The Influence of ChatGPT on AI-Related Crypto Assets: Evidence from Synthetic Control Analysis

2.1 Synthetic Difference-in-Difference

The study utilizes synthetic difference-in-difference (SDID) methodology, which combines elements of synthetic control and difference-in-difference approaches. This method constructs a weighted combination of control units that closely matches the treatment unit's pre-treatment characteristics.

The SDID estimator can be represented as:

$\\hat{\\tau}_{sdid} = \\left(\\sum_{t=T_0+1}^T Y_{1t} - \\sum_{t=T_0+1}^T \\hat{Y}_{1t}^{syn}\\right) - \\left(\\sum_{t=1}^{T_0} Y_{1t} - \\sum_{t=1}^{T_0} \\hat{Y}_{1t}^{syn}\\right)$

where $Y_{1t}$ represents the observed outcome for the treatment unit, $\\hat{Y}_{1t}^{syn}$ is the synthetic control prediction, and $T_0$ marks the intervention point (ChatGPT launch).

2.2 Data Collection

The analysis includes:

• Daily price data for AI-related cryptocurrencies
• Google search volume for AI-related keywords
• Market capitalization and trading volume metrics
• Control group of non-AI cryptocurrencies

Data spans 6 months pre-launch and 2 months post-launch to capture both baseline and treatment effects.

3.1 ChatGPT Effects on Returns

The analysis reveals significant positive effects on AI-related crypto assets:

• One-month post-launch: Average returns of 10.7% to 15.6%
• Two-month post-launch: Average returns of 35.5% to 41.3%
• Statistical significance: p < 0.01 across all models

These effects persisted after controlling for general market trends and cryptocurrency-specific factors.

3.2 Google Search Volume Analysis

Google search volume for AI-related terms emerged as a critical pricing indicator post-ChatGPT launch:

• Search volume increased 247% for "AI cryptocurrency"
• Strong correlation between search volume and price appreciation (r = 0.78)
• Search volume predicted 61% of return variance in the post-treatment period

The results suggest investor attention mediated the ChatGPT effect on market valuations.

4.1 Mathematical Framework

The synthetic control weights are determined by minimizing the distance between pre-treatment characteristics:

$\\min_{w} \\sqrt{(X_1 - X_0w)'V(X_1 - X_0w)}$

subject to $w_j \\geq 0$ and $\\sum_{j=2}^{J+1} w_j = 1$, where $X_1$ contains pre-treatment characteristics of the treated unit, $X_0$ contains pre-treatment characteristics of control units, and $V$ is a diagonal matrix with feature weights.

4.2 Code Implementation

import numpy as np
import pandas as pd
from scipy.optimize import minimize

class SyntheticControl:
    def __init__(self, treatment_unit, control_units, pre_periods):
        self.treatment = treatment_unit
        self.control = control_units
        self.pre_periods = pre_periods
    
    def fit(self):
        # Pre-treatment characteristics
        X1 = self.treatment[:self.pre_periods].mean()
        X0 = self.control[:self.pre_periods].mean(axis=1)
        
        # Optimization to find weights
        def objective(w):
            return np.sqrt((X1 - X0 @ w).T @ (X1 - X0 @ w))
        
        constraints = [{'type': 'eq', 'fun': lambda w: np.sum(w) - 1}]
        bounds = [(0, 1) for _ in range(len(self.control))]
        
        result = minimize(objective, 
                         x0=np.ones(len(self.control))/len(self.control),
                         bounds=bounds,
                         constraints=constraints)
        
        self.weights = result.x
        return self.weights
    
    def predict(self, post_periods):
        synthetic_control = self.weights @ self.control[post_periods]
        return synthetic_control