Rolling dependency
For each Bitcoin pair, Pearson correlation is calculated over 14, 30, 60 and 90 days, then Fisher transformed to form a stable regression target.
z_t = arctanh(rho_t)M.Sc. Artificial Intelligence · 2026
When Bitcoin moves with equities, diversification can disappear.
This thesis tests whether the changing dependency between Bitcoin and conventional assets can be forecast one day ahead, and whether that forecast can serve as a modest risk input for an investor.
- 3,053 daily observations
- 6 BTC asset pairs
- 4 rolling windows
- strict walk-forward testing
01 / MODEL DEMONSTRATOR
A two-layer view of market dependence
The first layer forecasts tomorrow's 30-day BTC-S&P 500 dependency. The second layer converts the forecast and crypto-market inputs into a stress probability. Move the inputs to inspect the mechanics.
BTC-S&P 500 · 30-day windowresearch demonstrator
Latest calculated model snapshot
Loading dated market snapshot.
----
Market conditions
Scenario inputs explain the model mechanics. The dated snapshot above uses the saved calculation rather than a live browser feed.
diversifyingmoving together
-20%+20%
calmextreme
downtrenduptrend
Model readout
AR(1) forecasts the Fisher-z transformed dependence. A stress score demonstrates the second-layer logic.
Tomorrow's dependency forecast
0.36
The model expects the current co-movement regime to persist.
Fisher-z now0.3654
Fisher-z forecast0.3582
Stress probability20.0%
Interpretationlow
Low estimated stress probability. In the thesis, this output is a prompt to monitor risk, not a buy or sell instruction.
The AR(1) persistence coefficient comes from the thesis demonstrator. The stress score is an explanatory approximation of the researched logistic-regression layer, not a live calibrated model or financial advice.
02 / METHOD
Forecast the relationship, then interpret the risk
The project separates a measurable forecasting problem from a deliberately conservative investor-use case. All predictions are generated in temporal order, without training on the future.
Walk-forward forecast
Naive, AR(1), HAR, regularised linear models, tree ensembles and a leakage-safe DCC-GARCH benchmark are re-estimated through time.
z_hat(t+1) = alpha + beta z_tStress-day probability
A logistic model combines dependency and crypto features to estimate whether next-day equity returns enter a stress definition below -0.75 trailing standard deviations.
P(stress_t+1 | crypto_t, z_hat_t+1)03 / EVIDENCE
The simple model is the important result
The strongest result is not that a complex algorithm won. It is that the rolling dependency is highly persistent, so AR(1), Ridge and HAR form a practically indistinguishable top tier.
What the thesis supports
0.0656
Best average RMSE, Ridge
AR(1) is almost identical at 0.0659. The ranking within this top tier is not economically meaningful.
0.885
R² from a single dependency lag
The target's own history contains most of the forecastable information.
24
Pair-window experiments
Tree ensembles have higher RMSE than the naive persistence baseline in every experiment.
0.5308
Best balanced accuracy, signal layer
This is an exploratory maximum among 20 configurations, so it is not evidence for a standalone trading rule.
04 / INVESTOR USE
An extra lens for risk decisions
The investor-facing part of the work does not predict market direction or promise returns. It highlights periods when Bitcoin and broad equities may be moving into a more connected, fragile regime.
Check co-movement
A rising BTC-equity dependency suggests that an investor should not assume Bitcoin will diversify an equity-heavy portfolio in a stress event.
Raise risk awareness
An elevated probability can prompt a review of gross exposure, stop levels, liquidity and existing hedges before a scheduled risk review.
Test an overlay
The next research step is a portfolio backtest with transaction costs, position sizing and a tail-risk objective. This thesis does not claim that result yet.
Important limitation. The signal is strongest for broad equity pairs and weakens for gold, silver and the dollar index. The evaluated daily one-step horizon, overlapping rolling windows and exploratory hyperparameter selection mean that the tool should remain an educational risk overlay until it is independently validated.