* University of Amsterdam, The Netherlands. This paper was written

Transcription

1 A TEST FOR MODEL SPECIFICATION IN THE ABSENCE OF ALTERNATIVE HYPOTHESIS ", by Herman J. Bierens * Discussin Paper N , Octber 1981 * University f Amsterdam, The Netherlands. This paper was written while I was spending a three-mnth term at the University f Minnesta, whse ind hspitality is gratefully acnwledged. This visit is supprted by a grant frm the Netherlands Organisatin fr the Advancement f Pure Science (Z.W.O.). Center fr Ecnmic Research Department f Ecnmics University f Minnesta Minneaplis, Minnesta 55455

2 Abstract In recent papers ([6], [4]) several tests have been prpsed fr testing the truth f a nnlinear specificatin f a regr.essin mdel against ne r mre well-specified alternative hypthesis. In this paper we first shw by an example that these tests may be defective if all the hyptheses are false, and then we prpse a new test that des nt need alternative hyptheses in the sense that we nly test whether r nt a given specificatin is true. Crrespndence shuld be addressed t: Herman J. Bierens Stichting vr Ecnmisch Onderze der Universiteit van Amsterdam Jdenbreestraat NH Amsterdam The Netherlands

3 1. INTRODUCTION In recent Ecnmetrica articles Pesaran and Deatn [6] and Davidsn and Mac Kinnn [4] prpse several tests fr testing a (nn) linear specificatin f a regressin mdel against ne r mre alternatives. These tests seem t perfrm well if either the null hypthesis r ne f the alternative hyptheses is true. Hwever, althugh these tests are able t reject all the hyptheses, it is pssible that if all the hyptheses are false the least wrst hypthesis will be accepted with relative high prbability. We shw this by applying the P-test f Davidsn and Mac Kinnn [4] t the fllwing example. Cnsider an independent sample [(xl,l' x 2,1' 1)'., (xl,n' x 2,n' en)} frm the trivariate standard nrmal distributin and let fr j = 1,2,.,n, Suppse we have specified this relatinship as where and In rder t test the truth f this hypthesis by the P - test prpsed in [4 J we need a nnnested alternative hypthesis, "thugh nt ne in which we need have any faith" (quting (4), fr example where is as befre and Clearly bth hyptheses are false, but H is clser t the truth than H l. Fllwing [4], we dente:

4 ft A.,. (f(x 1, ~),..., AT A g.,. (g(x p y), I.. M = I - X(X T X)-l XT, 0 T Y =..., (Y1 ' Y n ), f (X, ~» n g(x n, y» where and A yare the OLS estimates f ~ and y, respectively, and X is the 2 X n matrix with elements x.. (i = 1, 2; j = 1, 2, 1.,J..., n) The test statistic f the P - test is (1.1) where (1.2) ~ T.... (g - r) M (y - f) and " (1.3) A,.,.,. A 2 a = n 1 1\ y - f - Ct p M (g - f) II As is shwn in [4] the as N(O, 1) if H is true.,. test statistic t p H nr H1 is true, but nevertheless we have is asympttically distributed In the case under review, hwever, neither (1.4) t... N(O, 4) in distributin as n... c, p as will be prved in Appendix 1. Thus if we prceed the test at the five percent Significance level the prbability that we accept the false null is 1 2 -'2 u du.,. J e ~ du~0.673

5 - 3 - This example shws that the P - test prpsed by Davidsn and Mac Kinnn is nt apprpriate if there is a pssibility that nne f the hyptheses invlved is true. The same applies t their J - test, because fr the example under review the J - test and the P - test yield identical test statistics. We have nt checed thrugh the perfrmance f the test prpsed by Pesaran and Deatn [6J fr the abve case, but it seems t be liely that we wuld find similar results. Our argument maes clear that the prblem f testing the functinal. frm f a regressin mdel is nt yet cnclusively slved, hence there is still a need fr an alternative specificatin test that is mre watertight than thse we just have discussed. In this paper we shall prpse such a-'". r. test. This test has the advantage that it des nt need any specified t alternative hypthesis; we nly test whether r nt a given specificatin is true. In sectin 2 we shall give a review f this test prcedure. In sectin 3 we shall deal with the asympttic thery.

6 IDENTIFICATION OF THE NULL HYPOTHESIS AND SUMMARY OF!HE TEST PROCEDURE The Hypthesis Cnsider an i.i.d. stchastic prcess (Yl' Xl)'.., (Y n ' xn) in It X It, where the satisfy (2.1) The cnditin (2.1) is sufficient fr the existence f a Brel measurable functin g ( ) n It such that (2.2) (See Chung [3, Therem 9.1.2]. Defining u j = Y j - g(x j ) we then get the fllwing tautlgical regressin mael (2.3) j = 1, 2,..., n, where bviusly the u. J satisfy (2.4) a. s., j = 1,2,..., n,... Suppse we have specified the regressin functin g(x) as f (x, e ), where f(x, e) is a nwn real valued Brel measurable functin n R X 9 and e is a parameter space cntaining the unnwn parameter e if this specificatin is true. Testing the truth f this specificatinwe thus have t test the null hypthesis (2.5) against the alternative hypthesis that H fr sme is false: e e e (2.6) fr all e e e

7 - 5 - Identificatin But hw can we identify H versus Hl frm the distributin f the data, r with ther wrds, hw d H and H l, respectively, crrespnd with distinct characteristics f the distributin f (Yj' x j )? The answer t this questin is suggested by the fllwing fundamental therem. THEOREM 1. Elvl < a) (I) Let v be a real valued randm variable satisfying and let z be a randm vectr in P[E(v\z) = 0] < 1 if and nly if R Then we have: Eve ittz :F 0 fr sme nnrandm vectr t e 'R. (II) If in additin. z is bunded then p[e(vlz) = 0] < 1 ittz if and nly if Eve 0 ~ 0 fr sme nnrandm vec tr t e'r in an arbitrarily small neighbrhd f t = 0 Prf: Appendix 2. Thus frm part I f therem 1 it fllws that prbabili ty 1 fr sme fr all t e E,, hence - f(x., a)lxjj = 0 with J T it X if and nly if E(Yj - f(x j, 9 0» e J = 0 COROLLARY l! H is true if and nly if fr sme 9 e : T 0 it x. E(y. - f(x.,a» e J - 0 fr all t E'R J J 0 Als part II f therem 1 can be used fr identifying the hyptheses, as we nw shw. Let ~ be a bunded Brel measurable mapping frm R int such that x. J fr example and ~ (x j ) generate the same Euclidean Brel field, atg (Xl.),J (2.7) atg (x.),j

8 - 6 - Then (2.8) hence, applying part II f therem 1, we have: COROLLARY 2: Let ~ be any bunded Brel measurable mapping frm R int such that and x. generate the same Euclidean J Brel field. Then Hl is true if and nly if fr every e e there is a t in an arbitrarily small neighbrhd f t = 0 such that itaf(xj) E (y j - f (x r e)) e '" 0. Of curse, if H is true then fr sme fr all t e R but it is sufficient t verify this nly fr an arbitrarily small neighbrhd f t = O. A direct cnsequence f crllary 2 is: COROLLARY 3: Let ~ be defined as in crllary 2. Let N = X [- e:" e:. 1, t=l.",.", where t > 0 (t = 1, 2,, ) are arbitrarily chsen. Put (2.9) n(e) = IE(Yj - f(x j, itt~(xj)12 e)) e dt 0 Then H is true if.,,(9 0 ) = 0 fr sme e e and Hl is true if inf n(9) > 0. eee The Test This result suggests t test H against Hl by using a test statistic f the frm: (2.10)... where N and ~ are as befre and e is the nnlinear least squares estimatr f e 1.' e 0'

9 - 7 - (2.11) e ee a.s., Nte that if we put ~ 2 n 2 f(x, j e» = inf ~ (y. - f(x., s» see j=l 3 3 (2.12) and (2.13) ~. y. - f(x., e), J 3 3 ~ can be written as (2.14) In sectin 3 we shw that under H,. n 11 cnverges in distributin t a nnnegative randm variable with first mment I.l, say, and that we can estimate this mment cnsistently by (2.15),.,.2 IJ :a 2 a [IT It-t, - "'-1 -A } tr (A.D), -t,=l where,. a is the usual estimate f the variance f the u., 3 (2.16) and (2.17) ~ In n T ~,. B=2 l: ~ [(/a9 )f(x.],e)} [C%S)f(x., S)} n j =1 j = sin[!,(z,.-z,.)] J """,J1 """,J2 II t=l S by applying Chebishev's inequality (fr first mments) we cnclude that fr any Ci g (0, 1),

10 - 8 - (2.18) A 1 A limsupp[n'l1>~~} ~a n-d) if H is true. Mrever, fr every a > 0 we have (2.19) A 1 A lim Pen '11 > -!J.] = 1 n-~ a if Hl is true, as als will be shwn in the next sectin. Thus, prceeding the test at the a' 100 percent cnfidence level we accept H if and we reject H if nt. The result (2.18) is smewhat crude, as it is based n Chebishev's inequality, which is nt a very sharp inequality. It wuld, f curse, be better t base the test directly n the limiting distributin f n'l1, but this limiting distributin appears t have a t cmplicated structure fr that. Hwever, if we assume that the distributin f the u j i. 1. d. 2 N(O,Q' ) and that the sequence Cu,) is independent f the. J sequence (x, ) we can by Mnte Carl simulatin estimate a J 1... sharper lwer bund than the bund ~]J used J.' "" n.. (2 18) We shall discuss this pint t in the next sectin. is The abve test is nly ne example ut f a large class f similar tests that can be derived n the ba"sis f therem 1. Fr example, we als may use a test statistic f the frm A 1 n T it x. 2 1, =ll-!: (Yj - f(x j, e))e J I wet) dt n. 1 J= where wet) is a psitive weight functin, fr example a -variate nrmal density, and fr this test statistic we can, n basis f part I f therem 1, derive similar results as (2.18) and (2.19).

11 ASYMPTOTIC THEORY The Assumptins In this sectin we shall set frth cnditins such that (2.18) and (2.19) are true. The cnditins invlved are just thse fr wea cnsistency and asympttic nrmality f the nnlinear least squares estimatr as can be fund in Jennrich [5] and Bierens [1, 2]. Our first assumptin cncerns the distributin f the data. ASSUMPTION 1: The bservatins (y l' Xl)'..., (y, x ) are Ll.d. randm vectrs in R a > 0 x JR Yj The satisfy EIYjI2+~ < fr n n c sme The i.i.d. assumptin is, f curse, very restrictive and rules ut mst f the ecnmetric applicatins. It is, hwever, merely made fr cnvenience. Using the results in [1] and [2] it is pssible t extend the results in this paper t nnstatinary and nnlinear time series regressins, but this will be ut f the scpe f the present paper. The cnditin n the abslute mments f is strnger than (2. 1), but needed in rder t assure that the errrs u. J defined by (3.1) have finite abslute mments f rder slightly larger than 2. Thus we have (3.2) E U j = cr < c, E I u j I < c fr sme 0 > a. Mrever, we shall limit ur attentin t specificatins f(x, e) satisfying the fllwing cnditins.

12 ASSUMPTION 2: The functin f(x, e) and its first and secnd partial derivatives and i2 = 1, 2,., m), are cntinuus real functins n R X 8l, Tl1here tel is a cnvex cmpact subset f Rm. If H is true then e is 0 an interir pint f tel. This assumptin implies that if H. 0 is true then the true regressin functin g is cntinuus n R, but under H1 g may be any Brel measurable functin n R. Furthermre, f1rwing [lj we shall tmpse sme wea cnditins n the mments f f (x., e) and its first and secnd partia 1 deri va ti ves t e, 1. e., J ASSUMPTION 3: There exists a a > 0 such that: In rder t estimate the true parameter e cnsistently by least squares estimatin it shuld be unique in the sense that it is the nly pint in e such that (3.3 ) = E[g(x.) - f(x., e )}2 = inf E[g(x. - f(x., e)}2 J J 0 eee J J ' prvided H is true. But even if H is false we can define a pint 0 e in tel by the right equality in (3.3), and if such a pint is unique - it can be estimated cnsistently by least squares. Therefre we assume:

13 ASSUMPTION 4: Under H as well as under Hl there exists a unique pint 9* in e such that Of curse, under H we have e... e. Mrever we nte that by 0 * assumptin 4 the alternative hypthesis can be restated as (3.4) Frm the assumptins 1 thrugh 4 and Therem in [1) it nw fllws that the leas't squares estimatr e is wealy cnsistent: (3.5) plim e... 9 under H, ~ 0 0 but using the argument in Sectin 3.1 in [1) it is nt hard t shw " that als (3.6) plim e... 9* under Hl. n-'- Next suppse: ASSUMPTION 5: The matrix (3.7) is psitive definite and 2... (J a.s.. Frm the argument in Sectin in [1) it then fllws that under H (3.8) '- '" 1 n -1 T plim l\~n(e - e) - r-!: uja (/ee )f(x.,e)\i = 0 n_a:» 0 \In j=l J 0 s that by the central limit therem (3.9), " n (9-90) - Nm [0, a A ] in distributin

14 Asympttic Thery under H Befre we cntinue ur argument we shall first intrduce sme additinal ntatin. Thus we dente (3.10) (3.11) (3.12) (3.l3) say, where (3.14) E3.15) and (3.16) ~ ( 1 ) (t).. t (y j j=l ~(2) (t)." 1 ~ [u j + (e - n j"l T (3) 1 n it Zj 1 ~ (t)--e u.(e -(/e)f(xj,e)a- S (Tn n. 1 J 0 0 J= 1 n ittz j 1 1 n ~*(t) = n t ujee - (/e) f(x j, e )A- b(t)}.. Ii E U p.(t) j=l 0 j=l j J z. J is defined by (2.12), ittz. P (t).. e J - (ale) f(x., e )A -1 b(t) J J 0 T 1 n T lt z. b (t).. ii E (0/ 09) f (x ' 9 ) e J j=l j 0 ittz b(t).. E S (t).. E (/091f(x., 9 ) e j J 0 Nte that frm (2.10) and (3.10), (3.17),., = J 1 ~(1) (t) \2 dt Nw put 0,..* (3.18),.. * 12,.,- J I ~ (t) dt N We then have THEOREM 2. then p1im n_(x) I If H is true and if the assumptins 1 thrugh 5 hld,..,..*\ n,.,-n,., =0.

15 Prf: (3.19) - Using the mean value therem we can write T ~(l) 1 n it z. ~ (t) = Ii!: [u. + f(x, e) - f(x, j j e)]e J j=l J. T 1 n,. T T _ l.t z. = ii!: (U - (e - e) (~/~e) f(x, e(t)]e J j=1 j j where Set) is a mean value satisfying (3.20) a.s. fr all t e JR Using Therem in II] it is nt hard t shw (3.21) p1im sup I 'In ~(l\t) - rn~(2) (t) I = 0 n-+"o ten and c.mbining (3.8) and (3.11) we see that (3.22) plim sup Irn~(2) (t) _rn~(3) (t) I = 0 ~ ten Again using Therem in 11] we may cnclude that (3.23) and cnsequently that plimsup Ib (t) - bet) 1= 0 n-+<» ten 0 (3.24) plim sup 1m ~(3)(t) - Vn~* (t) I.. 0 n-+<» ten Cmbining (3.21), (3.22) and (3.24) we btain (3.25) plim sup Ivn~(l) (t) - Vn~*(t) I = 0 Il-+<D ten and applying Lemma in II] we cnclude frm (3.25) that als (3.26) This prves the therem. Q.E.D.

16 Therem 2 implies that under H A n't1 and A n 11 * have the same limiting distributin. But what is the limiting distributin f n ~ *? If we substitute (3.13) in (3.18) we get (3.27) where is the cmplex cnjugate f If we wuld be able t as a prduct f Li.d. randm variables,.* n't1 wuld cnverge in distributin t but it appears imp~sible t split up the integral A invlved in this way. S the limiting distributin f n't1* is prbably f an unnwn type. There are, hwever, tw ways ut f this prblem. A* The first 'way ut is t cmpute the expectatin f n'l1 and t apply Chebishev's inequality. This expectatin is: (3.28) ij. = En'T1* = -,. ln i: Eu. E r P.(t)"j (t) dt = a E r p.(t) p.(t) dt n j=l J i J & J J 0 which is bviusly independent f the sample size n. We then have by Chebishev's inequality (3.29) and.cnsequently by therem 2 A E nn* 1 :=0 a -1J Ct (3.30),. 1 lijnsup pen 'T1 > - ij.] ~ Ct a S if we can find a cnsistent estimate ij.' say, f ij. then (3.31) We can (3.32) limsup n- cnstruct such an n b (t) = 1 i: n j=l estimate ij. as fllws. Put. T T,. ~t z. (010 e ) f (x., 9) e J J

17 and (3.33) T,.. it z., P"j(t) - e J - (alae) f(x j, e)a 6(t), where A is defined by (2.16). Then (3.34) because 1 n,..,.. 1 n 1 sup 1- t ~. (t) p. (t).. - i: ~. (t) 1'. (t) ten n j=l J J n j-l J J ~ sup I b(t)t A- 1 bet) - b(t)t A- 1 bet) I ten in prb. (3.35) p1im A = A, n-+where A is defined in assumptin 5, and (3.36) p1im sup Ib(t) - bet) I = p, n- ten, () as is nt hard t verify by using Therem in [1]. Mrever, (3.37) 1 n p1im sup 1- i: ".(t) p.(t) - E p.(t) "j(t) I = 0 n- ten n j=l J J J as is als easily verified by using Therem in [1]. cmbine (3.34) and (3.37) we then may cnclude S if we (3.38) p1im J 1 ~ p.(t) 6.(t)dt-jE {p.(t) I'j(t)}dt = E j p.(t) Pj(t) dt ~ N n j-1 J J N J N J 0 and hence,..,..2 1 n,.. """A~~ (3.39) ~ = a - i: r p.(t) p.(t) dt n j=l ~ J J is a cnsistent estimatr f IJ.. We leave it t the reader t verify that the estimatr ~ defined by (3.39) can be written as (2.15). S we have prved by nw

18 THEOREM 3. If H is true and if the assumptins 1 thrugh 5 hld then fr every a e (O~ 1), > limsup "" 1 "" P (n T'l > ] ~ a. a n~ This is ur first way ut f the prblem that the limiting distributin "* f n T'l is unnwn. The secnd way is the fllwing. If we mae the additinal assumptin that the sequence (u.) f J disturbances (U j = y - E(Y.lx.» is independent f the sequence j J J f regressrs and mrever that these u.' s are distributed as J then by replacing the Uj's in (3.13) by ther independent randm drawings frm N(O, a 2 ) the resulting integral f the type (3.18) has the same distributin as the riginal ne. Thus if we draw an artificial randm sample (Wl~.., W n } frm the standard nrmal distributin and if we put (3.40) (3.41) 1 n ~*(t) =- E aw.p.(t), n j=l J J Ti*(t) = J 1 ~(t) \2 dt, N then n;* and n~* have the same distributin. Hwever, a and Pj(t) are nt bservable. Therefre we replace a in (3.40) by its estimate ; and Pj(t) by ~j(t) (defined by (3.33». Thus, we put: (3.42) ~ 1 n" A s (t)=;- E aw.f).(t), j=l J J (3.43) Ti**". J I ret) 12 dt. N Nw similar t therem 2 it can be shwn that (3.44) p lim InTi ** - n 11 * I = 0 n-+- and hence tha t n T'l -** and n T'l ""* have the same limi ting dis tribu tin Therefre by cmputing (3.43) fr a large number f artificially drawn

19 randm samples [WI", W n } frm the standard nrmal distributin we can establish a (randm) number Pa - such that a x 100 percent f the Ti** 's are larger than p - We then have apprximately a P[n~ > PaJ. :=::$ a fr large n, prvided H is true. Asympttic Thery under HI We have seen that if HI is true then under the assumptins 1 thrugh 4, plim. e =- e*, where e* is defined by assumptin 4. Therefre, if we put (3.45) ittz ~(l) (t) =-1 t (YJ' - f(x J " e*» e j *. n j=-l then frm (3.10), (3.6) and (3.45) (3.46) plim sup I,(l) (t) - ~.il) (t) I = 0 n-+cd ten and mrever, (3.47) where (3.48) plim sup I ~(;) (t) - ~(t) I.. 0 ll'+'d ten, T (1) ~t z ~*(t) = E ~* (t) = E[g(x,) - f(x., e») e j J J * as is nt hard t verify by applying Therem 2.3,4 in [lj. Thus we have under HI (3.49) plim. sup I~(l)(t) - ~(t)1 = 0 n-+- ten and cnsequently (3.50),. plim T1 = 'T'1* = n-+-

20 But frm crllary 3 it fllws that ~ > 0, hence: THEOREM 4. If li is true and if the assumptins 1 thrugh 4 hld l,. then plim n T1 = (l) n... co Since it is nt hard t shw that als under li l the estimatr ~ cnverges in prbability t a finite number, 7herem 4 implies that (2.19) hlds.

21 Prf f (l.4) In additin t the ntatin intrduced in sectin 1 we put (Al.l) (Al.2) z ~ 3 Xz,l 3 x 2,n (Al.3)./2; d e q,. 1,2, Using the ntatins (Al.l) and (Al.2) and realizing that M f =,. M X a = 0 identica lly we may rewrite (l.l) thrugh (l. 3) as: (Al.4) (Al.5) (A1.6) a,.,.,..", 2. (1/ (n - - 1» H y - XS - 0/ M Z y II p 0 We shall need the fllwing results, which are nt hard t prve: (Al.7) &llm (Al. 8) ~im (lin) (lin) x~ = I, T Z Z,. m6 I = lsi, (Al.9) gl..im (lin) Z~,. m l I = 3I, (Al.1O).. g1..im (lin) XTy,. (i) (Al.ll) gum (lin) ZT y = m 4 (i) = 3(i) and cnsequently

22 (Al.12) (Al.D) These results will nw be used fr shwing that asympttically nrmally distributed with zer mean. is First, bserve that (Al.14) Since EZIu"EXI~.. O and since (~i~) isasumf Li.d. randm vectrs in R4 we have by the central limit therem (Al.IS) where (Al.16) z'l' (l/vn( IU) - N4 (0, a) in distributin, X u 1.. -E n ( (ZIu)I (ZIu ), (ZIu)I (XIu) I (Z~u), (XIu) I m 2 + m 4, 0 m 4 + m 6, 0 m 2 + m , 0, 18, 0, 0, 120, 0, 18, 18, 0, 4, 0, Mrever it fllws frm (Al.8), (Al.9) and (Al.13) that (A.17) AI AI T T -1 ( ) plim (y, - y «lin) Z X) «lin) X X)) ).. 5' 5' - 5' ~ tt+"'d say. Thus frm (Al.14) thrugh (Al.17) it fllws

23 (Al.1S)... T T'" T / 96) (l/,fo.)y Z MU-N(O,~ n~) = N\O, 25 in distributin. Next we bserve that we may write (A 1. 19) s that frm (Al.20) (Al.7), (Al.a), (Al.9) and (Al.l3) we have (1) {m~ m:} 12 plim (l/n)\\m Z y\l = (1, 1) 1 2' -"2' = 25 n-+- m6 m6 Cmbining (A l. 5), (Al.1S) and (Al.20) we nw cnclude: (Al.21) in distributin. Mrever, since nw (Al.22) plim a = 0, we have n-+- p plim '2,2 '"' plim (lin) II y- X 112 = plim n- n- n- T 2 (lin) u u = 1I1z +m 2 = 2 and cmbining this result with (Al.20) we get (Al.23) The desired result (1.4) fllws nw easily frm (Al.4), (Al. 21) and (Al.23) Q. E.D.

24 APPENDIX 2 Prf f Therem 1 Prf f Part I Frm Chung [3, Therem 9.1.2] it fllws that there exists a Brel measurable real functin r, say, n R such that (A2.l) Put E(vlz) = r(z) a. s.. (A2.2) Then bviusly r l and r 2 are nnnegative Brel measurable real functins n R satisfying (A2.3) r = Nw assume fr the mment (A2.4) Then we can define prbability measures field ~ as fllws:!jl and n the Euclidean Brel (A2.5)!Jj(B) = J r/x) d!j(x)/c j, B j = 1, 2, where!j is the prbability measure generated by the randm vectr z and B is an arbitrary Brel set in~. Then we may write: (A2. 6) T ittz tt E ve lt z = E r(z) e = J rex) e l x djj(x) say, where

25 (AZ.7) (j = 1, 2) are the characteristic functins f the prbability measures ~j (j = 1, Z), respectively. T If E ve it z == fr every t e R then it fllws frm (AZ.6) that (AZ.8) t E R Hence, substituting t = 0, we get (AZ.9) s that frm (A2.4), (AZ.8) and (AZ.9) (AZ.10) "l(t) = "2(t) fr every t er But (AZ.10) implies that the prbability measures ~l and ~Z are r equal, Le., (AZ.ll) ~l (B) = ~ (B) fr eve ry Bre 1 se t B. Frm (AZ.S), (AZ.9) and (AZ.ll) we nw btain (AZ.12) S r (x) dll 1 ex) f r z (x) d~ ex} fr every Bre 1 se t B and cnsequently B B (AZ.13) But CA2.l4) S rex) dj.j.(xl=o fr every Brel set B. B Bl = (x e It : rex) > 0 } is a Brel set, and thus: CA2.l5) = I rex) dj.j.(x}, 1

26 which is nly pssible if Bi is a null set with respect t!j.. Similarly we cnclude that the Brel set (A2,.16) B2 = [x e]i : rex) < OJ is a null set with respect t!j., and hence (A2.17) is a null set with respect t!j.. This means that r(z) = O. a.s. Thus T we have prved by nw that if (A2.4) hlds and if E ve it z = 0 fr all t e]i then E (vi z) = 0 a. s.. Hwever, if (A2.4) des nt hld then ur cnclusin still hlds, as is nt hard t prve. This cmpletes the "nly if" part f part I f therem 1. Since the "if" part is trivial, part I f therem 1 is prved by nw. r Q.E.D. Prf f Part II Since nw z is bunded we may write (A2.18) S if Eve. T <0 EveJ.tz=Ev1: j=o +0 fr sme i j (ttz)j = CD i j E v (ttz)j j.' 1:., j=o J. t* e:lt then there exists a nnnegative integer j* such that CA2.19) ASSuming that j* is minimal, we therefre have (A2.20) which implies that Eve',., T ].",t* z :f 0 fr an arbitrarily $mall A > 0, say this prves part II f the therem. Q.E.D.

27 REFERENCES [1] BIERENS, H. J.: Rbust Methds and Asympttic Thery in Nnlinear Ecnmetrics, Lecture Ntes in Ecnmics and Mathematical Systems, vl. 192, Heidelberg: Springer-Verlag, [2] BIERENS, H. J.: "A Unifrm Wea Law f Large Numbers under cp - Mixing with Applicatin t Nnlinear Least Squares Estimatin," Statistica Neerlandica, 36 (1982) (t appear). [3] CHUNG, K. L.: A Curse in Prbability Thery, New Yr: Academic Press, [4] DA VInSON, R. AND L. G. MACKINNON: "Several Tests fr Mdel Specificatin in the Presence f Alternative Hyptheses," Ecnmetrica, 49 (1981), r [5] JENNRICH, R. I:: "Asympttic Prperties f Nnlinear Least, Squares Estimatrs;'The Annals f Mathematical Statistics, 40 (1969), [6] PESARAN, M. H. AND A. S. DEATON: IrTesting Nn-nested Nnlinear Regressin Mdels," Ecnmetrica, 46 (1978),