Quantitative Analysis

Purpose

Apply statistics to financial data without being fooled by it. Financial time series violate nearly every assumption of standard statistical methods, and the tools will produce confident, well-formatted, wrong answers without complaint.

When to Use

• Testing whether a signal has predictive value.
• Modeling returns, volatility, or correlations.
• Evaluating a quantitative claim someone else has made.
• Building a factor model or a risk model.

Capabilities

• Return distributions and their fat tails.
• Stationarity, unit roots, and spurious regression.
• Correlation, causation, and confounding.
• Multiple-testing correction and data snooping.
• Volatility modeling and clustering.

Inputs

• The data, and an honest account of how it was selected.
• The hypothesis — stated before the data was examined.
• The number of hypotheses that have already been tested on this data.

Outputs

• A result with its assumptions stated.
• A significance level corrected for the number of tests actually run.
• An honest statement of what the result does and does not support.

Workflow

1. Look at the distribution before assuming one — Financial returns are not normal. They have fat tails and skew, and every method that assumes normality will underestimate the probability of a large move — which is the only probability that matters for survival.
2. Test for stationarity — Regressing one non-stationary series on another produces a high R-squared and a significant t-statistic between two entirely unrelated things. This is spurious regression, and it is the most common error in financial statistics.
3. Work in returns, not prices — Prices are non-stationary by construction. Returns are approximately stationary. Almost every meaningful analysis operates on returns.
4. Count the hypotheses you have tested — If you tried a hundred signals and one has a p-value of 0.01, you have found exactly what pure chance predicts. A p-value not adjusted for the search is not evidence.
5. Distinguish correlation from causation, and both from coincidence — With enough series, something will correlate with anything. A mechanism stated in advance is what separates a finding from a coincidence.
6. Validate out of sample — On data that was not available when the hypothesis was formed. Everything else is description.

Best Practices

• Financial returns have fat tails. A "six-sigma" event under a normal distribution should occur once in several million years; in markets it occurs every few years. Any model built on normality is quietly understating tail risk.
• Non-stationarity produces spurious regressions with astonishing reliability. Two random walks will regress on each other with a significant t-statistic roughly 70% of the time. Always difference or test first.
• Correlation between financial series is unstable. A correlation estimated over three years describes those three years, and it will not hold in the crisis you are trying to protect against.
• The multiple-testing problem is the single largest source of false discovery in quantitative finance. The published anomaly literature is substantially a record of it.
• In-sample fit is not evidence. Any sufficiently flexible model fits any dataset. The only test that means anything is out of sample.
• Report the effect size, not just the p-value. A statistically significant edge of 0.02% per trade is not tradable after costs.

Examples

Spurious regression — the error that produces the most confident wrong answers:

import numpy as np, statsmodels.api as sm

# Two independent random walks. There is no relationship whatsoever.
np.random.seed(1)
a = np.cumsum(np.random.randn(500))
b = np.cumsum(np.random.randn(500))

model = sm.OLS(a, sm.add_constant(b)).fit()
print(f"R-squared: {model.rsquared:.3f}   t-stat: {model.tvalues[1]:.2f}")
# R-squared: 0.681   t-stat: 32.71
#
# An R-squared of 0.68 and a t-statistic of 33 between two series with NO
# relationship at all. This is not a fluke of the seed; it happens most of the
# time with non-stationary series. Any conclusion drawn from this is fiction.

# The correct procedure: test for stationarity, and work in returns.
from statsmodels.tsa.stattools import adfuller

for name, series in [("a", a), ("b", b)]:
    p = adfuller(series)[1]
    print(f"{name}: ADF p={p:.3f} -> {'non-stationary' if p > 0.05 else 'stationary'}")
# a: ADF p=0.712 -> non-stationary
# b: ADF p=0.884 -> non-stationary
#
# Differencing to returns removes the spurious relationship entirely.
model_returns = sm.OLS(np.diff(a), sm.add_constant(np.diff(b))).fit()
print(f"R-squared: {model_returns.rsquared:.4f}   t-stat: {model_returns.tvalues[1]:.2f}")
# R-squared: 0.0004   t-stat: -0.44   <- the truth: no relationship

Multiple testing — why most published anomalies are not real:

def corrected_significance(p_values: list[float], n_tested: int) -> Report:
    """A p-value of 0.01 means nothing if you tested 100 hypotheses."""
    best_p = min(p_values)

    # Bonferroni: the family-wise error rate.
    bonferroni = min(1.0, best_p * n_tested)

    # The probability that the best of n random signals looks this good by chance.
    prob_by_chance = 1 - (1 - best_p) ** n_tested

    return Report(
        raw_p=best_p,
        bonferroni_p=bonferroni,
        prob_at_least_one_by_chance=prob_by_chance,
        verdict=(
            "Not distinguishable from chance."
            if bonferroni > 0.05
            else "Survives correction. Now validate out of sample."
        ),
    )

# Tested 120 signal variants. The best has p = 0.008.
#   raw p                      : 0.008     "significant!"
#   Bonferroni-corrected p     : 0.96
#   P(at least one this good
#     by chance among 120)     : 0.62
#   Verdict: not distinguishable from chance.
#
# The signal is exactly as good as the best of 120 random signals would be
# expected to be. It is a discovery about the search, not about the market.

Notes

• The spurious-regression demonstration above is the single most important thing in this skill. An R-squared of 0.68 between two unrelated random walks is not a corner case — it is the typical result, and it has produced a great many published findings.
• Fat tails mean that variance is a poor risk measure for financial returns. Maximum drawdown and expected shortfall describe what actually hurts.
• This is educational material about quantitative methodology, not financial advice, and none of it constitutes a recommendation.