What is the importance of hat matrix, $H=X(X^{\prime}X )^{-1}X^{\prime}$, in regression analysis?

Is it only for easier calculation?

  • $\begingroup$ Also, could you please be more specific? $\endgroup$
    – Steve S
    Commented Jul 20, 2014 at 13:43
  • $\begingroup$ @SteveS Actually i want to know why we need hat matrix? $\endgroup$
    – user 31466
    Commented Jul 20, 2014 at 13:53
  • $\begingroup$ Are you asking why we need to have a special name/symbol (i.e. "hat matrix", "H") for the matrix or are you asking more about the importance of the matrix product on the righthand side? $\endgroup$
    – Steve S
    Commented Jul 20, 2014 at 14:43

3 Answers 3


In the study of linear regression, the basic starting point is the data generating process $ \textbf{y= XB + u} \quad $ where $ \textbf{u} \sim N(0,\sigma^2 \boldsymbol I) $ and $\textbf{X}$ deterministic. After minimizing the least squares criterion, one finds an estimator $ \widehat {\textbf{B} }$ for $\textbf{B}$, i. e. $ \widehat {\textbf{B}}= ( \textbf{X} ' \textbf{X})^{-1}\textbf{X} '\textbf{y}$. After plugging in the estimator in the initial formula, one gets $\widehat {\textbf{y}}=\textbf{X}\widehat {\textbf{B}}$ as a linear model of the data generating process. Now, one can substitute the estimator for $\widehat {\textbf{B}}$ and gets $\widehat {\textbf{y}}=\textbf{X}( \textbf{X} ' \textbf{X})^{-1}\textbf{X} '\textbf{y}.$

So, $\textbf{H} = \textbf{X}( \textbf{X} ' \textbf{X})^{-1}\textbf{X} '$ is actually a projection matrix. Imagine you take all variables in $\textbf{X}$. The variables are vectors and span a space. Hence, if you multiply $\textbf{H}$ by $\textbf{y}$, you project your observed values in $\textbf{y}$ onto the space that is spanned by the variables in $\textbf{X}$. It gives one the estimates for $\textbf{y}$ and that is the reason why it is called hat matrix and why it has such an importance. After all, linear regression is nothing more than a projection and with the projection matrix we cannot only calculate the estimates for $\textbf{y}$ but also for $\textbf{u}$ and can for example check whether it is really normally distributed.

I found this nice picture on the internet and it visualizes this projection. Please note, $\beta$ is used instead of $\textbf{B}$. Moreover, the picture emphasizes the vector of the error terms is orthogonal to the projection and hence not correlated with the estimates for $\textbf{y}$

enter image description here


The hat matrix is very useful for a few reasons:

  1. Instead of having $\widehat{y}=Z\widehat{\beta}$, we get that $\widehat{y}=Py$ where $P$ is the hat matrix. This gives us that $\widehat{y}$ is a linear mapping of the observed values.
  2. From the hat matrix $P$, it is easy to calculate the residuals $\widehat{\epsilon}$. We see that $\widehat{\epsilon}=y-\widehat{y}=y-Py=\left(I_n-P\right)y$.

It's nothing more than finding the "closest" solution for Ax = b where b is not in the column space of A. We project b onto the column space, and solve for Ax(hat) = p where p is the projection of b onto column space.

  • 2
    $\begingroup$ All this can be done without ever computing $H$. $\endgroup$
    – whuber
    Commented Nov 2, 2015 at 14:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.