Not-for-Publcaton Aendx to Otmal Asymtotc Least Aquares Estmaton n a Sngular Set-u Antono Dez de los Ros Bank of Canada dezbankofcanada.ca December 214 A Proof of Proostons A.1 Proof of Prooston 1 Ts roof closely follows Peñaranda and Sentana (212), were furter detals can be found. Let te sectral decomoston of V g ( ) be gven by V g ( ) = T 1 T 2 T 1 T 2 = T 1 T 1 were s a ( S) ( S) ostve de nte dagonal matrx and, wtout loss of generalty, let V + g ( ) be te Moore-Penrose 1 generalzed nverse of V g ( ) V + g ( ) = T 1 1 T 1 In order to smlfy te notaton, t s convenent to rearameterze te arameter sace nto te alternatve K arameters (S 1) and ((K S) 1) suc tat R() = were te rst S elements of R() are suc = r(). In artcular, we can coose R() to be a regular transformaton of on an oen negbourood of. Furter, let q [R()] = be te corresondng nverse transformaton of R() tat recovers back. Let te Jacobans of te nverse transformaton be gven by q( ) Q( ) = ( ) = Q ( ) Q ( ) 1 As noted by Peñaranda and Sentana (212), t s ossble to sow tat te results n ts rooston old for any generalzed nverse of V g ( ) Wle a smlar argument would aly ere, we focus on te Moore-Penrose generalzed nverse for smlcty. 1
Ts transformaton allows us to mose te arametrc restrctons r() = = by smly workng wt te smaller set of arameters and te dstance functons g [b q( )]. Tus te otmal ALS estmator can be de ned as b = q( b ) were b = arg mn T g [b q( )] V + g ( )g [b q( )] () Snce (T 1 T 2 ) s an ortogonal matrx, and te rank [Q( )] = K gven tat R() s a regular transformaton of on oen negbourood of, we ave by te nverse functon teorem tat rank ( ) T = rank 1 ( )Q ( ) T 1 ( )Q ( ) T 2 ( )Q ( ) T 2 ( = K )Q ( ) (A.1) Note now tat Assumtons 1 and 2 mly tat [l( )] T g [b q( )]! for all n te negbourood. So, by d erentatng ts random rocess wt resect to and evaluatng te dervatves at te true value we ave, by te contnuous mang teorem, tat T g b q( ) vec q( ) I S + q( ) T g b q( ) T! g b q( ) vec q( ) I S + q( ) T g b q( )! snce 1= T!. Usng te can rule, te revous exresson can be wrtten as g b q( ) vec q( ) I S Q ( )+ q( ) b q( ) Q ( ) wc mles tat q( ) l( ) Q ( ) = wt () = [() ] and were we ave used tat g b q( )! g q( ) = g( ) =, and tat b q( )! ( ) q( ) = q( ). Fnally, note tat snce T 2V g ( ) = ten T 2 must be a full-column rank lnear transformaton of (). Terefore, t as to be tat T 2 q( ) Q ( ) = wc mles tat rank Q 1 ( )Q ( ) = K S for (A.1) to be true. Tus, after mosng tat =, te reduced system of dstance functons Q 1g [b q( )] wll rst-order dentfy at. () Snce te transformaton from to ( ) s regular on an oen negbourood of, a rst-order exanson system of dstance functons delvers 1 T ( b ) = Q ( ) ( )V + g ( ) ( )Q ( ) Q ( ) ( )V + g ( ) T g(b ) + o (1) (A.2) 2
were Terefore, V = T ( b ) d! N [ V ] Q ( ) ( )V + g ( ) ( )Q ( ) 1 (A.3) In addton, note tat snce te otmal ALS estmator s gven by b = q( ), b we can use te Delta metod to comute ts asymtotc dstrbuton T ( b d )! N Q ( )V Q ( ) (A.4) We now comare te asymtotc covarance matrx of ts otmal estmator wt te ALS estmator tat uses W as a wegtng matrx and does not mose te restrctons r() =. In artcular, te asymtotc covarance matrx of suc an estmator s gven by ( )W ( ) 1 ( )WV g ( )W ( ) ( )W ( ) 1 Terefore, for b to be otmal, we need ( )W ( ) 1 ( )WV g ( )W ( ) ( )W ( ) 1 Q ( )V Q ( ) to be ostve semde nte, wc n turn requres ( )WV g ( )W ( ) ( )W ( ) Q ( )V Q ( ) ( )W ( ) to be ostve semde nte as well. It can be sown tat ts s te case gven tat ts matrx s te asymtotc resdual varance of te lmtng least squares rojecton of T ( )Wg b on T Q ( ) ( )V g + ( )g(b ). In artcular lm V ar T!1 T ( )Wg b T Q ( ) ( )V g + ( = )g(b ) ( )WV g ( )W ( ) ( )W ( )Q ( ) Q ( ) ( )W ( ) V 1 Alternatvely, we can consder te varance of a trd ALS estmator tat uses W as wegtng matrx but moses te restrctons r() = Q ( ) ( )W ( )Q ( ) 1 Q ( ) ( )WV g ( )W ( )Q ( ) Q ( ) ( )W ( )Q ( ) 1 and te varance of a fourt estmator tat uses te generalzed nverse of V g ( ) as a wegtng matrx but does not mose r() = ( )V + g ( ) ( ) 1 ( )V + g ( ) ( ) ( )V + g ( ) ( ) 1 = ( )V + g ( ) ( ) 1 3
Agan, t s ossble to rove tat te d erence between any of tese two matrces and Q ( )V Q ( ) s ostve semde nte. () Usng a Taylor exanson of T g b q( ) b and equaton (A.2), we ave tat T g b q( ) b = T g(b ) + ( )Q ( ) T ( b ) + o (1) = I ( )Q ( )V Q ( ) ( )T 1 1 T1 T g(b ) + o (1) and rearrangng te revous exresson as T g b q( ) b = T 1 1=2 I S H(H H) 1 H T 1=2 T 1g(b )+o (1) were H = 1=2 T 1 ( )Q ( ). Terefore, te crteron functon evaluated at te otmal ALS estmator s T g b q( ) b V + g ( )g b q( ) b = bz I S H(H H) 1 H bz+o (1) were bz = 1=2 T 1g(b ) s asymtotcally dstrbuted as a standard multvarate normal, wc mles tat te crteron functon converges to a c-square dstrbuton wt K degrees of freedom, gven tat te matrx I S H(H H) 1 H s demotent wt rank ( S) (K S) = K. A.2 Proof of Prooston 2 As n te roof of Prooston 1, we wll work wt te alternatve set of K arameters of nterest (S 1) and ((K S) 1) suc tat R() = were te rst S elements of R() are suc tat = r(). Agan, let q [R()] = be te nverse transformaton of R() tat recovers back, and let ts Jacobans be denoted by Q( ) =q( )=( ). As noted earler, ts (regular) transformaton allows us to mose te arametrc restrcton r() = by smly settng =. In artcular, te asymtotc dstrbuton of te ML estmate of subject to te restrcton tat = s gven by were ( ) = T ( ML b ) 1 E 2 log L() T d! N 1 ( ) s te relevant block of te nformaton matrx. Smlarly, snce te ML estmator of tat moses te restrcton r() = s gven by b ML = q( b ML ) we can use te Delta metod to comute ts asymtotc dstrbuton T ( b ML ) d! N Q ( ) 1 ( )Q ( ) 4
In artcular, te otmal ALS estmate of wll be asymtotcally equvalent to ML f tey ave te same asymtotc varance. Comarng ts exresson wt equaton (A.4), t s stragtforward to see tat ts wll only occur wen V = 1. In order to rove ts result, we wll work on an alternatve set of auxlary arameters (S 1) and (( S) 1) suc tat M [()] = () () were te rst S elements of M() are suc tat = r(). Let l [M()] = be te corresondng nverse transformaton of M() tat recovers back. Let te Jacobans of te nverse transformaton be gven by l( ) L( ) = ( ) = L ( ) L ( ) Note tat ts second (regular) transformaton of te auxlary arameters allows us to mose te arametrc restrcton r() = on bot te estmaton of te auxlary and arameters of nterest. Sec cally, we ave tat () = r [q( )] = for all. Furter, te asymtotc dstrbuton of te ML estmate of subject to te restrcton tat = s gven by were ( ) = T (bml ) 1 E 2 log L() T d! N 1 ( ) s te relevant block of te nformaton matrx. Note tat =. Moreover, snce te ML estmator of tat moses te restrcton r() s gven by b ML = l( b ML ) we can use te Delta metod to comute ts asymtotc dstrbuton T (bml d )! N L ( )( 1 )L ( ) (A.5) Fnally, note tat, snce te system s comlete, and te fact tat bot R() and M() are regular mly tat Q( ) and L( ) ave full rank, we can wrte tat =! 1 1 = L( ) Q Q = L L Q 1 ( ) wc, snce () = r [q( )] = for all mles tat = = we ave tat! Q = L (A.6) 5
Substtutng equatons (A.5) and (A.6) evaluated at = n te exresson for V n (A.3) we ave tat V 1 = + L ( ) ( ) ( )L ( ) 1 ( )L ( ) ( ) ( )L ( ) Let D be te term nsde te curly brackets. Premultlyng D by ( )L ( ) 1 ( ), and ostmultlyng t by 1 ( )L ( ) ( ) we nd tat ( )L ( ) 1 ( )D 1 ( )L ( ) ( ) = ( )L ( ) 1 ( )L ( ) ( ) were we ave used te fact tat a generalzed nverse must satsfy WW + W = W. Tus, D = for te last equaton to be true. Ts mles tat, V = 1 = 1 Terefore, te otmal ALS estmator tat uses a generalzed nverse of V g ( ) as te wegtng matrx and tat, smultaneously, moses te restrcton r() = r [()] = s asymtotcally equvalent to te ML estmator tat moses tat restrcton. 6