We have N companies with incomplete monthly data. For our research, we aim to run monthly cross $-$ sectional regressions using two different approaches:

$1$

$.$

$$In Approach $1$:

$$we include data for all companies regardless of whether we have complete data for them. If data for a company is missing, we assign a value of $0$ t both the independent and dependent variables for that company, while using the actual data for all other companies. The provided screenshot shows the data for Approach

$1$

$$for a particular month, where rows represent different companies. Vector

$1($

V

$1)$

$$is the dependent variable, and all other vectors are independent variables for the regression.

$2$

$.$

$$In Approach

$2$

$,$

$$we include only those companies each month for which we have complete data

$($

Data for independent and dependent variables

$)$

$.$

After running the cross

$-$

sectional regressions for the approach, we obtain the betas for the first independent variable and then perform a

$($

Stage

$2)$

$$time

$-$

series regression, regressing the betas from the cross

$-$

sectional regressions on the Fama

$-$

French factors to check for alpha. The approach for stage

$2$

$$is same for both approach

$1$

$$and approach

$2$

$.$

We observe that the results

$($

the intercept from time series regression

$)$

$$and the coefficients for the Fama

$-$

French factors differ between the two approaches. Specifically, in Approach

$2$

$,$

$$the coefficients are around

$2$

$$for the monthly time

$-$

series data

$($

See regression results for approach

$2)$

$.$

Question: Which of the two approaches is acceptable and robust, and why? Additionally, do the regression coefficients for the Fama

$-$

French factors being as large as $2$ or $2\mathrm{.}10$

$$make sense given that we have monthly data?