📊 Random Walk Hypothesis Research Project

🎯 Objective

This project tests the Random Walk Hypothesis on Bitcoin (BTC-USD) using rigorous statistical methods across 13 years of daily price data.

We investigate whether price movements are:

• Random
• Independent
• Unpredictable

Verdict: RWH REJECTED — Bitcoin is NOT weak-form efficient.

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🧠 Theory

The Random Walk Hypothesis states that:

Future price movements cannot be predicted using past price data.

It is closely related to the Efficient Market Hypothesis (Weak Form).

Mathematical formulation:

P_t = P_(t-1) + ε_t      where ε_t ~ IID(0, σ²)
r_t = ln(P_t / P_(t-1))  ← log return

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⚙️ Methodology

1. ✅ Data Collection — btc-usd-max.csv (CoinGecko, 2013–2026)
2. ✅ Data Preprocessing — datetime parsing, sorting, NaN removal
3. ✅ Log Return Calculation — 4,735 observations
4. ✅ Exploratory Data Analysis — 5-panel chart generated
5. ✅ Statistical Testing:
   • ✅ Augmented Dickey-Fuller (ADF)
   • ✅ Autocorrelation / Ljung-Box Q
   • ✅ Wald-Wolfowitz Runs Test
   • ✅ Variance Ratio (Lo-MacKinlay 1988)
6. ✅ Extensions:
   • ✅ GARCH(1,1) volatility modelling
   • ✅ Rolling window ADF + Runs analysis

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📂 Project Structure

Random Walk Hypothesis/
│── btc-usd-max.csv          ← Raw data (CoinGecko)
│── rw_btc_test.py           ← Main statistical test pipeline
│── rw_extensions.py         ← GARCH + rolling window analysis
│── btc_eda_plots.png        ← EDA visualisations
│── btc_variance_ratio.png   ← VR(k) profile chart
│── btc_garch_plots.png      ← GARCH volatility + rolling analysis
│── Research_Report.md       ← Full academic report
│── implementations.md       ← Code documentation
│── results.md               ← Test results and tables
│── analysis.md              ← Final verdict and insights
│── skills.md                ← Skills developed
│── README.md                ← This file

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🚀 How to Run

# Step 1 — Run core statistical tests
python rw_btc_test.py

# Step 2 — Run GARCH + rolling window extensions
python rw_extensions.py

Requirements:

pip install pandas numpy matplotlib statsmodels arch yfinance

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📊 Key Results

Test	Result	RWH Verdict
ADF (prices)	p = 0.7450	✔ Supports
ADF (returns)	p = 0.0000	✔ Supports
Ljung-Box ACF	Sig. lags 6–30	✘ Rejects
Runs Test	Z=3.04, p=0.0024	✘ Rejects
Variance Ratio	VR=1.32 at k=30	✘ Rejects
Final Score	2/5	❌ REJECTED

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📝 Tasks Roadmap

🔹 Phase 1: Setup

• Create project structure
• Load dataset (BTC-USD from btc-usd-max.csv)
• Inspect columns and data types

🔹 Phase 2: Data Preprocessing

• Convert timestamps to datetime
• Sort data chronologically
• Handle missing values
• Extract closing price

🔹 Phase 3: Feature Engineering

• Compute log returns
• Drop NaN values
• Validate return distribution

🔹 Phase 4: Exploratory Data Analysis

• Plot price series
• Plot log returns
• Generate histogram
• Compute summary statistics
• Generate rolling volatility chart

🔹 Phase 5: Statistical Testing

• Autocorrelation / Ljung-Box Q Test
• Augmented Dickey-Fuller Test
• Wald-Wolfowitz Runs Test
• Variance Ratio Test (Lo-MacKinlay)

🔹 Phase 6: Interpretation

• Analyze each test result
• Compare outcomes across tests
• Evaluate randomness vs. momentum

🔹 Phase 7: Conclusion

• Accept or reject Random Walk Hypothesis → REJECTED
• Document findings in results.md + analysis.md
• Suggest improvements (GARCH, rolling analysis)

🔹 Phase 8: Extensions

• GARCH(1,1) volatility modelling
• Rolling window ADF stationarity analysis
• Rolling Runs Test across time

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📈 Expected Outcome

✅ Determined that Bitcoin does not follow a random walk
✅ Identified market inefficiencies (momentum at 6–30 day horizons)
✅ Built statistical foundation for quantitative trading strategy research