How to run the regression with daily data for each stock per month

Running a regression with daily data for each stock per month involves several steps. Here's a detailed guide:

### Step-by-Step Guide

#### 1. Data Collection
- **Daily Stock Returns**: Collect daily closing prices for each stock and calculate the daily returns.
- **Market Return**: Collect the daily returns of the market index (e.g., CRSP Value-Weighted Market Index).
- **VIX Data**: Collect daily VIX data from the Chicago Board Options Exchange (CBOE).

#### 2. Data Preparation
- **Stock Returns**: Calculate the excess returns for each stock by subtracting the risk-free rate from the daily returns.
- **Market Excess Returns**: Calculate the excess returns for the market index similarly.

#### 3. Define the Regression Model
The regression model you want to run is:
\[ r_{i,t} = \alpha_i + \beta_{M,i} \text{MKT}_t + \beta_{V,i} \text{VIX}_t + \epsilon_{i,t} \]
where:
- \( r_{i,t} \) is the excess return of stock \( i \) on day \( t \).
- \( \text{MKT}_t \) is the excess return of the market on day \( t \).
- \( \text{VIX}_t \) is the change in VIX on day \( t \).
- \( \alpha_i \) is the intercept.
- \( \beta_{M,i} \) is the market beta.
- \( \beta_{V,i} \) is the VIX beta.
- \( \epsilon_{i,t} \) is the error term.

#### 4. Run the Regression for Each Stock
For each stock \( i \) and each month \( t \):
1. **Extract Daily Data**: Extract the daily excess returns of the stock, market, and VIX for the month.
2. **Calculate VIX Changes**: Compute the daily changes in VIX (\( \Delta \text{VIX}_t \)).
3. **Run the Regression**: Use a statistical software package (e.g., Python with `statsmodels`, R, or MATLAB) to run the regression.

### Example in Python

Here is an example using Python and the `statsmodels` library:

import pandas as pd
import numpy as np
import statsmodels.api as sm

# Assume you have a DataFrame `data` with columns: 'stock_returns', 'market_returns', 'vix'
# and a 'date' column with daily dates.

# Convert date to datetime format
data['date'] = pd.to_datetime(data['date'])

# Set the date as the index
data.set_index('date', inplace=True)

# Define the start and end dates for the month
start_date = '2023-01-01'
end_date = '2023-01-31'

# Filter the data for the specified month
monthly_data = data.loc[start_date:end_date]

# Calculate daily changes in VIX
monthly_data['delta_vix'] = monthly_data['vix'].diff().fillna(0)

# Prepare the independent variables
X = monthly_data[['market_returns', 'delta_vix']]
X = sm.add_constant(X)  # Add a constant term for the intercept

# Prepare the dependent variable
y = monthly_data['stock_returns']

# Run the regression
model = sm.OLS(y, X)
results = model.fit()

# Print the regression results
print(results.summary())

### Example in R

Here is an example using R:

# Assume you have a data frame `data` with columns: 'stock_returns', 'market_returns', 'vix'
# and a 'date' column with daily dates.

# Convert date to Date format
data$date <- as.Date(data$date)

# Define the start and end dates for the month
start_date <- as.Date("2023-01-01")
end_date <- as.Date("2023-01-31")

# Filter the data for the specified month
monthly_data <- subset(data, date >= start_date & date <= end_date)

# Calculate daily changes in VIX
monthly_data$delta_vix <- c(0, diff(monthly_data$vix))

# Prepare the independent variables
X <- model.matrix(~ market_returns + delta_vix, data = monthly_data)

# Prepare the dependent variable
y <- monthly_data$stock_returns

# Run the regression
model <- lm(y ~ X - 1)  # -1 to remove the intercept already included in X

# Print the regression results
summary(model)

### Automation for Multiple Stocks and Months

To automate the process for multiple stocks and months, you can loop through the stocks and months. Here is an example in Python:

# Assume `all_data` is a DataFrame with columns: 'stock_id', 'date', 'stock_returns', 'market_returns', 'vix'

# Function to run the regression for a single stock and month
def run_regression(stock_id, start_date, end_date, all_data):
    monthly_data = all_data[(all_data['stock_id'] == stock_id) & 
                           (all_data['date'] >= start_date) & 
                           (all_data['date'] <= end_date)]
    
    monthly_data['delta_vix'] = monthly_data['vix'].diff().fillna(0)
    
    X = monthly_data[['market_returns', 'delta_vix']]
    X = sm.add_constant(X)
    y = monthly_data['stock_returns']
    
    model = sm.OLS(y, X)
    results = model.fit()
    
    return results.params

# List of unique stock IDs and months
unique_stocks = all_data['stock_id'].unique()
months = pd.date_range(start='2023-01-01', end='2023-12-31', freq='MS')

# Store results
results_df = []

# Loop through each stock and month
for stock_id in unique_stocks:
    for month_start in months:
        month_end = month_start + pd.offsets.MonthEnd(1)
        params = run_regression(stock_id, month_start, month_end, all_data)
        params['stock_id'] = stock_id
        params['month'] = month_start
        results_df.append(params)

# Convert list of results to DataFrame
results_df = pd.DataFrame(results_df)

# Save results to CSV
results_df.to_csv('regression_results.csv', index=False)

### Key Points
- **Data Alignment**: Ensure that the dates align correctly for the stock returns, market returns, and VIX.
- **Handling Missing Data**: Fill or drop missing values appropriately.
- **Performance Considerations**: Running regressions for a large number of stocks and months can be computationally intensive. Consider optimizing your code or using parallel processing techniques.

By following these steps, you can systematically run the required regressions for each stock and each month using daily data.