Residual Sum of Squares - Matrix Expression For The OLS Residual Sum of Squares

Matrix Expression For The OLS Residual Sum of Squares

The general regression model with n observations and k explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is

where y is an n × 1 vector of dependent variable observations, each column of the n × k matrix X is a vector of observations on one of the k explanators, is a k × 1 vector of true coefficients, and e is an n× 1 vector of the true underlying errors. The ordinary least squares estimator for is

The residual vector is, so the residual sum of squares is, after simplification,

Read more about this topic:  Residual Sum Of Squares

Famous quotes containing the words matrix, expression, residual, sum and/or squares:

    “The matrix is God?”
    “In a manner of speaking, although it would be more accurate ... to say that the matrix has a God, since this being’s omniscience and omnipotence are assumed to be limited to the matrix.”
    “If it has limits, it isn’t omnipotent.”
    “Exactly.... Cyberspace exists, insofar as it can be said to exist, by virtue of human agency.”
    William Gibson (b. 1948)

    But the wise know that foolish legislation is a rope of sand, which perishes in the twisting; that the State must follow, and not lead the character and progress of the citizen; the strongest usurper is quickly got rid of; and they only who build on Ideas, build for eternity; and that the form of government which prevails, is the expression of what cultivation exists in the population which permits it.
    Ralph Waldo Emerson (1803–1882)

    The volatile truth of our words should continually betray the inadequacy of the residual statement. Their truth is instantly translated; its literal monument alone remains.
    Henry David Thoreau (1817–1862)

    I was brought up to believe that the only thing worth doing was to add to the sum of accurate information in the world.
    Margaret Mead (1901–1978)

    An afternoon of nurses and rumours;
    The provinces of his body revolted,
    The squares of his mind were empty,
    Silence invaded the suburbs,
    —W.H. (Wystan Hugh)