- One way is to see the effect of your independent variables on some specific characteristics like: the value of the time series at a given time (chosen or special like the first peak value), the maximum value in the first year, the slope or intercept of the time series.....
- Another is to apply a certain model like SARIMA and predict its coefficients using the independant variables.
- Sometimes your independant variables are changing with time, in this case you may consider some time varying coefficient models. generally you should construct the model your self and try to fit it using simpler available models.
I think the data could be used but the firm size needs to be the most recent and temporally close to your daily stock returns. Obviously, when you evaluate your models you should acknowledge these limitations.
Let me try to outline the concept and the analysis techniques which might help here.
The combination of a time series dependent variable and cross-sectional independent variables is known as panel data or longitudinal data.
Panel data analysis techniques can be used to model this type of data structure, which allows you to study how changes in the independent variables affect the dependent variable over time.