Stock return variance decomposition

Question

Variance Decomposition for Stock Returns: Issues with Weighted Returns and Risk-Free Rate Adjustments

I am attempting to replicate the variance decomposition of stock returns following the procedure described in Campbell et al. (2000) "Have Individual Stocks Become More Volatile?" using the CRSP dataset. However, I am encountering difficulties in understanding the correct steps regarding the calculation of variances and returns.

My process:

1. Variance Calculation for Market Returns (Equation 19 in the reference):

   I compute the market return using the weighted average of the firm-level returns, based on market capitalization. However, I am unclear whether this calculation requires dividing the monthly return variance by the number of trading days in the month. Should this variance represent daily or monthly volatility, and how should the trading days factor into the calculation?

   $$ \text{Market Return Variance:} \quad \sigma_{m,t}^2 = \sum_{i \in \mathcal{C}_t} (R_{i,t} - \mu_{m,t})^2 $$

   Should I divide the result by the number of trading days?

2. Subtraction of Risk-Free Rate:

   When calculating excess returns, I subtract the risk-free rate from the weighted returns of individual stocks. Is this correct for each step in the decomposition, or does the procedure differ for calculating the volatility components? Should I also subtract the risk-free rate when calculating industry returns or the residual returns?

   $$ R_{i,t}^{\text{excess}} = R_{i,t} - R_{f,t} $$

   Should I apply this uniformly at all stages of the decomposition?

3. Use of Log vs Simple Returns:

   I’m unsure whether to use simple or log returns when computing individual, industry, and market-level returns. My current approach is using simple returns, but Campbell et al. do not explicitly mention which form to use for each step in the decomposition.

   Is there a general preference for using log returns in such decomposition exercises?

4. Computation of Industry-Level Volatility:

   For industry-level returns, I grouped firms by industry, subtracted the market excess return, squared the residuals, and summed them within each industry. However, my results are significantly different from the figures reported in the paper.

   My steps were as follows:

   $$ \varepsilon_{i,t} = R_{i,t}^{\text{excess}} - R_{m,t}^{\text{excess}} $$

   Then, I squared the residuals and summed them for each industry:

   $$ \sigma_{\text{industry},t}^2 = \sum_{i \in \mathcal{I}_t} \varepsilon_{i,t}^2 $$

   Should the returns at the industry level also be weighted by market capitalization?

Key Confusion Points:

1. Should I calculate monthly variances for individual stocks and industries without dividing by the number of days in the month?

2. At what points do I need to subtract the risk-free rate, and do I apply it to the market, industry, and individual returns uniformly?

3. Should I be using weighted returns for each step, and if so, how do I incorporate the risk-free rate when weighting the returns?

Any guidance on the correct steps for the variance decomposition process or advice on common mistakes when replicating this procedure would be greatly appreciated.