Blockwise Formula
Suppose the design matrix can be decomposed by columns as . Define the Hat operator as . Similarly, define the residual operator as . Then the Hat matrix of can be decomposed as follows:
There are a number of applications of such a partitioning. The classical application has a column of all ones, which allows one to analyze the effects of adding an intercept term to a regression. Another use is in the fixed effects model, where is a large sparse matrix of the dummy variables for the fixed effect terms. One can use this partition to compute the hat matrix of without explicitly forming the matrix, which might be too large to fit into computer memory.
Read more about this topic: Hat Matrix
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