Exact multicollinearity means that there is at least one exact linear relation between the column vectors of the n × k data matrix of n observations on k variables. More commonly, multicollinearity means that the variables are so intercorrelated in the data that the relations are ‘almost exact’. The term was used by Frisch (1934) mainly in the context of attempts to estimate an exact relation between the systematic components of variables whose observed values contained disturbances or errors of measurement but where there might also be other relations between the systematic components which made estimates dangerous or even meaningless. In more recent work the data matrix has usually been the matrix X of regressor values in the linear regression model Y = X ß + ε with no measurement errors. Confusion between the two cases led at one time to some misunderstanding in the literature. Other terms used for the same phenomenon are collinearity and ill-conditioned data — although the latter may contain aspects of the scaling of variables which are irrelevant to multicollinearity.
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