hash != result.hash:
    raise RuntimeError(f"reloading the causal request for {label} changed its identity")

print(f"Registered causal identity: {result.hash}")

# %% [markdown]
# ## 3. Statistical Assessment
#
# Confounding bias is defined as:
#
# $$\text{Bias \%} = \frac{\hat{\theta}_{\text{naive}} - \hat{\theta}_{\text{DML}}}{|\hat{\theta}_{\text{DML}}|} \times 100$$
#
# Positive values mean the naive coefficient is more positive than the adjusted
# coefficient; interpret the sign alongside both coefficients.
#
# The two diagnostics answer different questions. Driscoll-Kraay inference asks whether the
# adjusted coefficient is distinguishable from zero after allowing for panel dependence. The
# block permutation asks whether its magnitude is unusual after disrupting the treatment-outcome
# timing while preserving each entity's short-run treatment dependence. Neither test validates
# the unobserved-confounding assumption.
#
#
# Two numbers below are populations and they are not the same one. The **analysis population** is
# what the request resolved: every row admissible under the estimand and the holdout cutoff.
# **Cross-fitted observations** is the subset carrying an out-of-fold residual, which is what the
# coefficient is computed on - a row outside every fold's validation window contributes nothing
# to it by construction.
#
# The refutation verdict is derived from the p-value **and** the number of placebo refits that
# succeeded, because a p-value alone cannot say whether the draws could have rejected at all.
#
# **Read the refutation against the warning the run prints above it**, which reports the share of
# treatment rows the permutation could not move. A row is frozen when its symbol's segment is too
# short to hold two blocks, and a 20-session block needs 40 clean sessions to have anywhere to go,
# so a daily option panel freezes a large share: every gap in a symbol's quotes starts a new
# segment. A frozen row keeps its observed treatment in every placebo, which pulls the placebo
# distribution towards the observed estimate and biases the p-value **towards 1**. The bias works
# against rejecting, so a refutation that passes with a large frozen share has passed a harder
# test than the number suggests - which is the opposite of the reading a frozen share usually
# warrants, and the reason it is worth checking rather than assuming.

# %%
metrics = result.metrics
dml_effect = metrics["dml_effect"]
se_hac = metrics["dml_se_hac"]
refutation_p = metrics["refutation_p"]

print(f"Analysis population: {computation['analysis_population']['n_rows']:,} rows")
print(f"Cross-fitted observations: {metrics['n_obs']:,}")
print(f"Naive OLS effect:  {metrics['naive_effect']:+.6f}")
print(f"Adjusted (DML):    {dml_effect:+.6f}  (HAC SE {se_hac:.6f})")
print(f"95% interval:      [{dml_effect - 1.96 * se_hac:+.6f}, {dml_effect + 1.96 * se_hac:+.6f}]")
print(f"Confounding bias:  {metrics['confounding_bias_pct']:+.2f}%")
print(f"p-value (HAC):     {metrics['p_value_hac']:.4f}")
print(f"  Significant at 5%: {'Yes' if metrics['p_value_hac'] < 0.05 else 'No'}")
print(
    f"Refutation:        {metrics['refutation_class']} "
    f"(p={refutation_p:.4f}, {metrics['refutation_n_successful']} successful draws)"
)

# %% [markdown]
# **Interpretation**: The DML coefficient is an adjusted conditional estimate,
# not proof that changing the IV-RV spread would change returns. Its causal
# interpretation still depends on the identifying assumptions above. Compare
# the naive and adjusted coefficients to see how observed confounders change the
# estimated association.
#
# Compare this result with the S&P 500 Options case study to see why confounding
# must be assessed for each treatment-outcome pair rather than inferred from the
# market alone.
#
# > **When should you be suspicious of large DML corrections?** A naive-to-DML
# > amplification exceeding 5x warrants scrutiny. Possible causes:
# > (1) nuisance models overfitting and stripping outcome-relevant variation,
# > (2) weak instrument-like behavior where the treatment residual has low variance,
# > (3) genuine massive confounding that naive OLS entirely misses.
# > The refutation test helps distinguish (3) from (1-2): if placebos also show
# > inflated effects, the DML correction may be unreliable.

# %% [markdown]
# ### Permutation Distribution
#
# The adjusted estimate against the distribution of placebo draws from within-entity block
# permutations of the treatment, read on the t-statistic rather than on the effect. The
# permutation frees the treatment from the confounders, so the residual it leaves the
# second stage keeps its variance and every placebo effect divides by a larger number than
# the observed one does. On the effect scale that shrinks the placebo distribution toward
# zero whether or not there is anything to find; dividing each draw by its own standard
# error removes it.

# %%
placebo_t_arr = np.asarray(metrics.get("placebo_t_stats") or [], dtype=float)
if placebo_t_arr.size == 0:
    # The registry stores the draws beside the p-value, so an empty array here is a row written
    # before that column existed rather than a refutation that did not run. Say which, instead
    # of showing an empty axis.
    print(
        "This causal row predates the stored placebo t-statistics, which are the scale the "
        "p-value is computed on; the p-value above is the test."
    )
else:
    observed_t = dml_effect / se_hac
    fig, ax = plt.subplots(figsize=FIGSIZE["single"])
    ax.hist(
        placebo_t_arr,
        bins=30,
        color=COLORS["silver_muted"],
        edgecolor=COLORS["neutral"],
        linewidth=0.5,
        label="Placebo t-statistics",
    )
    ax.axvline(
        observed_t,
        color=COLORS["negative"] if observed_t < 0 else COLORS["positive"],
        linewidth=2,
        label=f"Observed t-statistic ({observed_t:+.2f})",
    )
    relation = "outside" if refutation_p < 0.05 else "inside"
    add_message_title(
        ax,
        f"The adjusted estimate falls {relation} the central placebo range",
        subtitle=(
            f"Within-entity permutation in blocks of "
            f"{computation['refutation']['block_size']} sessions"
        ),
    )
    ax.set_xlabel("Driscoll-Kraay t-statistic")
    ax.set_ylabel("Count")
    ax.legend()
    plt.show()

# %% [markdown]
# ## Key Takeaways
#
# 1. **The holdout is never read**: the request drops every observation whose outcome window
#    could reach past the holdout endpoint cutoff, so exploratory causal diagnostics cannot
#    influence or consume the final evaluation window.
#
# 2. **Read the coefficient against its panel-aware standard error, not against zero**:
#    the DML estimate is a five-day return per one annualized unit of IV-RV spread, and the
#    Driscoll-Kraay standard error beside it is the one that allows for both serial and
#    cross-sectional dependence.
#
# 3. **Compare naive OLS and DML on the same rows**: `confounding_bias_pct` is the gap
#    between them as a share of the adjusted estimate. Both are computed on the cross-fitted
#    observations - the smaller of the two counts reported above, not the analysis population
#    - so the difference between them is adjustment rather than sample.
#
# 4. **The estimand is part of the identity**: treatment, confounders, temporal geometry and
#    refutation design all enter the hashed specification, so a run at a different block size,
#    placebo count, seed, fold count or population registers as its own result rather than
#    overwriting the one before it.
#
# 5. **A refutation is evidence about timing, not about confounding**: a causal reading still
#    requires conditional ignorability, overlap, and SUTVA, none of which any test here can
#    establish.

```