On Standard Inference for GMM with Seeming Local Identi cation Failure

Transcription

1 On Standard Inference for GMM with Seeming Local Identi cation Failure Ji Hyung Lee y Zhipeng Liao z First Version: April 4; This Version: December, 4 Abstract This paper studies the GMM estimation and inference problem that occurs when the Jacobian of the moment conditions is known to be a matrix of zeros at the true parameter values. Dovonon and Renault (3) recently raised a local identi cation issue stemming from this type of degenerate Jacobian. The local identi cation issue leads to a slow rate of convergence of the GMM estimator and a non-standard asymptotic distribution of the over-identi cation test statistics. We show that the zero Jacobian matrix contains non-trivial information about the economic model. By exploiting such information in estimation, we provide GMM estimator and over-identi cation tests with standard properties. The main theory developed in this paper is applied to the estimation of and inference about the common conditionally heteroskedastic (CH) features in asset returns. The performances of the newly proposed GMM estimators and over-identi cation tests are investigated under the same simulation designs used in Dovonon and Renault (3). Keywords: Degenerate Jacobian, Conditionally Heteroskedastic Factors, GMM, Local Identi cation Failure, Non-standard Inference, Over-identi cation Test, Asymptotically Exact Inference Introduction The generalized method of moments (GMM) is a popular method for empirical research in economics and nance. Under some regularity conditions, Hansen (98) showed that the GMM estimator has standard properties, such as p T -consistency and asymptotic normal distribution. The overidenti cation test (J-test) statistics has an asymptotic Chi-square distribution. On the other hand, when some of the regularity conditions are not satis ed, the GMM estimator may have non-standard We acknowledge useful comments from Don Andrews, Xu Cheng, Denis Chetverikov, Yanqin Fan, Jinyong Hahn, Guofang Huang, Ivana Komunjer, Rosa Matzkin, Peter Phillips, Shuyang Sheng, Ruoyao Shi, Yixiao Sun, and participants in Econometrics seminar at UC Davis, UCLA, UCSD, UIUC, Yale and the Seattle-Vancouver Econometrics Conference. Any errors are the responsibility of the authors. y Department of Economics, University of Washington, 336 Savery Hall, Box 35333, Seattle, WA jihyung@uw.edu z Department of Economics, UC Los Angeles, 8379 Bunche Hall, Mail Stop: 4773, Los Angeles, CA zhipeng.liao@econ.ucla.edu

2 properties. For example, when the moment conditions only contain weak information, the GMM estimator may be inconsistent and have mixture normal asymptotic distribution (see, e.g., Andrews and Cheng,, Staiger and Stock, 997 and Stock and Wright ). Dovonon and Renault (3, hereafter DR) recently pointed out an interesting issue that occurs due to the violation of one regularity condition. When testing for common conditionally heteroskedastic (CH) features in asset returns, DR showed that the Jacobian of the moment conditions is a matrix of zeros at the true parameter value. This causes a slower than p T rate of convergence of the GMM estimator. A new limit theory was then developed to investigate the non-standard asymptotic distribution of the J-test. When H and p are the number of moment conditions and parameters, respectively, the J-test was shown to have an asymptotic distribution that lies between (H p) and (H). Their results extend the ndings in Sargan (983) and provide an important empirical caution - the commonly used critical values based on (H p) lead to oversized J tests under the degeneracy of Jacobian moments. This paper revisits the issue raised in DR. We also consider moment functions for which the Jacobian of the moment conditions is known to be a matrix of zeros at the true parameter values due to the functional forms of the moment conditions. We provide alternative GMM estimation and inference using the zero Jacobian matrix as additional moment conditions. These additional moment restrictions contain extra information of the economic model. This additional information is exploited to achieve the rst-order local identi cation of the unknown structural parameters. We construct GMM estimators with p T -consistency and asymptotic normality by adding the zero Jacobian as extra moment conditions. The J-test statistics based on the new set of moments are shown to have asymptotic Chi-square distributions. We apply the newly developed theory to the main example - inference on the common feature in the common CH factor model. When using J tests for the existence of the common feature in this model, DR suggests using the conservative critical values based on (H) to avoid the over-rejection issue. We show that, under the same su cient conditions of DR, the common feature is not only rst order locally identi ed, but also globally identi ed by the zero Jacobian moment conditions. As a result, our GMM estimators of the common feature have p T -consistency and asymptotic normality. Our J-test statistic for the existence of the common feature have asymptotic Chi-square distribution, which enables non-conservative asymptotic inference. Moreover, the Jacobian based GMM estimator of the common feature has the closed form expression, which makes it particularly well suited to empirical applications. The rest of this paper is organized as follows. Section describes the key idea of our methods in the general GMM framework. Section 3 applies the main results developed in Section to the common CH factor models. Section 4 contains simulation studies and Section 5 concludes. Tables, gures, main proofs are given in the Appendix, while selected proofs and further technical arguments are available from the supplemental Appendix.

3 Degenerate Jacobian in GMM Models We are interested in estimating some parameter R p which is uniquely identi ed by H (H p) many moment conditions: ( ) E [ (X t ; )] E [ t ( )] = ; (.) where X t is a random vector which is observable in period t. As illustrated in Hansen (98), the global identi cation together with other regularity conditions can be used to show standard properties of the GMM estimator of. For any, de ne E [ t ()] : Then () is an H p matrix of functions. The standard properties of the GMM estimator, such as p T -consistency and asymptotic normality, rely on the condition that ( ) has full rank. When ( ) = Hp, the properties of the GMM estimator are nonstandard. For example its convergence rate is slower than p T and the associated over-identi cation J-test statistic has a mixture of asymptotic Chi-square distributions. DR have established these nonstandard properties of GMM inference when ( ) = Hp, in the context of testing for the common feature in the common CH factor models. In this section, we discuss the same issue and propose an alternative solution in a general GMM context. De ne g () = vec( () ), then ( ) = Hp implies that g ( ) = ph. The zero Jacobian matrix provides ph many extra moment conditions: g () = ph when = : (.) The new set of moment restrictions ensures the rst order local identi cation of, when the Jacobian of g () (or essentially the Hessian of ()) evaluated at has full column rank. We de ne the corresponding Jacobian matrix of g () as H () where H () is now a ph p matrix g () ; When H ( ) H has full column rank the rst order local identi cation of could be achieved, which makes it possible to construct GMM estimators and J tests with standard properties. We next provide a Lemma that enables checking the rank condition of the moment conditions based on the Jacobian matrix. Lemma. Let h () be the h-th (h = ; : : : ; H) component function in (). Suppose that (i) belongs to the interior of ; and (ii) for any, ( ( ) ( ) = (.3) hh 3

4 if and only if =. Then the matrix H has full rank. Condition (ii) in Lemma. is the second order local identi cation condition of based on the moment conditions in (.). This condition is derived as a general result in DR (see, their Lemma.3), and is used as a high-level su cient assumption in Dovonon and Gonalves ¾ (4). Lemma. shows that when the moment conditions in (.) are used under the condition (ii), the rst order local identi cation of is achieved. The moment conditions in (.) alone may not ensure the global/unique identi cation of. However, as is globally (uniquely) identi ed by the moment conditions in (.), we can use the moment conditions in (.) and (.) in GMM to ensure both the global identi cation and the rst order local identi cation of. Let t t () and g t () = vec( t () ). We can de ne the GMM estimator of using all moment conditions as b m;t = arg min X T T X m t () W m;t m t () where m t () = ( t () ; gt ()) is an H(p + ) dimensional vector of functions, and W m;t is an (Hp + H) (Hp + H) weight matrix. Similarly, we de ne the GMM estimator of using only the moment conditions in (.) as b g;t = arg min where W g;t is an Hp Hp weight matrix. X T T X g t () W g;t g t () Assumption. (i) The Central Limit Theorem (CLT) holds: T P T m t( )! d N(; m ) where m is a positive de nite matrix with m = g where is an H H matrix and g is a ph ph matrix; (ii) W m;t! p m and W g;t! p W g, where W g is a symmetric, positive de nite matrix. We next state the asymptotic distributions of the GMM estimators b m;t and b g;t. Proposition. Under the conditions of Lemma., Assumption. and Assumption A. in the Appendix, we have (i) g g! ; p T ( b m;t )! d N(; ;m ) (.4) (.5) where ;m = H ( g g g) H; (ii) moreover if is uniquely identi ed by (.), then p T ( b g;t )! d N(; ;g ) 4

5 where ;g = (H W g H) H W g g W g H(H W g H). In the linear IV models, Condition (.3) does not hold under the zero Jacobian, and hence is not identi ed. Phillips (989) and Choi and Phillips (99) showed that if the Jacobian has insu cient rank but is not entirely zero in the linear IV models, some part of (after some rotation) is identi ed and p T -estimable. Proposition. shows that in some non-linear models, the degenerate Jacobian together with Condition (.3) and other regularity conditions can be used to derive a p T -consistent GMM estimator of. Our results therefore supplement the earlier ndings in Phillips (989) and Choi and Phillips (99). When W g = g, we have ;g = H g H ;m, which implies that b m;t is preferred to b g;t from the e ciency perspective. In some examples (e.g., the common CH factor model in the next Section), the computation of b g;t may be easier than b m;t. We next propose an estimator which is as e cient as b m;t, and can be computed similarly to b g;t. Let b g;t, b g ;T and b ;T be the consistent estimators of g, g and m respectively. These variance matrix estimators can be constructed using b g;t, for example. The new GMM estimator is de ned as b g ;T = arg min X T T X bg ;t() W g ;T bg ;t() where bg ;t() = g t () b g ;T b ;T t( b g;t ) and W g ;T = b g;t b g ;T b ;T b g;t. Theorem. Under the conditions of Proposition., we have p T ( b g ;T )! d N(; ;m ) (.6) where ;m is de ned in Proposition.. From Theorem., b g ;T has the same asymptotic variance as b m;t. Moreover, it is essentially computed based on the moment conditions (.), hence it bene ts computational simplicity whenever b g;t is easy to calculate. When the GMM estimators have the standard properties, it is straightforward to construct the over-identi cation test statistics and show their asymptotic distributions. As the model speci cation implies both the moment conditions in (.) and (.), one can jointly test their validity using the following standard result: T X J m;t T m t ( b m;t ) ^ m;t X T m t ( b m;t )! d (Hp + H p): (.7) When is identi ed by (.), it may be convenient to use the J-test based on (.5) in practice: T X T J g;t T g t ( b X g;t ) ^ g t ( b g;t )! d (Hp p); (.8) g;t 5

6 where b g;t denotes the GMM estimator de ned in (.5) with weight matrix W g;t = ^ g;t. One interesting aspect of the proposed J-test in (.7) is that it has standard degrees of freedom, i.e. the number of moment conditions used in estimation minus the number of parameters we estimate. Among the (Hp + H) many moment restrictions, H moments from (.) have degenerate Jacobian moments. By combining this information on H moments with the extra information provided by the Hp Jacobian moments, we avoid the issue of rank de ciency. Stacking the additional moments from (.) provides enough sensitivity of the J-test statistic to parameter variation. As a result, the standard degrees of freedom show up in the asymptotic Chi-square distribution in (.7). Without incurring greater computation costs, we prefer to have a more powerful test by testing more valid moment restrictions under the null. For this purpose, one can use the following test statistics: T X T J h;t T m t ( b X g;t ) W h;t m t ( b g;t ) where W h;t is an H(p + ) H(p + ) real matrix. Let m() = E [m t ()] for any. The following theorem provides the asymptotic distribution of J h;t. Theorem. Suppose that the conditions of Proposition. hold and W h;t! p W h where W h is non-random real matrix. Then we have (.9) where J h;t! d B H(p+) m P W h P m B H(p+) P I ( ph ;(H W g H) H W g ) and B H(p+) denotes an H(p + ) standard normal random vector. Theorem. indicates that the asymptotic distribution of J h;t is not pivotal. However, its critical values are easy to simulate in practice. The performance of the test statistics J m;t, J g;t and J h;t are investigated in the simulation study below in the common CH factor model. 3 Application to Common CH Factor Model Multivariate volatility models commonly assume fewer number of conditionally heteroskedastic (CH) factors than the number of assets. A small number of common factors may generate CH behavior of many assets, which can be motivated by economic theories (see, e.g., the early discussion in Engle et al., 99). Moreover, without imposing the common factor structure, there may be an overwhelming number of parameters to be estimated in the multivariate volatility models. From these theoretical and empirical perspectives, common CH factor models are preferred and widely used. Popular examples include the Factor Gaussian generalized autoregressive conditional heteroskedastic (GARCH) models (Silvennoien and Terasvirta; 9, Section.) and Factor Stochastic Volatility models (see, e.g., Section. of Broto and Ruiz (4) and references therein). It 6

7 is therefore important to test whether a common CH factor structure exists in multivariate asset returns of interest. Engle and Kozicki (993) proposed to detect the existence of common CH factor structure, or equivalently the common CH features, using the GMM over-identi cation test (Hansen, 98). Consider an n-dimensional vector of asset returns Y t+ = (Y ;t+ ; : : : ; Y n;t+ ) which satis es Var (Y t+ jf t ) = D t + ; where Var(jF t ) denotes the conditional variance given all available information F t at period t, is an n p matrix (p n), D t = diag( ;t ; : : : ; p;t) is a p p diagonal matrix, is an n n positive de nite matrix and ff t g t is the increasing ltration to which fy t g t and f k;t g t;kp are adapted. The Assumptions 3., 3., 3.3, 3.4, 3.5 and 3.6 below are from DR. Assumption 3. Rank() = p and V ar [Diag (D t )] is non-singular. Assumption 3. E [Y t+ jf t ] =. Assumption 3.3 We have H many F t -measurable random variables z t such that: (i) V ar (z t ) is non-singular, and (ii) Rank [Cov (z t ; Diag (D t ))] = p. Assumption 3.4 The process (z t ; Y t ) is stationary and ergodic with E[kz t k ] < and E[kY t k 4 ] <. We also allow weak dependence structure so that (z t ; vec (Y t Y t )) ful ll a central limit theorem. When p < n, e.g., p = n, there exists a nonzero R n such that =. The real vector is called the common CH feature in the literature. In the presence of the common CH feature, we have Var Y t+ jf t = D t + = = Constant. (3.) Note the CH e ects are nulli ed in the linear combination Y t+, while the individual return Y i;t+ s (i = ; : : : ; n) are showing CH volatility. The equations in (3.) lead to the following moment conditions E (z t z ) Y t+ Y t+ = H when = R n ; 6= : (3.) where z denotes the population mean of z t. Given the restrictions in (3.), one can use GMM to estimate the common feature and conduct inference about the validity of the moment conditions. DR have shown that GMM inference using (3.) is subject to the issue of zero Jacobian moments. The GMM estimator based on (3.) can therefore be as slow as T =4 with a nonstandard limiting distribution. As explained earlier, the J-test based on (3.) has an asymptotic mixture of two di erent chi-square distributions, (H p) and (H). Following DR s empirical suggestion - using critical values based on (H) rather than (H p) - provides conservative size control. We show that it is possible to construct p T -consistent and asymptotically normally distributed GMM 7

8 estimators and non-conservative J tests by applying theory developed in Section to this common CH factor model. Following DR, we assume that exclusion restrictions characterize a set R n that contains at most one unknown common feature up to a normalization condition denoted by N. Assumption 3.5 We have R n such that = \ N is a compact set and and = p, = : (3.3) The non-zero restriction on could be imposed in several ways. For example, the unit cost condition N = f R n ; P n i= i = g can be maintained without loss of generality. To implement this restriction, we de ne an n (n ) matrix G as G = C A n(n ) = I (n ) (n )! n(n ) where I (n ) is an (n ) (n ) identity matrix and (n ) is an (n ) (row) vector of ones. Then for any, = ; : : : ; n ; X n i i= = ; X n i i= : = G + l n (3.4) where = ( ; : : : ; n ) is an (n )-dimensional real vector and l n = (; ; ; ) is an n vector. Hence, we can write where is a non-degenerated subspace of R n = f : = G + l n, 8 g; (3.5). The following lemma studies the restriction that the existence and uniqueness of impose on the factor loading matrix. Lemma 3. Suppose that Assumption 3. holds. Then there exists a unique such that (3.3) holds if and only if (i) p = n ; and (ii) G is invertible. In such case, satis es: = ; X n i= ;i where = ( G ) l n is an (n ) vector. = G + l n In the rest of this section, we assume that there is a unique satisfying the restrictions in (3.) and (3.4). This means that the moment conditions in (3.4) together with the unit cost restriction We use this normalization in the rest of the paper because the main results in DR are also derived under this restriction. However, the issue in DR and our proposed solutions are irrespective of a speci c normalization condition. 8

9 (3.) identify a unique common feature. The moment conditions in (3.) become () E (z t z ) [Y t+ Y t+ E(Y t+ Y t+)] = H (3.6) when = R p, where the relation between and is speci ed in (3.4). Assumption 3.6 The vector belongs to the interior of. Following the notations introduced in Section, we de ne t () (z t z ) [Y t+ Yt+ E(Y t+ and g t () vec t (3.7) for any and any t. Then by de nition, we have () = E [ t ()] and g () = E [g t ()] for any : Under the integrability condition in Assumption 3.3, the nullity of the moment Jacobian occurs at a true common feature in (3.6) because ( ) = E (z t z ) Y t+ Y t+g = Hp : (3.8) We consider using both restrictions in (3.6) and (3.8) by stacking them: m ( ) E [m t ( )] ( ) g ( ) = (ph+h) : (3.9) As discussed in the previous section, the rst order local identi cation of could be achieved in (3.9), if we could show that the following matrix has full column ( = t t( = Hp H php Lemma 3. Under Assumptions 3., 3.3 and (3.3), the matrix H has full rank. Lemma 3. shows that is ( rst-order) locally identi ed by the stacked moment conditions. The source of the local identi cation is from the zero Jacobian matrix, which actually contains more information than that needed for the local identi cation. We next show that if is uniquely identi ed by (3.6), it is also uniquely identi ed by (3.8). Lemma 3.3 Under Assumptions 3., 3.3 and (3.3), is uniquely identi ed by (3.8). As t () is quadractic in, we de ne the Jacobian moment functions as the partial derivative of t () divided by to simpify notations. : 9

10 From Lemmas 3. and 3.3, we see that is not only ( rst-order) locally identi ed, but also globally identi ed by the moment conditions in (3.8). As a result, one may only use these moment conditions to estimate the common feature. It is clear that the moment conditions in (3.8) are linear in, which makes the corresponding GMM estimators easy to compute. The GMM estimator based on the stacked moment conditions may be more e cient, as illustrated in Proposition.. However, its computation may be costly, particularly when the dimension of is high. Using the sample average z of z t as the estimator of z, we construct the feasible moment functions as bm t () bt () bg t () (z t z) Y t+ Yt+ ((z t z) I p ) G Y t+yt+ The GMM estimator b m;t is calculated using (.4) by replacing m t () with bm t () and using the weight matrix W m;t constructed by a rst-step GMM estimator with identity weight matrix 3. In the supplemental appendix (Lee and Liao, 4), we show that under Assumptions 3., 3., 3.3, 3.4, 3.5 and 3.6, Proposition.(i) holds for the GMM estimators b m;t. From = G + l n, we can write where we de ne T T X bg t () = H T + S T ; : H T P T ((z t z) I p ) G Y t+y t+ G T and S T P T ((z t z) I p ) G Y t+y T t+ l n : (3.) Given the weight matrix W g;t, we can compute the GMM estimator b g;t as b g;t = H T W g;t H T H T W g;t S T : (3.) Let b g;t, b g ;T and b ;T be the estimators of g, g and respectively. The modi ed GMM estimator b g ;T as in (.6) can be also obtained as b g ;T = H T W g ;T H T H T W g ;T (S T F T ) : (3.) where W g ;T = b g;t b g ;T b ;T b g;t as earlier, and F T = b g ;T b ;T A T, and X T A T = T (z t z) G b g;t + l n Yt+Y t+ G b g;t + l n : In the supplemental appendix, we also show that under Assumptions 3., 3., 3.3, 3.4 and (3.3), 3 One can use the rst step estimator, for example, b T = (H T H T ) H T S T to construct the weight matrices W m;t, W g;t and W g ;T, and the estimators of g, g and in this Section.

11 Proposition.(i) and Theorem. holds for b g;t and b g ;T respectively. The closed form expressions of b g;t and b g ;T enable us to show their p T asymptotic normality without the compactness assumption on. After the GMM estimators b m;t, b g;t and b g ;T are obtained, we can use the J-test statistic de ned in (.7), (.8) and (.9) to conduct inference about the existence of common feature. It is clear that the test based on J g;t is the easiest one to use in practice because b g;t has a closed form solution and J g;t has an asymptotically pivotal distribution. The test using J h;t is also convenient, although one has to simulate the critical value. The test using J m;t is not easy to apply, when the dimension of parameter is high. 4 Simulation Studies In this section, we investigate the nite sample performances of the proposed GMM estimators and J tests using the Monte Carlo experiments D3, D4 and D5 from DR 4. Speci cally, Y t+ = (Y ;t+ ; Y ;t+ ; Y 3;t+ ) is generated from the following model: Y t+ z } 3 { 6 4 Y ;t+ Y ;t = k 4 F t+ z } 3 { f ;t+ f ;t u t+ z } 3 { u ;t+ u ;t+ 7 5 Y 3;t+ f 3;t+ u 3;t+ for k = ; ; 3, where k contains the factor loadings of asset returns Y t+ in the k-th simulation design, f l;t+ (l = ; ; 3) is generated from a GARCH model: f l;t+ = l;t l;t+ and l;t =! l + l f l;t + l l;t with 6 4!!! = 4 : : :6 : :4 :4 : : :8 t = ( ;t ; ;t ; 3;t ) are independent with u s for any t and s and are i.i.d. from N(; I 3 ) and N(; :5I 3 ) respectively. The simulation designs D3, D4 and D5 are de ned via their factor loadings: 3 = , 4 = ; 7 5 and 5 = I 3 4 We have also investigated our estimators and inference methods in their simulation design D and D and found similar results. The simulation results in D and D are available upon request. We thank Prosper Dovonon for sharing the Fortran Codes for simulation.

12 respectively. For each simulated sample with sample size T, we generate T + observations and drop the rst observations to reduce the e ect of the initial conditions of the data generating mechanism on the GMM estimation and inference. We consider the portfolio = ( ; ; ) where ( ; ) is a real vector. There are two sets of moment conditions for estimating. The moment conditions proposed in DR are: h E (z t+ z ) jy 3;t+ + (Y ;t+ Y 3;t+ ) + (Y ;t+ Y 3;t+ )j i =, (4.) where z t+ = (z ;t+ ; z ;t+ ; z 3;t+ ) = (Y ;t ; Y ;t ; Y 3;t ), and the moment conditions de ned using the Jacobian of the moment functions in (4.) are: E [(z t+ z )(Y j;t+ Y 3;t+ ) [Y 3;t+ + (Y ;t+ Y 3;t+ ) + (Y ;t+ Y 3;t+ )]] =, (4.) for j = ;. Four GMM estimators are studied in D4: (i) the GMM estimator b ;T based on (4.); (ii) the GMM estimator b m;t based on (4.) and (4.); (iii) the e cient GMM estimator b g ;T based on the modi ed moment conditions of (4.); and (iv) the GMM estimator b g;t based on (4.). The nite sample properties of these four GMM estimators in D4 are summarized in Table C.. In D4, both the moment conditions in (4.) and (4.) identify the unique common feature = (; ; ). Hence, we can evaluate the bias, variance and MSE of the GMM estimators in this design 5. From Table C., we see that: (i) with the growth of the sample size, the bias in b m;t, b g ;T and b g;t goes to zero much faster than b ;T ; (ii) the variance of b g ;T is smaller than b g;t which shows the gain in e ciency of using modi ed moment conditions with strong IVs; (iii) the nite sample properties of b m;t and b g ;T are very similar when the sample size becomes large (e.g., T = 5; ), and their variances are almost identical when the sample size is larger than,; (iv) the MSE of b ;T goes to zero very slowly when compared with b m;t, b g ;T and b g;t. We next investigate the properties of the J tests in D3, D4 and D5. We let W g;t and W h;t be the identity matrix when constructing J h;t. In addition to the J tests J m;t, J g;t and J h;t proposed in this paper, we also consider the J-test based on b ;T and the moment conditions in (4.). Following DR, we consider two critical values: (H) and (H p) for the last two J tests at the nominal size. The curves of the empirical rejection probabilities at each sample size of tests based on (H) and (H p) are denoted as Ori-GMM and Ori-GMM respectively. The empirical rejection probabilities of the J tests J m;t, J g;t and J h;t are denoted as E -GMM, Jac-GMM and Sim-GMM respectively. The empirical rejection probabilities of the J tests in D3 and D4 are depicted in Figure C.. In the simulation D3, we see that all the J tests we considered are undersized. As noted in DR, this undersized phenomenon may be explained by the lack of unique identi cation, and/or the fact that the IVs used in constructing the moment conditions (4.) and (4.) are weak in nite samples. On the other hand, the empirical size properties of the J tests are well illustrated in D4. From Figure 5 As noted in DR, there is no uniquely identi ed common feature in D3. As a result, the stacked moment conditions can not ensure a uniquely identi ed common feature either.

13 C., we see that the over-identi cation tests based on b g;t have nice size control. The test based on b m;t has slight over-rejection for each nominal size we considered, and its size converges to the nominal level with the growth of the sample size. Moreover, it is clear that for the J-test statistic based on b ;T, the test using (H) is conservative and undersized, and the test using (H) is over-sized. From the simulation results in D3 and D4, we see that the J tests based on b g;t have good size control. On the other hand, the tests based on b ;T with critical values from (H) is undersized. It is easy to see that the undersized test based on (H) su ers from poor power, while the test based on (H p) leads to over-rejection. It is interesting to check: (i) how much power the J tests based on b m;t and b g;t gain when compared with the test based on b ;T and (H); and (ii) whether they are less powerful than the tests based on b ;T and (H p). The empirical rejection probabilities of the J tests in D5 are depicted in Figure C.. From Figure C., we see that the tests based on b g;t are much more powerful than the tests based on b ;T and (H). The J-test J h;t is more powerful than the test J g;t which only uses the moment conditions (4.). Moreover, the J-test J h;t is as powerful as the test based on b ;T and (H p), which has large size distortion in the nite samples as we have seen in Figure C.. 5 Conclusion This paper investigates the GMM estimation and inference when the Jacobian of the moment conditions is degenerate. We show that the zero Jacobian contains non-trivial information about the unknown parameters. When such information is employed in estimation, one can possibly construct GMM estimators and over-identi cation tests with standard properties. Our simulation results in the common CH factor models support the proposed theory. In particular, the GMM estimators using the Jacobian-based moment conditions show remarkably good nite sample properties. Moreover, the J tests based on the Jacobian GMM estimator have good size control and better power than the commonly used GMM inference which ignores the information contained in the Jacobian moments. References [] Andrews, D.W.K., and X. Cheng (): Estimation and Inference With Weak, Semi-Strong and Strong Identi cation, Econometrica, 8, 53. [] Broto, C., and Ruiz, E. (4): Estimation Methods for Stochastic Volatility Models: A Survey, Journal of Economic Surveys, 8(5), [3] Dovonon and Gonalves ¾ (4): Bootstrapping the GMM Overidenti cation Test Under Firstorder Underidenti cation, Working Paper, Concordia University. [4] Dovonon, P., and Renault, E. (3): Testing for Common Conditionally Heteroskedastic Factors, Econometrica, 8(6),

14 [5] Engle, R. F., and Kozicki, S. (993): Testing for Common Features, Journal of Business and Economic Statistics, (4), [6] Engle, R. F., Ng, V. K., & Rothschild, M. (99): Asset Pricing with A Factor-ARCH Covariance Structure: Empirical Estimates for Treasury Bills, Journal of Econometrics, 45(), [7] Hansen, L.P. (98): Large Sample Properties of Generalized Method of Moments Estimators, Econometrica, 5, [8] Choi, I. and Phillips, P.C.B. (99): Asymptotic and Finite Sample Distribution Theory for IV Estimators and Tests in Partially Identi ed Structural Equations, Journal of Econometrics, 5(), 3-5. [9] Newey, W.K. and D. F. McFadden (994): Large Sample Estimation and Hypothesis Testing,, in R.F. Engle III and D.F. McFadden (eds.), Handbook of Econometrics, Vol. 4. North- Holland, Amsterdam. [] Phillips, P.C.B. (989): Partially Identi ed Econometric Models, Econometric Theory, 5, 8-4. [] Sargan, J.D. (983): Identi cation and Lack of Identi cation, Econometrica, 5, [] Silvennoinen, A. and Teräsvirta, T. (9): Multivariate GARCH Models, In Handbook of Financial Time Series, -9, Springer Berlin Heidelberg. [3] Staiger, D., and J.H. Stock (997): Instrumental Variables Regression With Weak Instruments, Econometrica, 65, [4] Stock, J.H., and J.H. Wright (): GMM With Weak Identi cation, Econometrica, 68, APPENDIX A Proof of the Main Results in Section Proof of Lemma.. We rst notice that by de nition, H = H ( ) ( = (A ; : : : ; A H ) ; (A.) where A h ( ) for h = ; :::; H. Because is an interior point in, the condition (ii) of lemma is equivalent to 8x R p ; x A h x hh = if and only if x = : (A.) Now, suppose that Rank (H) < p. Then there exists a non-zero ~x R p such that ~x H = ~x A ; : : : ; ~x A H =ph ; 4

15 which implies that ~x A h ~x = for all h. This contradicts (A.) and hence, we have rank (H) = p. P Assumption A. (i) m() = E [m t ())] is continuous in ; (ii) sup T T [m t() m()] = O p (T P i ); (iii) sup T = o p is continuous in. The proof of Proposition. is standard (see, e.g., Newey and McFadden, 994) and thus is omitted. We next present the proof of Theorem.. Proof of Theorem.. By de nition, T T X bg ;t() = T T X g t () b g ;T b ;T T t( b g;t ): (A.3) Using the consistency of b g ;T and b ;T, the p T -consistency of b g;t, Assumptions A. (iii) and (iv), we deduce that b g ;T b ;T T T X t( b g;t ) t( ) = O p (T ); (A.4) which together with T P T t( ) = O p (T ), Assumptions. (ii) and A. (ii) implies that T T X bg ;t() W g ;T T T X bg ;t() g() ;gg() = o p () (A.5) uniformly over, where ;g = g g g. It is clear that g() ;gg() is uniquely minimized at, because ;g is positive de nite and is identi ed by g() =. This together with the uniform convergence in (A.5) and the continuity of g() implies the consistency of b g ;T. Next, we note that b g ;T satis es the rst order condition T T t ( b g ;T W g;t T bg ;t( b g ;T ) = : (A.6) Applying the mean value theorem and using the consistency of b g ;T and b ;T, the p T -consistency of b g;t, (A.4), Assumptions. and A., we get T bg ;t( b g ;T ) = T +T T X h i g t ( ) g t( t ( e g ;T ) hp i T ( g ;T ) + o p (); (A.7) 5

16 where e g ;T denotes a p Hp matrix whose j-th (j = ; : : : ; Hp) column represents the mean value (between and b g ;T ) of the j-th moment function in g t (). Under Assumption.(ii), T h i g t ( ) g t( )! d N ; g g g : (A.8) Using Assumptions A.(iii) and (iv), and the p T -consistency of b g;t, we have T T t ( b g ;T p H and t ( e g ;T p H which together with (A.6), (A.7), (A.8), and the full rank of H and ;g proves the claimed result. Proof of Theorem.. Applying the mean value theorem, we get T m t ( b g;t ) = T m t ( ) + t ( e g;t ( b g;t ) (A.9) where e g;t denotes a p H(p + ) matrix whose j-th (j = ; : : : ; H(p + )) column represents the mean value (between and b g;t ) of the j-th moment function in m t (). Using Assumptions A.(iii) and (iv) and the p T -consistency of b g;t, we get t ( e g;t ( b g;t ) ) hp i T ( g;t ) + o p (): (A.) Using the standard arguments of showing Proposition., we have hp T ( b g;t )i = (H W g H) H W g T g t ( ) + o p () which together with (A.9) and (A.) implies that T m t ( b g;t ) = P T m t ( ) + o p (): (A.) Plugging the above expression in the de nition of J h;t W h;t! p W h, we immediately prove the result. and then applying Assumption. and B Proof of the Main Results in Section 3 Proof of Lemma 3.. First, we note that for any in (3.4), we can write = G + l n : (B.) 6

17 Under condition 3.3, there exists a unique if and only if the linear equations = p have one and only one solution. Using (B.), we can rewrite these linear equations as G = l n : (B.) It is clear that the above equations have a unique solution if and only if G is invertible. Moreover, the unique solution is = ( G ) l n. Proof of Lemma 3.. Let = ( ; : : : ; p ), where j s (j = ; : : : ; p) are n real vectors. It is easy to see that () = Cov (z t ; Diag (D t )) Diag( ) = G Diag( ) (B.3) where G Cov (z t ; Diag (D t )) and Diag( ) ( ) ; : : : ; ( p ) : Note that G is an H p matrix with full rank by Assumption 3.3. Hence, there is () = G ( ) ; : : : ; ( p ) p G where G is de ned in the main text. Using the Kronecker product, we get g () vec( () ) = (G G ) C A p p (B.4) which further implies that H =(G G ) C A G : (B.5) p p Let G ;hj (h = ; : : : ; H and j = ; : : : ; p) be the h-th row and j-th column entry of G. Then we can rewrite the equation (B.5) as H = P p j= G ;jg j jg. P p j= G ;HjG j jg C A ; where P p j= G ;hjg j jg is a p p matrix for any h = ; : : : ; H. Suppose that there is an ~x R p such that H~x = P p j= G ;jg j jg ~x. P p j= G ;HjG j jg ~x C A = Hp: (B.6) 7

18 But G has full column rank, which means that H~x = Hp if and only if G j jg ~x = p for all j. (B.7) The condition in (B.7) implies that = px G j jg ~x = G G ~x: j= We have shown in Lemma 3. that G is invertible, which implies that G G is also invertible. Hence there must be ~x =. Proof of Lemma 3.3. Recall the equation (B.4) in the proof of Lemma 3.: g () = (G G ) C A : p p (B.8) As the common feature satis es = p, we immediately have j = for any j = ; : : : ; p, which implies that g ( ) = ph. This shows that is one possible solution of the linear equations g () = ph. By the relation between = G + l n, and the de nition of the matrix H, we can write the linear equations g () = ph as l n H = (G G ) C A p pl n which together with the fact that H is a full rank matrix implies that is the unique solution. C Tables and Figures 8

19 Table C.. Finite Sample Properties of the GMM Estimators in D4 b g;t () b g ;T () b m;t () b ;T () b g;t () b g ;T () b m;t () b ;T () T=5 Bias Variance MSE T=, Bias Variance MSE T=5 Bias Variance MSE T=5, Bias Variance MSE T=, Bias Variance MSE T=5, Bias Variance MSE T=5, Bias Variance MSE Notes:. The simulation results are based on, replications;. the probability limits of b ;T, b m;t, b g;t and b g ;T are (,-); (iii) For the GMM estimators b m;t, b g;t and b g ;T, the weight matrices are constructed using the equation () in DR and the GMM estimator de ned in (3.) with identity matrix; (iv) for the GMM estimator b ;T, a rst-step estimator based on the moment conditions in (4.) and the identity matrix is calculated and then used to construct the e cient weight matrix. 9

20 Figure C.. The Empirical Rejection Probabilities of the Over-identi cation Tests in D3 and D4 Notes:. The simulation results are based on, replications;. to estimate the empirical size of the tests in di erent sample sizes, we start with T = 5 and move to T = 5; we then add 5 more observations each time until T = 6,.

21 Figure C.. The Empirical Rejection Probabilities of the Over-identi cation Tests in D5 Notes:. The simulation results are based on, replications;. to estimate the empirical size of the tests in di erent sample sizes, we start with T = 5 and move to T = 5; we then add 5 more observations each time until T = 5,.