# Mathematical (matrix) notation for a regression model with several dummy variables

I have spent a significant amount of time on this, but my knowledge in Matrix Algebra is still limited. So, I am not sure ...

I have a linear regression model based on $$n$$ observations. The dependent variable is $$y$$.

As predictors, there are 10 independent continuous variables + 9 dummy variables (based on a 10-category scale) + 5 dummy variables (based on a 6-category response). The model looks like: $$y = \beta_0+\beta_1 \text{C}_1+\beta_2 \text{C}_2+ ...+\beta_{11}\text{dummy}_1+\beta_{12}\text{dummy}_2+ ...+\beta_{19}\text{dummy}_9+\beta_{20}\text{dummy}_{10}+...\beta_{24}\text{dummy}_{14}+\epsilon$$

Would this be an accurate matrix notation of the above model? $$\textbf{y} = \textbf{C}{\beta}+\textbf{S} \gamma + \mathbf{R}\lambda + \textbf{e}$$

where $$\textbf{C}$$ is n $$\times$$ 10+1 design matrix with 1s in the first column and 10 other columns containing the continuous predictors , $$\textbf{S}$$ is the n $$\times$$ 9 matrix containing dummy variables for the 10-category scale, $$\textbf{R}$$ is the n $$\times$$ 5 matrix containing dummy variables for the 6-category response, $${\beta}$$, $${\gamma}$$ and $${\lambda}$$ are the vectors of parameters, and $$\textbf{e}$$ is the error vector.

Is this notation sufficiently accurate? Any references on matrix notation for dummy variables are also very much appreciated.

Sincerely,

Jacob

• Welcome to CV.SE. Yes, you are fine. Good job (+1) Please see my answer below I expand on this a bit more. Dec 13, 2020 at 2:10

Yes, you are fine. Please note that all your categorical vectors will have their first level absorbed within the intercept $$\beta_0$$, therefore take note of the way the encoding of dummy variables is done. I would say that if anything your notation lends itself nicely to group lasso regularisation; standard lasso solution would only select individual dummy variables instead of whole factors. (check Meier et al.(2007) The group lasso for logistic regression for more information on that aspect.)