Verwaltete ETF-Faktoren mit Post-Double-Selection-LASSO testen

# 11_factor_zoo_validation.py


```py
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# %% [markdown]
# # Factor Zoo Validation via Post-Double-Selection LASSO
#
# **Chapter 15: Causal Machine Learning**
#
# This notebook uses post-double-selection LASSO to ask whether managed-portfolio
# factor returns add incremental time-series explanatory power for SPY after ten
# principal-component controls. It is a compact factor-spanning application of the
# Belloni-Chernozhukov-Hansen selection logic discussed in this chapter.
#
# ## Learning objectives
#
# - distinguish a naive single-factor association from a conditional factor loading;
# - implement both LASSO selections with time-ordered cross-validation;
# - carry the selected union into an intercept-inclusive OLS regression with HAC inference;
# - interpret what the exercise does and does not establish about the factor zoo.
#
# **Book references**: Chapter 14, Section 14.1 (Making the case for latent factors), and
# Chapter 15, Section 15.4 (Isolating factor effects with DML).
#
# **Prerequisites**: [`01_pca_equity_sectors`](../14_latent_factors/01_pca_equity_sectors.ipynb)
# and [`03_econml_dml`](03_econml_dml.ipynb).

# %% [markdown]
# ## Setup
#
# The ETF data are daily, so the cube-root Newey-West bandwidth provides a transparent
# default for serial dependence. The bandwidth is computed after the 60-day warm-up.

# %%
"""Factor-spanning validation with post-double-selection LASSO."""

import matplotlib.pyplot as plt
import numpy as np
import polars as pl
import statsmodels.api as sm
from sklearn.decomposition import PCA
from sklearn.linear_model import Lasso
from sklearn.model_selection import GridSearchCV, TimeSeriesSplit
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import StandardScaler

from data import load_etfs
from utils.reproducibility import set_global_seeds
from utils.style import COLORS, FIGSIZE, add_message_title, show_with_alt, zero_line

# %% tags=["parameters"]
START_DATE = "2006-01-01"
END_DATE = "2024-12-31"
MIN_OBSERVATIONS = 252
N_PCA_FACTORS = 10
N_CV_SPLITS = 5
N_ALPHAS = 60
SIGNIFICANCE_LEVEL = 0.05
MAX_SYMBOLS = 0  # 0 = all eligible symbols
OUTCOME_SYMBOL = "SPY"
WARMUP = 60
SEED = 42

# %%
set_global_seeds(SEED)

# %% [markdown]
# ## 1. Build a balanced, disjoint ETF panel
#
# SPY is excluded from the factor-building universe and retained only as the outcome.
# The source contains ETFs with different inception dates. Replacing pre-inception
# observations with zero would manufacture returns, so the analysis retains the
# long-history symbols observed on every date in the requested sample.
#
# This eligibility rule uses the current curated ETF list and is not a point-in-time
# historical universe. The result is an in-sample teaching exercise, not a
# survivorship-free backtest, and nothing here is held out.

# %%
etf_data = load_etfs(start_date=START_DATE, end_date=END_DATE).sort(["symbol", "timestamp"])

duplicate_keys = etf_data.select(pl.struct("symbol", "timestamp").is_duplicated().sum()).item()
if duplicate_keys:
    raise ValueError(f"ETF input contains {duplicate_keys} duplicate symbol-timestamp keys")

etf_returns = etf_data.with_columns(
    pl.col("close").pct_change().over("symbol").alias("return")
).drop_nulls(subset=["return"])

n_source_dates = etf_returns["timestamp"].n_unique()
symbol_coverage = etf_returns.group_by("symbol").len().sort("symbol")
eligible_symbols = symbol_coverage.filter(
    (pl.col("len") == n_source_dates) & (pl.col("len") >= MIN_OBSERVATIONS)
)["symbol"].to_list()

if OUTCOME_SYMBOL not in eligible_symbols:
    raise ValueError(f"Outcome {OUTCOME_SYMBOL!r} lacks complete sample coverage")
if MAX_SYMBOLS > 0:
    eligible_symbols = eligible_symbols[:MAX_SYMBOLS]
    if OUTCOME_SYMBOL not in eligible_symbols:
        eligible_symbols.append(OUTCOME_SYMBOL)

# %% [markdown]
# Pivoting after the coverage filter produces a genuinely balanced panel. The null
# and finite-value assertions make the no-imputation contract executable.

# %%
return_wide = (
    etf_returns.filter(pl.col("symbol").is_in(eligible_symbols))
    .pivot(on="symbol", index="timestamp", values="return")
    .sort("timestamp")
)

symbols_all = [column for column in return_wide.columns if column != "timestamp"]
if len(symbols_all) <= N_PCA_FACTORS + 1:
    raise ValueError("The balanced universe is too small for the requested PCA basis")
if return_wide.select(pl.sum_horizontal(pl.exclude("timestamp").null_count())).item() != 0:
    raise ValueError("Balanced return panel contains missing values")

returns_all = return_wide.select(symbols_all).to_numpy().astype(np.float64)
if not np.isfinite(returns_all).all():
    raise ValueError("Balanced return panel contains non-finite values")

outcome_idx = symbols_all.index(OUTCOME_SYMBOL)
outcome_return = returns_all[:, outcome_idx]
zoo_mask = np.ones(returns_all.shape[1], dtype=bool)
zoo_mask[outcome_idx] = False
zoo_returns = returns_all[:, zoo_mask]
zoo_symbols = [symbol for symbol in symbols_all if symbol != OUTCOME_SYMBOL]
T, N = zoo_returns.shape

print(
    f"Analysis window: {return_wide['timestamp'].min()} to {return_wide['timestamp'].max()} "
    f"({T:,} trading days)"
)
print(f"Outcome: {OUTCOME_SYMBOL}; factor-building universe: {N} long-history ETFs")
print("Missing returns imputed: 0")

# %% [markdown]
# ## 2. Extract principal-component controls
#
# Standardization prevents high-volatility ETFs from dominating the covariance
# structure. The final basis is fitted on the full post-warm-up inference sample
# because these are explicitly in-sample controls; uncertainty is conditional on
# this estimated basis. Selection below refits both transformations inside each fold.

# %%
pca_scaler = StandardScaler()
zoo_returns_inference = zoo_returns[WARMUP:]
zoo_returns_scaled = pca_scaler.fit_transform(zoo_returns_inference)
pca = PCA(n_components=N_PCA_FACTORS, random_state=SEED)
factor_returns = pca.fit_transform(zoo_returns_scaled)

factor_names = [f"PC{i + 1}" for i in range(N_PCA_FACTORS)]
variance_table = pl.DataFrame(
    {
        "factor": factor_names,
        "variance_explained": pca.explained_variance_ratio_,
        "cumulative_variance": np.cumsum(pca.explained_variance_ratio_),
    }
).with_columns(pl.selectors.numeric().round(4))

print(
    f"Cumulative variance explained by {N_PCA_FACTORS} PCs: {pca.explained_variance_ratio_.sum():.1%}"
)
variance_table

# %% [markdown]
# ## 3. Construct lagged managed-portfolio factors
#
# Each portfolio uses information ending at $t-1$ to set long and short baskets,
# then records their return at $t$. The four candidates differ only in the ranking
# statistic and lookback window.

# %%
momentum_20d = np.zeros(T)
for t in range(20, T):
    momentum = zoo_returns[t - 20 : t].sum(axis=0)
    top = momentum >= np.percentile(momentum, 80)
    bottom = momentum <= np.percentile(momentum, 20)
    momentum_20d[t] = zoo_returns[t, top].mean() - zoo_returns[t, bottom].mean()

momentum_60d = np.zeros(T)
for t in range(60, T):
    momentum = zoo_returns[t - 60 : t].sum(axis=0)
    top = momentum >= np.percentile(momentum, 80)
    bottom = momentum <= np.percentile(momentum, 20)
    momentum_60d[t] = zoo_returns[t, top].mean() - zoo_returns[t, bottom].mean()

# %% [markdown]
# Low-volatility and mean-reversion portfolios reverse the ranking direction: they
# buy the low-volatility or recent-loser quintile and sell the opposite quintile.

# %%
low_vol = np.zeros(T)
for t in range(60, T):
    volatility = zoo_returns[t - 60 : t].std(axis=0)
    low = volatility <= np.percentile(volatility, 20)
    high = volatility >= np.percentile(volatility, 80)
    low_vol[t] = zoo_returns[t, low].mean() - zoo_returns[t, high].mean()

mean_reversion = np.zeros(T)
for t in range(5, T):
    recent_return = zoo_returns[t - 5 : t].sum(axis=0)
    losers = recent_return <= np.percentile(recent_return, 20)
    winners = recent_return >= np.percentile(recent_return, 80)
    mean_reversion[t] = zoo_returns[t, losers].mean() - zoo_returns[t, winners].mean()

# %% [markdown]
# The common warm-up removes the initialized zeros before any inference. Annualized
# Sharpe ratios here are descriptive summaries, not selection criteria.

# %%
candidate_names = ["Mom_20d", "Mom_60d", "LowVol", "MeanRev"]
candidates = np.column_stack([momentum_20d, momentum_60d, low_vol, mean_reversion])[WARMUP:]
outcome_trimmed = outcome_return[WARMUP:]
controls_trimmed = factor_returns
HAC_LAGS = max(1, int(len(outcome_trimmed) ** (1 / 3)))

candidate_summary = pl.DataFrame(
    {
        "factor": candidate_names,
        "annualized_sharpe": [
            candidates[:, i].mean() / candidates[:, i].std(ddof=1) * np.sqrt(252)
            for i in range(len(candidate_names))
        ],
        "daily_volatility": [candidates[:, i].std(ddof=1) for i in range(len(candidate_names))],
    }
).with_columns(pl.selectors.numeric().round(4))

print(f"Inference sample: {len(outcome_trimmed):,} days; Newey-West bandwidth: {HAC_LAGS} lags")
candidate_summary

# %% [markdown]
# ## 4. Match the naive and conditional estimands
#
# A factor-mean test and a regression loading answer different questions. The
# comparison below holds the target fixed: the naive slope comes from SPY on one
# candidate, while the post-selection slope adds selected PCA controls. Both models
# include an intercept and use the same Newey-West covariance estimator.


# %%
def ols_hac_test(outcome: np.ndarray, design: np.ndarray, hac_lags: int) -> dict[str, float]:
    """Estimate the first slope in an intercept-inclusive OLS-HAC regression."""
    design_2d = design.reshape(-1, 1) if design.ndim == 1 else design
    model = sm.OLS(outcome, sm.add_constant(design_2d)).fit(
        cov_type="HAC", cov_kwds={"maxlags": hac_lags}
    )
    return {
        "coef": float(model.params[1]),
        "se": float(model.bse[1]),
        "t_stat": float(model.tvalues[1]),
        "p_value": float(model.pvalues[1]),
    }


# %%
naive_results = []
for index, name in enumerate(candidate_names):
    result = ols_hac_test(outcome_trimmed, candidates[:, index], HAC_LAGS)
    naive_results.append({"factor": name, **result})

naive_table = pl.DataFrame(naive_results).with_columns(
    pl.col("coef", "se").round(4),
    pl.col("t_stat").round(2),
    pl.col("p_value").round(4),
)
naive_table

# %% [markdown]
# ## 5. Select controls without leaking future folds
#
# Each LASSO uses an expanding time-series split. Its pipeline refits the input
# scaler, PCA basis, and control scaler on each training fold, so later observations
# cannot change an earlier validation prediction. A fixed broad alpha grid avoids
# using the full target series to calibrate the candidate penalties.


# %%
def fit_lasso_selector(target: np.ndarray, pca_inputs: np.ndarray) -> dict:
    """Tune a fold-local PCA-LASSO pipeline and return its full-sample support."""
    alpha_grid = np.geomspace(1e-8, 1e-2, N_ALPHAS)
    pipeline = make_pipeline(
        StandardScaler(),
        PCA(n_components=N_PCA_FACTORS, random_state=SEED),
        StandardScaler(),
        Lasso(max_iter=20_000, random_state=SEED),
    )
    search = GridSearchCV(
        pipeline,
        {"lasso__alpha": alpha_grid},
        cv=TimeSeriesSplit(n_splits=N_CV_SPLITS),
        scoring="neg_mean_squared_error",
        # The grid is small and the design is a few thousand rows; a worker per core would
        # take the whole machine from every other notebook executing beside this one.
        n_jobs=1,
    )
    search.fit(pca_inputs, target)
    coefficients = search.best_estimator_.named_steps["lasso"].coef_
    return {
        "selected": np.flatnonzero(np.abs(coefficients) > 1e-10),
        "alpha": float(search.best_params_["lasso__alpha"]),
    }


# %% [markdown]
# The first selection finds PCA controls that explain SPY. The second finds PCA
# controls related to the candidate. Their union protects the candidate slope from
# controls that one predictive equation alone might omit.


# %%
def double_selection_test(
    outcome: np.ndarray,
    candidate: np.ndarray,
    controls: np.ndarray,
    pca_inputs: np.ndarray,
    control_names: list[str],
    hac_lags: int,
) -> dict:
    """Run post-double-selection and HAC inference for one candidate slope."""
    outcome_selection = fit_lasso_selector(outcome, pca_inputs)
    candidate_selection = fit_lasso_selector(candidate, pca_inputs)
    selected_union = sorted(
        set(outcome_selection["selected"]) | set(candidate_selection["selected"])
    )

    final_design = candidate.reshape(-1, 1)
    if selected_union:
        final_design = np.column_stack([candidate, controls[:, selected_union]])
    inference = ols_hac_test(outcome, final_design, hac_lags)

    return {
        **inference,
        "n_outcome": len(outcome_selection["selected"]),
        "n_candidate": len(candidate_selection["selected"]),
        "n_union": len(selected_union),
        "outcome_alpha": outcome_selection["alpha"],
        "candidate_alpha": candidate_selection["alpha"],
        "selected_names": [control_names[i] for i in selected_union],
    }


# %% [markdown]
# ## 6. Run post-double-selection for the four candidates

# %%
post_results = []
for index, name in enumerate(candidate_names):
    result = double_selection_test(
        outcome_trimmed,
        candidates[:, index],
        controls_trimmed,
        zoo_returns_inference,
        factor_names,
        HAC_LAGS,
    )
    post_results.append({"factor": name, **result})

post_table = pl.DataFrame(post_results).select(
    "factor",
    pl.col("coef").round(4),
    pl.col("se").round(4),
    pl.col("t_stat").round(2),
    pl.col("p_value").round(4),
    "n_outcome",
    "n_candidate",
    "n_union",
)
post_table

# %% [markdown]
# ## 7. Compare uncertainty and selection breadth
#
# The left panel compares the same SPY loading before and after PCA conditioning, with
# Newey-West intervals at the conventional two-sided level. The right panel shows what each
# of the two LASSO steps selected, since the union that enters the final regression is only
# as interesting as the two selections behind it: an outcome equation that keeps the whole
# basis makes the union the whole basis whatever the candidate equation chose, and
# post-double-selection then reduces to controlling for everything.

# %%
factor_positions = np.arange(len(candidate_names))
naive_coef = np.array([result["coef"] for result in naive_results])
naive_se = np.array([result["se"] for result in naive_results])
post_coef = np.array([result["coef"] for result in post_results])
post_se = np.array([result["se"] for result in post_results])
outcome_selected = np.array([result["n_outcome"] for result in post_results])
candidate_selected = np.array([result["n_candidate"] for result in post_results])
figure_subtitle = (
    f"HAC estimates; {N}-ETF factor zoo; {return_wide['timestamp'].min()} "
    f"to {return_wide['timestamp'].max()}"
)

# %% [markdown]
# The coefficient panel uses position and marker shape as well as color, so the
# comparison remains legible in grayscale. The selection bars start at zero.

# %%
fig, axes = plt.subplots(1, 2, figsize=FIGSIZE["dual_h_tall"], sharey=True)

axes[0].errorbar(
    naive_coef,
    factor_positions - 0.12,
    xerr=1.96 * naive_se,
    fmt="o",
    color=COLORS["neutral"],
    capsize=3,
    label="Naive",
)
axes[0].errorbar(
    post_coef,
    factor_positions + 0.12,
    xerr=1.96 * post_se,
    fmt="s",
    color=COLORS["blue"],
    capsize=3,
    label="Post-double-selection",
)
zero_line(axes[0], axis="x")
axes[0].set_yticks(factor_positions, candidate_names)
axes[0].set_xlabel("SPY loading (slope)")
_ = axes[0].legend(loc="best")

axes[1].barh(
    factor_positions - 0.18,
    outcome_selected,
    height=0.34,
    color=COLORS["blue"],
    alpha=0.85,
    label="Selected for SPY",
)
axes[1].barh(
    factor_positions + 0.18,
    candidate_selected,
    height=0.34,
    color=COLORS["amber"],
    alpha=0.85,
    label="Selected for the candidate",
)
axes[1].set_xlim(0, N_PCA_FACTORS)
axes[1].set_xlabel("Selected PCA controls (count)")
# The outcome LASSO retains all ten components for every candidate, so every navy bar spans
# the full axis and no corner inside the panel is free. A key placed in one sits on a bar of
# its own colour and cannot be read; it goes above the panel instead.
axes[1].legend(loc="lower left", bbox_to_anchor=(0.0, 1.0), ncol=2, frameon=False, fontsize=8)
axes[1].invert_yaxis()

add_message_title(
    axes[0],
    "SPY loading before and after PCA conditioning",
    subtitle=figure_subtitle,
)
show_with_alt(
    fig,
    "Two panels sharing a vertical axis of candidate factor names. The left panel plots each "
    "factor's SPY loading twice, the naive estimate and the post-double-selection estimate at "
    "slightly offset heights with different marker shapes, each with a horizontal "
    "Newey-West interval and a vertical line at zero. The right panel is a grouped "
    "horizontal bar chart of how many of the ten PCA controls each LASSO selected for that "
    "factor, one bar for the SPY equation and one for the candidate equation, with a legend "
    "naming them; the regression uses their union.",
)

# %% [markdown]
# **Interpretation**: the naive slopes mix each managed factor's association with
# SPY and its correlation with broad co-movement. Once the PCA basis enters, the
# coefficient magnitudes contract toward zero and their HAC intervals include zero.
# The outcome-selection LASSO retains the full ten-component basis, so this
# low-dimensional example effectively becomes a conservative all-PC spanning test.
#
# This is not a direct replication of Feng, Giglio, and Xiu (2020). Their target is
# a cross-sectional SDF loading estimated from many test assets. Here the target is
# a conditional time-series loading for one disjoint ETF, which isolates the
# post-double-selection mechanics without claiming an SDF or causal estimand.

# %% [markdown]
# ## Key takeaways
#
# 1. **Compare like with like**: both columns test the SPY loading on a candidate;
#    the conditional version differs only by the selected PCA controls.
# 2. **Temporal validation matters inside selection**: expanding folds and
#    fold-local scaling and PCA keep later observations out of earlier validation.
# 3. **Serial dependence changes uncertainty**: intercept-inclusive Newey-West
#    inference replaces both IID mean tests and a manual HC1 calculation.
# 4. **Missing data are part of the estimand**: the analysis uses a balanced
#    long-history panel instead of converting pre-inception observations to zeros.
# 5. **Scope remains limited**: the coefficients are in-sample associations,
#    conditional on estimated PCs and a current curated ETF universe. Cross-sectional
#    SDF inference and point-in-time asset-pricing validation require a richer design.
#
# **Connection to Section 15.3**: post-double-selection protects one target
# coefficient by taking the union of controls predictive of the outcome and of the
# focal factor. The next step for a production study would add a point-in-time
# universe, cross-sectional test assets, and inference designed for estimated SDF
# loadings.

```