mha-finance 0.14.8

Multi-Horizon Statistical Modeling for Financial Time Series

mha-finance

mha-finance is a free Python framework for Multi-Horizon Statistical Modeling of Financial Time Series. It focuses on regime characterization, risk-return estimation, and horizon-dependent dynamics.

Objectives of the project:

1. Statistically characterize horizon-wise asset returns (A statistical estimation, not prediction).
2. Statistically characterize horizon-wise asset volatility.
3. Statistically Characterize Horizon-Wise Market Regimes.

Problem statement definition for each Objective

1. Statistically characterize horizon-wise asset returns (A statistical estimation, not prediction).

1 The objective is to statistically characterize horizon-specific asset returns using historical price data, without predicting future prices.

Given historical daily price data over a sufficiently long period (e.g., 5 years for monthly estimation), the system estimates:

• The average realized monthly return
• The dispersion of horizon-specific returns
• The uncertainty associated with estimated mean

All estimates are descriptive and inferential, not predictive.

2 Scope of the Objective

In scope

• Statistical estimation
• Rolling window analysis
• Return Characterization
• Uncertainty quantification

Out of scope

• Price prediction
• Trading strategies

3 Horizon based data span selection

• Intra-Day estimation:- A 5 Minute interval data of 50 Days

• Weekly estimation:- A daily interval data of 2 years

• Monthly estimation:- A daily interval data of 5 years

• Annual estimation:- A daily interval data of 15 years

4 Steps for achieving this objective

• Step 1-- Data Ingestion

  Data Based on horizon is loaded and cleaned

  • Incomplete current-day records are removed.

  • Data is sorted chronologically.

• Step 2-- Return Construction

  Horizon-based log returns are computed from closing prices for each

  interval specified by the horizon selection (see 3):

  • Let P_t be the closing price on day t.

  • Declare horizon (H) based on selection of user (eg. H=21 for Monthly horizon by user)

  • The return ending at at time t is: r_t^(H) = log(P_t) - log(P_{t-H})

  This produces a time series of realized monthly returns.

• Step 3-- Rolling estimation window

  Rolling estimation window

  • W = Data Span for the horizon / Horizon Length.

  • Returns inside the window are assumed locally stationary.

  • This Window defines the data used for estimation.

  • Each rolling window defines the data used for statistical estimation.

• Step 4-- Statistical estimation

  Within Rolling Window

  • Mean horizon return (μ̂^(H)) μ̂^(H) = (1 / W) * Σ r_t^(H) Represents the average realized Horizon return.

  • Return dispersion (sample Variance) D̂^(M) = (1 / (W − 1)) * Σ (r_t^(M) − μ̂^(M))²

    Represents the empirical dispersion of monthly returns (not volatility modeling)

5 Output Structure

Final output format for 1 month as example horizon

ReturnSummary(
    mean,                 # Estimated mean monthly return
    variance,             # Sample dispersion of monthly returns
    confidence_interval,  # Uncertainty of the mean
    window_used={
        "horizon": "monthly",
        "frequency": "daily",
        "lookback": "5 years",
        "effective_samples": 60
    }
)

6 Concepts Involved

Concepts Of Finance

    Log returns
    Time horizons
    Rolling windows
    Non-stationarity
    Risk vs return (descriptive)

Concepts of Statistics

    Sample Mean
    Sample Variance
    Confidence Interval
    Bootstrap inference
    Effective sample size

Machine learning is intentionally not used to avoid unjustified prediction

2. Statistically characterize horizon-wise asset volatility (Statistically estimation of conditional variabilty).

1 The objective is to statistically characterize horizon-specific asset volatility using historical price data, without predicting future volatility or market movements.

Given historical daily price data over a sufficiently long period and at an appropriate sampling frequency, depending on horizon we must have long lookback window. The system estimates the recent conditional variability of returns, along with measures of uncertainty and stability.

Specifically, the framework estimates:

• The realized or conditional volatility at a given horizon
• The dispersion and variability of volatility estimates across rolling windows
• Diagnostic measures indicating the stability or degradation of volatility estimatprs

All estimates are descriptive and inferential, not predictive.

2 Scope of the Objective

In scope

• Statistical volatility estimation
• Rolling and window-based estimators
• Horizon specific volatility Characterization
• Uncertainty and stability diagnostics
• Conditional and regime-aware volatility

Out of scope

• Volatility prediction or forecasting
• Profit or risk optimization
• Trading, hedging, or portfolio construction strategies
• Automated decision-making or alerts

3 Horizon based data span selection

• Intra-Day estimation:- A daily interval data of 70 Days

• Weekly estimation:- A daily interval data of 2 years

• Monthly estimation:- A daily interval data of 5 years

• Annual estimation:- A daily interval data of 15 years

4 Steps for achieving this objective

• Step 1-- Data Ingestion

  Data Based on horizon is loaded and cleaned

  • Incomplete current-day records are removed.

  • Data is sorted chronologically.

• Step 2-- Return Series construction

  Volatility estimation is performed on returns, not prices.

  • Let Pt denote the closing price at time t.

  • Log returns are consrtructed at the base sampling frequency: rt = log(Pt)-log(Pt-1) This produces a time series of realized returns, which serves as the input for volatility estimation.

  (Note: The return frequency is decoupled from estimation horizon. Higher frequency returns may be used to characterize lower-frequency volatility.)]

• Step 3-- Rolling estimation window

  A rolling estimation window is defined to support conditional and adaptive estimation

  • Let W denote the number of observations in the rolling window.
  • Window length is determined by:
    • hoizon selection
    • required historical depth
  • Returns within each window are assumed locally stationary.

• Step 4-- Statistical volatility estimation

  Within each rolling window,, volatility is estimated using statistical estimators, not predictive models

  Typical estimators include:

  • Sample varince σ^2 = (1(W-1))*∑(rt-rˉ)^2
  • Rolling standard deviation σ = (σ^2)^(1/2)
  • Exponentially Weighted Moving Average (EWMA) σ(t)^2 = λσ(t-1)^2 + (1-λ)r(t)^2

  These estimators characterize recent conditional variability. not future risk.

• Step 5-- Volatility uncertainty and dispersion analysis

  Beyond point estimation, the framework evaluates uncertainty and robustness of volatility estimates.

  • Dispersion of volaility estimates across windows
  • Sensitivity to window length
  • Temporal smoothness or clustering behavior

  This step quantifies how stable or unstable volatility estimates are over time.

• Step 6-- Stabilty diagnostics

  Stability diagnostics are computed to assess whether volatility assumptions remain valid.

  • Detection of abrupt changes in variability

  These diagnostics flag instability but do not trigger automated actions.

5 Output construction and delivery

For each horizon and evaluation point, the framewok outputs a structured volatility summary:

- Estimated volatility level
- Associated uncertainty measures
- Stability indicators
- Metadata:
    - horizon
    - window size
    - estimator used
    - data span

This output is descriptive, interpretable, and reproducible.

6 Concepts Involved

Concepts Of Finance

        Log-returns  
        Realized volatility  
        Conditional volatility  
        Volatility clustering  
        Volatility persistence  

Concepts of Statistics

        Squared returns  
        Rolling variance  
        Rolling standard deviation  
        Exponentially weighted moving averages (EWMA)  
        Sensitivity to window length  

3. Statistically Characterize Horizon-Wise Market Regimes.

1 The objective is to statistically characterize horizon-specific market regime using historical price data, without predicting future volatility or market movements.(“This module provides a foundation for future extensions such as regime persistence analysis and transition summaries.”)

Given horizon-wise statistical summaries of asset returns and volatility computed over rolling windows (Objectivea 1 and 2), the system identifies distinct statistical regimes that describe recurring market conditions at a given horizon.

Regimes are defined ex post as periods during which the joint statistical behaviour of returns and volatility remains approximately stable.

No attempt is made to forecast regime changes or optimize decisions based on regimes.

2 Scope of the Objective

In scope

• Descriptive regime identification
• Horizon-wise return–volatility characterization
• Uncertainty quantification

Out of scope

• Prediction or forecasting
• Trading or decision systems

3 Horizon based data span selection

• Intra-Day estimation:- A daily interval data of 70 Days

• Weekly estimation:- A daily interval data of 2 years

• Monthly estimation:- A daily interval data of 5 years

• Annual estimation:- A daily interval data of 15 years

4 Steps for achieving this objective

• Step 1-- Data Ingestion

  Data Based on horizon is loaded and cleaned

  • Incomplete current-day records are removed.

  • Data is sorted chronologically.

• Step 2-- Statistical feature construction

  Regime identification is performed on statistical estimates, not raw prices.

  • Horizon-wise return estimates are constructed.
  • Horizon-wise volatility estimates are constructed.

  These estimates represent the statistical behavior of the market at each time index.

• Step 3-- Horizon-wise regime identification

  Regimes are identified by grouping time periods that exhibit similar statistical behavior at the selected horizon.

  • Each group corresponds to a distinct regime
  • No predictive interpretation is attached to regime labels

  Each historical time index is assigned a regime identifier.

5 Output construction and delivery

For each horizon and evaluation point, the framework outputs a structured regime identification result:

- Regime label for each historical time index
- Number of identified regimes
- Metadata:
    - horizon
    - lookback window
    - identification method

This output is descriptive, interpretable, and reproducible.

6 Concepts Involved

Concepts Of Finance

        Market regimes  
        Horizon-dependent behavior  
        Risk–return states   

Concepts of Statistics

        Statistical similarity  
        Unsupervised partitioning

Data Source and Usage:

1. Market Data Source
2. Data Handling Policy
3. Use of Derived Insights
4. Responsibility and Compliance
5. Design Philosophy

Details

1. Market Data Source

mha-finance retrieves historical market price data at runtime using publicly accessible endpoints provided by Yahoo Finance, via the open-source yfinance Python library.

The framework does not bundle, store, cache, or redistribute any financial datasets. All market data:

• is fetched on demand
• is downloaded directly by the end user
• remains subject to Yahoo Finance’s terms of service

2. Data Handling Policy

The framework follows a non-redistributive data usage model:

• No raw OHLCV or intraday data is stored on disk
• No historical datasets are included in the repository
• No cached price data is persisted across sessions
• No market data is served to third parties

Raw price data exists only transiently in memory during computation and is discarded after statistical processing.

3. Use of Derived Insights

mha-finance operates exclusively on derived statistical quantities, including:

• horizon-specific log returns
• rolling statistical moments (mean, variance)
• volatility estimators
• regime descriptors

All outputs are descriptive statistical summaries or aggregated analytical results. The framework does not expose or reconstruct raw historical price series.

4. Responsibility and Compliance

By running mha-finance, users fetch market data directly from Yahoo Finance and are responsible for ensuring their usage complies with the data provider’s terms.

The project provides:

• analytical methodology
• statistical estimation tools
• reproducible computation logic

but does not act as a data provider.

5. Design Philosophy

This data access model is intentionally chosen to support:

• academic reproducibility
• educational use
• license-aware open-source distribution

while avoiding unauthorized redistribution of proprietary financial data.