[bashtage]

Target 9.1: merge after v9.0.0 is tagged. xref #904. This implements the five standard
backtests from the proposal. max_undetectable_understatement is left out, pending a
decision on the issue.

Adds the new module arch/univariate/backtest.py, exported from arch.univariate:

• var_exceptions(returns, var): a bool array of -returns > var. VaR is a positive loss,
  and a loss equal to the VaR is not an exception.
• kupiec_pof(returns, var, p): the Kupiec (1995) proportion-of-failures likelihood-ratio
  test, chi2(1).
• christoffersen_independence(returns, var): Christoffersen's (1998) Markov independence
  test, chi2(1).
• christoffersen_conditional_coverage(returns, var, p): LR_uc + LR_ind, chi2(2).
• traffic_light(returns, var, p=0.01, window=250): the Basel Committee (1996) zone,
  returned as a frozen TrafficLightResult with the zone, the count, the cumulative
  probability, nobs and p.

The three tests return WaldTestStatistic.

Inputs:

• The data must be finite and 1-d.
• Two Series are aligned on the index. Otherwise the lengths must match.
• 0·log 0 = 0 through xlogy.

Validation:

• R 4.6.1, rugarch 1.5.6 VaRTest, on 11 cases: largest difference in the statistics
  1.2e-14.
• Degenerate series (zero or all exceptions, first-only or last-only, 2 observations)
  cannot go through rugarch, which indexes a 2x2 table. They are checked against an
  independent R likelihood implementation, cross-checked with rugarch.
• The traffic-light probabilities match scipy and R pbinom to 1e-12, and the published
  Table 2 of the BIS 1996 backtesting framework, including the zone boundaries at 5 and
  10.
• The Kupiec 99% non-rejection regions (255/510/1000 observations) match the usually quoted
  values.
• The references are stored as constants in the tests.

Tests:

• test_backtest.py (198) and test_validation_backtest.py (299 table rows).
• Each of 8 deliberately introduced bugs is caught.
• Fast suite 27,472 passed. The new tests pass in random orders. numpydoc lint 0, and the
  docs build has no new warnings (a new "Backtesting Value-at-Risk" section with an
  example).

For review:

1. traffic_light raises for a series shorter than window, rather than using what is
   there.
2. Zero exceptions is always green.
3. p is required except in traffic_light.
4. TrafficLightResult is public.
5. var_exceptions returns an ndarray, not a Series.
6. WaldTestStatistic.pval is 1 - cdf, which is imprecise below about 1e-15. A survival
   function would fix it, as a separate change.

Thanks to @sarthakguptaquant for the proposal.

🤖 Generated with Claude Code

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 98.97%. Comparing base (d2c90b4) to head (aac6fce).
⚠️ Report is 6 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #965      +/-   ##
==========================================
+ Coverage   98.95%   98.97%   +0.01%     
==========================================
  Files         136      139       +3     
  Lines       30043    30596     +553     
  Branches     2331     2360      +29     
==========================================
+ Hits        29729    30282     +553     
  Misses        256      256              
  Partials       58       58              

Flag	Coverage Δ
adder	98.91% <100.00%> (+0.01%)	⬆️
subtractor	98.91% <100.00%> (+0.01%)	⬆️

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