This repository analyzes the relationship between price delay and stock/cryptocurrency market returns across multiple regions (cryptocurrency, U.S., and international markets). It employs regression models to quantify the impact of price delay—a measure of market inefficiency—on asset returns.

---

What is Price Delay?

Price delay measures the extent to which a stock's price incorporates new information. It reflects market frictions such as liquidity constraints, information asymmetry, or limited investor recognition. Price delay is crucial in understanding asset pricing anomalies and improving trading strategies.

Why Do We Need It?

1. Market Efficiency: Delayed price responses can highlight inefficiencies in information dissemination.
2. Predictive Power: Assets with higher delays often offer a return premium due to risks or frictions.
3. Portfolio Optimization: Identifying delayed assets can enhance investment strategies.

Metrics Used

Three main delay measures are derived using regression on lagged returns:

1. (D_1): Fraction of return variance explained by lagged market returns.
2. $$D_1 = 1 - \frac{R^2_{\text{restricted}}}{R^2_{\text{unrestricted}}}$$ Where $R^2_{\text{restricted}}$ excludes lagged returns and $R^2_{\text{unrestricted}}$ includes them.
3. (D_2): Weighted lag coefficients:
4. $$D_2 = \frac{\sum_{n=1}^4 n \beta_{-n}}{\beta_0 + \sum_{n=1}^4 \beta_{-n}}$$
5. (D_3): Weighted lag coefficients normalized by standard error:
6. $$D_3 = \frac{\sum_{n=1}^4 n \frac{\beta_{-n}}{\text{SE}(\beta_{-n})}}{\frac{\beta_0}{\text{SE}(\beta_0)} + \sum_{n=1}^4 \frac{\beta_{-n}}{\text{SE}(\beta_{-n})}}$$

These metrics quantify how much and how quickly prices respond to market-wide news.

---

Repository Overview

Structure

• crypto-pricedelay-regression: Analysis of price delay in cryptocurrency markets.
• international-exceptUS-pricedelay-regression: Focus on international equity markets (excluding U.S.).
• usa-pricedelay-regression: Price delay analysis for U.S. markets.

---

Code Explanation

1. Data Preprocessing

Each analysis begins with data cleaning and preparation:

• Remove Missing Values: Handles missing return observations.
• Standardization: Normalizes variables for regression consistency.
• Feature Engineering: Creates lagged return features for regression.

Example snippet from preprocessing (pseudocode):

# Create lagged market returns
for lag in range(1, 5):
    data[f'market_return_lag{lag}'] = data['market_return'].shift(lag)

---

2. Regression Analysis

Model

The regression model captures the relationship between individual stock returns and lagged market returns: [ r_{i,t} = \alpha + \beta_0 R_{m,t} + \sum_{n=1}^4 \beta_{-n} R_{m,t-n} + \epsilon ] Where:

• (r_{i,t}): Return of stock (i) at time (t).
• (R_{m,t-n}): Lagged market return.
• (\beta_{-n}): Coefficients on lagged returns indicate delay.

The regression outcomes help compute delay metrics:

• High (\beta_{-n}) values indicate significant lag, contributing to higher price delay.

Implementation

• Crypto Analysis (crypto_analysis.ipynb): Implements the regression using pandas and statsmodels.
• U.S. Analysis (usa_analysis.ipynb): Similar structure but tailored to U.S. data.

Example code:

import statsmodels.api as sm

X = data[['market_return', 'market_return_lag1', 'market_return_lag2', 'market_return_lag3', 'market_return_lag4']]
y = data['stock_return']
X = sm.add_constant(X)  # Add intercept

model = sm.OLS(y, X).fit()
print(model.summary())

---

3. Portfolio Formation

To improve regression precision and reduce noise, stocks are grouped into portfolios based on size and delay. Portfolio-level delay is computed as the average of individual delays.

---

4. Results and Insights

Cryptocurrency

• High transaction delays in cryptocurrencies lead to slower information incorporation, evidenced by significant (\beta_{-n}) coefficients.

U.S. Markets

• Delayed firms (small, illiquid) show a 1-2% monthly return premium.

International Markets

• Delayed stocks, especially in developing markets, exhibit strong post-announcement drift.

---

Math Behind Price Delay

The regression decomposes return variance:

• Immediate response: Captured by (\beta_0).
• Delayed response: Distributed across (\beta_{-1}) to (\beta_{-4}).

The metrics $D_1$, $D_2$, and $D_3$ summarize:

• Variance fraction explained ($D_1$).
• Weighted delay impact ($D_2$).
• Normalized precision-weighted delay ($D_3$).

---

Why It Matters

Price delay uncovers inefficiencies in markets and offers actionable insights for:

1. Hedge Funds: Identify mispriced assets.
2. Regulators: Assess market health.
3. Academia: Explore links between delay and asset characteristics.

---

Getting Started

Dependencies

• pandas
• numpy
• matplotlib
• statsmodels

Install via:

pip install pandas numpy matplotlib statsmodels

---

Contributing

Fork and submit a pull request for improvements.

---

License

MIT License.

---

This repository provides a robust framework to study price delay, its causes, and its impact, empowering traders, researchers, and policymakers to navigate market inefficiencies.