2_xxx_white.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 3, 2010, 371-381 ISSN 1307-5543 – www.ejpam.com SPECIAL ISSUE ON GRANGER ECONOMETRICS AND STATISTICAL MODELING DEDICATED TO THE MEMORY OF PROF. SIR CLIVE W.J. GRANGER Bootstrapping the Shrinkage Least Absolute Deviations Esti- mator Tae-Hwan Kim1∗ and Halbert White2 1 School of Economics, Yonsei University, 134 Shinchon-dong, Seodaemun-gu, Seoul, 120-749, Korea 2 Department of Economics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0508 Abstract. Kim and White [13] studied a James-Stein type estimator that shrinks towards a data- dependent point rather than a fixed point. This was subsequently extended and applied to combining the OLS and 2SLS estimators by [12, 14]. This approach can be used to combine any two estimators in an optimal way. While the risk dominance properties of the new shrinkage estimator have been well established, a clear prescription for how to conduct inference and hypothesis testing has been missing. In this paper, we close this gap using a bootstrap approach. 2000 Mathematics Subject Classifications: 62 Key Words and Phrases: Shrinkage, James-Stein type estimators 1. Introduction In commemorating Clive W.J. Granger’s life and work, it is fitting to note his delight in tackling interesting issues from unconventional angles, frequently opening up new areas of research and new ways of thinking about important topics in econometrics. As Jim Stock noted in his discussion of Granger’s work in a memorial session at the 2010 American Eco- nomic Association meetings, perhaps one of the most interesting developments in modern econometrics has been the very different asymptotic distribution theory required to treat the estimators emerging from Granger’s Nobel Prize-winning work on cointegration [6, 5], com- pared to the standard asymptotic normality results that previously prevailed. ∗Corresponding author. Email addresses: tae-hwan.kim�yonsei.a .kr (T. Kim), hwhite�u sd.edu (H. White) http://www.ejpam.com 371 c© 2010 EJPAM All rights reserved. T. Kim and H. White / Eur. J. Pure Appl. Math, 3 (2010), 371-381 372 Here we offer analysis that also involves unusual asymptotics, growing in a perhaps unan- ticipated way that we hope Clive would find gratifying, from distribution theory developed by [18], designed to better describe finite sample estimator distributions. The estimators we study are shrinkage estimators that have themselves been taken in a novel direction. Indeed, the seminal work of [17] and [11] transformed statistical thinking by showing the inadmissi- bility of the ordinary least squares (OLS) estimator and initiating an entire field devoted to the study of shrinkage techniques designed to gain the "optimal trade-off" between bias and vari- ance. At the outset, most research in this area focused exclusively on shrinking a given base estimator towards a fixed point, such as zero. As is well understood, this shrinkage vanishes asymptotically. Subsequently, researchers’ understanding of the issues involved in shrinkage deepened, and a different shrinkage approach emerged, one that shrinks estimators towards a data-dependent point. See, for example, [15, 16]. In contrast to fixed point shrinkage, data-dependent shrinkage does not vanish asymptotically. This approach has been extended to a fairly general context by [13], where the data- dependent point can be any other potentially biased estimator for the same parameter of interest and is allowed to be correlated with the base estimator itself. The method proposed in [13] can thus be viewed as a general way to combine two different estimators to improve estimation risk and prediction precision. A finite sample counterpart of the Kim and White estimator has been developed by [12]; in [14], this approach was used to combine the biased OLS estimator and the unbiased 2SLS estimator in the presence of endogeneity. Nevertheless, a drawback of the shrinkage estimators developed in [13] has been the lack of a measure of precision. The usual Gaussian asymptotics do not apply, as the limiting distribution of these shrinkage estimators is, as alluded to above, not normal. Instead, it is a nonlinear function of a normal random vector. Here we propose bootstrap methods to fill this gap. Using the bootstrap to obtain the sampling distribution of shrinkage estimators is a famil- iar approach in the literature. Vinod and Raj [20] apply a bootstrap method to a ridge-type shrinkage estimator to investigate economic issues in the Bell System divestiture. Brown- stone [3] bootstraps two shrinkage estimators: Mundlak’s restricted principal-components estimator and a Stein-rule estimator that shrinks the OLS estimator toward the Mundlak esti- mator. Brownstone uses non-pivotal statistics, i.e., the percentile method, to get the sampling distributions. He shows that the non-parametric bootstrap provides a good estimate of the es- timator’s risk and standard errors. Vinod [19] provides a solution to the non-pivotal problem for ridge regression by applying Beran’s double bootstrap. This method involves a bootstrap within a bootstrap, which is computationally intensive. The importance of using pivotal statistics to get a better bootstrap confidence interval is well known. See, for example, [2, 8, 9]. In this paper, we focus on bootstrapping James-Stein- type estimators that shrink towards a data-dependent point, working with pivotal statistics obtained by using lemmas 1 and 2 of [18]. We then use these pivotal statistics for the boot- strapping approximation to obtain standard errors and confidence intervals for our James- Stein-type shrinkage estimators. T. Kim and H. White / Eur. J. Pure Appl. Math, 3 (2010), 371-381 373 2. The Shrinkage Estimator and its Asymptotic Moments Suppose that one is interested in estimating a k-dimensional parameter vector β0 from data generated as yt = X ′tβ 0 + ǫt , t = 1,2, . . . , n, where we assume that (i) (X ′t ,ǫt) is inde- pendent and identically distributed (IID); and (ii) E(ǫt) = 0, 0< σ2 ≡ var(ǫt)<∞. Suppose that there is an estimator bn of β0 based on n observations of yt and X t . Traditional James- Stein estimators shrink towards a fixed point, usually zero when there is no prior information. Extending earlier proposals, [13] propose shrinking bn towards a data-dependent point, gn, where the base estimator bn and the data-dependent point gn are assumed to satisfy the following mild condition: � n1/2(bn− β 0) n1/2(gn− β 0) � d → � U1 U2 � ∼ N(ξ,Σ), (1) where ξ ≡ � 0 θ � , Σ≡ � A ∆ ∆′ B � , and A, B, and Σ are symmetric positive definite matrices. In their application, Kim and White used the least absolute deviations (LAD) estimator for bn and the OLS estimator for gn. For concreteness, we will use the same estimators here. The "Optimal Weighting Scheme" (OWS) estimator is δOW (bn, gn) = � 1−λ1 − λ2 (bn− gn) ′Qn(bn− gn) � (bn− gn) + gn, (2) where λ1 and λ2 are weights to be optimally chosen [see 13] and Qn is a random symmetric positive definite matrix defining the quadratic loss of estimation, (bn − β 0)′Qn(bn − β 0). We assume that n−1Qn converges in probability to a nonstochastic symmetric positive definite matrix Q. The limiting distribution of the OWS estimator is not a member of a location-scale family. If it were, then the usual studentization would be enough to yield a pivotal statistic. Neverthe- less, the result of [1] shows that the bootstrap can still estimate the true sampling distribution up to second-order terms, justifying studentization for non-location-scale families. With the same assumptions and notations used in [13] and using lemmas 1 and 2 of [18], it is straightforward to show that n1/2(δOW (bn, gn)− β 0) d → � 1−λ1 − λ2 (U1 − U2) ′Q(U1 − U2) � (U1 − U2) + U2. (3) We let h(U) denote the limiting distribution in (3). Note that there is no asymptotic bias in this case since the data-dependent point (the OLS estimator) is consistent. Since Σ is positive definite, there is a matrix P, such that Σ = PP ′. Let Z = P−1U . Then Z is normally distributed with mean vector µ = � 0k×1 0k×1 � and identity covariance matrix. The limiting random variable, h(U), can be rewritten as h(U) = � 1−λ1− λ2 Z ′P ′J ′1QJ1PZ � J1PZ + J2PZ ≡ H(Z), (4) T. Kim and H. White / Eur. J. Pure Appl. Math, 3 (2010), 371-381 374 where J1 ≡ [Ik;−Ik] and J2 ≡ [0k; Ik]. Let M1 ≡ P ′J ′1QJ1P, M2 ≡ J1P, and M3 ≡ J2P. Then we can further simplify H(Z) to H(Z) = M Z − � λ2M2Z Z ′M1Z � , (5) where M ≡ (1−λ1)M2+M3. Using lemmas 1 and 2 of [18], the following new results provide the first and second moments of the limiting distribution of the OWS estimator. Theorem 1. Let Hi(Z) be the ith component of H (Z). Then, under regularity conditions allow- ing the exchange of limit and integral, E(Hi(Z)) = 0. The next theorem shows the asymptotic second moment of the OWS estimator with the following definitions; let Mi j and M2i j be the (i, j)th element of M and M2 respectively, and let the indicator matrix Ii j be the zero matrix except with element (i, j) equal to one. Theorem 2. Let vi j be the (i, j)th element of E(H(Z)H(Z)′). Then, under the same conditions as in Theorem 3 in [13], we have vi j = ai j − bi j − ci j + di j, where 1. ai j is the (i, j)th element of M M ′; 2. bi j = λ2 2k∑ a=1 2k∑ b=1 MiaEabM2 j b, with Eab = Γ(1) −1 ∞∫ 0 ��N0t ��−1/2 tr(IabN−1 0t ) d t and N0t = I + 2tM1; 3. ci j = b ji; 4. di j = λ 2 2 2k∑ a=1 2k∑ b=1 M2iaFabM2 j b, with Fab = Γ(2) −1 ∞∫ 0 t ��N0t ��−1/2 tr(IabN−1 0t )d t. The diagonal terms vii are the asymptotic variances of the OWS estimates. We use these to construct a studentized statistic for bootstrapping. The only input needed to estimate vii is a consistent estimator of Σ. The diagonal sub-matrices A and B of Σ are easily estimated, since these are the covariance matrices of the LAD and the LS estimators. A consistent estimator of the off-diagonal matrix ∆ is also provided in [13] as follows: b∆=  bf (0)n−1 n∑ t=1 X tX ′ t   −1 n−1 n∑ t=1 S1t ,S ′ 2t    n−1 n∑ t=1 X t X ′ t   −1 , (6) where S1t = −X t(1[ǫt≤0]−0.5), S2t = X tǫt , and bf (0) is a kernel estimator of the density of ǫt evaluated at zero. T. Kim and H. White / Eur. J. Pure Appl. Math, 3 (2010), 371-381 375 3. Bootstrap Confidence Intervals In this section, we show how the results of the previous section can be used in bootstrap- ping the OWS estimator. Bootstrap methods for the LAD estimator (the base estimator in our case) are not new. [7] bootstraps the quantile estimator and shows that the bootstrap distribution converges weakly to the limiting distribution of the quantile estimator. However, bootstrapping a shrinkage LAD estimator (the OWS estimator in our case) has not previously been studied. We generate artificial data as y = Xβ0 + ǫ, where ǫ ∈ Rn and β0 ∈ Rk, with n = 80 and k = 3. We set β0 = 0 without loss of generality. We draw errors from the standard normal distribution. Each row of X is drawn from the joint normal distribution†, N(1,Ω), where the covariances are each 0.8 and the variances are one. Once the first set of data is generated, we take it to be our original data and pretend not to know the true value of β0. The OWS estimator, δni, and its asymptotic standard deviation, si , are computed using the original data and the method described in the previous section. We consider (i) the equal tail percentile−t method (studentized); (ii) the equal tail per- centile method (unstudentized); (iii) the naïve percentile method; and (iv) the “normal approximation” method for constructing confidence intervals for β0. For the percentile−t method‡, the population equation is given by Pr[t p L < n1/2(δni − β 0 i )/si < t p U] = 1−α. (7) The ideal (1−α)% confidence interval is (t p L , t p U ), but we cannot obtain this interval because the exact distribution of δni is not known. The population equation in (7) can be approxi- mated by the following sample equation: Pr[ts L < n1/2(δ∗ni − δni)/s ∗ i < ts U] = 1−α (8) where δ∗ni , s ∗ i are bootstrap estimates. Even though the sample equation in (8) can be solved in principle, in most cases this is intractable because the empirical distribution from which the bootstrap re-sampling is taken is not continuous. Hence, we approximate the solution to the sample equation using bootstrap re-sampling. By bootstrapping (yt , X t) pairs§, we generate {γi : i = 1, . . . , m}, where γi = n1/2(δ∗ ni − δni)/s ∗ i and m is the number of bootstrap re-samples. We take the α/2 percentile (bts L) and (1 − α/2) percentile (bts U) to be approximations for ts L and ts U , respectively. The (1−α)% bootstrap confidence interval is given by [δni −bts Usi/n 1/2,δni −bts Lsi/n 1/2]. (9) †We use the multivariate normal random vector generator, DNRVG, which is a FORTRAN subroutine in NSWC (Naval Surface Warfare Center) Library. We set the seed to be 3833981. ‡For the percentile method the population equation is Pr [t p L < δni − β 0 i < t p U] = 1− α and the sample equation is Pr[t s L < δ∗ ni − δni < t s U ] = 1− α. We compute the naïve percentile confidence intervals by taking the α/2 and (1− α/2) percentiles of {δ∗ ni ; i = 1, 2, . . . , m}. The normal approximation interval is given by [δni − 1.96si/n 1/2, δni + 1.96si/n 1/2]. §[3, 19] bootstrap residuals. We prefer bootstrapping pairs because this is more robust to assumptions on the error term. See [4]. T. Kim and H. White / Eur. J. Pure Appl. Math, 3 (2010), 371-381 376Table 1: Standard Errors and t-values. Estimator Variable Coefficient Std. Error t-value LS x1 0.077618 0.172397 0.450227 x2 −0.005561 0.169351 −0.032840 x3 0.015524 0.190734 0.081391 LAD x1 −0.001885 0.221519 −0.008510 x2 0.050000 0.217604 0.229775 x3 0.045658 0.245080 0.186299 OWS x1 0.072559 0.171099 0.424076 x2 −0.002030 0.168950 −0.011990 x3 0.017442 0.189935 0.091832 Table 1 summarizes the regression results using the original data. The LAD estimates have uniformly higher standard errors than the OLS estimates as we should expect, since the OLS estimator is the maximum likelihood estimator here. The standard errors of the OWS estimator are quite comparable to those of the OLS estimator. Table 2¶ shows the bootstrap 95% confidence intervals and some descriptive statistics such as the length and shape of the intervals‖. The percentile−t bootstrap confidence intervals are [-.243, .434], [-.336, .296], and [-.349, .406]. The confidence intervals cover the true β0 correctly. Note that these are not symmetric intervals, according to the reported shape statis- tics. The non-symmetry can be visualized using the histograms of the standardized bootstrap estimates given in Figures 1-3. In many cases, enforcing symmetry will cause size distortions. Allowing for asymmetry can be considered an advantage of using the bootstrap method. The percentile confidence intervals are fairly comparable to the percentile−t confidence intervals. They also are not symmetric and are a little bit shorter. As expected, the naïve percentile intervals have the exact same length as the percentile intervals, because the naïve percentile intervals are a shifted version of the percentile intervals. ¶Let ’cl’ be the lower bound and ’cu’ be the upper bound of a confidence interval. Then ’Length’ and ’Shape’ are defined as follows. (1) Length = cu - cl. (2) Shape = (cu - b)/(b - cl) where b is the JSLAD estimate computed from the original data set. Shape measures how asymmetric the bootstrap confidence interval is around its center (b). If Shape > 1, then (cu - b) > (b - cl). If Shape < 1, then (cu - b) < (b - cl). ‖We use the DQAGI subroutine in NSWC Library which allows us to compute the required integrals. T. Kim and H. White / Eur. J. Pure Appl. Math, 3 (2010), 371-381 377Table 2: Bootstrap 95% Con�den e Intervals. Confidence Interval Method Lower Bound Upper Bound Length Shape Percentile-t x1 −0.24321 0.434032 0.677245 1.144725 (Studentized) x2 −0.33606 0.296353 0.632408 0.893269 x3 −0.34916 0.406528 0.755685 1.061344 Percentile-t x1 −0.25978 0.395289 0.655070 0.971079 (Unstudentized) x2 −0.33165 0.301504 0.633151 0.920841 x3 −0.32338 0.383159 0.706539 1.073044 Naïve x1 −0.25017 0.404900 0.655070 0 (Percentile) x2 −0.30556 0.327596 0.633151 0 x3 −0.34828 0.358264 0.706539 0 Normal x1 −0.26280 0.407913 0.670709 1 (Approximation) x2 −0.33317 0.329116 0.662283 1 x3 −0.35483 0.389714 0.744544 1 Figure 1: Histogram of Standardized Bootstrap OWS Estimates for β1. T. Kim and H. White / Eur. J. Pure Appl. Math, 3 (2010), 371-381 378 Figure 2: Histogram of Standardized Bootstrap OWS Estimates for β2. Figure 3: Histogram of Standardized Bootstrap OWS Estimates for β3. REFERENCES 379 4. Conclusion We have used results of [18] to obtain the asymptotic moments of the OWS estimator shrinking to a data-dependent point. This permits us to use a consistent estimator for the asymptotic moments to construct pivotal non-parametric bootstrap statistics. We demonstrate their use by showing how to calculate bootstrap standard errors and confidence intervals for the OWS estimator. ACKNOWLEDGEMENTS This work was supported by National Research Foundation of Ko- rea - Grant funded by the Korean Government (NRF-2009-327-B00088). References [1] Babu, G.J. and Singh, K., “Inference on Means using the Bootstrap,” Annals of Statistics, 11, 999-1003. 1983. [2] Beran, R., “Prepivoting to Reduce Level Error of Confidence Sets,” Biometrika, 74, 457- 468. 1987. [3] Brownstone, D., “Bootstrapping Improved Estimators for Linear Regression Models,” Journal of Econometrics, 44, 171-187. 1990. [4] Efron, B. and Tibshirani, R.J., An Introduction to the Bootstrap, Chapman & Hall. 1993. [5] Engle, R. and Granger, C.W.J., “Co-Integration and Error Correction: Representation, Estimation, and Testing,” Econometrica, 55, 251-276. 1987. [6] Granger, C.W.J., “Co-Integrated Variables and Error-Correcting Models,” Discussion Pa- per 83-13, University of California at San Diego. 1983 [7] Hahn, J., “Bootstrapping Quantile Regression Estimators,” Econometric Theory, 11, 105- 121. 1995. [8] Hall, P., “On the Bootstrap and Likelihood-based Confidence Regions,” Biometrika, 74, 481-493. 1987. [9] Hall, P., “Theoretical Comparison of Bootstrap Confidence Intervals,” Annals of Statistics, 16, 927-953. 1988. [10] Hall, P., The Bootstrap and Edgeworth Expansion, Springer-Verlag. 1992. [11] James, W. and Stein C., “Estimation with Quadratic Loss,” Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability (vol. 1), Berkeley, CA: University of California Press, pp. 361-379. 1960. REFERENCES 380 [12] Judge, G. and Mittelhammer R.C., “A Semi-parametric Stein-like Basis for Combining Estimators under Quadratic Loss,” Journal of the American Statistical Association, 99, 479-487. 2004. [13] Kim, T.H. and White H., “James-Stein-Type Estimators in Large Samples with Application to the Least Absolute Deviations Estimator,” Journal of the American Statistical Associa- tion, 96, 697-705. 2001 [14] Mittelhammer, R.C. and Judge G., “Combining Estimators to Improve Structural Model Estimation under Quadratic Loss,” Journal of Econometrics, 128, 1-30. 2005. [15] Saleh, A.K.M.E. and Sen P.K., “On Some Shrinkage Estimators of Multivariate Location,” The Annals of Statistics, 13, 272-281. 1985. [16] Sen, P. K. and Saleh A.K.M.E., “On Preliminary Test and Shrinkage M-estimation in Lin- ear Models,” The Annals of Statistics, 15, 1580-1592. 1987 [17] Stein, C., “Inadmissibility of the Usual Estimator for the Mean of a Multivariate Dis- tribution,” Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability (vol. 1), Berkeley, CA: University of California Press, pp. 197-206. 1955. [18] Ullah, A., “Finite Sample Econometrics: A Unified Approach,” in Contributions to Econo- metric Theory and Application: Essays in Honour of A.L. Nagar, eds. R.A.L. Carter, J. Dutta, and A. Ullah. New York: Springer-Verlag, pp. 242-292. 1990. [19] Vinod, H.D., “Double Bootstrap for Shrinkage Estimators,” Journal of Econometrics, 68, 287-302. 1995. [20] Vinod, H.D. and B. Raj, “Economic Issues in Bell System Divesture: A Bootstrap Appli- cation,” Applied Statistics, 37, 251-261. 1988. Appendix Proof. (Theorem 1) Let f (Z) ≡ M Z and g(Z) ≡ λ2M2Z/Z ′M1Z . Let fi(Z) and gi(Z) be the ith elements of f (Z) and g(Z) respectively. Then Hi(Z) = fi(Z)− gi(Z). First, we note that E( fi(Z)) = E(Mi Z) = Mi E(Z) = 0 where Mi is the ith row of M . By defining g1i(Z) ≡ M2i Z and g2i(Z) ≡ λ2/Z ′M1Z , we have that gi(Z) = g1i(Z)g2i(Z). By definition, g1i(d) = M2id = M2i h µ+ ∂ ∂ µ i where M2i is the ith row of M2; and it can be shown that E(g2i(Z)) = λ2Γ(1) −1 ∞∫ 0 ��N0t ��−1/2 exp(−1/2µ′N1tµ) d t where N0t ≡ I + 2tM1, N1t ≡ 2tM1N−1 0t , and Γ is the gamma function. We define W (µ)≡ Γ(1)−1 ∞∫ 0 ��N0t ��−1/2 exp(−1/2µ′N1tµ) d t. Therefore, using lemmas 1 and 2 of [18], we have that E(gi(Z)) = E(g1i(Z)g2i(Z)) = g1i (d)E(g2i(Z)) = REFERENCES 381 M2i h µ+ ∂ ∂ µ i λ2W (µ)|µ=0 = 0. Finally, we note that ∂ ∂ µ W (µ)|µ=0 = Γ(1) −1 ∞∫ 0 ��N0t ��−1/2 exp(−1/2µ′N1tµ)(−N1tµ) d t |µ=0 = 0. Hence, we have the desired result: E(Hi(Z)) = 0. Proof. (Theorem 2) We first note the following: H(Z)H(Z)′ = f (Z) f (Z)′− f (Z)g(Z)′− g(Z) f (Z)′+ g(Z)g(Z)′. The expected value of the first term is E( f (Z) f (Z)′) = E(M Z Z ′M ′) = M E(Z Z ′)M ′ = M M ′. Hence ai j is the (i, j)th element of M M ′. The second term is given by E( f (Z)g(Z)′) = E(M Z h λ2M2Z Z ′M1Z i′ ) = λ2E h M ZZ ′M ′2 Z ′M1 Z i = λ2M E h ZZ ′ Z ′M1Z i M ′2. Let Eab be the (a, b) element of E h ZZ ′ Z ′M1Z i . Then Eab can be computed using Ullah’s lemmas as follows: Eab = E � Z ′ IabZ Z ′M1Z � = Γ(1)−1 ∞∫ 0 ��N0t ��−1/2 (tr(IabN−1 0t ) +µ ′N2tµ) exp(−.5 µ′N1tµ) d t |µ=0 = Γ(1)−1 ∞∫ 0 ��N0t ��−1/2 (tr(IabN−1 0t ) d t, where Iab is defined as before and where N2t = N−1 0t IabN−1 0t . Hence, the (i, j)th element of E( f (Z)g(Z)′) is given by bi j = λ2 ∑2k a=1 ∑2k b=1 MiaEbaM2 j b. By symmetry, the (i, j)th element of f (Z)g(Z)′ is equal to the (i, j)th element of E(g(Z) f (Z)′). Hence, ci j = g ji. The expected value of the last term is E(g(Z)g(Z)′) = E �h λ2 M2Z Z ′M1Z ih λ2M2 Z Z ′M1Z i′� = λ2 2E h ZZ ′ (Z ′M1 Z)2 i . Let Fab be the (a, b) element of E h ZZ ′ (Z ′M1 Z)2 i . Then Fab can be computed using Ullah’s lemmas as follows: Fab = E � Z ′ IabZ (Z ′M1Z)2 � = Γ(2)−1 ∞∫ 0 t ��N0t ��−1/2 (tr(IabN−1 0t ) +µ ′N2tµ) exp(−.5 µ′N1tµ) d t|µ=0 = Γ(2)−1 ∞∫ 0 t ��N0t ��−1/2 (t r(IabN−1 0t ) d t. Hence, the (i, j)th element is di j = λ 2 2 ∑2k a=1 ∑2k b=1 M2iaFabM2 j b which completes the proof.