# Adaptive Lag Transform Experiment: M2 → BTC

## Question

Which currently implemented transformation, if any, gives a sufficiently
stable positive M2-leading-BTC relationship to become the default estimator
input?

## Fixed experimental contract

- Input: `data/COINBASE_BTCUSD, 360-IndicatorM2LI_DebugArchive.csv`
- Source: `M2 US EU CN`; target: BTC `close`
- Grid: daily UTC; source last-observation then forward-filled; target daily
  final close, never forward-filled.
- Lag definition: source at `t - L` pairs with target at `t`.
- Lag search: 30–120 daily intervals, step 1.
- Evaluation window: 365 days; minimum observations: 240.
- Correlation: Pearson; selection: strongest positive.
- Dates: 80 month-end-like daily-grid endpoints from January 2020 through the
  archive end.

This is a causal in-sample characterization of rolling estimates, not an OOS
trading test and not evidence of causation.

## Rolling summary

| Transform | Median lag | Lag IQR | Median selected r | Positive selected r | Median peak separation | Distinct selected lags |
| --- | ---: | --- | ---: | ---: | ---: | ---: |
| Levels | 46 | 30–72 | 0.626 | 77.5% | 0.001 | 26 |
| First difference | 56 | 49–80 | 0.123 | 100.0% | 0.101 | 29 |
| Percentage change | 57 | 49–83 | 0.124 | 100.0% | 0.107 | 28 |
| Log return | 57 | 53–83 | 0.123 | 100.0% | 0.112 | 26 |

## Representative endpoints

At the 2026-08-21 endpoint, levels selected lag 46 with `r = -0.803` and
relative peak separation `0.0005`: the candidate spectrum was a very broad,
negative plateau. First difference, percentage change, and log return all
selected lag 46 with a modest positive `r ≈ 0.13`; their separation was also
low (`0.026–0.051`) because lags 45 and 46 were nearly tied.

At the 2022-03-19 endpoint, returns/differences selected lag 37 with a modest
positive relationship (`r ≈ 0.16–0.19`) and materially clearer separation.
At the 2024-06-04 endpoint, these transforms selected lags 34–57 with
`r ≈ 0.10–0.12`.

## Decision

Do **not** set a universal default transformation from this experiment.

Levels yield larger coefficients but are not reliable for the intended signal:
their sign can reverse, their lag peak is almost flat, and their magnitude is
likely affected by shared non-stationary trends. Differences and returns are
more consistently positive and have more distinct peaks, but their
correlations are currently too modest and their estimated lags too variable to
justify a trading rule or a fixed lag.

For the next experiment, preserve all transformations as explicit settings and
compare them in a walk-forward prediction/backtest protocol. Evaluate a
training-period lag estimate only against subsequent target data; do not score
the same trailing window used to select the lag.