Lecture 6 Testing Nonlinear Restrictions 1. The previous lectures prepare us for the tests of nonlinear restrictions of the form:

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1 Eco 75 Lecture 6 Testig Noliear Restrictios The previous lectures prepare us for the tests of oliear restrictios of the form: H 0 : h( 0 ) = 0 versus H : h( 0 ) 6= 0: () I this lecture, we cosier Wal, Lagrage multiplier (LM), a quasi-likelihoo ratio (QLR) test. The Delta Metho will be useful i costructig those tests, especially the Wal test. The Delta Metho The elta metho ca be use to the asymptotic istributio of h( b ), suitably ormalize, if ( b 0 ) Z: Theorem (-metho): Suppose ( b 0 ) Y where b a Y are raom k-vectors, 0 is a o-raom k-vector, a f : g is a sequece of scalar costats that iverges to i ity as. Suppose h() : R k R` is i eretiable at 0, i.e., h() = h( 0 ) h( 0 0 )( 0 ) o(jj 0 jj) as 0. The, If Y N(0; V ), the 0 h( 0 )Y N 0; (h( b ) h( 0 )) 0 h( 0)Y: h( 0 0 )V ( h( 0 0 )) 0. Proof: By the assumptio of i eretiability of h at 0, we have (h( b ) h( 0 )) = 0 h( 0) ( b 0 ) o(jj b 0 jj): The rst term o the right-ha sie coverges i istributio to h( 0 0 )Y: So, we have the esire result provie o(jj b 0 jj) = o p (). This hols because o(jj b 0 jj) = jj ( b 0 )jjo() = O p ()o() = o p () a the proof is complete. The otes for this lecture is largely aapte from the otes of Doal Arews o the same topic I am grateful for Professor Arews geerosity a elegat expositio. All errors are mie. Xiaoxia Shi Page:

2 Eco 75 2 Tests for Noliear Restrictios The R r -value fuctio h() e ig the restrictios a the matrix 0 (e e i Assumptio CF) are assume to satisfy: Assumptio R: (i) h() is cotiuously i eretiable o a eighborhoo of 0. (ii) H = 0 h( 0 ) has full rak r ( ; where is the imesio of ): (iii) 0 is positive e ite. For example, we might have h() = 2 ; h() = 2 3, or h() = : (2) 2 3 The requiremet that 0 is positive e ite esures that the asymptotic covariace matrix, B0 0B0, of p ( b 0 ) is positive e ite. The Wal statistic for testig H 0 is e e to be W = h( b ) 0 ( H b B b b B b b H) 0 h( b ), where bh = 0 h(b ) (3) a a b are cosistet estimators of B 0 a 0 respectively. See Lecture 5 for the choice of bb a b : As e e, the Wal statistic is a quaratic form i the (urestricte) estimator h( b ) of the restrictios h( 0 ). The quaratic form has a positive e ite (p) weight matrix. If H 0 is true, the h( b ) shoul be close to zero a the quaratic form W i h( b ) also shoul be close to zero. O the other ha, if H 0 is false, h( b ) shoul be close to h( 0 ) 6= 0 a the quaratic form W i h( b ) shoul be i eret from zero. Thus, the Wal test rejects H 0 whe its value is su cietly large. Large sample critical values are provie by Theorem 6. below. The LM a QLR statistics epe o a restricte extremum estimator of 0. The restricte extremum estimator, e, is a estimator that satis es e 2 ; h( e ) = 0, a ^Q ( e ) iff ^Q () : 2 ; h() = 0g o p ( =2 ): (4) Xiaoxia Shi Page: 2

3 Eco 75 The LM statistic is e e by LM = ^Q 0 ( e ) B e H e ( 0 H e B e e B e e H) 0 H e B e ^Q ( e ), where eh = 0 h(e ) (5) a e B a e are cosistet estimators uer H 0 of B 0 a 0, respectively, that are costructe usig the restricte estimator e rather tha b : The LM statistic is a quaratic form i the vector of erivatives of the criterio fuctio evaluate at the restricte estimator e. The quaratic form has a p weight matrix. If the ull hypothesis is true, the the urestricte estimator shoul be close to satisfyig the restrictios. I this case, the restricte a urestricte estimators, e a b, shoul be close. I tur, this implies that the erivative of the criterio fuctio at e a b shoul be close. The latter, ^Q ( b ), equals zero by the rst-orer coitios for urestricte miimizatio of ^Q () over. (More precisely, ^Q ( b ) is oly close to zero, viz., o p ( =2 ), by Assumptio EE2, sice b is ot require to exactly miimize ^Q () just approximately miimize it.) The, ^Q ( e ) shoul be close to zero uer H 0 a the quaratic form, LM, i ^Q ( e ) also shoul be close to zero. O the other ha, if H 0 is false, the b a e shoul ot be close, ^Q ( e ) shoul ot be close to zero, a the quaratic form, LM, i ^Q ( e ) also shoul ot be close to zero. I cosequece, the LM test rejects H 0 whe the LM statistic is su cietly large. Asymptotic critical values for LM are etermie by Theorem 6. below. Note that LM ca be compute without computig b. This ca have computatioal avatages if the criterio fuctio is simpler whe the restrictios h() are impose tha whe urestricte. For example, if the urestricte moel is a oliear regressio moel a the restricte moel is a liear moel, the the LS estimator has a close-form solutio uer the restrictios, but ot otherwise. Next, we cosier the QLR statistic. It has the proper asymptotic ull istributio oly whe the followig assumptio hols: Assumptio QLR: (i) 0 = cb 0 for some scalar costat c 6= 0: (ii) bc p c for some sequece of o-zero raom variables fbc : g: For example, i the ML example with correctly speci e moel, the iformatio matrix equality yiels 0 = B 0, so Assumptio QLR hols with c = bc =. I the LS example with coitioally homoskeastic errors (i.e., E(U i jx i ) = 0 a.s. a E(U 2 i jx i) = 2 a.s.), Assumptio QLR hols with c = 2 a bc = P i= (Y i g(x i ; b )) 2 : (6) Xiaoxia Shi Page: 3

4 Eco 75 I the GMM, MD, a TS examples with optimal weight matrix (e.g., A 0 A = V 0 for the GMM a MD estimators, see Lecture 5), Assumptio QLR hols with c = bc =, sice 0 = B 0 = 0 0V 0 0. The QLR statistic is e e by QLR = 2( ^Q ( e ) ^Q ( b ))=bc : (7) Whe H 0 is true, the restricte a urestricte estimators, e a b, shoul be close a, hece, so shoul the value of the criterio fuctio evaluate at these parameter values. Thus, uer H 0, the statistic QLR shoul be close to zero. Uer H, QLR shoul be oticeably larger tha zero, sice the miimizatio of ^Q () over the restricte parameter space, which oes ot iclue the miimum value 0 of Q(), shoul leave ^Q ( e ) oticeably larger tha ^Q ( b ). I cosequece, the QLR test rejects H 0 whe QLR is su cietly large. Asymptotic critical values are provie by Theorem 6. below. For the ML example, the QLR statistic equals the staar likelihoo ratio statistic, viz., mius two times the logarithm of the likelihoo ratio. Whe Assumptio QLR hols, the LM statistic simpli es to LM = ^Q 0 ( e ) B e ^Q ( e )=bc or LM = ^Q 0 ( e ) e ^Q ( e ) (8) (sice i this case oe ca take e = bc B e a ^Q ( e ) = H e 0 e for some vector e of Lagrage multipliers). For the LM a QLR tests, we use the followig assumptio. Assumptio REE: e p 0 uer H 0 : This assumptio ca be veri e usig the results of Lecture 3 with the parameter space replace by the restricte parameter space e = f 2 : h() = 0g: For the Wal a LM tests, we assume that cosistet estimators uer H 0 of B 0 a 0 are use i costructig the test statistics: p Assumptio COV: For the Wal statistic, B b B0 a b p 0 uer H 0. For the LM statistic, B e p B0 a e p 0 uer H 0 : Estimators that satisfy this assumptio are iscusse i Lecture 5. Xiaoxia Shi Page: 4

5 Eco 75 Theorem 6.: (a) Uer Assumptios EE2, CF, R, a COV, W 2 r uer H 0 ; where 2 r eotes a chi-square istributio with r egrees of freeom. (b) Uer Assumptios EE2, CF, R, COV, a REE, LM 2 r uer H 0 : (c) Uer Assumptios EE2, CF, R, COV, REE, a QLR, QLR 2 r uer H 0 : Oe rejects H 0 usig the Wal test if W > k r, where k r is the quatile of the 2 r istributio. The LM a QLR tests are e e aalogously. These tests have sigi cace level approximately equal to for large : Uer sequeces of local alteratives to H 0 : h() = 0, the Wal, LM, a QLR test statistics have ocetral chi-square istributios with the same ocetrality parameter, see ext Lecture Thus, the three tests, Wal, LM, a QLR, have the same large sample power fuctios. Oe caot choose betwee these tests base o ( rst orer) large sample power. The choice ca be mae o computatioal grous. It also ca be mae base o the closeess of the true ite sample size of a test to its omial asymptotic size. The best test accorig to the latter criterio epes o the particular testig cotext. The folklore (backe up by various simulatio stuies), however, is that the Wal test over-rejects i may cotexts a the LM a QLR tests ofte are preferable i cosequece. Proof of Theorem 6.: First we establish part (a). Elemet-by-elemet mea value expasios of h( b ) about 0 give p h( b ) = p h( 0 ) 0 h( ) p ( b 0 ) = H p ( b 0 ) o p () p Z 0 N(0; HB0 0B0 H0 ); (9) where lies betwee b a 0 (a, hece, satis es p 0 by Assumptio EE2(i)), the seco equality hols because h( 0 ) = 0 uer H 0 a a the covergece i istributio uses Theorem h( ) p 0 h( 0 ) = H usig Assumptio R, Xiaoxia Shi Page: 5

6 Eco 75 By Assumptios COV, EE2, a R, Combiig (9) a (0) gives ( b H b b H ) p (HB 0 0B 0 H0 ) : (0) W = p h( b ) 0 ( b H b b H ) p h( b ) Z 0 0(HB 0 0B 0 H0 ) Z 0 ; () where Z = (HB 0 0B 0 H0 ) =2 Z 0 N(0; I r ) a Z 0 Z 2 r: Next, we establish part (b). Elemet-by-elemet mea value expasios of ^Q ( e ) a h( e ) about 0 yiel p ^Q ( e ) = p ^Q ( 0 ) 2 ^Q 0 ( ) p ( e 0 = p h( e ) = p h( 0 ) 0 h( ) p ( e 0 ) = h( 0 ) p ( e 0 ); 0 ) a (2) The, pre- where p 0 a p 0. Let B () = 2 ^Q 0 () a H() = h(). 0 multiplicatio of the rst equatio of (2) by H( )B ( ) gives H( )B ( ) p ^Q ( e ) = H( )B ( ) p ^Q ( 0 ) H( ) p ( e 0 ) = H( )B ( ) p ^Q ( 0 ) p Z 0 N(0; HB 0 0B 0 H0 ) (3) usig the seco equatio of (2) a Assumptios CF(iii), CF(iv), REE, a R. Combiig (3), the fact that p ^Q ( e ) = O p () by (7) below, H( ) p H, e H p H, B ( ) p B 0, a e B p B 0 gives By Assumptios COV, REE, a R, p eh B e ^Q ( e ) Z 0 N(0; HB0 0B0 H0 ): (4) ( e H e B e e B e H 0 ) p (HB 0 0B 0 H0 ) : (5) Combiig (4) a (5) gives the esire result LM Z 0 0 (HB 0 0B 0 H0 ) Z 0 = Z 0 Z 2 r (6) Xiaoxia Shi Page: 6

7 Eco 75 uer H 0, where Z is as above. We ow show that p ^Q ( e ) = O p (): (7) With probability that goes to oe as, e is i the iterior of a there exists a raom vector e of Lagrage multipliers such that Equatios (3) a (8) combie to give ^Q ( e ) H( e ) 0 e = o p ( =2 ): (8) H( )B ( ) H( e ) 0p e = O p (): (9) Sice H( )B ( ) H( e ) 0 p HB0 a HB0 is osigular, (9) implies p e = O p (): This a (8) imply (7). Next, we establish part (c). A two-term Taylor expasio of ^Q ( e ) about b gives QLR = 2( ^Q ( e ) ^Q ( b ))=bc = 2 ^Q ( b )( e b )=bc ( e b ) 0 2 = o p () ( e b ) ^Q ( )( e b )=bc ; 0 ^Q ( )( e b )=bc (20) where lies betwee e a b (a, hece, satis es p 0 ) a the thir iequality hols because =2 ^Q ( b ) = o p () by EE2(ii) a =2 ( e b ) = O p () by (22) below. Elemet-by-elemet mea-value expasios of (=) ^Q ( e ) about b gives p ^Q ( e ) = p ^Q ( b ) 2 0 ^Q ( ) p ( e b ) = o p () (B 0 o p ()) p ( e b ); (2) where p 0. Sice B 0 is osigular, (2) implies Substitutio of (22) ito (20) gives p ( e b ) = B0 p ^Q ( e ) o p (): (22) QLR = o p () ^Q ( e ) 0 B0 B ( )B0 ^Q ( e )=bc = o p () ^Q ( e ) 0 B e ^Q ( e )=bc = o p () LM ; (23) Xiaoxia Shi Page: 7

8 Eco 75 where the seco equality hols by (7) a B B ( )B 0 = e B 0 o p () a the thir iequality hols by the simpli e expressio for LM give i (8). Part (c) ow hols by (23) a part (b). 3 Oe-Sie Hypotheses Sometimes, oe-sie hypotheses are of iterest: H 0 : h ( 0 ) 0 vs. H : h ( 0 ) > 0. (24) We oly cosier oe-imesioal hypotheses, i.e., those with real-value h () :Testig multiple imesioal iequality costraits is much trickier, as explaie i Wolak (99). We use the same assumptios as above. We cosier Wal Statistic oly. Derivig the asymptotic properties of LM a QLR statistics requires erivig those for iequality-costrait extremum estimators, which is ot covere by the previous lectures. The Wal statistic for testig H 0 is e e to be W = h h( b i 2 ) bh b H b, where [a] 0 = maxf0; ag. (25) As e e, the Wal statistic is a quaratic form i the positive part of the (urestricte) estimator h( b ) of the restrictios h h( 0 ). The quaratic form has a positive e ite (p) weight matrix. If H 0 is true, the h( b i h ) shoul be close to zero a the quaratic form W i h( b i ) also shoul be close to zero. Oh the other ha, if H 0 is false, h( b ) shoul be close to h( 0 ) > 0 a the quaratic form W i h( b i ) shoul be i eret from zero. Thus, the Wal test rejects H 0 whe its value is su cietly large. Large sample critical values are provie by Theorem 6.2 below. Uer the oe-sie ull hypothesis, there are two cases: h ( 0 ) < 0 a h ( 0 ) = 0. asymptotic istributios of the Wal statistic are i eret i the two cases. Theorem 6.2 summarize the result. De e the mixe chi-square istributio The 2 = [N(0; )] 2. (26) Let "wpa" eote "with probability approachig oe". Theorem 6.2: (a) Uer Assumptios EE2, CF, R, a COV, ( = 0 wpa if h ( 0 ) < 0 W : 2 if h ( 0 ) = 0 Proof of Theorem 6.2: Whe h ( 0 ) < 0, h( b ) < 0 wpa because h( b ) p h ( 0 ) by Assump- Xiaoxia Shi Page: 8

9 Eco 75 tios R(i) a EE2(i). Therefore, W = 0 wpa. Whe h ( 0 ) = 0, the proof is the same as that for Theorem 6.(a) except for (). The oe-sie couterpart of ().is: W = h ph( b )i 2 bh b b H [Z 0 ] 2 q 2 HB0 0B0 = HB H0 0 0B0 H0 Z 0 = 2. (27) Xiaoxia Shi Page: 9