Errors-in-variables models: a generalized functions approach

Errors-in-v ariables mo dels: a generalized functions approac h Victor ia Zinde-W alsh ∗ McGill Universit y a nd CIREQ Revised version of September 29, 20 09 Abstract Iden tification in e rr ors - in-v ariables reg ression mo dels w as recen tly extended to wide mod e ls classes by S. Sc hennach (Econometrica, 2007) (S) via use of generalized functions. In this pap er the problems of non- and semi- parametric iden tification in such mo dels are re-examined. Nonparametric iden tification holds under w eak er assump tio ns than in (S); the p roof here do es not rely on decomp osition of generalized functions in to ord inary and s i ngular p a rts, which ma y not hold. C o n- ditions f o r con tin uit y of the identificatio n mappin g are p ro vided and a consisten t nonparametric plug-in estimator for regression fun c tions in th e L 1 space constructed. Semiparametric iden tification via a fi- nite set of moment s is sh o wn to hold for classes of functions that are ∗ The suppo r t of the So cial Sciences and Humanities Research Council of Canada (SSHR C), the F onds qu´ eb e c ois de la re cher che sur la so ci´ et´ e et la cultur e (FRQSC) is grate- fully ackno wledged. The a uth or tha nk s par ticipa n ts in the Cowles F oundatio n conference, the UK E SG, CEA 2009, Sta ts in the Chateau and C E SG meetings and P .C.B.Phillips, X. Chen, D. Nekipe lo v, L.W a ng and T.W o otersen for illuminating discuss io ns and sugges- tions. An ano nymous referee provided a very thoroug h rep ort cont aining many impor t ant po in ts a nd sug gestions. The re m aining error s are all mine. 1 explicitly c haracterized; un li ke (S) existence of a m oment generating function for th e measurement error is not required. Keyw ords: errors-in-v ariables mo del, generalize d functions 2 1 In tro ductio n The familiar errors in v ariables mo del with an unkno wn regression function, g , and measuremen t error in the scalar v a riable has the f orm Y = g ( X ∗ ) + ∆ Y ; X = X ∗ + ∆ X , where v ariables X and Y are observ able; X ∗ and ∆ X , ∆ Y are not observ a ble. A widely used a pp roach mak es use of instrumen t a l v ariables. Supp ose that instrumen ts are av aila ble and Z represen ts an identified pro jection of X on the instrumen ts so that additionally X ∗ = Z − U ; assume tha t U is indep e nden t of Z (Berkson-t yp e error from using the instrumen ts) a n d that E [∆ Y | Z , U ] = 0; E [∆ X | Z , U, ∆ Y ] = 0; E ( U ) = 0 . These assumptions w ere made by e.g. Hausman et al. (1991) who examined p olynomial regression. Newe y (200 1 ) added another momen t condition for es- timation in sem iparametric regression leading to t w o equations f or unkno wn g and F , the measuremen t error distribution (all inte grals o v er ( −∞ , ∞ )) : E ( Y | Z = z ) = Z g ( z − u ) dF ( u ); E ( Y X | Z = z ) = Z ( z − u ) g ( z − u ) dF ( u ) . 3 Define W y ( z ) ≡ E ( Y | Z = z ); W xy ( z ) ≡ E ( Y ( Z − X ) | Z = z ) . The mo del assumptions then can b e considered in terms of classes of functions W y , W xy , g and distribution F that satisfy the equations: W y ( z ) = Z g ( z − u ) dF ( u ); (1) W xy ( z ) = Z ( z − u ) g ( z − u ) dF ( u ); w e sa y that these functions W y , W xy , g , F satisfy mo del assumptions. The functions g and F en ter in con v olutions; this motiv ates using F ourier tr a ns - forms (F t). F ourier tra ns forms: ε y ( ζ ) = F t ( W y ( · )); (2) ε xy ( ζ ) = F t ( W xy ( · )); γ ( ζ ) = F t ( g ( · )); the characteristic function is obt a ine d as φ ( ζ ) = R e iζ u dF ( u ) . Pro vided that for some sub class of functions F ourier transforms are w ell defined, deriv ative s exist and the con v olution theorem applies, (1) is equiv a- len t to a system with tw o unkno wn functions, γ , φ : ε y ( ζ ) = γ ( ζ ) φ ( ζ ); (3) iε xy ( ζ ) = · γ ( ζ ) φ ( ζ ) , (4) where · γ = dγ dζ . S. Sche nnac h (2007) (S) suggested that these equations can b e justified for a wide class of functions if one uses generalized functions, sp e cifically , t ho se in the space of temp ered distributions, T ′ (defined b elo w in section 2.1) 1 . Assumption 1. The functions g , W y , W xy that satisfy the mo del as- 1 A referee p o in ted out that the usual notation for the space of temp ered distributions is S ′ , but here w e follow the nota tio n in (S). 4 sumptions are suc h that eac h r epresen ts an elemen t in the space of temp ered distributions, T ′ . Some examples of suc h functions are the class considered in (S, As- sumption 1): functions suc h that | g ( x ∗ ) | , | W y ( z ) | , | W xy ( z ) | are defined and b ounded b y p olynomials on R. How eve r, the assumption here allows for v ery wide classes of functions. This class ma y b e difficult to c haracterize explicitly; the Ass umption 1’ b elo w pro vides an imp ortan t subclass of lo cally integrable functions in T ′ . Consider functions b ( t ) for t ∈ R that satisfy Z (1 + t 2 ) − l | b ( t ) | dt < ∞ for some l ≥ 0 . (5) Assumption 1’. The functions g , W y , W xy that satisfy the mo del a s - sumptions a r e suc h that each satisfies (5). The functions g , W y , W xy that satisfy (5) satisfy Assumption 1. Any func- tion in the space L 1 of absolutely in tegrable functions satisfies Assum ption 1’ here but not the Assumption 1 in (S) unless the function is ev erywhere b ounded. W hile the assu mption in (S) extends to p olynomial regress ion functions or distribution functions for binar y choice models (where Ft do not exist in t he ordinary sense), a regression function that is unbounded at some p oin t s is not allow ed. There are cases where suc h prop erties may a ris e, e.g. for some hazard f unctions, f or liquidit y trap; the more general assumption here accommo dates suc h cases. F ourier tra nsfor m is a contin uous inv ertible op erator in T ′ , all temp ered distributions are differen tiable in T ′ (th us · γ is defined). F ourier transform of an ordinary function of the t yp e considered here ma y no longer b e an ordinary f u nction (e.g. F t ( const ) = δ , the D irac delta-function that cannot b e represen ted as a n or din ary f unction), and thus is not defined p oin t -wis e; th us the notation γ ( ζ ) , etc. for the F t in e.g. (3 ,4) whic h w e k eep here for conv enience refers just to the generalized function γ without necessarily 5 giving meaning to v alues at a p oin t. In the class of functions t ha t satisfies t he mo del assumptions denote by A the sub class of functions ( g , F ), by A ∗ the subclass of functions ( g ); the mapping P : A → A ∗ is giv en by P ( g , F ) = g . Denote by B the class o f functions ( W y , W xy ). E quations ( 1 ) of mo del assumptions map A in to B (mapping M : A → B ); F ourier transforms map B in to F t ( B ) , the class of functions that are F ourier transforms o f functions from B ; if equations (3,4) could be solved they would pro vide solutions φ ∗ = φI ( γ 6 = 0) (where I ( A ) = 1 if A is true, zero otherwise) a nd γ ∗ if φ 6 = 0; applying inv erse F ourier transform w ould g ive g ∗ = F t − 1 ( γ ∗ ) . This sequenc e of mappings can b e represen t ed as fo llo ws: A ( g, F ) M → B ( W y , W xy ) F t → F t ( B ) ( ε y ,ε xy ) S → F t ( ˜ A ) ( φ ∗ ,γ ∗ ) → F t ( A ∗ ) ( γ ∗ ) F t − 1 → A ∗ ( g ∗ ) . (6) If (6 ) pro vides the same result as P so tha t g ∗ ≡ g (and γ ∗ ≡ γ ) then g can b e iden tified from the functions ( W y , W xy ) with the iden tification mapping M ∗ : B → A ∗ (7) giv en by comp osition of the last fiv e mappings in (6). The most c hallenging part is in solving the equations to establish the mapping (for γ ∗ ≡ γ ) S : F t ( B ) ( ε y ,ε xy ) → F t ( ˜ A ) ( φ ∗ ,γ ) (8) Tw o additio na l assumptions are similar to those in (S) and are standard. Assumption 2 . The function φ ( ζ ) is contin uous, contin uously differen- tiable o n R ; and φ ( ζ ) 6 = 0 . In terms of the mo del this implies a further conditio n tha t a bsolute mo- men t of U exist. Assumption 3. Supp ort of generalized function γ coincides with | ζ | ≤ ¯ ζ where ¯ ζ > 0 and could b e infinite. 6 Under Assumptions 1-3 iden tification is p ossible as sho wn in Theorem 1 of this pap er; the theorem in (S) asserts an analytic form ula (S, (13)) that relies o n a decomp osition that may not hold. When the erro r s- in- v ariables problem is examined in t he space of tem- p ered distributions the corresp onding (weak ) top ology is that of the space T ′ ; in that to p ology the mappings Ft, Ft − 1 are kno wn to b e contin uous, ho w ev er, the mapping (8) ma y b e discontin uous, rendering the iden tifica- tion mapping (7) discontin uous as w ell th us implying ill-p osedness of the problem. One reason f o r this is t ha t a to o thin-tailed characteristic func- tion ma y mask high-frequency comp onen ts in the F ourier transform of the regression f unction. Theorem 1 here prov ides a condition under whic h con- tin uit y o btains . When iden tification is provided b y a con tin uous mapping nonparametric plug-in estimation is po s sible as long a s F ourier tr a ns forms of the conditional momen t functions can be consisten tly estimated in T ′ ; this applies e.g. if the regression function is in the L 1 space; Prop osition 2 establishes t h is result. Iden tification in classes of parametric functions requires that the mapping from the parameter space to the function space b e (at least lo cally) inv ert- ible. (S) uses generalized functions to widen classes of parametric functions for w hic h iden tification is pro vided b y a finite n um b er of momen t condi- tions; in par tic ular she expands classes of L 1 functions to whic h t he results of W ang and Hsiao (1995 , 2009) a p ply and also allow s sums of suc h func- tions with p olynomial functions, where b efore p olynomial functions w ere considered o n ly b y themselv es in Hausman et al. (1 991). Her results rely on existence of a moment g e nerating function for the measuremen t error and use sp ecial we ighting functions (some of whic h are improp erly defined). Here g eneral classes of functions where suc h iden tification is a chiev able are explicitly c haracterized rather than via existence of momen ts conditions ( a s in S, Assumption 6), the requiremen t of a momen t generating function for measuremen t error is av oided; appropriate w eigh ting functions ar e giv en. 7 Section 2 deals with identific ation and w ell-p osedness in the non-parametric case. Section 3 examines iden tification for the semiparametric mo del. Pro ofs are in Appendix A. App endix B pro vides an explanation of the claims ab out the main errors in (S). 2 Non-parametric id e n tificatio n In the first part of this section kno wn results on generalized functions that confirm the exis tence and con tin uit y of some of the mappings in (6) are pro vided, in particular, for the F our ier transform and its in v erse . Other map- pings, suc h as (8) require sp ecial treatment b ecaus e they inv olve multiplica- tion of generalized f unctions. Multiplication in spaces of generalized functions cannot b e defined (Sc h w artz’s imp ossibilit y result, 19 54, see also Ka min ski and Rudnic ki (1991) f o r examples) although there are cases when sp ecific pro ducts are kno wn to exist. Here conditions under whic h some generalized functions can b e m ultiplied by some contin uous functions to obtain general- ized functions are pro vided. With this additional insigh t the existence and con tin uit y of the mappings can be examined. In t he second part of this section the iden tification result is prov ed and sufficien t conditions for the iden tification mapping to b e con tinuous are provided. A prop osition ab out consisten t (in t opology of T ′ ) nonparametric estimation that in particular applies to functions in space L 1 completes this section. 2.1 Results ab out generalized functions and existence and con tin uit y of mappings All the kno wn results in this section a r e in Sc h w artz (1966 ), Gel’fand and Shilo v’s monograph (vol.1 and 2, 1 964) - (G S) and in Ligh thill (1959)- (L); they are listed for the con v enience of the reader. The sequen tial approach of Mikusinski in Antonisek et al (1973) is also referred to here. A somew hat 8 distinct approach to multiplic ation by a contin uous function in the space of generalized functions is dev elop e d at the end of this section to explain the v alidity of some of the mappings in (6). Definitions o f generalized function spaces usually start with a top ological linear space of w ell-b eha v ed ”test functions”, G . Two most widely used spaces are G = D and G = T (usually denoted S in the literature). The linear top ological space of infinitely differen tia ble functions with finite supp ort D ⊂ C ∞ ( R ) , where C ∞ ( R ) is t he space of all infinitely differen tia ble functions; con v ergence is defined for a sequence of functions that are zero outside a common b ounded set and con v erge uniformly tog ether with deriv at ives of all orders. The space T ⊂ C ∞ ( R ) of test functions is defined as: T =  s ∈ C ∞ ( R ) :     d k s ( t ) dt k     = O ( | t | − l ) as t → ∞ , for in teger k ≥ 0 , l > 0  , k = 0 corresp onds to t he f un ction itself; | ·| is the absolute v alue; these functions go to zero f as ter tha n an y p o w er. A sequence in T con v erges if in ev ery b ounded region the pro duct of | t | l (for an y l ) with any order deriv ativ e con v erges uniformly . A generalized function, b, is defined b y an equiv alence class of w eakly con v erging sequences of test functions in G : b =  { b n } : b n ∈ G, suc h that fo r an y s ∈ G, lim n →∞ Z b n ( t ) s ( t ) dt = ( b, s ) < ∞  . An alternativ e equiv alen t definition is that b is a linear con t inuous functional on G with v alues defined b y ( b, s ) 2 . The linear top ological space o f generalized functions is denoted G ′ ; the top ology is that of con v ergence of v alues of functionals for any conv erging sequence of test functions ( w eak to pology); G ′ is complete in that top ology . F o r G = D or T the spaces are D ′ and T ′ , 2 As a refere e pointed out this is the more commonly used definition of generalized func- tion; the one ab o ve (used b y S) is a neces sary and sufficient co nd ition a nd th us repr esen ts an equiv alent character ization. 9 corresp ondingly . It is easily established t h at D ⊂ T ; T ′ ⊂ D ′ and that D ′ has a w eake r top ology tha n T ′ , meaning that any sequ ence that con v erges in T ′ con v erges in D ′ , but there are sequences that con v erge in D ′ , but not in T ′ . The space T ′ is also called the space o f temp ered distributions. An y generalized function b in T ′ or D ′ is (w eakly) infinitely differen- tiable: the generalized function b ( k ) is the k − th order generalized deriv at ive defined b y ( b ( k ) , s ) = ( − 1) k ( b, s ( k ) ) . The differen tiation op erator is contin u- ous in these spaces. F or any probability distribution function F on R k the densit y function exists as a generalized function (see e.g., Zinde-W a ls h, 2008) and contin uo usly dep ends on the distribution function, thus the generalized deriv ativ e of F , g e neralized densit y function f , is in T ′ . An y lo cally summable (integrable on any b ounded set) function b ( t ) de- fines a generalized function b in D ′ b y ( b, s ) = Z b ( t ) s ( t ) dt ; (9) an y suc h function that additio na lly satisfies (5) similarly b y (9 ) defines a gen- eralized function b in T ′ . A distinction b et w een functions in the ordinary sense (a p oin tw ise mapping fro m the domain o f definition in to the reals or complex n um b ers) and generalized functions is that generalized functions are not de- fined p oin t wise. G en eralized functions defined via (9) b y ordinary functions b ( t ) are called regular functions; w e can refer t o them as or d inary regula r functions in G ′ . The functions F , g , W y , W xy are ordinary regula r functions in T ′ (and th us in D ′ ) if they satisfy Assumption 1’. If a generalized function b is suc h that a represen tation (9) do es not hold, it is said that b is singular, so an y b ∈ G ′ is either regular or singular. A w ell- kno wn singular generalized function is the δ − function: δ : ( δ , s ) = s (0) . An y generalized function in D ′ or T ′ is a generalized finite order deriv ativ e of a con tin uous function. An ordinary function that defines a generalized function is regular if it integrates to a contin uo us function and singular otherwise. 10 F or example, the ordinary function b ( t ) = | t | − 3 2 defines a singular generalized function; it do es not integrate to a con tin uous function; it do es not satisfy (9), in fact (see G S, v.1, p.51) it defines a generalized function b y ( b, s ) = Z ∞ 0 t − 3 2 { s ( t ) + s ( − t ) − 2 s (0) } dt. (10) No sp ecial treatmen t is needed to consider complex-v alued generalized functions; all the same prop erties hold. F or s ∈ T or D F ourier transform F t ( s ) = R s ( t ) e itζ dt exists and is in T . F or b ∈ T ′ : ( F t ( b ) , s ) = ( b, F t ( s )) , so F t ( b ) ∈ T ′ . F ourier transform defines a con tinu ous and con tinuously in- v ertible linear op erator in T ′ (but not for D ′ ). Th us F ourier transforms of W y , W xy , g , and of the generalized deriv ativ e, f , of F exist in T ′ and their in v erse F ourier transforms coincide with the original functions. Since all the functions are differentiable as generalized functions · γ exists in T ′ . By As- sumption 2 the characteristic function φ is contin uous, as is its deriv ative, · φ ; they are regular ordinary functions in T ′ . Since G ′ do es not ha v e a mu ltiplicative structure, pro ducts and con v olu- tions can b e define d for sp ecific pairs only and generally exist only fo r sp ecial classes. The pro duct b et we en a generalized function in T ′ and a function from C ∞ with all deriv ativ es b ounded b y p olynomial functions exists. This class of multipliers is denoted by O M . Conv olution of F o ur ier transforms o f generalized functions with F ourier transforms o f functions from O M exists and the con volution theorem applies. Pro ducts and conv olutions ma y ex- ist for other sp ecific pairs of generalized functions. When suc h conv olutions and pro ducts of their F ourier tr a ns forms exist as generalized functions t he con v olution theorem similarly applies to such pairs. Con v olution Theorem. If for b 1 , b 2 ∈ T ′ , con v olution b 1 ∗ b 2 ∈ T ′ , pro duct F t ( b 1 ) · F t ( b 2 ) ∈ T ′ , then F t ( b 1 ∗ b 2 ) = F t ( b 1 ) · F t ( b 2 ) . The pro o f o f this theorem uses exactly the same seq uen tial a rgume nt as in Antonise k et al (1973), where it utilized the specific delta- con v ergen t 11 sequence s; the only difference here is that the a r gume n t can b e applied to an y sequen ce in the equiv alence class that defines ev ery giv en generalized function. T o consider the pro duct of a generalized function with a con tin uous func- tion that ma y not b e infinitely differen tiable, the prop ert y that the pro duct do es not dep e nd o n the sequenc e that defines the generalized function has to b e made a requiremen t. W e th us say that ab for b ∈ G ′ and contin uo us a is defined in G ′ if fo r any sequence b n from the equiv a le nce class of b t h ere exists a sequence ( ab ) n in G suc h that fo r an y ψ ∈ G lim Z a ( x ) b n ( x ) ψ ( x ) dx exists and equals lim Z ( ab ) n ( x ) ψ ( x ) dx. (11) Denote b y 0 n a zero-con v ergen t sequence that b elongs to the equiv alence class defining the function that is identically zero in G ′ . Prop osition 1 F or the pr o duct ab b etwe en a c ontinuous function a and b ∈ G ′ to b e defi ne d in G ′ it is ne c essary and sufficie n t that (i) (11) hold for some se quenc e ˜ b n in the cl a ss that defines b and (ii) for any zer o-c onver gent se quenc e, 0 n ( x ) , lim Z a ( x )0 n ( x ) ψ ( x ) dx = 0 . (12) Pro of. An y sequence b n differs from a sp ecific ˜ b n b y a zero-con v ergen t seq uence.  Here we consider functions tha t stem from the mo del assumptions. Ad- ditionally , w e distinguish the following cases. Case 1 . Supp ort of γ is a b ounded set: ¯ ζ < ∞ . Case 2 . The function φ − 1 satisfies (5). Case 3 . The function φ ∈ O M . Lemma 1 Under Assumptions 1-3 (i) the pr o ducts γ φ a nd γ ˙ φ ar e defi n e d in T ′ and in D ′ ; (ii) for ˜ φ − 1 = φ − 1 ( ζ ) I ( | ζ | < ¯ ζ ) the pr o duct ( γ φ ) · ˜ φ − 1 is always define d in D ′ ; 12 (iii) if either c ase 1 app l i e s or b oth c ase s 2 and 3 apply the pr o duct ( γ φ ) · ˜ φ − 1 is define d in T ′ ; (iv) if neither c ase 1 nor c a s e 2 applies the pr o duct may n ot b e d e fine d i n T ′ . Pro of. See App endix . F rom Lemma 1 existence of pro ducts to justify the con v olution theorem and th us equations (3 ,4) follows. The mapping (8) in v olv es solving equations (3,4) for the unkno wn functions and requires multiplic ation b y φ − 1 ; as one can see from Lemma 1 existence of such pro ducts in T ′ is not guara n teed. 2.2 The nonparametric iden tification theorem This section contains t w o results. The first is Theorem 1 that pro v es t he existence of the iden tification mapping M ∗ under Assumptions 1-3 . It differs from the statemen t in (S, Theorem 1) in three w a ys: first, Ass umption 1 (and ev en the more restrictiv e Assumption 1’) of this pap er is more general; second, it do es not rely on decomp osition of generalized functions 3 ; third, it pro vides the condition under whic h the mapping is obtained via o perations in the space T ′ and discusses the con tin uit y of the identific ation mapping. The second result is Prop osition 2 that show s that when con tin uity holds, consisten t (in the top ology of T ′ ) plug- in non-parametric estimation of the regression f un ction is p ossible , e.g. for functions in space L 1 . Theorem 1 F o r functions satisfying mo del assumptions and Assumptions 1-3 the mappin g M ∗ in (7) exists and pr ovides identific ation for g ; if c on di- tions of (iii) of L emma 1 ar e satisfie d the mapping i s define d via op er ations in T ′ ; it c an b e dis c ontinuous under c ondition (iv) of L emma 1. Pro of. See App endix . 3 There is no known decomp osition in the space of genera liz e d functions into g eneralized functions c o rrespo nd ing to or dinary functions and to singular functions, cla im ed in (S); the p oin twise ar gumen t provided in (S, 20 07, Supplementary Ma t eria l) is inco rrect (see Appendix B ). 13 The implication of this Theorem is that the iden t ificatio n result holds under the general assumptions 1-3 . If φ is to o thin-ta iled, ho w ev er, the map- ping whereb y the iden tificatio n is ac hiev ed ma y no t b e con tin uous: this po in t is illustrated b y the example in the proo f of Theorem 1 where high frequency comp onen ts b n are magnified b y m ultiplication with φ − 1 from a thin-tailed distribution; this pr o duces inv erse F ourier T ransforms tha t div erge. Con tin uit y requires that t he mapping M given by the mo d el assumptions b e con tin uous in T ′ . Contin uity in T ′ allo ws for a consisten t (in T ′ ) plug- in estimator; the follow ing Prop osition pro vides sufficie nt conditions. Denote b y → T ′ con v ergence in t opology of T ′ . F ollowing Gel’fand and Vilenkin (19 6 4) w e define a random generalized function as t he random con tin uous functional on the space of test functions. Prop osition 2 (a) Under the c ond it ions of The or em 1 supp ose that W y n , W xy n ar e r andom gener alize d function s (estima tors) that satisfy mo del assumptions and Assump t ions 1-3 to g e t her with some (unknown) functions g n , F n ; c ondition (iii) of L emma 1 is s a t isfie d; φ n ∈ O M ; fo r ε y n = F t ( W y n ) , ε xy n = F t ( W xy n ) as su me that the F ourier tr ansforms satisfy: ε y n ( ζ ) is c on- tinuous and non -zer o a.e . on supp( γ ) and iε xy n ( ζ ) − ˙ ε y n ( ζ ) is c ontinuous and that Pr( ε y n ( ζ ) → T ′ ε y ( ζ )) → 1; (13) Pr( ε xy n ( ζ ) → T ′ ε xy ( ζ )) → 1 , then it is p oss ible to find a s e quenc e g n ( x ) such that Pr( g n ( x ) → T ′ g ( x )) → 1 . (b) Supp ose that the function g ( x ) ∈ L 1 . Then there exists a sequence of step function estimators, g n , suc h that Pr( g n ( x ) → T ′ g ( x )) → 1 . Pro of. See App endix . Con v ergence of the estimators is in the w eak top ology of space T ′ of 14 generalized f unctions, not in L 1 . If g n ( x ) → T ′ g ( x ) and g ( x ) is a con tin uous function then there is p oin tw ise con vergenc e and unifor m con v ergence on b ounded sets. 3 Semiparametric sp ecification and iden tifi- cation Semiparametric mo dels with measuremen t erro r w ere examined for p olyno- mial regression functions b y Hausman et a l (1991), for regr ession function in the L 1 space b y W ang and Hsiao (1 995, 2009). (S) significan tly widened the classes of semiparametric mo dels with errors-in- v ariables where iden tification can b e achie v ed via momen t conditions b y utilizing generalized functions, but did no t explicitly characterize the class of functions which she conside red: v erifying momen t conditions of (S, Assumption 6) is needed. In con trast, the class of parametric functions is characterize d directly in Assumptions 5 and 6 of this pap er; the assumptions giv e sufficien t conditions for iden tification via momen ts. The results in (S) rely on existence of a moment generating f un c- tion for the measuremen t error; this restriction in not imp osed here. In this pap er as w ell as in (S) some moment conditions inv olv e limits for sequence s of w eighting functions; the limits are explicitly given here. Assumption 4. T he function g ( x ∗ ) is in a parametric class of lo cally in tegrable f un ctions g ( x ∗ , θ ) where θ ∈ Θ; Θ is an op en set in R m ; for some θ ∗ ∈ Θ mo del assumptions and (1) hold. Denote all the F ourier transforms of the pa r ame tric functions in the mo del assumptions as γ ( θ ); ε y ( θ ); ε xy ( θ ) . The following assumption restricts the generalized function γ ( θ ) to hav e no more than a finite n um b er of sp ecial p oin t s: ∆ p oin ts of singularit y and J of ”jump” discon t inuit y in some region | ζ | < ¯ ζ < ∞ . Notation [ x ] is fo r integer part of x ; δ ( ζ − a ) denotes a shilted δ − f u n ction : ( δ ( ζ − a ) , ψ ) = ψ ( a ) for ψ ∈ G. Assumption 5. The F ourier transform, γ ( θ ) , of the real function g ( x ∗ , θ ) 15 in the region | ζ | < ¯ ζ < ∞ that b elongs to its supp ort (and ma y coincide with it) can b e represen ted as γ ( θ ) = γ o ( θ ) + γ s ( θ ) , (14) where (i) if ∆ = 0 , γ s ( θ ) ≡ 0 ; if ∆ ≥ 1 γ s ( θ ) = 2 π L X l =0 γ s l ( θ ) , where L =  ∆ 2  and (15) γ s l ( θ ) = ¯ k l X k =0  γ k ( s l , θ ) δ ( k ) ( ζ − s l ) + γ k ( s l , θ ) δ ( k ) ( ζ + s l )  ; for l = 0 , 1 , ...L ; (ii) γ o ( θ ) ≡ γ o ( ζ , θ ) is defined b y a lo cally integrable function of ζ contin- uous except p ossibly in a finite n um b er of p oin ts and suc h t ha t its generalized deriv ativ e, · γ o ( θ ) , is o f the form · γ o ( θ ) = · γ oo ( θ ) + · γ os ( θ ) , where if J = 0 , then · γ os ( θ ) = 0 , a n d if J > 0 , then fo r p oin ts b j , j = 1 , ...  J 2  , · γ os ( b j , θ ) = γ os 0 (0 , θ ) δ ( ζ ) I ( J is o dd) + Σ [ J 2 ] j =1  γ osj ( b j , θ ) δ ( ζ − b j ) + γ osj ( b j , θ ) δ ( ζ + b j )  , · γ oo ( θ ) ≡ · γ oo ( ζ , θ ) is an ordinary lo cally in tegrable function con tin uous exce pt p ossibly in a finite n um b er of p oin ts; (iii) γ o ( ζ , θ ) 6 = 0 except p ossibly for a finite n um b er of p oin ts in ( − ¯ ζ , ¯ ζ ); (iv) A t an y non- zero singularity p oin t : s l 6 = 0 , γ o ( ζ , θ ) is contin uous and non-zero. Under Assumptions 1 and 5 g could b e in L 1 , or a sum of a function from L 1 and a p olynomial (singularit y p oin t ζ 0 = 0) and a ls o p ossibly a p erio dic 16 function, e.g. sin ( · ) or cos ( · ) with singularities at some p oin ts ± s, s 6 = 0. Here t h e parameters, γ · ( · , θ ) , are allow ed to take complex v a lue s, otherwise one w ould need to b e more specific ab out the f unctions with singular F o urier transforms; since the functions are assumed kno wn it is easy in eac h sp ecific case to separate out the imaginary parts as in the case of p olynomials. Assumption 5 p ermits to write momen t conditions; ho w ev er, to get a suf- ficien t condition for iden tification of all para m eters additionally the following Assumption 6 is made. If ∆ > 0 define the matrices Γ y ( s l , θ ) and Γ xy ( s l , θ ) for eac h s l ≥ 0 (similarly to (S) for the case s l = 0 ) b y t heir elemen ts: Γ y , i +1 ,k +1 ( s l , θ ) = k + i i ! γ k + i ( s l , θ ) I ( k + i ≤ ¯ k l ); Γ xy ,i +1 ,k +1 ( s l , θ ) = k + i + 1 i + 1 ! γ k + i ( s l , θ ) I ( k + i ≤ ¯ k l ) , i, k = 0 , 1 , ... ¯ k l . Denote by { A } 11 the first matrix elemen t of a mat rix A. Assumption 6. The function γ satisfies Assumption 5. Additionally all γ o ( ζ , θ ) , · γ oo ( ζ , θ ) , γ s l ( θ ) a re contin uo us ly differen tiable with resp ec t to the parameter, θ , in some neighborho o d of θ ∗ . The m × 1 parameter v ector can b e partitio ned as θ T = [ θ T I ; θ T I I ] . F or any comp onen t, θ i , of m I × 1 v ector θ I (where m ≥ m I ≥ 0) either γ o ( ζ , θ ∗ ) ∂ ∂ θ i · γ oo ( − ζ , θ ) | θ ∗ + · γ oo ( ζ , θ ∗ ) ∂ ∂ θ i γ o ( − ζ , θ ) | θ ∗ 6 = 0 (16) a.e., or if (16) do es not hold for some i ∗ , then ∂ ∂ θ i ∗ γ o ( − ζ , θ ) | θ ∗ 6 = 0 . If m I I > 0 the matrix that stac ks for all s l , l ≥ 0 matrices ∂ ∂ θ T I I [Γ y ( s l , θ )] − 1 | θ ∗ Γ y ( s l , θ ∗ ) + ∂ ∂ θ T I I [Γ xy ( s l , θ )] − 1 | θ ∗ Γ xy ( s l , θ ∗ ) ∂ ∂ θ T I I  [Γ y ( s l , θ )] − 1 | θ ∗ Γ y ( s l , θ ∗ )  11 | θ ∗ ! 17 is of rank m I I . By c heck ing w e can see that all the examples provid ed in (S) satisfy assumptions 5 and 6 here and t h us sufficie n t conditio ns for iden tification hold. If the same parameters en t er in to b oth the ordinary and singular parts (S, assumption 6) ma y b e viola t e d, eve n though iden t ifi cation is po s sible and the results of this pa p er hold. Additional assumptions 7 and 8 b elo w are needed. 4 Assumption 7. The densit y function p ( z ) exists and is p ositiv e. Assumption 8. The c haracteristic function of measuremen t erro r, φ ( ζ ) , is suc h that (i) φ ( ζ ) 6 = 0 for | ζ | < ¯ ζ where it is con tin uously differen tiable; (ii) it is ¯ k l times contin uo u sly differen tiable at ev ery s l . Theorem 2 b e low establishes t ha t moment conditions for the parame- ters θ of γ ( ζ , θ ) hold and Theorem 3 that the a ssumptions are sufficien t for iden tification. The nota t ion Re( x ) refers to the real part of a complex x. Theorem 2. Under mo de l assumption and Assumptions 1’, 4, 7, 8 (i) if Assumption 5 (i,ii) holds ther e exist r e al functions r y ( z , θ ) , r xy ( z , θ ) such that the moment E  Y r xy ( z , θ ) + X Y r y ( z , θ ) p ( z )  (17) exists for θ in some neig h b orho o d of θ ∗ and e quals zer o for θ = θ ∗ ; (ii) if 5(i-iii) holds ther e ar e functions r y 1 n ( z , θ ) such that lim n →∞ E  Y r y 1 n ( z , θ ) p ( z ) − 1  (18) exists for θ in some neig h b orho o d of θ ∗ and e quals zer o for θ = θ ∗ ; (iii) If ∆ > 0 and 5( i -ii) hold then for e ach s l ≥ 0 ther e exist ve ctor functions r y s l ( z , θ ) , r y s l,n ( z , θ ) , r xy sl ( z , θ ) , r xy sl,n ( z , θ ) , and a diagonal invert- 4 Assumptions 7 and 8(ii) are also implicit in the pro ofs in (S). 18 ible matrix M l such that lim n →∞ Re[Γ − 1 y ( s l , θ ) M − 1 l E  Y r y s ,l,n ( z , θ ) p ( z )  (19) +Γ − 1 xy ( s l , θ ) M − 1 l E  X Y r xy s,l,n ( z , θ ) p ( z )  ] exists for θ in some neig h b orho o d of θ ∗ and e quals zer o for θ = θ ∗ ; (iv) If ∆ > 0 and 5(i-iv) hold then for e ach s l ≥ 0 ther e exist f unc tion s r y s l, 1 ,n ( z , θ ) , r y s lo, 1 ,n ( z , θ ) such that for s 0 = 0 lim n →∞ E  Y r y s 0 , 1 ,n ( z , θ ) p ( z ) − 1  (20) and lim n →∞ Re E  Y ( r y s l, 1 ,n ( z , θ ) − r y s lo, 1 ,n ( z , θ )) p ( z )  (21) exist for θ in s o me neigh b orho o d of θ ∗ and e qual zer o for θ = θ ∗ . Pro of. See App endix . The functions r · ( z , θ ) and matrices M l are pro vided there. Some of the momen t conditions can b e redundan t. Differen t sets of w eigh ting functions could b e appropriate; similarly to reasoning in ( S) the w eigh ting functions are designed in a w ay t h at isolates different compo nents of the γ function: the ones in (i) are for the ordinary function comp onen t and are supplemen ted b y momen ts in (ii) for the case of a scale m ultiple for the o r dina r y component, the ones in (iii) are for the co efficien t s of the singular part with (iv) for the p ossible scale factor at eac h singularity . If only (17) a pplie s then the we ighting functions prop osed in (S) can b e used, but for the other comp onen ts the w eigh ts prop osed here solv e the pro ble m without additional requiremen ts that momen t generating function for errors exist and av oid the problematic f unction µ in (S, Definition 2): µ (0) and an y deriv ativ es of µ at 0 are zero (see App endix B). Define b y E Q ( θ ) the v ector with comp onen ts pro vided b y the stac k ed 19 expressions (whic hev er are defined) fr o m ( 1 7, 18, 19, 21). Theorem 3. Under the c onditions o f The or em 2 and Assumption 6 the functions r · ( z , θ ) c an b e sele cte d in such a way that the matrix ∂ ∂ θ E Q ( θ ∗ ) exists and has r ank m . Pro of. See App endix . Theorem 3 provides sufficien t conditions under whic h the equations E Q ( θ ) = 0 fully iden tify the parameter v ector θ ∗ . 4 App endix A 4.1 Pro ofs Pro of of Lemma 1. (i) By mo del assumption and Assumption 1 the con v olutions in (1) pro- vide elemen ts in T ′ ; this implies that an y of the sequen tial definitions of con v olution in Kamiski (1982) hold, therefore by his theorem 9, the ”ex- c hange fo rm ula” implies that t he pro ducts for some sequence s in the equiv- alence classes defining the F t ′ s exist. By Prop osition 1 it follows that since φ, · φ ∈ T ′ , (12) holds for con tin uous functions φ and · φ then γ φ ∈ T ′ , · γ φ ∈ T ′ and additionally (b y applying the pro duct rule to (3,4)) γ ˙ φ ∈ T ′ . Since T ′ ⊂ D ′ , the pro ducts are defined in D ′ as we ll. (ii) No w consider a sequence ( γ φ ) n defined as f o llo ws: select some se- quence ˜ γ n for γ from D ; then eac h ˜ γ n has finite supp ort; for a sequence of n um b ers ε n → 0 select ˜ φ n in D suc h that    ˜ φ n − φ    < ε n sup | ˜ γ n φ − 1 | on compact supp ort of γ n . Then for the sequence ( γ φ ) n = ˜ γ n ˜ φ n and a n y ψ ∈ D Z ˜ γ n ˜ φ n φ − 1 ψ = Z ˜ γ n ψ + Z ˜ γ n ( ˜ φ n − φ ) φ − 1 ψ → Z γ ψ . No w w e c hec k that (12) holds for a = φ − 1 . In D supp ort of any ψ is b ounded, on that compact set φ − 1 is b ounded thus (12) will hold and the pro duct is 20 defined in D ′ . (iii) F or Case 1 the pro duct with φ − 1 ( ζ ) I ( | ζ | < ¯ ζ ) is similarly to (ii) defined in T ′ since it is sufficien t to consider ψ ∈ T with b ounded supp ort (con taining supp ort of γ ) . If cases 2 and 3 hold it is straightforw ard t o verify that the function φ − 1 is in O M , th us the pro d uct is defined (contin uously) in T ′ . (iv) W e construct a coun terexample. The function φ ( x ) = e − x 2 do es not b elong to either case 1 or case 2 . The pro duct of function b ( x ) ≡ 0 and φ ( x ) − 1 do es not exist in T ′ . Consider b n ( x ) =      e − n if n − 1 n < x < n + 1 n ; 0 ≤ b n ( x ) ≤ e − n if n − 2 n < x < n + 2 n ; 0 otherwise. (22) This b n ( x ) conv erges to b ( x ) ≡ 0 in T ′ . Indeed for an y ψ ∈ T Z b n ( x ) ψ ( x ) dx = Z n +2 /n n − 2 /n b n ( x ) ψ ( x ) dx → 0 . But the sequence b n ( x ) ˜ φ ( x ) − 1 do es no t conv erge in the space T ′ of tem- p ered distributions. Indeed if it did t h en R b n ˜ φψ w ould con v erge for a ny ψ ∈ T . But for ψ ∈ T suc h that ψ ( x ) = exp( − | x | ) for, e.g. | x | > 1 Z n +2 /n n − 2 /n b n ( x ) e x 2 ψ ( x ) dx ≥ e − n Z n +1 /n n − 1 /n e x 2 − x dx ≥ 2 n e − 2 n +( n − 1 /n ) 2 . This diverges .  Pro of of Theorem 1. The pro of mak es use of different spaces of generalized functions and ex- ploits relatio ns b e tw een them. It pro ceeds in tw o parts. First in part one, it is sho wn that fro m equations (3 ,4) the contin uous 21 function κ = ˙ φφ − 1 can b e uniquely p oin t wise determined on the in terv al [ − ¯ ζ , ¯ ζ ] (whic h is in the supp ort of γ and consequen tly o f ε 1 ); this requires additionally considering the g e neralized functions spaces , D ′ and also D 0 ( U ) ′ whic h is defined on the space of test functions that are con tin uous with supp ort con ta ine d in U. The function φ is uniquely defined on the in terv al ( − ¯ ζ , ¯ ζ ) a s the solution of the corresp onding differen tial equation that sat- isfies the condition φ (0) = 1 . D efin e ˜ φ = φI ( | ζ | < ¯ ζ ) ; define ˜ φ − 1 to equal φ − 1 I ( | ζ | < ¯ ζ ). Of course, when ¯ ζ = ∞ , ˜ φ = φ and ˜ φ − 1 = φ − 1 on R. Next in part t w o, γ is defined as ε y ˜ φ − 1 . By Lemma 1 this pro duct can alw a ys b e uniquely defined as a generalized function in D ′ ; b y construction γ defines a generalized function in T ′ ⊂ D ′ ; this pro vides the required mapping M ∗ b y applying in v erse F ourier T ransform to γ . If condition (iii) of Lemma 1 applies the pro duct ε y ˜ φ − 1 is defined in T ′ ; in this case all the mappings that define the mapping M ∗ are defined in T ′ . The pro of concludes with an example that demonstrates that the mapping can b e discon tin uous if (iv) of Lemma 1 applies. P art one. Consider the space of g e neralized functions D ′ . By Assumption 2 φ is non-zero and contin uously differen tiable, then by differen tiating (3) , substituting (4) and making use of Lemma 1 to m ultiply by φ − 1 in D ′ w e get that the generalized function ε y φ − 1 ˙ φ − ( ˙ ε y − iε xy ) equals zero in the sense of generalized function in D ′ . Denoting κ = ˙ φφ − 1 w e can c haracterize κ as a con tin uous at ev ery p oin t (by Assumption 2) f unction in D ′ that satisfies the equation ε y κ − ( ˙ ε y − iε xy ) = 0 . (23) If (23) holds in D ′ , it holds also for any test functions with supp ort limited to U : ψ ∈ D ( U ) ⊂ D , and thus holds in any D ( U ) ′ . 22 W e sho w tha t the function κ is uniquely determined in the class of con- tin uous functions on on [ − ¯ ζ , ¯ ζ ] by (23) holding in D ( U ) ′ for an y interv al U ⊂ ( − ¯ ζ , ¯ ζ ) . Pro of is b y contradiction. Supp ose t h at there are tw o distinct con tin uous functions on [ − ¯ ζ , ¯ ζ ] , κ 1 6 = κ 2 that satisfy ( 23), then κ 1 ( ¯ x ) 6 = κ 2 ( ¯ x ) for some ¯ x ∈ ( − ¯ ζ , ¯ ζ ); by con tin uit y κ 1 6 = κ 2 ev erywhere for some in terv al U ⊂ ( − ¯ ζ , ¯ ζ ) . Consider now D ( U ) ′ ; we can write ( ε y ( κ 1 − κ 2 ) , ψ ) = 0 for an y ψ ∈ D ( U ) . A generalized function that is zero for all ψ ∈ D ( U ) coincides with the ordinary zero function on U and is also zero for all ψ ∈ D 0 ( U ), where D 0 denotes the space of contin uous test functions. F or the space of test function D 0 ( U ) m ultiplication b y con tin uous ( κ 1 − κ 2 ) 6 = 0 is an isomorphism. Then from (2 3) w e can write 0 = ([ ε y ( κ 1 − κ 2 )] , ψ ) = ( ε y , ( κ 1 − κ 2 ) ψ ) implying that ε y is defined and is a zero generalized function in D 0 ( U ) ′ . If that w ere so ε y w ould b e a zero generalized function in D ( U ) ′ since D ( U ) ⊂ D 0 ( U ); this con tradicts Assumption 2. This concludes the first part of the pro of since from κ the function φ ( ζ ) = exp Z ζ 0 κ ( ξ ) dξ that solv es o n [ − ¯ ζ , ¯ ζ ] ˙ φφ − 1 = κ ; φ (0 ) = 1 is uniquely determined on [ − ¯ ζ , ¯ ζ ] and ˜ φ (a nd ˜ φ − 1 ) defined ab o ve are uniquely determined. P art t w o. Consider tw o cases. 23 Case 1.Part (iii) o f Lemma 1 applies. Multiplying ε y (= γ ˜ φ ) by ˜ φ − 1 pro- vides a temp ered distribution b y Lemma 1; it is equal to γ . The inv erse F ourier T ransform provide s g . The theorem holds and mor eov er, all the op- erations by whic h the solution w as o btained w ere defined in T ′ . Case 2. The condition (iii) of Lemma 1 may no t hold, so multiplic ation b y ˜ φ − 1 = φ − 1 ma y not lead to a temp ered distribution. Consider now D ′ ; T ′ ⊂ D ′ . Multiplication b y φ − 1 is a contin uous op eration in D ′ ; define the same differen tial equations, solv e to obta in φ and get via m ultiplication ( γ φ ) · φ − 1 in D ′ the function γ ∈ D ′ . Since γ is the F ourier transform of g (a temp ered distribution) it also belongs to T ′ , and it is p ossible to reco ve r g b y a n in v erse F ourier T ransfor m. In the follo wing example the mapping M ∗ in (7) is not contin uo u s. Define β n = F t − 1 ( b n ) where b n is defined b y (22); b n ∈ T , th us β n ∈ T . In w as sho wn in pro of of Lemma 1 t ha t b n ( x ) conv erges to b ( x ) ≡ 0 in T ′ . Supp ose that the mo del mapping M in (6) is defined for functions in L 1 and is con tinu ous in L 1 (and thus in T ′ ) . Supp os e that W y n = W y + β n ; from b n → 0 in T ′ and the con tinuit y of the F ourier T ransfor m mapping in T ′ , it follo ws that β n → 0 and th us W y n → W y in T ′ . Then ε y n = ε y + b n . Supp ose that φ is prop ortionate to e − x 2 . Then b y the pro of in part 1 each γ n = ε y n φ − 1 ∈ D ′ , but as a Ft of a function in T ′ (ev en in L 1 here) is defined in T ′ ; the in v erse F o urier transform, ˜ g n = F t − 1 ( γ n ) , exists. How ev er, ˜ g n do es not conv erge to g in T ′ . Indeed, if it did so con v erge, then that w ould imply con v ergence γ n → γ in T ′ , but b n ( x ) e x 2 do es not con v erge in the space T ′ of temp ere d distributions a s w as sho wn in the pro of of Lemma 1.  Pro of of Prop osition 2. (a) W e establish that the mapping from ( W y n , W xy n ) to g n is con tin uous. Consider ζ ∈ supp( γ ) . Similarly t o pro of in Theorem 1 a pplied to eve ry n a con tin uous function κ n ( ζ ) , that satisfies the equation κ n ( ζ ) ε y n ( ζ ) + ( iε xy n ( ζ ) − ˙ ε y n ( ζ )) = 0 (24) 24 in generalized functions, exists (defined as ˙ φ n φ − 1 n ) a nd is unique. Moreov er, from Lemma 1 it follo ws that the pro duct with κ n = ˙ φ n φ − 1 n ∈ O M alw a ys exists. Since all functions in (24) are contin uous it represen ts an equalit y of con tin uous functions and since ε y n is non-zero a.e. w e ha v e κ n ( ζ ) = ( iε xy n ( ζ ) − ˙ ε y n ( ζ ))( ε y n ( ζ )) − 1 . The g e neralized functions κ n ε y n − κ ε y = i ( ε xy n − ε xy ) + ( ˙ ε y n − ˙ ε y ) and κ n ( ε y − ε y n ) con v erge to zero as generalized functions in T ′ ; as a result, so do es ( κ n − κ ) ε y n , but since this is a con tin uous function this implies p oin tw ise con v ergence. Supp ose that on some b ounded in terv al κ n − κ is separated a w a y fro m zero for some subsequenc e { n i } , this implies then that on that set ε y n i con v erges to zero p oin tw ise, thus the limit in T ′ (= ε y ) is zero on this interv al whic h b elongs to supp ort of γ , and thus of ε y . This contradiction establishes that κ n → κ p oin t wise and uniformly o n an y b ounded set. F rom the differen tial equation φ − 1 n ˙ φ n = κ n with the condition φ n (0) = 1 the function φ n is uniquely determined; and φ n → φ where φ − 1 ˙ φ = κ , φ (0) = 1 . Then also since φ is non- zero, φ − 1 n → φ − 1 p oin twise ; φ − 1 n satisfies (5) so that ε 1 n φ − 1 n can b e defined as a temp ered distribution. F inally consider ε y n φ − 1 n − ε y φ − 1 = ε y n ( φ − 1 n − φ − 1 ) + ( ε y n − ε y ) φ − 1 . The con tin uous function ε y n ( φ − 1 n − φ − 1 ) → 0; t h e g e neralized function ( ε y n − ε y ) φ − 1 → 0 in T ′ (as a temp ered distribution) ε y n φ − 1 n con v erges to γ in T ′ , and it s in v erse F ourier T ransform conv erges to g as a temp ere d distribution (b y contin uity of inv erse F ourier T ransform in T ′ ). F rom contin uity of the mapping the result follows. (b) F or an y g ∈ L 1 there exists a sequen ce of step-functions g n ∈ L 1 suc h t h at k g n − g k L 1 → 0 (implying g n → T ′ g ); for F there is a sequence of 25 step-functions F n suc h that sup | F n − F | → 0 ( im plying F n → T ′ F ) . Sp ecifi cally , g n ( x ) = N X k =1 a k I ( b k ≤ x < b k +1 ) for b 1 < ... < b N ; F n ( x ) = N X j =1 c j I ( d j ≤ x ) with c j > 0; Σ c j = 1; d 1 < ... < d N , where all the parameters dep end o n n. The generalized deriv ativ e of F n is f n ( x ) = Σ c j δ ( x − d j ) where δ ( x − d j ) is a shifted δ − function: R δ ( x − d j ) ψ ( x ) dx = ψ ( d i ) for ψ ∈ T . Then φ n ( ζ ) = Σ c j e iζ d j . The function φ n is not inte grable (otherwise f w ould b e contin uous), t hus φ − 1 n satisfies (5). All the para meters dep end on n. Then W y n ( v ) = N 2 X m =1 α m I ( | v − t m | < τ m ); W xy n ( v ) = N 2 X m =1 α m ( v − ε m ) I ( | v − t m | < τ m ) , where m corresp onds to a pair ( k , j ) and α m = a k c j ; t m = d j + b k + b k +1 2 , ε m = d j , τ m = b k +1 − b k 2 . This represen ts W y n as a step-function and W xy n as a piece- wise linear function . The conditional mean function W y can b e consisten tly estimated in L 1 b y step functions implying existe nce of a sequence W y n ( v ) suc h that Pr( W y n → T ′ W y ) → 1 , similarly , for some piece-wise linear W xy n ( v ) Pr( W xy n → T ′ W xy ) → 1 implying (13), moreov er, we can write (using know n 26 F ourier tra ns forms) ε y n ( ζ ) = N X k =1 2 τ k α k χ k ( ζ ) sinc ( τ k ζ π ); ε xy n ( ζ ) = − i N X k =1 2 τ k d dζ  α k χ k ( ζ ) sinc ( τ k ζ π )  − N X k =1 2 τ k α k ε k χ k ( ζ ) sinc ( τ k ζ π ) , where the si n c ( x ) function is defined as sin π x π x and χ k ( ζ ) = e it k ζ . The condi- tions ab out con tinuit y and ε y n ( ζ ) non-zero a.e., required in (a) a r e satisfied; (13) follo ws f rom the con tin uity of the F ourier t r ans form op erator in T ′ .  Prior t o pro of of Theorem 2 we mak e t w o preliminary o bserv a tions. Firstly , under Assum ption 5 a nd 8 ( t ha t j ustifies pro ducts of γ s and · γ s with φ ) and b y Lemma 1 equations (3,4) in T ′ lead to ( i 2 = − 1): ε y = ε y o + ε y s (25) with ε y o ( ζ ) = γ o ( ζ , θ ∗ ) φ ( ζ ); ε y s = γ s ( θ ∗ ) φ ( ζ ); i ε xy = i ε xy o + i ε xy os + i ε xy s (26) with i ε xy o ( ζ ) = · γ oo ( ζ , θ ∗ ) φ ( ζ ) , i ε xy os ( ζ ) = · γ os ( ζ , θ ∗ ) φ ( ζ ) , and i ε xy s = · γ s ( θ ∗ ) φ, where · γ s ( θ ) is the generalized deriv ativ e of γ s ( θ ) . Second, to construct weigh ting functions some w ell-kno wn functions are used. Denote b y T R ⊂ T t h e space of real test functions t h at are Ft of real- v alued functions from T ; t h ey satisfy ψ ( − ζ ) = ψ ( ζ ) . A smo oth cut-off (o r ”sm udge”) f u nction is defined (e.g. in GS or L) as f cut ( ζ ) = exp( − 1 1 − ζ 2 ) I ( | ζ | < 1); 27 ”bump function” is f bump ( ζ ) = f cut ( ζ ) R 1 − 1 f cut ( ζ ) dζ . Consider sets V , U defined a s V = ∪ ([ a i , b i ] ∪ [ − b i , − a i ]) ⊂ ∪ ( a i − ε, b i + ε ) ∪ ( − b i − ε, − a i + ε ) = U, (27) where a i 6 = b i and the in terv als and ε are suc h tha t the only tw o in terv a ls in U that could in tersect w ould corr esp ond to some i with b i = − a i ; define the function f V ( ζ ) = I ( | ζ | ∈ V ) ∗ f bump ( 2 ζ ε ) 2 ε . This function has the prop ert y that it equals 1 on V , 0 outside of U and tak es v alues b et w een 0 and 1. F or an y ξ ∈ R, p ≥ 0 , ε > 0 consider a closed set V ξ = [ ξ − α, ξ + α ] ∪ [ − ξ − α, − ξ + α ] and the function f V ξ ( ζ ) , defined ab o v e. Define f ξ ,p ( ζ ) = ( ζ − ξ ) p f U ξ ,V ξ ( ζ ) . This function has the prop ert y that d l f ξ, p,ε dζ l ( ξ ) = ( − 1) l d l f ξ ,p,ε dζ l ( − ξ ) = ( p ! if l = p ; 0 otherwise. All the functions, f bump , f V , f ξ, p,ε are in T R . Pro of of Theorem 2. (i) L et e b e small enough that closed e − neigh b orho ods o f all t he points of singularit y and discon tinuit y of γ o and · γ o do not interse ct in ( − ¯ ζ , ¯ ζ ). Define the union of op en in terv als that is t he complimen t t o this set in ( − ¯ ζ , ¯ ζ ) b y U. Construct for a small enough ε a corresp onding union of closed interv als, V ⊂ U that can b e defined b y (27 ) . Define µ ( ζ ) = f V ( ζ ) . Then d p µ dζ p ( s l ) = 0 for all s l , p ; in tegrals R · γ o ( ζ , θ ) µ ( ζ ) dζ and R γ o ( ζ , θ ) µ ( ζ ) dζ are defined for any θ . The inv erse Ft’s r y ( z , θ ) = F t − 1 ( γ o ( − ζ , θ ) µ ( − ζ )) and r xy ( z , θ ) = F t − 1 ( i · γ o ( − ζ , θ ) µ ( − ζ ) exist. Since ε y o ( ζ ) = γ o ( ζ , θ ∗ ) φ ( ζ ), ε xy o ( ζ ) = − i · γ o ( ζ , θ ∗ ) φ ( ζ ) , · γ o ( ζ , θ ) and ε xy o ( ζ ) are ordinary lo cally inte grable 28 functions in T ′ and ε y o ( ζ ) and γ o ( ζ , θ ) are con tinuous and satisfy (5), the pro ducts ε y o ( ζ ) · γ o ( − ζ , θ ) and ε xy o ( ζ ) γ o ( − ζ , θ ) are w ell defined in T ′ . Th us the integral (where µ ∈ T R ) Z h ε y o ( ζ ) i · γ o ( − ζ , θ ) µ ( − ζ ) + ε xy o ( ζ ) γ o ( − ζ , θ ) µ ( − ζ ) i dζ exists. Since ε y o ( ζ ) = γ o ( ζ , θ ∗ ) φ ( ζ ), ε xy o ( ζ ) = − i · γ o ( ζ , θ ∗ ) φ ( ζ ) the v alue of the in tegral is zero for θ = θ ∗ . Moreo v er, b ecause the functions µ are zero together with all the deriv ativ es at singularity p oin ts, ε y o can b e replaced by ε y pro viding: Z h ε y o ( ζ ) i · γ o ( − ζ , θ ) µ ( − ζ ) + ε xy o ( ζ ) γ o ( − ζ , θ ) µ ( − ζ ) i dζ = ( i ε y · γ o ( − ζ , θ ) , µ ( − ζ )) + ( ε xy γ o ( − ζ , θ ) , µ ( − ζ )) By applying P arsev al iden tit y to generalized f un ctions this leads to ( W y , r xy ( θ )) + ( W xy , r y ( θ )) = Z [ W y ( z ) r xy ( z , θ ) + W xy ( z ) r y ( z , θ )] dz where the f un ctionals are expresse d via in tegr a ls f o r ordinary lo cally inte - grable functions. Multiplying and dividing by the non-zero function p ( z ) do e s not c hange the integral. Then b y law of iterated exp ectations Z 1 p ( z ) E | z ( Y r xy ( z , θ ) + X Y r y ( z , θ )) p ( z ) dz = E  Y r xy ( z , θ ) + X Y r y ( z , θ ) p ( z )  . This concludes the pro of of (i). (ii) By Assumption 5(iii) there exists a sequence ξ n → 0 suc h that γ o ( ζ , θ ) 6 = 0 for ζ : | ζ − ξ n | < ε n < | ξ n | and is con tinuous in those inte r- v als; without loss of generality a ssume that ζ ∈ U defined in (i) . Consider 29 the function µ n ( ζ ) = 1 2 { f bump ( ζ − ξ n ε n ) + f bump ( ζ + ξ n ε n ) } The function µ n ( − ζ ) γ o ( − ζ ,θ ) is a con tin uous function with b ounded supp ort. Set r 1 yn ( z , θ ) = F t − 1 ( µ n ( − ζ ) γ o ( − ζ ,θ ) ) . Then f or an y n w e get Z ε y o ( ζ ) γ o ( − ζ , θ ) − 1 µ n ( − ζ ) d ζ = ( ε y · γ o ( − ζ , θ ) − 1 , µ n ( − ζ )) = Z E ( Y | z ) F t − 1 ( γ o ( − ζ , θ ) − 1 µ n ( − ζ )) d z = E  Y r y 1 n ( z , θ ) p ( z )  , where the first equalit y follows from the f a c t that ( ε y s γ o ( − ζ , θ ) − 1 , µ n ( − ζ )) = 0 (since ζ ∈ U ) , the second by P arsev a l iden tity and the third b y multiplying and dividing by p ( z ) > 0 and iterated exp ec tation; the in tegral exists for eac h n . F or θ ∗ w e get ( γ o ( ζ ) = γ o ( − ζ )) E  Y r y 1 n ( z , θ ∗ ) p ( z )  = Z ε y o ( ζ ) γ o ( ζ , θ ∗ ) − 1 µ n ( − ζ ) d ζ = Z φ ( ζ ) µ n ( − ζ ) dζ = 1 2 [ φ ( ξ n ) + φ ( − ξ n )] + O ( ε n ) This conv erges to φ (0) = 1 . (iii) Consider a n y s l ≥ 0. Below all relev ant functions are subscripted b y l . F or ε as defined in (i) define the function µ l,i ( ζ ) = f s l ,i,ε ( ζ ) ∈ T R , then µ ( i ) l,i (0) 6 = 0 , but µ ( k ) l,i (0) = 0 , k = 0 , ...i − 1 , i + 1 , ... ¯ k + 1 and supp ort of µ l,i is give n by I ( | ζ − s l | < ε ) + I ( | ζ + s l | < ε ); denote the deriv ative of µ l,i b y µ ′ l,i . F or a seque nce ε n → 0 consider f V n ( ζ ) for U n = { ζ : | ζ − s l | < ε n } ∪ { ζ : | ζ + s l | < ε n } ; V n = { ζ : | ζ − s l | ≤ ε n 2 } ∪ { ζ : | ζ + s l | < ε n 2 } 30 and define µ l,i,n ( ζ ) = µ li ( ζ ) f U n ,V n ( ζ )) . The functions µ · are in T R . D e note b y r xy s,l,i,n ( z ) the in verse Ft: F t − 1 ( µ l,i,n ( − ζ )) and b y r y s ,l,i,n ( z ) the in verse Ft: F t − 1 ( i µ ′ l,i,n ( − ζ )); they exist in T . The v ector r xy s,l,n ( z ) is defined to ha v e r xy s,l,i,n ( z ) as its i − th comp onen t; v ector r y s ,l,n ( z ) is defined similarly . Define b y M l the diagonal matrix with no n- ze ro dia g onal en tries { M l } ii = µ ( i ) l,i,n (0) ≡ µ ( i ) l,i (0), i = 0 , ... ¯ k . Consider no w the v ector ( ε y , µ ′ l,n ) with comp onen ts ( ε y , µ ′ l,i,n ( − ζ )) and ( ε xy , µ l,n ) with ( ε xy , µ l,i,n ( − ζ )). Since the mat r ices Γ y ( s l , θ ), Γ xy ( s l , θ ) and M l are in v ertible the expression Γ y ( s l , θ ) − 1 M − 1 l ( ε y , µ ′ l,n ) + Γ xy ( s l , θ ) − 1 M − 1 l ( ε xy , µ l,n ) (28) is finite for ev ery n . By P arsev al identit y ( ε y ( ζ ) , µ ′ l,i,n ( − ζ )) = ( W y ( z ) , r y , l,i,n ( z )) = Z W y ( z ) r y , l,i,n ( z ) dz , the last equality follows since W y is lo cally in tegrable. Th us b y argumen ts similar to those in (i) and (ii) this in tegral is E Y r y ,l,i,n ( z ) p ( z ) so that ( ε y , µ ′ l,n ) = E Y r y ,l,n ( z ) p ( z ) and ana logously ( ε xy , µ l,n ) = E X Y r xy ,l,n ( z ) p ( z ) . W e need t o establish that limits as n → ∞ exist. First, note tha t     Z ε y o ( ζ ) µ ′ l,i,n ( − ζ ) d ζ     =     Z ε y o ( ζ ) f ζ l ,i,ε ( − ζ ) f U n ,V n ( − ζ ) dζ     ≤ max U   ε y o ( ζ ) f ζ l ,i,ε ( − ζ )   2 ε n and go es to zero; ( ε y s , µ ′ l,i,n ( − ζ )) = Σ ¯ k k ≥ i γ ( s l , θ ∗ )( − 1) i k + i − 1 i − 1 ! µ ( i ) l,i ( s l ) φ ( k − i + 1) ( s l ) (29) and do es not depend on n, finally , ε y = ε y o + ε y s , so the limit exis ts. By a 31 similar represen t ation for − ( ε xy s , µ l,i,n ( − ζ )) existence of (19) is established. F or θ = θ ∗ using (25 ,26 ) for ε y s and ε xy s in (2 8 ) leads to Γ y ( s l , θ ∗ ) − 1 M − 1 l ( ε y s , µ ′ l,n ) + Γ xy ( s l , θ ∗ ) − 1 M − 1 l ( ε xy s , µ l,n ) = 0 . Note that the same considerations apply to singularit y a t − s l with the dif- ference that the Γ . ( − s l , θ ) matr ices no w are complex conjugate to Γ . ( s l , θ ) . Com bining prov ides the real part in (19). (iv) Consider the first compo ne nt of Γ y ( s l , θ ) − 1 M − 1 l ( ε y s , µ ′ l,n ) with µ ′ l,n , M l defined in (iii); this comp onen t is { Γ y ( s l , θ ) − 1 M − 1 l } 11 ( ε y s , µ ′ l, 1 ,n ) . Note that µ ′ l, 1 ,n = µ l, 0 ,n , recall that µ ∈ T R . W e see that φ ( s l ) equals lim { Γ y ( s l , θ ∗ ) − 1 M − 1 l } 11 R ε y s µ l, 0 ,n dζ . Define r y s l, 1 ,n ( z , θ ) = { Γ y ( s l , θ ) − 1 M − 1 l } 11 F t − 1 ( µ l, 0 ,n ( − ζ )) . Similarly to ab o v e by Parse v a l identit y { Γ y ( s l , θ ) − 1 M − 1 l } 11 ( ε y s , µ l, 0 ,n ) = E ( Y r y sl, 1 ,n ( z ,θ ) p ( z ) ) and φ ( s l ) = lim E ( Y r y sl, 1 ,n ( z ,θ ) p ( z ) ) . F or s 0 = 0 w e ha v e φ (0) = 1. Th us (20) fol- lo ws. Consider now for s l 6 = 0 the function µ l,n ( ζ ) = 1 2 { f bump ( ζ − s l − ξ n ε n ) + f bump ( ζ − s l + ξ n ε n ) } similar to the one in (ii) and define r slo,n ( z ) = F t − 1 ( µ l,n ( − ζ ) γ o ( − ζ ,θ ) ) . F or this function E ( Y r slo,n ( z ,θ ) p ( z ) ) = R ε y ( ζ ) µ l,n ( − ζ ) γ o ( − ζ ,θ ) dζ ex ists and at θ ∗ con v erges to φ ( s l ). Th us (21) fo llo ws.  Pro of of Theorem 3. Let the v ector Q ( z , θ ) denote the ve ctor of functions for whic h exp ecta- tions are ta ken in E ( Q ); partition Q ( z , θ ) in to Q I ( z , θ ) corresp onding t o ex- pressions in (17, 18) and Q I I ( z , θ ) for (19, 21). Then the matrix ∂ ∂ θ T E ( Q ( z , θ )) 32 is a blo c k matr ix ∂ ∂ θ T I E ( Q I ( z , θ )) · · ∂ ∂ θ T I I E ( Q I I ( z , θ )) ! and it is sufficien t to sho w that ∂ ∂ θ T I E ( Q I ( z , θ )) has rank m I and ∂ ∂ θ T I I E ( Q I I ( z , θ )) has rank m I I . F or θ I first note that in terc hange of differentiation with resp ect to the parameter and inte grat io n (taking exp ec ted v alue) for ∂ ∂ θ T I E Q I ( z , θ ) follow s from contin uity in ζ o f all the functions in the in tegrals and their con tin uous differen tiabilit y with resp ect to θ , so that ∂ ∂ θ I E ( Q I ( z , θ )) = E ( ∂ ∂ θ I Q I ( z , θ )) . One can c ho ose m I functions µ defined in pro ofs of The orem 2(i,ii) that are functionally indep enden t and under Assumption 6 the corresp onding m I conditions of t yp e E ( ∂ ∂ θ I Q I ( z , θ )) will pro vide a ra nk m I submatrix. If for t he functions µ in expressions Q I I ( z , θ ) in ( iii,iv ) of Pro of o f Theorem 2 the matrix ∂ ∂ θ I I E ( Q s ( z , θ )) has rank less than m I I consider v arying the functions µ · for all p ossible v alues of non-zero deriv ativ es at the p oin ts s l ; the rank cannot b e deficien t o v er all suc h c hoices without viola tion of Assu mption 6.  App endix B Three main problems with (S) a r e listed b elo w. 1. The issue of decomp osition. (S) claims t h at any generalized function in T ′ can b e decomp osed into a sum of an ordinary function and a singular function; this decomp osition is used in formu la (13) of S., Theorem 1. There is no pro of of existence o f suc h a decomp osition in the literature, as the example of the function in Section 2.1 in (10) shows no unique de- comp osition in to a singular and regular ordinary function exists in T ′ (nor in D ′ ). The at t empted pro of in (S, Supplemen tary material, p.3) is incorr ect. Indeed it states: ”The result directly follo ws from the fact that ev ery generalized function can b e written as the deriv ative o f order k ∈ N of some 33 con tin uous function c ( t ) (Theorem I I I in T emple (1963 ) establishe s this for a class of generalized functions including those considered here as a particular case). [-my c omment: this is c orr e ct] A t ev ery p oin t t where c ( t ) is k times differen tiable in the usual sense , the generalized function can b e written as an ordinary function, while at ev ery p oin t where c ( t ) is not k times differen- tiable, a delta f unction deriv ative is created in the differentiation pro ce ss. [my c omme nt : this is inc orr e ct, s e e (a) b elow] The fact that the tw o pieces are additiv ely separable follo ws from the linear nature of the space of generalized functions. [my c omment: this is inc orr e ctly applie d: se e (b) b elow]” (a). Consider the function b ( ζ ) = | ζ | − 3 2 ; it is a ” weak” (or generalized) second deriv ativ e of the contin uous function c ( ζ ) = − 4 | ζ | 1 2 . The first w eak deriv ativ e, − 2 | ζ | − 1 2 sig n ( ζ ) , is an ordinary function that is summable and so gives a regular f unctiona l and is an ordinary function that is a t the same time a g eneralized function. At p oin t ζ = 0 it is not differen tiable in the ordinary sense; y et no delta-function o r its derivative app e ars . (b) An y generalized function is either regular or singular; the ” ordinary” function b ( ζ ) ab o v e is a singular generalized function (see (10)). Since g e n- eralized f un ctions are generally not defined p oin t wise a p oin t wise argumen t cannot b e helpful. An additional assumption w ould hav e to b e used to establish form ula (13) in Theorem 1 of (S). 2. V alidit y of pro ducts. V alidit y of pro ducts in the spac e of generalized f un ctions needs to b e established to pro vide a correct pro of of the iden t ificatio n result. Neither the pap er nor the suppleme ntary material in (S) provide s a complete correct pro of indicating in whic h space of generalized functions the multiplic ation op erations ar e v alid; in fa c t a s Lemma 1 here sho ws multiplication ma y not b e v alid in T ′ under the Assumptions (despite the claim in (S)). 3. Definition 2 in (S) leads to inappropriate w eigh ts. 34 Def. 2 prop oses the function µ ( ζ ) = ∞ X k =0 1 k ! d k ( ζ k λ ( ζ )) /d ζ k for λ that satisfies S, D e f. 1 ( λ is an analytic function). W e hav e ∞ X k =0 1 k ! d k ( ζ k λ ( ζ )) /d ζ k = ∞ X k =0 1 k ! k X i =0 C k i  d i ζ k /dζ i   d k − i λ ( ζ ) /dζ k − i  = ∞ X k =0 1 k ! k X i =0 C k i k ! ( k − i )! ζ k − i  d k − i λ ( ζ ) /dζ k − i  = ∞ X k =0 k X i =0 C k i 1 i ! ζ i  d i λ ( ζ ) /dζ i  . 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