minimum can be written as: A geometrical interpretation of these equations is that the vector of residuals, is orthogonal to the column space of X, since the dot product is equal to zero for any conformal vector, v. This means that is the shortest of all possible vectors , that is, the variance of the residuals is the minimum possible. This is illustrated at the right. Introducing and a matrix K with the assumption that a matrix is non-singular and KT X = 0 (cf. Orthogonal projections), the residual vector should satisfy the following equation: The equation and solution of linear least squares are thus described as follows: Another way of looking at it is to consider the regression line to be a weighted average of the lines passing through the combination of any two points in the dataset. Although this way of calculation is more computationally expensive, it provides a better intuition on OLS. Maximum likelihood The OLS estimator is identical to the maximum likelihood estimator (MLE) under the normality assumption for the error terms.[proof] This normality assumption has historical importance, as it provided the basis for the early work in linear regression analysis by Yule and Pearson. From the properties of MLE, we can infer that the OLS estimator is asymptotically efficient (in the sense of attaining the Cramér–Rao bound for variance) if the normality assumption is satisfied. Generalized method of moments In iid case the OLS estimator can also be viewed as a GMM estimator arising from the moment conditions These moment conditions state that the regressors should be uncorrelated with the errors. Since xi is a p-vector, the number of moment conditions is equal to the dimension of the parameter vector β, and thus the system is exactly identified. This is the so-called classical GMM case, when the estimator does not depend on the choice of the weighting matrix. Note that the original strict exogeneity assumption implies a far richer set of moment conditions than stated above. In particular, this assumption implies that for any vector-function , the moment condition will hold. However it can be shown using the Gauss–Markov theorem that the optimal choice of function is to take , which results in the moment equation posted above. Properties Assumptions There are several different frameworks in which the linear regression model can be cast in order to make the OLS technique applicable. Each of these