Article-20589-Volatility-Modeling-And-Forecasting-In-MQL5

## Overview

This project documents and organizes the concepts presented in an MQL5 article about a native library for univariate volatility modeling and forecasting. The article describes a modular framework inspired by Python's `arch` package, where a complete model is composed of:

- a mean process,
- a volatility process,
- an error distribution.

The implementation centers on a mean-model base class (`HARX`) that manages initialization, parameter estimation, simulation, and forecasting for the full model.

## Original Article

- Article title: Volatility Modeling and Forecasting in MQL5
- Article ID: 20589
- Author: Francis Dube
- Publication date: 2026.01.14
- Category: Articles / MQL5 programming / optimization / quantitative finance
- URL: https://www.mql5.com/en/articles/20589

## Repository Purpose

This repository should be treated as a reference or reconstruction project based on the article text.

Its purpose is to capture:

- the described library architecture,
- the exposed data structures and APIs,
- the fitting and forecasting workflow,
- the ARCH-LM diagnostic procedure,
- the included demonstration of HAR equivalence via ARX with exogenous averages.

## Key Concepts

- Modular volatility modeling in MQL5
- Mean/volatility/distribution separation
- `ArchParameters` as a unified configuration struct
- `ArchModelResult` and `ArchForecast` as result containers
- HAR, HARX, AR, and ARX mean processes
- Constant variance, ARCH, and GARCH-family process selection
- Joint parameter estimation through `fit()`
- Forecast generation via analytic, simulation, or bootstrap methods
- Residual diagnostics using Engle's ARCH-LM test
- Equivalence of HAR and ARX under averaged exogenous regressors

## Algorithm / Architecture Summary

The article describes a class hierarchy where `HARX` acts as the base interface for mean models and also orchestrates the full volatility model.

### Model composition

A model is formed from three components:

1. Mean process
2. Volatility process
3. Error distribution

The mean process is the central object and owns the estimation workflow.

### Configuration

Model setup is performed through an `ArchParameters` struct, which includes:

- sample observations,
- optional exogenous regressors,
- lag specification,
- mean model selection,
- volatility model selection,
- distribution selection,
- holdout and scaling options,
- GARCH order parameters,
- bootstrap settings,
- distribution initialization parameters,
- RNG seeds.

The `initialize()` method validates and normalizes these inputs, adjusts incompatible options where possible, builds lag matrices, and instantiates the selected volatility process and distribution.

### Estimation

The article states that estimation uses ALGLIB's nonlinear constrained optimization via `CMinNLC`, minimizing a log-likelihood objective exposed by the model class.

The returned `ArchModelResult` includes:

- estimated parameters,
- in-sample conditional volatility,
- residuals,
- covariance matrix,
- fit range,
- log-likelihood,
- R-squared and adjusted R-squared helpers,
- t-values, p-values, and standard errors.

### Forecasting

Forecasts are returned in an `ArchForecast` struct with:

- `mean`
- `variance`
- `residual_variance`

The article describes support for:

- analytic forecasts,
- Monte Carlo simulation forecasts,
- bootstrap forecasts.

For models with exogenous variables, future exogenous matrices must be supplied.

### Validation

Model adequacy is checked using `archlmtest()`, which performs a regression-based ARCH-LM test on squared residuals, optionally using standardized residuals. The result is encapsulated in `WaldTestStatistic`, providing:

- test statistic,
- degrees of freedom,
- null and alternative hypotheses,
- p-value,
- chi-square critical values.

### Explicitly attached files in the article text

- `MQL5/scripts/HAR_as_ARX_Demo.mq5` — demonstration script comparing HAR and ARX formulations

## Tags

`mql5`, `volatility-modeling`, `arch`, `garch`, `har`, `arx`, `econometrics`, `time-series`, `forecasting`, `alglib`

## Difficulty

Advanced

## Reference

- MQL5 article: "Volatility Modeling and Forecasting in MQL5"
- Demonstrated script: `HAR_as_ARX_Demo.mq5`
- Related concepts referenced in the article:
- Python `arch` package
- ALGLIB `CMinNLC`
- Engle's ARCH-LM test