Question

Consider a linear regression model with 100 data points (n = 100). The model is represented as:

y_i = β_0 + β_1 * x_1,i + β_2 * x_2,i + e_i

We construct a matrix X to include the regressor data:

X = [1 ... 1]
[x_1,1 ... x_1,100]
[x_2,1 ... x_2,100]

If we compute Z = X'X, what is the dimension of matrix Z?

a. Z is a 2x100 matrix.
b. Z is a 2x2 matrix.
c. Z is a 100x100 matrix.
d. Z is a 1x100 matrix.
e. Z is a 2x1 matrix.

Answer 1

The dimensions of matrix Z, obtained by computing X'X in a linear regression model with n = 100 and two predictors (x1 and x2), result in a 3x3 matrix. This is because the original matrix X includes an intercept and columns for each predictor, making it a 3x100 matrix, and the transpose of X is a 100x3 matrix.

Step-by-step explanation:

You've asked about the dimensions of the matrix Z which is obtained by computing X'X in a linear regression model with two predictors.

The original matrix X is constructed with a row of 1's to represent the intercept (β_0), and one row for each of the predictor variables x_1 and x_2. Given there are 100 data points (n = 100), matrix X is of dimension 3x100, including the intercept row.

When you transpose matrix X (denoted as X'), it becomes a 100x3 matrix. The multiplication of the transposed matrix X' by the original matrix X (X'X) results in a square matrix whose dimension is the number of rows of X' by the number of columns of X. Therefore, the matrix Z is a 3x3 matrix, which matches none of the options provided in your question.