Lecture 7 Testing Nonlinear Inequality Restrictions 1

Transcription

1 Eco 75 Lecture 7 Testig Noliear Iequality Restrictios I Lecture 6, we discussed te testig problems were te ull ypotesis is de ed by oliear equality restrictios: H : ( ) = versus H : ( ) 6= : () We sowed tat te Wald, QLR ad LM statistics ave te same asymptotic distributio uder H ad te distributio is 2 r. Te rejectio rule for te Wald, QLR ad LM tests of sigi cace level ca tus be: reject H if T > 2 r;, (2) were T = W, QLR or LM ad 2 r; is te quatile of 2 r. A test de ed as suc as correct asymptotic size because: Asy:SZ : = lim sup! 2:( )= Pr T > 2 r; = Pr 2 r > 2 r; =. (3) We is oe-dimesioal, tese tests are ofte cosidered as "two-sided" test as violatios of ( ) = o bot sides are detected. I tis lecture, we discuss te testig problems were te ull ypotesis is de ed by oliear iequality restrictios: H : ( ) vs. H : ( ) >. (4) Testig situatios like tis arise we, for example, oe wises to test tat some matrices are positive semi-de ite. Tey also arise, sometimes i a bit disguise, i speci catio testig for models de ed by iequalities (e.g. momet iequalities). Notice tat, we testig (), te asymptotic ull distributios of W, QLR or LM do ot deped o. Test statistics wit tis property are called asymptotically pivotal. Wit (asymptotically) pivotal test statistics, tests ca be costructed easily. Te critical values simply ca be cose as te appropriate quatiles of te asymptotic distributio (tat is ivariat to te true parameter). Te asymptotic ull rejectio probabilities are exactly te sigi cace level we coose, o matter wat is as log as H olds. Aalogous test statistics for (4) are ot asymptotically pivotal. Te asymptotic distributios Te ote is made up from my ead based o Wolak (99), Adrews ad Soares (2), Gourieroux ad Mofort (995, capter 2). It does ot follow ay of tose sources closely. Accuracy is ot guarateed. Commets welcome. Xiaoxia Si Page:

2 Eco 75 of tem deped o, i particular, o weter some or all elemets of ( ) are zero. We de e te Wald ad QLR statistics rst ad te discuss te problems caused by te lack of pivotaless. Trougout, we maitai Assumptio QLR. Tus, p! d N(; cb ) = N(; c2 ). Te Wald-type statistic is de ed as W = mi t2[ ;] r t ^H t, (5) were ^H = (^ ), ^c! p c ad ^B = 2 ^Q(^ ). Wald statistic de ed tis way does ot require oe to compute te iequality costrait estimator ~ : ~ ad ^Q ~ if ^Q () + o p =2. (6) 2:() Tis as advatage we te iequality costraits make ~ ard to compute. If oe is willig to compute ~, tere are oter ways of de ig Wald-type statistics as well: ~W = ~ ^H ( ~ ) ad W = ~ ^c ^B ~. (7) Te secod statistic is a Hausma-Wald type statistic as it as te feature of a Hausma test: measurig te di erece betwee te urestricted ad te restricted estimators. A QLR test statistic ca be de ed i te same way as i Lecture 6: QLR = 2^c ^Q ~ ^Q : (8) Teorem 7.: Suppose Assumptios EE2, CF, R, COV, REE ad QLR old. Te, (a) uder H, W = W ~ + o p () = W + o p () = QLR + o p () ad (b) W! d J ; = mi t2[ ;] r (Z + t) (Z + t), were Z N(; ) ad = ( ;; :::; ;r) ( ;j = if j ( ) = if j ( ) < Before givig te (rater tedious) proof, I would like to discuss te implicatios of part (b). As predicted above, te asymptotic distributio of te test statistics are ot parameter free. If we wat Xiaoxia Si Page: 2

3 Eco 75 to costruct a test usig te same kid of critical value as i te equality costrait case, we ave to aswer te questio: wic J ; sould we use to take te quatile from? Di eret J ; ave di eret quatiles. Naturally, we would like to cotrol te size of te test. Tat is, we wat: sup Pr J ; > c =. (9) :( ) Tis idea requires us to d te least favorable J ; : te J ; tat as te igest quatile. We r =, J ; is oe dimesioal. Te least favorable J is straigtforward: it soud be te J ; suc tat ( ) =, wic is We r >, te task is way more complicated, as explaied i Wolak (99). Te type of critical value suggested above is called " xed critical-value". Tey are xed i a sese tat tey do ot vary wit data. Tere are oter ways of coosig critical values, wic are muc more feasible ta xed critical values i te cotext we are dealig wit. We de e two types ere. Te rst type is "plug-i asymptotics (PA)". Literally, it meas, we use te asymptotic distributio (ot parameter free) ad plug i te estimated values of te parameters. I J ;, ca be cosistetly estimated uder te ull. Terefore, we use te estimated value i place of. However, caot be cosistetly estimated because it takes i ite values sometimes. We get aroud tis by usig i place of. Usig istead of ( ) sifts te distributio of J ; to te rigt ad tus teds to make te critical values large. If we oly care about cotrollig te size, tis sould t cause ay problem. To sum up, te PA critical value is take as te coditioal (o data) quatile of J;, were J; = mi t2[ ;] r Z t Z t, () were ^ = ^c ^H ^B ^H. You ca sow tat J;! d J ;, ad te quatile of J; coverges i probability to tat of J ;, wic is o smaller ta tat of J ;, at ay uder te ull. Tis implies tat te size of te test is well cotrolled asymptotically. Te secod type is "geeralized momet selectio (GMS)". 2 Te PA procedure replaces by zero ad ca potetially produce critical values tat are too large. Large critical values a ects te power of te test. To improve power, GMS procedure uses te data to decide weter is i ite or ot. Te judgemet is ot perfect, but it elps reducig te critical value wile at te same time preservig te size property of te test. Te GMS procedure replaces by a momet 2 We use te te term "momet" eve toug ( ) are ot ecessarily momet of aytig. But te term is developed i momet iequality literature. It s coveiet to follow te traditio. Xiaoxia Si Page: 3

4 Eco 75 selectio fuctio: 8 < = : if { < if { > ; were { = o ( p ) is a sequece of positive umbers. Te GMS critical value is take to be te coditioal quatile of J;. It ca be sow tat te sizes of te GMS tests are well cotrolled asymptotically ad te tests ave better power ta PA tests. Notice tat PA ad GMS tests are developed oly recetly because tey require simulatig te quatiles of o-regular distributios. Eve toug tey are muc easier to use ta xed critical value tests, tey could ot be more feasible ta te latter witout powerful computers. Now, let s prove Teorem 7.. Proof of Teorem 7.: First we establis part (a). We start wit te asymptotic equivalece betwee W ad W ~. Let be te solutio to te quadratic miimizatio problem i (5). It su ces to sow tat p ^ ~ = p ^ + o p (): () We sow tis by comparig te Ku-Tucker coditios for te quadratic miimizatio problem ad tose for if 2:() ^Q (). By te Ku-Tucker teorem, q satis es: (KT q ): ^H ( q ), q = q ( q ) q =, (2) were te q are te Lagrage multipliers. Elimiatig q, we ave (KT q ): ^H t q ^H ^c ^B ^H =, (3) Xiaoxia Si Page: 4

5 Eco 75 By te Ku-Tucker Teorem, ~ satis es (KT c ): ^Q ~ + H ~ c = o p =2 ~, c ~ c =. (4) Mea-value expasio of ^Q ( ~ ) aroud ^ gives: ~H c = o p =2 ^Q = o p =2 + B + B ~ ~, (5) were B = 2 ^Q( ) for some lyig betwee ^ ad ~, ad te secod equality olds by Assumptio EE2(ii). Let H + be te matrix tat satis es ~ = H + ~. Suc a matrix always exists by te mea-value teorem ad H +! p H uder our assumptios (R(i), EE2(i), REE, CF(iv)). Premultiplyig H + (B) to bot sides of te equatio above ad we ave: H + (B) H ~ c = ~ + o p =2. (6) Tus c = H + (B) H ~ ~ + o p =2. (7) Usig tis to elimiate c i (4), we ave (KT c ): H + (B) H ~ ~, ~ H + (B) H ~ ~ + o p =2 ~ + o p =2 = (8) Te systems (KT q ) ad (KT c ) uiquely pi dow ad ~ ~ asymptotically equivalet. Terefore, p p ( ) ( )! d N(; H cb H ), we ave (). respectively. Te two systems are = p ( ( )) + o p (). 3 Because 3 More rigorous argumet uses te facts tat () eiter system implicitly de es a cotiuous mappigs from elemets wit kow asymptotic distributios to p ~ ( ) or p ( ), respectively, (2) Te cotiuous mappigs de ed i te two systems are te same; ad (3) te elemets wit kow asymptotic distributios i bot systems ave te same asymptotic distributios. Xiaoxia Si Page: 5

6 Eco 75 Next we establis te asymptotic equivalece betwee ~ W ad W. Combiig (5) ad (7), we ave ~ = (B) H ~ H + (B) H ~ Plug tat i W, we ave W = ~ + o p =2 (B ) ~ H H + (B ) ~ H = ~ ^H ^c ^B ^H ~ + o p =2. (9) H + (B) H ~ ~H (B) ^c ^B ~ + o p =2 ~ + o p () = ~ W + o p (). (2) Te secod equality olds because Assumptio COV olds ad p ( later of wic olds by te argumets used to establis (). ~ ) = O p (), te Next we establis te asymptotic equivalece betwee QLR ad W. Usig secod-order Taylor expasio of ^Q aroud ^, we ave QLR = 2^c " ^Q (^ ) = 2^c o p + 2 = ^ ~ + ^ ~ 2 ^Q ( ) ^ # ~ 2 ^ ~ ^B ^ ~ ^ ~ ^c ^B ^ ~ + o p () = W + o p (), (2) were te secod equality olds by Assumptio EE2(ii), COV ad ^ by (9) ad p ( ~ ) = O p ()). Now we establis part (b). De e ~ = O p ( =2 ) (wic olds S (m; ) = mi (m t2[ ;] t) (m t), (22) r Te proof is a direct applicatio of te cotiuous mappig teorem oce we observe tat p( W = S ad sow tat S is cotiuous i bot of its argumets o (R [ f ( )) + p ( ) ; ^H, (23) g) r, were is te space of positive de ite matrices. Te proof of tis is left as a exercise ad a versio of it ca be foud Xiaoxia Si Page: 6

7 Eco 75 i Adrews ad Soares (2). Note tat we eed te cotiuity to old o te exteded real space because we wat to allow to take i ite values. Xiaoxia Si Page: 7