Linear Regression Reversion

1. Introduction & Hook

In the fast-paced world of algorithmic trading, strategies that blend mathematical rigor with practical market insight stand out. Linear Regression Reversion is one such approach. It leverages statistical analysis to identify moments when price is likely to revert to its mean, offering traders a systematic edge. Whether you are a Pine Script enthusiast, a Python quant, or a Node.js developer, mastering this strategy can transform your trading outcomes. This comprehensive guide will walk you through every facet of Linear Regression Reversion, from theory to code, risk management, and advanced automation.

2. What is Linear Regression Reversion?

Linear Regression Reversion is a quantitative trading strategy that uses linear regression to estimate the fair value of an asset over a lookback period. When the current price deviates significantly from this regression line, the strategy anticipates a reversion—meaning the price will move back toward the regression line. This approach is rooted in the statistical concept of mean reversion, which posits that prices and returns eventually move back toward their historical averages.

3. Market Logic Behind the Strategy

The core logic of Linear Regression Reversion is simple yet powerful. Markets often overreact to news, rumors, or short-term imbalances, causing prices to stray from their equilibrium. By fitting a linear regression line to recent price data, traders can estimate the 'fair value' and identify when prices are stretched too far from this mean. The strategy then bets on a return to normalcy, capitalizing on the tendency of prices to revert to the mean after excessive moves.

• Overbought/Oversold Detection: When price is far above the regression line, it may be overbought; when far below, oversold.
• Statistical Edge: The use of regression provides a mathematically grounded way to define these conditions.
• Versatility: Works across timeframes and asset classes, from stocks to crypto.

4. Mathematical Foundation & Formula

Linear regression fits a straight line to a set of data points, minimizing the sum of squared differences between the observed values and the values predicted by the line. The formula for a simple linear regression line is:

y = a + b * x

Where:

• y = predicted value (regression line)
• a = intercept
• b = slope
• x = independent variable (e.g., time or bar index)

The slope and intercept are calculated as:

b = Cov(x, y) / Var(x)
a = mean(y) - b * mean(x)

In trading, y is typically the price, and x is the bar index or time.

5. Step-by-Step Calculation Example

Suppose you have the following closing prices over 5 bars: [100, 102, 101, 105, 107]. Assign bar indices [1, 2, 3, 4, 5].

• Calculate mean(x) and mean(y):
  mean(x) = (1+2+3+4+5)/5 = 3
  mean(y) = (100+102+101+105+107)/5 = 103
• Calculate Cov(x, y):
  Cov(x, y) = sum((xi - mean(x)) * (yi - mean(y))) / n
• Calculate Var(x):
  Var(x) = sum((xi - mean(x))^2) / n
• Compute b and a, then the regression line for each bar.

Once you have the regression line, measure the distance between the actual price and the regression value. If the distance exceeds a threshold (e.g., 2 standard deviations), a reversion trade is triggered.

6. Pine Script Implementation

Below is a robust Pine Script implementation of the Linear Regression Reversion strategy. This script calculates the regression line, measures price deviation, and generates buy/sell signals when price deviates beyond a configurable threshold.

//@version=6
// Linear Regression Reversion Strategy
// Author: The Wallstreet Bulls
// This script identifies mean reversion opportunities using linear regression.

strategy("Linear Regression Reversion", overlay=true, default_qty_type=strategy.percent_of_equity, default_qty_value=10)

// === Parameters ===
length = input.int(50, minval=5, title="Regression Length")
deviation_mult = input.float(2.0, minval=0.5, title="Deviation Multiplier")

// === Linear Regression Calculation ===
var float[] regression = array.new_float()
var float[] deviations = array.new_float()

float sum_x = 0.0
float sum_y = 0.0
float sum_xy = 0.0
float sum_x2 = 0.0
for i = 0 to length - 1
    sum_x := sum_x + i
    sum_y := sum_y + close[i]
    sum_xy := sum_xy + i * close[i]
    sum_x2 := sum_x2 + i * i

mean_x = sum_x / length
mean_y = sum_y / length
b = (sum_xy - length * mean_x * mean_y) / (sum_x2 - length * mean_x * mean_x)
a = mean_y - b * mean_x
regression_val = a + b * (length - 1)

// Calculate standard deviation of residuals
float sum_sq = 0.0
for i = 0 to length - 1
    y_hat = a + b * i
    sum_sq := sum_sq + math.pow(close[i] - y_hat, 2)
stddev = math.sqrt(sum_sq / length)

// Entry/Exit Logic
long_condition = close < regression_val - deviation_mult * stddev
short_condition = close > regression_val + deviation_mult * stddev

if (long_condition)
    strategy.entry("Long", strategy.long)
if (short_condition)
    strategy.entry("Short", strategy.short)

// Plotting
plot(regression_val, color=color.blue, linewidth=2, title="Regression Line")
plot(regression_val + deviation_mult * stddev, color=color.red, style=plot.style_line, title="Upper Band")
plot(regression_val - deviation_mult * stddev, color=color.green, style=plot.style_line, title="Lower Band")

7. Parameters & Customization in Pine Script

Key parameters for tuning this strategy in Pine Script:

• Regression Length: Number of bars for regression calculation. Shorter lengths react faster but may be noisy; longer lengths are smoother but lag more.
• Deviation Multiplier: Number of standard deviations from the regression line to trigger trades. Higher values reduce trade frequency but increase signal quality.
• Position Sizing: Adjust default_qty_value for risk management.
• Stop-Loss/Take-Profit: Add strategy.exit calls for automated exits.

8. Python & FastAPI + NoSQL Implementation

Python is ideal for backtesting and deploying regression reversion strategies at scale. Below is a simplified example using numpy for regression and FastAPI for serving signals. Data can be stored in a NoSql Database like MongoDB for persistence.

import numpy as np
from fastapi import FastAPI
from pydantic import BaseModel
from typing import List

app = FastAPI()

class PriceData(BaseModel):
    closes: List[float]
    deviation_mult: float = 2.0

@app.post("/signal")
def get_signal(data: PriceData):
    closes = np.array(data.closes)
    x = np.arange(len(closes))
    A = np.vstack([x, np.ones(len(x))]).T
    m, c = np.linalg.lstsq(A, closes, rcond=None)[0]
    regression_line = m * x + c
    residuals = closes - regression_line
    stddev = np.std(residuals)
    last_price = closes[-1]
    last_reg = regression_line[-1]
    if last_price < last_reg - data.deviation_mult * stddev:
        return {"signal": "buy"}
    elif last_price > last_reg + data.deviation_mult * stddev:
        return {"signal": "sell"}
    else:
        return {"signal": "hold"}

This API can be integrated with a NoSql Database for storing signals, trades, and historical data.

9. Node.js / JavaScript Implementation

Node.js is popular for real-time trading bots and web dashboards. Here is a basic implementation using JavaScript:

// Linear Regression Reversion in Node.js
function linearRegressionReversion(closes, deviationMult = 2.0) {
    const n = closes.length;
    const x = Array.from({length: n}, (_, i) => i);
    const meanX = x.reduce((a, b) => a + b, 0) / n;
    const meanY = closes.reduce((a, b) => a + b, 0) / n;
    let num = 0, den = 0;
    for (let i = 0; i < n; i++) {
        num += (x[i] - meanX) * (closes[i] - meanY);
        den += (x[i] - meanX) ** 2;
    }
    const b = num / den;
    const a = meanY - b * meanX;
    const regression = x.map(xi => a + b * xi);
    const residuals = closes.map((c, i) => c - regression[i]);
    const stddev = Math.sqrt(residuals.reduce((s, r) => s + r * r, 0) / n);
    const lastPrice = closes[n - 1];
    const lastReg = regression[n - 1];
    if (lastPrice < lastReg - deviationMult * stddev) return 'buy';
    if (lastPrice > lastReg + deviationMult * stddev) return 'sell';
    return 'hold';
}

10. Backtesting & Performance Insights

Backtesting is essential to validate the effectiveness of the Linear Regression Reversion strategy. In Pine Script, use the strategy framework to simulate trades and analyze metrics like win rate, profit factor, and drawdown. In Python, libraries like backtrader or zipline can be used for more advanced analysis.

• Key Metrics: Sharpe ratio, maximum drawdown, average trade duration, and expectancy.
• Walk-Forward Analysis: Test the strategy on out-of-sample data to avoid overfitting.
• Parameter Optimization: Use grid search or Bayesian optimization to find the best regression length and deviation multiplier.

11. Risk Management Integration

Effective risk management is critical. Integrate position sizing, stop-loss, and take-profit mechanisms directly into your strategy code.

• Position Sizing: Use a fixed percentage of equity or volatility-based sizing.
• Stop-Loss/Take-Profit: Set exits based on ATR, fixed points, or a percentage of price.

// Pine Script: Automated Exits
stop_loss = input.float(1.5, title="Stop Loss (%)")
take_profit = input.float(3.0, title="Take Profit (%)")
if (long_condition)
    strategy.entry("Long", strategy.long)
    strategy.exit("Long Exit", from_entry="Long", stop=close * (1 - stop_loss / 100), limit=close * (1 + take_profit / 100))
if (short_condition)
    strategy.entry("Short", strategy.short)
    strategy.exit("Short Exit", from_entry="Short", stop=close * (1 + stop_loss / 100), limit=close * (1 - take_profit / 100))

12. Combining with Other Indicators

Linear Regression Reversion can be enhanced by combining it with other technical indicators:

• RSI: Confirm mean reversion signals with overbought/oversold readings.
• MACD: Filter trades based on trend direction.
• Bollinger Bands: Use as an additional volatility filter.

// Pine Script: Combining with RSI
rsi = ta.rsi(close, 14)
long_condition = long_condition and rsi < 30
short_condition = short_condition and rsi > 70

13. Multi-Timeframe & Multi-Asset Usage

This strategy is adaptable across timeframes and asset classes:

• Timeframes: Apply on 1m, 15m, daily, or weekly charts. Shorter timeframes yield more signals but higher noise.
• Assets: Works on equities, forex, crypto, and even options (for mean-reverting underlyings).

// Pine Script: Multi-Timeframe Example
higher_close = request.security(syminfo.tickerid, "D", close)
long_condition = long_condition and close > higher_close

14. AI/ML Enhancements

Machine learning can boost the performance of regression reversion strategies:

• Feature Engineering: Use regression residuals, volatility, and volume as features for ML models.
• Reinforcement Learning: Train an RL agent to optimize parameters like regression length and deviation multiplier.

# Python: RL Agent Example (Pseudocode)
for episode in range(num_episodes):
    state = env.reset()
    done = False
    while not done:
        action = agent.select_action(state)
        next_state, reward, done, _ = env.step(action)
        agent.learn(state, action, reward, next_state)
        state = next_state

15. Automation with Playwright/Jest

Automated testing ensures your strategy scripts are robust and error-free. Use playwright for end-to-end browser automation or Jest for unit testing in JavaScript/TypeScript projects.

// Jest: Unit Test Example
const { linearRegressionReversion } = require('./strategy');
test('should return buy when price is below lower band', () => {
    const closes = [100, 102, 101, 105, 107, 95];
    expect(linearRegressionReversion(closes)).toBe('buy');
});

16. Advanced Variations

• Polynomial Regression: Fit higher-order curves for non-linear mean reversion.
• Adaptive Windows: Dynamically adjust regression length based on volatility.
• Multi-Asset Pairs: Apply regression to price spreads between correlated assets.

17. Common Pitfalls & Misconceptions

• Assuming All Markets Mean-Revert: Trending markets can cause persistent losses.
• Ignoring Regime Shifts: Sudden changes in volatility or trend can invalidate signals.
• Overfitting: Excessive parameter tuning may not generalize to live trading.
• Neglecting Transaction Costs: High-frequency reversion strategies can be eroded by fees and slippage.

18. Conclusion & Key Takeaways

Linear Regression Reversion is a powerful, statistically grounded strategy that can deliver consistent results when applied with discipline and robust risk management. Its adaptability across platforms and asset classes makes it a valuable tool for traders and developers alike. By understanding its mathematical foundation, coding robust implementations, and integrating automation and AI, you can identify new levels of trading performance.

Glossary of Key Terms

• Linear Regression: A statistical method for modeling the relationship between a dependent variable and one or more independent variables.
• Mean Reversion: The tendency of a price to move back toward its historical average.
• Standard Deviation: A measure of volatility or dispersion from the mean.
• Residual: The difference between the observed value and the value predicted by the regression line.
• Lookback Period: The number of bars used for regression calculation.
• Deviation Multiplier: The number of standard deviations used to define entry/exit thresholds.
• Backtesting: Simulating a strategy on historical data to evaluate performance.
• Reinforcement Learning: A type of machine learning where agents learn to make decisions by trial and error.

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