If n = p + 1, and the x matrix is invertible, show that the hat matrix h is given by the (p + 1) ∗ (p + 1) identity matrix. in this case, what are hii and yˆi?

Step-by-step explanation:

Consider that we have a multiple linear model given by this:

And we assume that we have p independent variables and p+1 is the dimension with the vector including the intercept, so then n =p+1.

For this case we want to minimize the sum of error

And we have this:

And when we find the partial derivate respect to b we got:

We are assuming that the matrix X is invertible(non singular). And applying the inverse matrix X'X on both sides we got:

The estimator for [tex\beta[/tex] is given by:

And the predicted values can be written like this:

And we can see that and on this case if we find the dimensions for H, assuming that X is nxn. Then H is a matrix nxn since only depends of the matrix X.

On this case the elements represent the elements of the diagonal for the matrix H and are used in order to find the estimated values

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step-by-step explanation: