In a regression with p predictors and an intercept, the trace of the hat matrix (sum of leverages) equals:

In a regression with p predictors and an intercept, the trace of the hat matrix (sum of leverages) equals:

• A. p
• B. p + 1
• C. n
• D. p + 2

Answer: (B)

The hat matrix H = X(X'X)^{-1}X' is the projection onto the column space of X. Its trace equals the dimension of that space, i.e., the rank of X. With p predictors plus an intercept, X has p+1 columns; if these columns are independent, rank(X) = p+1, so the trace (sum of leverages) is p+1. This reflects the number of parameters being estimated in the mean function: the intercept plus the p slopes. If the columns aren’t all independent, the trace would be less than p+1, but under full rank that’s the value.

The hat matrix H = X(X'X)^{-1}X' is the projection onto the column space of X. Its trace equals the dimension of that space, i.e., the rank of X. With p predictors plus an intercept, X has p+1 columns; if these columns are independent, rank(X) = p+1, so the trace (sum of leverages) is p+1. This reflects the number of parameters being estimated in the mean function: the intercept plus the p slopes. If the columns aren’t all independent, the trace would be less than p+1, but under full rank that’s the value.