Consider a sample of N individuals. Assume that y is the N × 1 binary dependentvariable and d1, d2, . . . , dM are the N × 1 dummy regressors for exhaustive andexclusive categories, meaning that each person in the population falls into one andonly one category (if for individual i dki = 1, all the other M − 1 regressors ofindividual i are equal to 0). Consider the linear regression, without the intercept,y on X, with X = [d1 d2 . . . dM ].Prove that:(a) The matrix X′X is diagonal.(b) The OLS estimates of the coefficients are all in the [0, 1] interval (0 ≤ ˆβj ≤ 1,for j = 1 . . . M ).(c) The fitted values from the OLS regression are in the [0, 1] interval for all indi-viduals (0 ≤ ˆyi ≤ 1, for i = 1 . . . N

To prove the given statements, we will follow the following steps:

(a) To prove that the matrix X'X is diagonal, we need to show that all off-diagonal elements are zero. Let's consider two different columns of X, say dj and dk, where j ≠ k. Since the dummy regressors are exhaustive and exclusive, it means that for any individual i, either dji or dki will be equal to 1, and the other will be equal to 0. Therefore, the dot product of column dj and column dk will always be zero, resulting in a zero off-diagonal element in X'X. Hence, X'X is diagonal.

(b) To prove that the OLS estimates of the coefficients are all in the [0, 1] interval, we need to show that 0 ≤ ˆβj ≤ 1 for j = 1, 2, ..., M. In the linear regression without the intercept, the OLS estimate of the coefficient βj is given by ˆβj = (X'X)^(-1)X'y. Since X'X is diagonal (as proved in part (a)), the inverse of X'X will also be diagonal. Therefore, the OLS estimate ˆβj will be equal to the dot product of the jth column of X'X and the dependent variable y, divided by the dot product of the jth column of X'X with itself. Since the dependent variable y is binary, taking values of 0 or 1, and the dot product of the jth column of X'X with itself will always be positive, it follows that 0 ≤ ˆβj ≤ 1 for j = 1, 2, ..., M.

(c) To prove that the fitted values from the OLS regression are in the [0, 1] interval for all individuals, we need to show that 0 ≤ ˆyi ≤ 1 for i = 1, 2, ..., N. The fitted values ˆyi are obtained by multiplying the OLS estimates ˆβj with the corresponding dummy regressors dj for each individual i, and summing them up. Since the OLS estimates ˆβj are all in the [0, 1] interval (as proved in part (b)), and the dummy regressors dj are binary variables taking values of 0 or 1, it follows that the fitted values ˆyi will also be in the [0, 1] interval for all individuals i = 1, 2, ..., N.

Therefore, we have proved that: (a) The matrix X'X is diagonal. (b) The OLS estimates of the coefficients are all in the [0, 1] interval. (c) The fitted values from the OLS regression are in the [0, 1] interval for all individuals.