Simple linear regression model and its column rank Consider the simple linear regression model $$Y_i = \beta_0 + \beta_1 X_i + \epsilon_i, i = 1, \dots, n$$ Give the order and column rank of the following for the model $Y:$ Order = ? Column Rank = ? $X:$ Order = ? Column Rank = ? $\beta$ Order = ? Column Rank = ? $X' X$ Order = ? Column rank = ? * * * Attempt I believe $Y$ would be a n x 1 matrix $X$ is going to be a n x 2 matrix I don't know what $\beta$ means $X'$ is 2 x n matrix so $X'X$ would be a 2 x 2 matrix therefore $Y:$ Order = $n \times 1$ Column Rank = $1$ $X:$ Order = $n \times 2$ Column Rank = $2$ $\beta$ Order = ? Column Rank = ? $X' X$ Order = $2 \times 2$ Column rank = $2$ Am I right? And how do I get $\beta's$?

I assume you are referring to the rows and columns of the matrices/vectors which is referred to as the format of a matrix.

As far as I can see you are trying to determine the linear regression for a data set $X$ and a weight vector $\beta$ normally. We use the data set $\tilde{X}$ and add an additional column to the left which only contains one entry. This manipulation is done to get a vector equation.

$$\mathbf{y}=\mathbf{X\beta}+\mathbf{\varepsilon}$$

Please note that $\beta$ and $\varepsilon$ are vectors it is a strange behavior of LaTeX/MathJax. The output vector $y$ has the format $n\times 1$ containing the outputs/predictions for each observation. If we have $p$ predictors then $\mathbf{X}$ has the format $n \times (p+1)$ (note that we have an additional column for the added column) and $\beta$ has the format $(p+1)\times 1$. The error has the format $n\times 1$.

Can you determine the format of $\mathbf{X}^T\mathbf{X}$ as $(p+1) \times (p+1)$.