Basic Assumptions: Further Considerations

Basic assumptions have been stated explicitly or implicitly. In many textbooks, one of these assumptions has the form (4.1)% MathType!MTEF!2!1!+-% feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaabm% aabaGaamiwaaGaayjkaiaawMcaaiabg2da9iaad2gaaaa!3B47!$$r\left( X \right) = m$$ which implies that the rank of this fixed data matrix X should be equal to the number of its columns m. If X is rank-deficient then the determinant of (X’X) is zero so that vector b̂ is undefined. However, depending on the computer program used, one may obtain some estimation results for multivariate models due to rounding errors, particularly when the numbers are stored in single precision memory allocation and not double precision. Hence, proper tests are required as it is explained in the Appendix of this Part, in order to overcome this trap. It is recalled that we stated (4.1) in a more rigid form, assuming that (X’X) is positive definite, since such a matrix is always invertible, whereas a nonsingular matrix is not necessarily positive definite.