Likelihood Ratio Based Test for the Exogeneity and the Relevance of Instrumental Variables

Likelihood Ratio Based est for the Exogeneity and the Relevance of Instrumental Variables Dukpa Kim y Yoonseok Lee z September [under revision] Abstract his paper develops a test for the exogeneity and the relevance of instrumental variables using Vuong s (989) model selection test. he test statistic is derived from the likelihood ratio of two competing models: one with exogenous and possibly relevant instruments and the other with irrelevant and even possibly endogenous instruments. he likelihood ratio is given as the di erence between the rst stage F test and the overidentifying restrictions test, which are commonly used tests for the relevance and the exogeneity of instrumental variables. he null hypothesis is that the second model is at least as close to the true distribution of the data as the rst model, while the alternative hypothesis is that the rst is strictly closer than the second. Hence, rejection of the proposed test implies that the given instruments are better described as relevant and exogenous than irrelvant and endogenous. It is also shown that when the instruments are irrelevant and exogenous, the true distribution is equally close to both models. However, the limiting distribution of the likelihood ratio is di erent from the one obtained by Vuong (989) because the rst model is not identi ed. Nonetheless, this limiting distribution is nuisance parameter free and the relevant quantiles are reported. Keywords: exogeneity, relevance, instrumental variables, likelihood ratio. JEL Classi cations: C; C3 First draft: March 9. y Department of Economics, University of Virginia, Monroe Hall, McCormick Rd., Charlottesville, VA 93; email: dukpa@virginia.edu z Department of Economics, University of Michigan, 6 appan Street, Ann Arbor, MI 489; email: yoolee@umich.edu

Introduction A set of instrumental variables is said to be relevant if they are correlated with the endogenous regressors and exogenous if uncorrelated with the errors. It is a common practice in empirical studies to check these two conditions since the standard inference results on the structural parameters hold only when these conditions hold. he overidentifying restrictions (OID) test (e.g., Anderson and Rubin, 949; Sargan, 958; Basmann, 96) is widely used for the exogeneity condition, and the rst stage F and Wald tests are typically used for the relevance condition (or for the weak instruments). Cragg and Donald (993), Hall, Rudebusch and Wilcox (996) and Stock and Yogo (5) are more recently developed relevance tests. While there is no test considering both conditions simultaneously to the best of our knowledge, it has not been discussed much in the econometrics literature how to combine and interpret these two types of tests, either. he null distribution of the OID test is approximated by the Chi-square distribution under the implicit assumption that the instruments are relevant and strong. As Staiger and Stock (997) point out, however, the Chi-square distribution is a good approximation only if the instruments are strongly correlated with the endogenous regressors. Without the knowledge of the relevance of the instruments, we cannot be sure about the legitimacy of the Chi-square approximation. One may consider a two-stage testing procedure testing for the relevance rst and continuing to test for the exogeneity if the rst stage relevance test rejects no or weak relevance. However, the distribution of the OID test conditional on the rejection of the relevance test can be quite di erent from the unconditional distribution and the literature has not gured out, even asymptotically, what the exact error probability is when we use the conventional critical values. We performed some Monte Carlo experiments to show the di culties arising when the relevance test and exogeneity test are used in the conventional manner. Let (^(k )) and min (G ) be the standard OID test and Stock and Yogo s (5) weak instruments test, respectively, where ^(k ) is the k-class estimator. he limit distributions of these statistics are given in Staiger and Stock (997) as (^(k )) d (z u (z V + C ) ()) (z u (z V + C ) ()) () + () () () min (G ) d g min ((z V + C ) (z V + C )). () ables.a and.b report the rejection probabilities of the OID test, which are conditional Recently, Moreira (3) among others, propose an inferential method that is robust to arbitrarily weak instruments. his weak instruments robust inference, however, requires exogenous instruments and the necessity to check the exogeneity of instruments still remains (e.g., Doko and Dufour, 8).

on the rejection of the rst stage weak-iv test (rej ) and unconditional (uncond). he cases of C C :5 and :8 correspond to somewhat weak instruments, while : to strong instruments. he rst observation in able.a is that the size distortion of increases (e.g., see the uncond columns) as the instruments get weaker, which shows the danger of applying the standard OID test without knowing the strength of the instruments. Note that the actual sizes vary from less than 4% to more than % when the instruments are weakly correlated with the endogenous variables. he second observation is that the sequential procedure creates more size distortion in that the rejection probabilities of conditional on the rejection of g is always greater than their unconditional counterparts. Also, the size of is very liberal near, while it is mildly conservative near. able.b reports the results obtained using the Fuller-k estimator instead of the SLS estimator. Overall, able.b exhibits a great deal of similarity to able.a and the general conclusions from able.a remain valid. he main goal of this paper is to develop a testing procedure considering the relevance and exogeneity conditions at the same time. We employ Vuong s (989) model selection test. From a structural equation and its associated reduced form equation, we consider two competing models: One imposing the instruments to be exogenous and the other imposing the instruments to be irrelevant. Assuming normality, we show that the likelihood ratio of these two models is equivalent to the di erence of the standard rst stage Wald statistic and the OID test statistic. he null hypothesis is that the second model is at least as close to the true distribution of the observations as the rst model and the alternative hypothesis is that the rst is strictly closer to the true distribution than the second. In Vuong s (989) model selection test, rejecting the null hypothesis in favor of one model does not imply that the true distribution belongs to the favored model. It only implies that the true distribution is closer to the favored model compared to the competing model. Similarly, rejection of our test in favor of the rst model implies that the given set of instruments are better described as relevant and exogenous than irrelevant and endogenous. Non-rejection is interpreted as lack of evidence supporting the view that the given set of instruments is relevant and exogenous. Among other things, Vuong (989) assumes that the expectation of the likelihood functions of the competing models under the true distribution are maximized at a unique value. We show that this condition is satis ed when the instruments are relevant and the test can be performed using the scaled likelihood ratio combined with the standard normal critical values. However, when the instruments are irrelevant, the rst model is not identi ed and thus Vuong s (989) condition does not hold. Nonetheless, we show that the scaled likelihood ratio is asymptotically pivotal under the null hypothesis, although the distribution is non-standard. We provide the critical values for this limiting distribution. Using the big-

ger of the critical value of this non-standard distribution and that of the standard normal distribution will control the size of the test. his paper is organized as follows. Section describes the model and the test. Section 3 presents the asymptotic results pertaining to the likelihood ratio test under the null and the alternative hypotheses. Section 4 concludes with some remarks. All the technical proofs and simulation results are provided in Appendix. Model and est We consider a structural equation with an associated reduced form equation given by y Y + " (3) Y Z + V, (4) where y is a vector, Y is a n matrix of n endogenous variables, and Z is a K matrix of (excluded) exogenous variables to be used as instruments. he number of instruments, K, satis es n < K < and it is assumed to be xed. " and V are, respectively, a vector and a n matrix of random disturbances. here is often a K matrix of (included) exogenous variables, say X, in both (3) and (4). We will consider, without loss of generality, a model without X, since it can always be projected out. he main interest is to devise a test procedure which provides statistical evidence supporting the view that the set of instrumental variables Z is uncorrelated with " while it is correlated with Y. In other words, we are looking for evidence that one zero moment condition holds while the other zero moment condition fails. Obviously, it is di cult to test such a hypothesis in the standard testing framework. he strategy used in this paper is the model selection test proposed by Vuong (989). We assume that the true distribution of (y t ; Y t ) satis es the following two conditions. Assumption he observed variables y and Y are generated by: y Y + Z + u (5) Y Z + V Assumption (u t ; V t ) jz t i:i:d:(; ), where uu uv V u with uv ( V u ) 6 and the partition is conformable with (u t; Vt ). 3

In this speci cation, the set of instrumental variables Z is exogenous if so that it is orthogonal to the structural error. Z is relevant if 6 or more precisely is of full column rank, and thus correlations between the instruments and the endogenous regressors are nonzero. A key idea of the Vuong (989) model selection test is to use the likelihood ratio between two competing models. We call these two models F and G respectively. he model F imposes a restriction that the instruments are exogenous though its relevance is left unspeci ed. Model F : ( y Y + u Y Z + V with (u t ; V t ) jz t i:i:n(; ): On the other hand, the model G imposes a restriction that the instruments are irrelevant and can be even endogenous. Model G : ( y Y ' + Z + u V Y V with (u V t ; V t ) jz t i:i:n ; Both models assume that the errors are normally distributed. Let f t () and g t () be the density function for each model conditional on the instruments Z, where (; ; ) and ('; ; ; ). Using Vuong s (989) notation, the model F is denoted as a collection of densities, ff t (); g and the model G as a collection of densities, fg t (); the parameter space and each model to be : g, where are compact sets. De ne and, pseudo true values for arg max E log f t () arg max E log g t () hen, each model s distance from the true density h(y t ; Y t ) is the minimum KLIC given by E log h(y t ; Y t ) E log f t ( ) and E log h(y t ; Y t ) E log g t ( ) where the expectation E is with respect to the true joint distribution of (y t ; Y t ). In Vuong s (989) model selection test, the null hypothesis is that the two models are 4

equally close to the true density so that H : E log f t ( ) E log g t ( ) and the alternative hypothesis is either the model F is closer to the true density than the model G, H f : E log f t ( ) > E log g t ( ) or the opposite H g : E log f t ( ) < E log g t ( ). he KLIC is not observed but can be consistently estimated by its sample analog and it is natural to use the likelihood ratio statistic for the testing problem just described. Let log F () P t log f t() and log G() P t log g t(), where we use the notation P t P t. he likelihood ratio between the models F and G is given by LR max log F () max log G() log I n + G log( + (^ LIML )), (6) where ^ LIML is the LIML estimator, G ^ Y P Z Y ^ and (^ LIML ) ^u P Z ^u ^u M Z ^u (7) with ^ Y M Z Y, ^u y Y ^ LIML. he detailed derivation of (6) is given in Appendix. Note that the rst component of (6), which is approximately tr (G ), is nothing but the Wald statistic from the reduced form equation in (4) testing for. commonly used rst stage F statistic is equivalent to this statistic when there is only one endogenous regressor; Hall, Rudebusch and Wilcox s (996), and Stock and Yogo s (5) statistics are its variants. On the other hand, the second component of (6), which is approximately (^ LIML ), is the standard overidentifying restrictions (OID) test statistic. For example, it is Anderson-Rubin (949) statistic when the true value of is used instead of ^ LIML and is the Basmann s (96) test when the two stage least squares estimator is used. It is an interesting point that the LR statistic can be interpreted as the di erence between a relevant instrument test statistic and the standard OID test statistic, and naturally our test procedure can be viewed as providing a formal way of interpreting the di erence between the two commonly used test statistics. In terms of Vuong s (989) terminology, the models F and G are overlapping. In this he 5

case, the null hypothesis should be further split into two cases: (i) log f t ( ) log g t ( ), (ii) log f t ( ) 6 log g t ( ) but E log f t ( ) E log g t ( ). He shows that LR O p () under (i) but O p ( ) under (ii). In order to discriminate (i) and (ii), he also proposes to use a sequential procedure. Let ^s " X t log f t(^) g t (^) # LR : hen, he shows ( ^s O p() under (i); ^s a:s: otherwise. Hence the sequential procedure is rst to test for the case (i) using ^s. Not rejecting the rst step test automatically implies the null hypothesis is not rejected and the test procedure ends. When the rst step test is rejected, Vuong (989) continues to test for the case (ii) above. If (ii) is true, Vuong (989) shows that LR q ^s d N(; ) One of the key assumptions that Vuong (989) makes is that the pseudo true values and are unique and in the interior of the parameter space. hen, it is well known that the maximum likelihood estimates are asymptotically normal around these values. his is the key ingredient in deriving the limiting distributions of the LR and the variance statistics. In the next section, we will focus on the four cases for the true distribution: (a) 6 and 6 (relevant but endogenous IVs), (b) 6 and (relevant and exogenous IVs), (c) and (irrelevant and exogenous IVs) and (d) and 6 (irrelevant and even endogenous IVs). We show that unique and exist for (a) and (b) and Vuong s (989) results directly apply. For (c) and (d), there is no unique due to the lack of identi cation in the model F while a unique exists. his means that Vuong s (989) results do not apply and the limiting distributions of the LR and the variance statistics are di erent from the ones obtained by Vuong (989). Lastly, note that distinguishing the model G from the null hypothesis is not important for our testing problem. hypothesis is We naturally consider only one directional test where the null H : E log f t ( ) E log g t ( ) (8) 6

and the alternative hypothesis is H : E log f t ( ) > E log g t ( ): (9) In other words, we only look for evidence supporting the view the model F is closer to the true density than the model G. It should be emphasized that rejecting the null in favor of a certain model never implies that the true density belongs to the favored model. It only implies that the favored model is closer to the true density compared to the competing model. Hence, in our case, rejecting H in favor of H does not imply that the instruments are relevant and exogenous. It only implies that the instruments are better described as relevant and exogenous than irrelevant and endogenous. 3 Main Results Recall that (; ; ) and ('; ; ; ). he true density h(y t ; Y t ) depends on ( ; ; ; ). In order to state the pseudo true values conveniently, we de ne E Z t Z t, ( ; ) ( ; ), b ( ) and J I n heorem (Relevant IVs) Suppose that Assumptions and hold with 6. For the model F, let k be the smallest eigenvalue of with respect to and b be the corresponding eigenvector with the rst element being normalized to unity. hen, the pseudo true values are given by + k k V u ( ; )b b b b J ( + k)b b ( + k)b J ( + k)j b J J + kj b (b b ) b J () and E log f t ( ; ; ) log( + k) log j j For the model G, the pseudo true values are given by n +. ' + ( ) V u uu uv ( ) V u + 7

and E log g t ' ; ; ; log log + n +. When the instruments are relevant ( 6 ), there exist pseudo true values for both models regardless of the exogeneity parameter ( ) in the interior of the parameter space and Vuong s (989) theory directly applies. Note that log f t ( ) 6 log g t ( ) and the scaled LR statistic can be used without pretesting the variance test. Also, the true densities that belong to the null hypothesis H in (8) are those satisfying log( + k) log I n + ( ) ( ). () When the true density yields exact equality in (), the scaled LR statistic converges in distribution to the standard normal distribution, and when it yields strict inequality in (), it diverges to negative in nity. Hence, the asymptotic size of the test is controlled by the standard normal critical values. On the other hand, when the true density does not satisfy the condition in (), the scaled LR statistic diverges to positive in nity. A leading case of this instance is a set of good instrumental variables, which are both relevant and exogenous (i.e., 6 and ). Given its importance, we present this case in the following corollary. Note that k when. Corollary (Good IVs) Suppose that Assumptions and hold with 6 and. hen, for the model F, the pseudo true values are given by and, therefore and E log f t ( ; ; ) > E log g t ' ; ; ; LR q ^s a:s: as. Now, we consider the case of irrelevant instrumental variables. De ne P () ( ) I 8

and () uu() uv () P () V u () uu V u uv P (). hen, for any given, it can be shown that log f t (; ; ()) log f t ( ; ; ) when, and the model F is not identi ed. We look at the model F conditional on and let () and () be the solution to the partial maximization problem for any given. max ; E log f t (; ; ) heorem 3 (Irrelevant and Exogenous IVs) Let be xed at an arbitrary value. Suppose that Assumptions and hold with and. hen, for the model F conditional on, the pseudo true values are given by () () () and, for the model G, the pseudo true values are given by ' + ( ) V u. herefore, log f t (; (); ()) log g t (' ; ; ; ). his theorem states that for any given, there exist pseudo true values for the model F conditional on and the model G, and the log likelihoods evaluated at the pseudo true values are exactly the same. Consequently, we also have max E log f t () max E log g t () which belongs to the null hypothesis in (8). However, the limiting distributions of the LR and the variance statistic under the null hypothesis obtained by Vuong (989) do not apply, 9

because there is no unique pseudo true value for the unconditional model F. For any, the maximized expectation of the log likelihood function is the same. We de ne the conditional likelihood, LR() by LR() max log F (; ; ) max ; log I n + G log G() log( + ()), and the variance estimator, ^s () by ^s () " # X log f t(; ^(); ^()) g t (^) LR() t o obtain the limiting distribution of these two statistics, we show that the conditional likelihood LR() is close to a quadratic form uniformly in B so that LR() h M()h + o p () where the middle matrix M() is a continuous function of and the vector h does not depend on and converges in distribution to a standard multivariate normal distribution. hen, the limiting distribution of ^ LIML is combined with this expression using the Continuous Mapping heorem. A similar argument applies to the variance statistic. Under Assumption, we have p (Z u; Z V ) d ( Zu; ZV ) as, where ( Zu; vec( ZV ) ) N(; ). De ne Gaussian random matrices z u Zu( uu) z V ZV ( ) so that (zu; vec(z V ) ) N ; I K with I n

where ( ) V u ( uu). hen, we let (z u z V ) p and we have (vec(z V ) ; ) N ; I (n+)k. We state the limiting distributions of the LR and variance statistics. heorem 4 (Irrelevant and Exogenous IVs) Suppose that Assumptions and hold with and. Let c (zv z V I) zv and be the smallest root satisfying j(; z V ) (; z V ) I n+ j. hen, LR d vec(z V ; ) Mvec(z V ; ) and where ^s d vec(z V ; ) M vec(z V ; ), M I n c( + c c) c ( + c c) c ( + c c) c I K. +c c hese limiting distributions are non-standard but free of nuisance parameters. able below presents the critical values for the scaled LR statistic. When there is only one endogenous regressor (n ), the critical values are smaller than those of the standard normal distribution. However, when there are multiple endogenous regressors (n ) except when n, K 5 with % asymptotic size, the critical values are bigger. In order to control the size of test properly, the bigger of the standard normal critical values and those in able should be used. Remark 5 he limiting distribution of LR can be written as LR d tr zv (I K z V (zv z z V I) z V ) V + zv (z V z V I) zv in which case the rst expression corresponds to the limiting distribution of the rst stage F statistic, G and the second expression to the OID statistic, (^ LIML ). he last case we consider is when and 6. Recall that and J are de ned in (). hey are.8,.65 and.3 for.9,.95 and.99 quantiles.

heorem 6 (Irrelevant and Endogenous IVs) Let be xed at an arbitrary value. Suppose that Assumptions and hold with and 6. Let s b b with b ( ) hen, for the model F conditional on, the pseudo true values are given by. () b b b J () ( + s)b b ( + s)b J ( + s)j b J J + sj b (b b) b J and E log f t (; (); ()) log( + s) log j j For the model G, the pseudo true values are given by n +. ' + ( ) V u. herefore, and E log g t (' ; ; ; ) log j j n + E log f t (; (); ()) < E log g t (' ; ; ; ). For any xed, the true distribution with and 6 is closer to the model G than the model F, and it belongs to the set of null distributions. In fact, the true distribution with and 6 belongs to the model G. he distance between the true density and the model F is measured by the value s. Note that the maximizer of E log f t is on the boundary of B, and Vuong s (989) results do not apply again. Since the maximizer is on the boundary, s can be made arbitrarily small as can be arbitrarily large. Indeed, it can be shown that ^LIML O p, which means that the distance between the model F and the true density collapses to zero. As a result the scaled LR statistic does not diverge to negative in nity, but its distribution shifts left from the distribution obtained when and, and the asymptotic size

of our test remains at the controlled level. 4 Concluding Remark A test for the relevance and the exogeneity conditions is proposed using Vuong s (989) model selection test. In particular, the test statistic is derived from two competing models: one imposing the instruments to be exogenous and the other imposing the instruments to be irrelevant. he likelihood ratio of these two models is the di erence of the standard rst stage Wald statistic and the OID test statistic. When the instruments are relevant, it is shown that Vuong s (989) regularity conditions are satis ed and the scaled likelihood ratio is asymptotically standard normal under the null hypothesis that the true distribution is at least as close to the second model as the rst model. When the instruments are irrelevant, it is shown that Vuong s (989) result do not apply. Nonetheless, it is further shown that the test is asymptotically pivotal under the null hypothesis. he critical values of this asymptotic distribution is reported. Using the bigger of the standard normal critical values and those we report in this paper will control the size of the test. 3

Appendix: Mathematical Proofs Derivation of the Likelihood Ratio (6) model F as First, write the log-likelihood function of the log F () log j uuj u() u() uu log V ju h tr V ju Y Z u() Y Z u() i, where u() y Y, V u uu and V ju V u uv uu. Let ^u y Y ^ LIML. From the rst order conditions, we obtain ^ uu ^u ^u, and ^ Y Z ^ ^u(^u^u) Y M Z ^u(^u M Z ^u) ^ (Z Z) Z (Y ^u ^) (Z Z) Z (Y ^u^u M Z Y (^u M Z ^u) ) ^ V ju Y Z ^ ^u^ Y Z ^ ^u^ Y M Z Y Y M Z ^u(^u M Z ^u) ^u M Z Y. Note that ^V ju n ^ Y M Z Y ^ ^u M Z ^u n Y M Z Y ^u M Z ^u, where ^ ^ LIML I n. We then have max log F () log j^ uu j ^ V ju (n + ) ^u^u log Y ^u M Z ^u n+ M ZY (n + ) log( + (^ LIML )) log Y M ZY (n + ) Now, write the log-likelihood function of the model G as log G() log (y Z Y ') (y Z Y ') log j j tr Y Y. 4

Similarly as the rst case, we have ^ Y Y ^ (Z Z) Z (y Y ^') (Z Z) Z y Y (Y M Z Y ) Y M Z y ^' (Y Y ) Y (y Z ^) (Y M Z Y ) Y M Z y ^ (y Z ^ Y ^') (y Z ^ Y ^') y M Z y y M Z Y (Y M Z Y ) Y M Z y Y M ZY jy M Z Y j. Moreover, ^ n Y Y n Y M Z Y + Y P Z Y n n (Y M Z Y ) (I n + G )(Y M Z Y ) I n + G Y M Z Y, which yields max log G() ^ log ^ log Y M ZY (n + ) log I n + G (n + ). herefore, the LR statistic is derived as Proof of heorem LR max log F () max log G() log I n + G log( + (^ LIML )) First, the log likelihood of the model F is " log f t () log jj tr y t Yt Y t Z t y t Y Y t t Z t # 5

for each t. Let and. Rewrite y t Y Y t t Z t t ( ) + Zt + u t ( ) Z t + V t Y Vt + Zt( ) + u t Z t + V t V t + u t V t I n u t V t + ( ) Z t Z t + ( ) Z t. It follows that E log f t (; ; ) log jj tr 6 4 B @ I n + ( ) I n ( ) 3 C7 A5. he rst order condition gives that + ( ) I n I n ( ) b b b J J b J + b ( ; ) ( ; )b b ( ; ) J ( ; )b b ( + )b b ( J ( ; ) ) (J ( ; ))b J J +, where ( ; ) ( ; ), b and J. I n Note that log j j b log ( + )b b ( J ( ; ) ) (J ( ; ))b J J + log b ( + )b J J + + log (J ( ; )) b (b ( + )b ) b ( J ( ; ) ). 6

Now, taking di erential with respect to yields d E log f t (; ; ) d J J + log (J ( ; )) b (b ( + )b ) b ( J ( ; ) ) h h tr + (J ( ; ))b b i i ( + )b b ( ; ) d V ju or herefore, Also, let then ( ; )b b b b b J b. J b b b b ( ; ). s b b b b, j b j ( + )b b ( J ( ; ) ) (J ( ; ))b J J + ( + s)b b ( + s)b J ( + s)j b J J + sj b (b b ) b J ( + s) b J J + sj b (b b b ) b J ( + s)j b (b b ) b J ( + s) b b J J J b b b b J (b ( + s) b ) b J J b J J ( + s) I n I n ( + s) j j. (A.) his shows that E log f t (; ; ) log n + j j log( + s) log j j n + and b is chosen to minimize s. From the de nition of s, b is an eigenvector corresponding to the smallest eigenvalue of with respect to with the rst element being normalized 7

to unity. Let k be the smallest root. hen, by de nition, k uu k uv k V u k and thus k k V u. he proof for the model G is relatively straightforward and omitted. Proof of heorem 3 Proof of heorem 4 [o be added] We work with the concentrated likelihood functions given by log F (; ) log ^ uu ^ uu u() u() log ^V ju (n + ) ^ V ju (Y Z) M u() (Y Z) u() y Y M u() I u()(u() u()) u() and log G() log ^ ^ (y Z) M Y (y Z) ^ Y Y M Y I Y (Y Y ) Y. log ^ (n + ) Note that @ log F (; ) S F (; ) ^ @vec V ju I K vec Z M u() (Y Z) H F (; ) @ log F (; ) @vec@(vec) ^ V ju Z M u() Z + o p () @ log G() S G () Z M Y (y Z) @ ^ H G () @ log G() @@ ^ Z M Y Z + o p () 8

and the maximum likelihood estimates from the two models are ^() Z M u() Z Z M u() Y and Recall that we de ne u t () V t ^ Z M Y Z Z M Y y. ( ) I u t V t P () u t V t and uu () V u () uv () P () uu V u uv P (). Let whose second moments are given by hen, uniformly in B, u V t () u t () Vt V u() Vt u () V t u t () V u ()uu () () uu () uv () V u(), V ju () V u ()uu () uv (). ^() p ^ p. Furthermore, p S F ( ; ) p S G ( ) V ju () I K vec p Z V u () + o p () Z u V ( ) + o p () and H F ( ; ) V ju () + o p () H G ( ) + o p(), where o p () is uniformly in B. 9

he likelihood ratio statistic is given by and we rst analyze LR(). Note that LR log F (^(^ LIML ); ^ LIML ) log G(^) h i max log F (^(); ) log G(^) max LR() log F ( ; ) log F (^(); ) + (vec^) H F ( ; )vec^ + o p () log G( ) log G(^) + ^ H G ( )^ + o p (), where o p () is uniformly in B. Hence, LR() log F (^(); ) log G(^) log F ( ; ) log G( ) (vec ^()) H F ( ; )vec ^() + ^ H G ( )^ + o p () vec^() ^ V ju () vec ^() + o p () ^ because log F ( ; ) log G( ) log u() u() log Y M u() Y log u() u() u() Y Y u() Y (n + ) Y log y y y Y I n Y y Y Y I n log y y y Y Y y Y (n + ), Y log y M Y y log Y Y (n + ) log y y y Y Y y Y (n + ). Y (n + ) (n + )

he aylor expansion of the scores gives, uniformly in B, p S F ( ; ) + H F ( ; ) p vec^() + o p () p S G ( ) + H G ( ) p ^ + o p (). Hence, p " vec^() ^ H F ( ; ) H G( ) # V ju () S F ( ; ) p S G ( ) S F ( ; ) p S G ( ) + o p () + o p () " # " V ju () V ju () # I K I n+ vec p Z [V u (); u V ( )] + o p () h I n+ i vec p Z [V u (); u V ( )] + o p () " () # vec p Z [V; u V ( )] because vec vec p Z [V u (); u V ( )] " p Z [V; u V ( )] () # I K vec p Z [V; u V ( )] vec p Z [V u (); u V ( )] + o p () (), where () I n V u I n + ( ) uv ()uu () V u uv ()uu. ()

Now, it follows that with LR() h " vec ^ ^ V ju () h vec () V ju () p Z [V; u V ( )] vec^ + o p () ^ () # I K h + o p (). his expression shows that the LR() statistic is uniformly close to a continuous function of h and. Hence, the limiting distribution of the LR statistic can be obtained using the continuous mapping theorem. h does not depend on and follows the Central Limit heorem: h d vec(z V ) N ; I (n+)k. Now, we consider the eigenvalues of the middle matrix: " () V ju () () # I K. Note that where ^ LIML d uu (), hen, it can also be shown that () + ( ) c c (z V z V I) z V. uu (^ LIML ) d uu ( )( + c c) V u (^ LIML ) d uu ( ) c V ju (^ LIML ) d I n c( + c c) c.

Hence, after some algebra, and nally (^ LIML ) V ju () d I n c( + c c) c ( + c c) c ( + c c) c +c c (^ LIML ) LR d vec(z V ; ) Mvec(z V ; ) tr z (I K z V (zv V z z V I) z V ) V + zv (z V z V I) zv, where M I n c( + c c) c ( + c c) c ( + c c) c I K. +c c Similarly, we nd the variance statistic is close to the following quadratic from uniformly in B, and ^s () vec^ ^ V ju () V ju () V u()uu () uv () V ju () uu () ^s d vec(z V ; ) M vec(z V ; ). vec ^ + o p () ^ Proof of heorem 6 [o be added] 3

able.a. Rejection Probabilities of the Overidentifying Restrictions est (SLS) n K 3 K 9 C C min rej uncond rej uncond.5..55.365.45.3.66.4.56.5.75.53.667.66.8.9.43.667.79..33.6.3333.88 C C min rej uncond rej uncond.8..49.395.44.465.3.498.438.549.58.5.63.547.869.69.8.97.985.63.948..38.494.35.347 C C min rej uncond rej uncond...473.49.486.475.3.56.455.54.58.5.6.54.663.587.8.996.84.46.8..88.6.69.58 Note: n is the number of endogenous regressors; K is the number of instruments; represents the degree of endogeneity; C C corresponds to the weak limit of the concentration matrix (i.e., Z Z p C C); and min is the boundary value of the minimum eigenvalue for the weak instruments set based on the % SLS bias. he column rej shows P( > K n;:5 jg rejects) and uncond P( > K n;:5 ). he rst-stage weak IV tests using () are based on the % SLS/Fuller-k bias (see Stock and Yogo, 5, for the precise de nition) at the 5% signi cance level. he second-stage OID tests using () are based on the Chi-square distribution. 4

able.b. Rejection Probabilities of the Overidentifying Restrictions est (Fuller-k) n K 3 K 9 C C min rej uncond rej uncond.5..7.739.887.83.3.655.85.33.888.5.4.789.48.779.8.94.449.578.97..753.74.54.78 C C min rej uncond rej uncond.8..96.44.667.4.3.39.4.47.39.5.666.77.334.6.8.5.9.447.47..939.7.338.578 C C min rej uncond rej uncond...933.9.5.3.3.83.9.44.84.5.4.945.495.55.8.57.665.67.796..466.56.7.5 Note: n is the number of endogenous regressors; K is the number of instruments; represents the degree of endogeneity; C C corresponds to the weak limit of the concentration matrix (i.e., Z Z p C C); and min is the boundary value of the minimum eigenvalue for the weak instruments set based on the % SLS bias. he column rej shows P( > K n;:5 jg rejects) and uncond P( > K n;:5 ). he rst-stage weak IV tests using () are based on the % SLS/Fuller-k bias (see Stock and Yogo, 5, for the precise de nition) at the 5% signi cance level. he second-stage OID tests using () are based on the Chi-square distribution. 5

q able. Critical Values for LR ^s test ( and ) n n n 3 :9 :95 :99 :9 :95 :99 :9 :95 :99 K.94..4 3..6.49.58.73. 4.3..54.7.87.5.3.7.54 5.6.4.57.84.98.5.3.44.7 6.7.6.6.94.8.35.46.6.87 7.9.8.63.3.7.44.6.74.99 8.9.9.65..5.5.73.86 3. 9..9.66.9.33.6.84.98 3.4..3.66.6.4.66.96 3.9 3.35..3.68.33.46.73 3.7 3.9 3.45..3.7.39.53.8 3.6 3.9 3.55 3..3.7.45.59.85 3.6 3.39 3.64 4..33.7.5.65.9 3.35 3.48 3.74 5..33.7.57.7.97 3.44 3.57 3.8 6.3.34.7.6.76 3. 3.5 3.65 3.9 7.4.34.73.67.8 3.8 3.6 3.74 3.98 8.3.33.73.7.86 3.3 3.68 3.8 4.6 9.3.34.73.78.9 3.7 3.76 3.89 4.4.4.35.75.83.96 3. 3.84 3.97 4. Note: n is the number of endogenous regressors; K is the number of instruments. 6

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