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I'm stuck learning the hat matrix and wondered if someone could help with a question. If I have the model $$Y_i =\beta_0+\beta_1X_i+\epsilon_i,i = 1,2,3 \dots n,$$ how can I calculate the hat matrix as

$$H = X(X^\prime X)^{-1}X^\prime$$

And what would the $h_{ij}$ element be?

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  • $\begingroup$ en.wikipedia.org/wiki/Hat_matrix This looks like it will help. $\endgroup$ Oct 28, 2015 at 20:25
  • $\begingroup$ (1) That minus sign does not belong in the expression for $H$. I guess you intend it to mean an inverse or a pseudo-inverse of the quantity within parentheses. (2) Are you perhaps asking what a matrix is and how to multiply matrices? $\endgroup$
    – whuber
    Oct 28, 2015 at 20:39
  • $\begingroup$ Sorry I have corrected it now. Yes please whatever is relevant to solve the question $\endgroup$ Oct 28, 2015 at 20:42
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    $\begingroup$ Assuming that your linear model is mod in R you would write something like: X = model.matrix(mod); (H = (X %*% solve(crossprod(X)) %*% t(X))). See this thread here for more details. $\endgroup$
    – usεr11852
    Oct 28, 2015 at 21:42
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    $\begingroup$ It's unclear what you are asking. You can calculate the hat matrix using the formula that you wrote down. Is there something particular about that formula that you do not understand? $\endgroup$ Oct 29, 2015 at 0:41

1 Answer 1

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We can write the model in matrix notation as

$$ Y = \beta X + \epsilon $$

The OLS solution for the vector of regression coefficients $\beta$ is:

$$ \hat{\beta} = (X'X)^{-1} X'Y $$

The hat matrix is the projection matrix that maps the response vector $Y$ to the vector of fitted values $\hat{Y}$ (hence the name "hat" matrix). That is:

$$ \hat{Y} = HY$$

Now, since
$$ \hat{Y} = X \hat{\beta} = X(X'X)^{-1} X'Y $$

it immediately follows that

$$ H = X(X'X)^{-1} X'$$

as required.

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