▁sum ▁of ▁squ ared ▁resid uals ▁of ▁the ▁model ▁subject ▁to ▁the ▁constraint ▁A . ▁The ▁const rained ▁least ▁squares ▁( CL S ) ''' ▁estim ator ▁can ▁be ▁given ▁by ▁an ▁explicit ▁formula : ▁▁▁ ▁This ▁expression ▁for ▁the ▁const rained ▁estim ator ▁is ▁valid ▁as ▁long ▁as ▁the ▁matrix ▁X TX ▁is ▁invert ible . ▁It ▁was ▁assumed ▁from ▁the ▁beginning ▁of ▁this ▁article ▁that ▁this ▁matrix ▁is ▁of ▁full ▁rank , ▁and ▁it ▁was ▁noted ▁that ▁when ▁the ▁rank ▁condition ▁fails , ▁ β ▁will ▁not ▁be ▁ident ifiable . ▁However ▁it ▁may ▁happen ▁that ▁adding ▁the ▁restriction ▁A ▁makes ▁ β ▁ident ifiable , ▁in ▁which ▁case ▁one ▁would ▁like ▁to ▁find ▁the ▁formula ▁for ▁the ▁estim ator . ▁The ▁estim ator ▁is ▁equal ▁to ▁▁▁ ▁where ▁R ▁is ▁a ▁p × ( p   −   q ) ▁matrix ▁such ▁that ▁the ▁matrix ▁ ▁is ▁non - sing ular , ▁and ▁. ▁Such ▁a ▁matrix ▁can ▁always ▁be ▁found , ▁although ▁generally ▁it ▁is ▁not ▁unique . ▁The ▁second ▁formula ▁coinc ides ▁with ▁the ▁first ▁in ▁case ▁when ▁X TX ▁is ▁invert ible . ▁▁ ▁Lar ge ▁sample ▁properties ▁ ▁The ▁least ▁squares ▁estim ators ▁are ▁point ▁estimates ▁of ▁the ▁linear ▁regression ▁model ▁parameters ▁ β . ▁However , ▁generally ▁we ▁also ▁want ▁to ▁know ▁how ▁close ▁those ▁estimates ▁might ▁be ▁to ▁the ▁true ▁values ▁of ▁parameters . ▁In ▁other ▁words , ▁we ▁want ▁to ▁construct ▁the ▁interval ▁estimates . ▁ ▁Since ▁we ▁haven ' t ▁made ▁any