Statistical Inference, Econometric Analysis and Matrix Algebra

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1 Statistica Inference, Econometric Anaysis and Matrix Agebra

2 Bernhard Schipp Water Krämer Editors Statistica Inference, Econometric Anaysis and Matrix Agebra Festschrift in Honour of Götz Trenker Physica-Verag A Springer Company

3 Editors Prof. Dr. Water Krämer Universität Dortmund LS Statistik und Ökonometrie Vogepothsweg Dortmund Germany Prof. Dr. Bernhard Schipp TU Dresden Fak. Wirtschaftswissenschaften Abt. Ökonometrie Mommsenstr Dresden Germany ISBN: e-isbn: Library of Congress Contro Number: c 2009 Physica-Verag Heideberg This work is subject to copyright. A rights are reserved, whether the whoe or part of the materia is concerned, specificay the rights of transation, reprinting, reuse of iustrations, recitation, broadcasting, reproduction on microfim or in any other way, and storage in data banks. Dupication of this pubication or parts thereof is permitted ony under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must aways be obtained from Springer. Vioations are iabe to prosecution under the German Copyright Law. The use of genera descriptive names, registered names, trademarks, etc. in this pubication does not impy, even in the absence of a specific statement, that such names are exempt from the reevant protective aws and reguations and therefore free for genera use. Cover design: WMXDesign GmbH, Heideberg Printed on acid-free paper springer.com

4 Optima Estimation in a Linear Regression Mode using Incompete Prior Information Hege Toutenburg, Shaabh, and Christian Heumann Abstract For the estimation of regression coefficients in a inear mode when incompete prior information is avaiabe, the optima estimators in the casses of inear heterogeneous and inear homogeneous estimators are considered. As they invove some unknowns, they are operationaized by substituting unbiased estimators for the unknown quantities. The properties of resuting feasibe estimators are anayzed and the effect of operationaization is studied. A comparison of the heterogeneous and homogeneous estimation techniques is aso presented. 1 Introduction Postuating the prior information in the form of a set of stochastic inear restrictions binding the coefficients in a inear regression mode, Thei and Godberger [3] have deveoped an interesting framework of the mixed regression estimation for the mode parameters; see e.g., Srivastava [2] for an annotated bibiography of earier deveopments and Rao et a. [1] for some recent advances. Such a framework assumes that the variance covariance matrix in the given prior information is known. This specification may not be accompished in many practica situations where the variance covariance may not be avaiabe for one reason or the other. Even if avaiabe, its accuracy may be doubtfu and consequenty its credibiity may be sufficienty ow. One may then prefer to discard it and treat it as unknown. Appreciating such circumstances, Toutenburg et a. [4] have introduced the method of weaky unbiased estimation for the regression coefficients and have derived the optima estimators in the casses of inear homogeneous as we as inear heterogeneous estimators through the minimization of risk function under a genera quadratic oss structure. Unfortunatey, the thus obtained optima estimators are not functions of Hege Toutenburg Institut für Statistik, Universität München, D München, Germany toutenb@stat.uni-muenchen.de 185

5 186 H. Toutenburg et a. observations aone. They invove the coefficient vector itsef, which is being estimated, besides the scaing factor of the disturbance variance covariance matrix. Consequenty, as acknowedged by Toutenburg et a. [4], such estimators have no practica utiity. In this paper, we appy a simpe operationaization technique for obtaining the feasibe versions of the optima estimators. The technique essentiay invoves repacement of unknown quantities by their unbiased and/or consistent estimators. Such a substitution generay destroys the optimaity and superiority properties. A study of the damage done to the optima properties is the subject matter of our investigations. It is found that the process of operationaization may often ater the concusions that are drawn from the performance of optima estimators that are not friendy with users due to invovement of unknown parameters. The pan of presentation is as foows. In Sect. 2, we describe the mode and present the estimators for the vector of regression coefficients. Their properties are discussed in Sect. 3. Some numerica resuts about the behaviour of estimators in finite sampes are reported in Sect. 4. Some summarizing remarks are then presented in Sect. 5. In the ast, the Appendix gives the derivation of main resuts. 2 Estimators for Regression Coefficients Consider the foowing inear regression mode: y = Xβ + ε, (1) where y is a n 1 vector of n observations on the study variabe, X is a n p matrix of n observations on the p expanatory variabes, β is a p 1 vector of regression coefficients and ε is a n 1 vector of disturbances. In addition to the observations, et us be given some incompete prior information in the form of a set of stochastic inear restrictions binding the regression coefficients: r = Rβ + φ, (2) where r is a m 1 vector, R is a fu row rank matrix of order m p and φ is a m 1 vector of disturbances. It is assumed that ε and φ are stochasticay independent. Further, ε has mean vector 0 and variance covariance matrix σ 2 W in which the scaar σ is unknown but the matrix W is known. Simiary, φ has mean vector 0 and variance covariance matrix σ 2 V. When V is avaiabe, the mixed regression estimator of β proposed by Thei and Godberger [3] is given by b MR =(S + R V 1 R) 1 (X W 1 y + R V 1 r) = b + S 1 R (RS 1 R +V) 1 (r Rb), (3)

6 Optima Estimation in a Linear Regression Mode 187 where S denotes the matrix X W 1 X and b = S 1 X W 1 y is the generaized east squares estimator of β. In practice, V may not be known a the time and then the mixed regression estimator cannot be used. Often, V may be given but its accuracy and credibiity may be questionabe. Consequenty, one may be wiing to assume V as unknown rather than known. In such circumstances, the mixed regression estimator (3) cannot be used. For handing the case of unknown V, Toutenburg et a. [4] have pioneered the concept of weaky unbiasedness and utiized it for the estimation of β. Accordingy, an estimator ˆβ is said to be weaky (R,r) unbiased with respect to the stochastic inear restrictions (2) when the conditiona expectation of R ˆβ given r is equa to r itsef, i.e., E(R ˆβ r)=r (4) whence it foows that the unconditiona expectation of R ˆβ is Rβ. It may be observed that the unbiasedness of ˆβ for β impies weaky (R,r) unbiasedness of ˆβ but its converse may not be necessariy aways true. Taking the performance criterion as R A ( ˆβ,β)=E( ˆβ β) A( ˆβ β), (5) that is, the risk associated with an estimator ˆβ of β under a genera quadratic oss function with a positive definite oss matrix A, Toutenburg et a. [4] have discussed the minimum risk estimator of β; see aso Rao et a. [1] for an expository account. The optima estimator in the cass of inear and weaky unbiased heterogeneous estimators for β is given by ˆβ 1 = β + A 1 R (RA 1 R ) 1 (r Rβ) (6) whie the optima estimator in the cass of inear and weaky unbiased homogeneous estimators is ˆβ 2 = β X W 1 [ ( y σ σ 2 + β β + A 1 R (RA 1 R ) 1 2 β )] Sβ Sβ β Sβ r Rβ. (7) Ceary, ˆβ1 and ˆβ 2 are not estimators in true sense owing to invovement of β itsef besides σ 2 which is aso unknown. As a consequence, they have no practica utiity. A simpe soution to operationaize ˆβ 1 and ˆβ 2 is to repace the unknown quantities by their estimators. Such a process of operationaization generay destroys the optimaity of estimators. If we repace β by its generaized east squares estimator b and σ 2 by its unbiased estimator ( ) 1 s 2 = (y Xb) W 1 (y Xb), (8) n p

7 188 H. Toutenburg et a. we obtain the foowing feasibe versions of ˆβ 1 and ˆβ 2 : β 1 = b + A 1 R (RA 1 R ) 1 (r Rb) (9) [ ( β 2 = b Sb s s 2 + b b + A 1 R (RA 1 R ) b )] Sb Sb b r Rb. (10) Sb It may be remarked that Toutenburg, Toutenburg et a. ([4], Sect. 4) have derived a feasibe and unbiased version of the estimator ˆβ 1 such that it is optima in the cass of inear homogeneous estimators. This estimator is same as β 1. It is thus interesting to note that when the optima estimator in the cass of inear heterogeneous estimators is operationaized, it turns out to have optima performance in the cass of inear homogeneous estimators. 3 Comparison of Estimators It may be observed that a comparison of the estimator β 1 with ˆβ 1 and β 2 with ˆβ 2 wi furnish us an idea about the changes in the properties due to the process of operationaization. Simiary, if we compare ˆβ 1 and ˆβ 2 with β 1 and β 2,itwirevea the changes in the properties of the optima estimator and its feasibe version in the casses of inear heterogeneous and inear homogeneous estimators. 3.1 Linearity First of a, we may observe that both the estimators ˆβ 1 and β 1 are inear and thus the process of operationaization does not ater the inearity of estimator. This is not true when we consider the optima estimator ˆβ 2 in the cass of inear homogeneous estimators and its feasibe version β 2. Further, from (9) and (10), we notice that β 2 = 1 s 2 + b Sb [b Sb β 1 + s 2 A 1 R (RA 1 R ) 1 r] (11) so that β 2 is a weighted average of β 1 and A 1 R (RA 1 R ) 1 r whie such a resut does not hod in case of ˆβ Unbiasedness From (9) and (10), we observe that R β 1 = R β 2 = r (12)

8 Optima Estimation in a Linear Regression Mode 189 whence it is obvious that both the estimators β 1 and β 2 are weaky (R,r) unbiased ike ˆβ 1 and ˆβ 2. Thus the operationaization does not disturb the property of weaky unbiasedness. Next, et us consider the traditiona unbiasedness property. It is easy to see that the optima estimator ˆβ 1 and its feasibe version β 1 in the cass of inear heterogeneous estimators are unbiased whie the optima estimators ˆβ 2 and its feasibe version β 2 in the cass of homogeneous estimators are generay not unbiased. This may serve as an interesting exampe to demonstrate that weaky unbiasedness does not necessariy impy unbiasedness. Thus, with respect to the criterion of unbiasedness, no change arises due to operationaization. 3.3 Bias Vector Let us examine the bias vectors of the estimators ˆβ 2 and β 2. It is easy to see that the bias vector of ˆβ 2 is given by where B( ˆβ 2 )=E( ˆβ 2 β) σ 2 = σ 2 + β Sβ A 1 Mβ, (13) M = A R (RA 1 R ) 1 R. (14) The exact expression for the bias vector of β 2 is impossibe to derive without assuming any specific distribution for the eements of disturbance vector ε. It may be further observed that even under the specification of distribution ike normaity, the exact expression wi be sufficienty intricate and any cear inference wi be hard to deduce. We therefore consider its approximate expression using the arge sampe asymptotic theory. For this purpose, it is assumed that expanatory variabes in the mode are at east asymptoticay cooperative, i.e., the imiting form of the matrix n 1 X W 1 X as n tends to infinity is a finite and nonsinguar matrix. We aso assume that ε foows a mutivariate norma distribution. Theorem I: If we write Q = n 1 S, the bias vector of β 2 to order O(n 1 ) is given by B( β 2 )=E( β 2 β) = σ 2 nβ A 1 Mβ + σ 2 [p n 2 β +(p 1) σ 2 ] β A 1 Mβ (15) which is derived in the Appendix.

9 190 H. Toutenburg et a. A simiar expression for the optima estimator to order O(n 2 ) can be straightforwardy obtained from (13) as foows: B( ˆβ 2 )= σ 2 (1 nβ + σ 2 ) 1 nβ A 1 Mβ = σ 2 σ 4 nβ A 1 Mβ + n 2 (β ) 2 A 1 Mβ. (16) If we compare the optima estimators ˆβ 2 and its feasibe version β 2 with respect to the criterion of bias to order O(n 1 ) ony, it foows from (15) and (16) that both the estimators are equay good. This impies that operationaization does not ater the asymptotic bias to order O(n 1 ). When we retain the term of order O(n 2 ) aso in the bias vector, the two estimators are found to have different bias vectors and the effect of operationaization precipitates. Let us now compare the estimators ˆβ 2 and β 2 according to the ength of their bias vectors. If we consider terms upto order O(n 3 ) ony, we observe from (15) and (16) that [B( ˆβ 2 )] [B( ˆβ 2 )] [B( β 2 )] [B( β 2 )] = 2σ 2 [p n 3 β +(p 2) σ 2 ] β β MA 2 Mβ. It is thus surprising that the feasibe estimator β 2 is preferabe to the optima estimator with respect to the criterion of the bias vector ength to the given order of approximation in the case of two or more expanatory variabes in the mode. If p = 1, this resut continues to hod true provided that β is greater than σ 2. Thus it is interesting to note that operationaization of optima estimator improves the performance with respect to the bias vector ength criterion. 3.4 Conditiona Risk Function From Toutenburg et a. ([4], p. 530), the conditiona risk function of ˆβ 1,givenr is R A ( ˆβ 1,β r) =E[( ˆβ 1 β) A( ˆβ 1 β) r] =(r Rβ) (RA 1 R ) 1 (r Rβ). (17) Simiary, the conditiona risk function of ˆβ 2 given r can be easiy obtained: R A ( ˆβ 2,β r) =E[( ˆβ 2 β) A( ˆβ 2 β) r] =(r Rβ) (RA 1 R ) 1 (r Rβ) σ 2 [ + σ 2 + nβ β Mβ + (1 + σ 2 ) ] nβ r (RA 1 R ) 1 r. (18)

10 Optima Estimation in a Linear Regression Mode 191 Using the resut we can express σ 2 σ 2 + nβ = σ 2 (1 nβ + σ 2 ) 1 nβ = σ 2 nβ σ 4 n 2 (β ) 2 + O(n 3 ), (19) R A ( ˆβ 2,β r) =(r Rβ) (RA 1 R ) 1 (r Rβ) + σ 2 nβ [β Mβ + r (RA 1 R ) 1 r] σ 4 β Mβ n 2 (β ) 2 + O(n 3 ). For the feasibe estimator β 1, it can be easiy seen that the conditiona risk function of β 1 given r is given by R A ( β 1,β r) =E[( β 1 β) A( β 1 β) r] =(r Rβ) (RA 1 R ) 1 (r Rβ)+ σ 2 (20) n trmq 1. (21) As the exact expression for the conditiona risk of the estimator β 2 is too compex to permit the deduction of any cear inference regarding the performance reative to other estimators, we consider its asymptotic approximation under the normaity of disturbances. This is derived in Appendix. Theorem II: The conditiona risk function of the estimator β 2 given r to order O(n 2 ) is given by R A ( β 2,β r) =E[( β 2 β) A( β 2 β) r] =(r Rβ) (RA 1 R ) 1 (r Rβ)+ σ 2 n trmq 1 σ 4 [ ( β n 2 β 2trMQ 1 )] Mβ 5 β. (22) It is obvious from (17) and (21) that the operationaization process eads to an increase in the conditiona risk. Simiary, comparing ˆβ 2 and β 2 with respect to the criterion of the conditiona risk given r to order O(n 1 ), we observe from (20) and (22) that the operationaization process resuts in an increase in the conditiona risk when trmq 1 > β Mβ β + r (RA 1 R )r β. (23)

11 192 H. Toutenburg et a. The opposite is true, i.e., operationaization reduces the conditiona risk when the inequaity (23) hods true with a reversed sign. If we compare the exact expressions (17) and (18) for the conditiona risk function given r, it is seen that the estimator ˆβ 1 is uniformy superior to ˆβ 2. This resut remains true, as is evident from (21) and (22), for their feasibe versions aso when the criterion is the conditiona risk given r to order O(n 2 ) and ( β trmq 1 ) Mβ < 2.5 β (24) whie the opposite is true, i.e., β 2 has smaer risk than β 1 when ( β trmq 1 ) Mβ > 2.5 β. (25) The conditions (24) and (25) have itte usefuness in actua practice because they cannot be verified due to invovement of β. However, we can deduce sufficient conditions that are simpe and easy to check. Let λ min and λ max be the minimum and maximum eigen vaues of the matrix M in the metric of Q, and T be the tota of a the eigenvaues. Now, it is seen that the condition (24) is satisfied so ong as T < 2.5λ min (26) which is a sufficient condition for the superiority of β 1 over β 2. Simiary, for the superiority of β 2 over β 1, the foowing sufficient condition can be deduced from (25): T > 2.5λ max. (27) We thus observe that the optima estimator ˆβ 1 is uniformy superior to ˆβ 2 with respect to both the criteria of conditiona and unconditiona risks. The property of uniform superiority is ost when they are operationaized for obtaining feasibe estimators. So much so that the superiority resut may take an opposite turn at times. Further, we notice that the reduction in the conditiona risk of ˆβ 1 over ˆβ 2 is generay different in comparison to the corresponding reduction in the conditiona risk when their feasibe versions are considered. The change in the conditiona risk performance of the optima estimators starts appearing in the term of order O(n 1 ). When their feasibe versions are compared, the eading term of the change in risk is of order O(n 2 ). This can be attributed to the process of operationaization. 3.5 Unconditiona Risk Function Now et us compare the estimators under the criterion of the unconditiona risk function.

12 Optima Estimation in a Linear Regression Mode 193 It can be easiy seen from (17), (18), (20), (21) and (22) that the unconditiona risk functions of the four estimators are given by R A ( ˆβ 1,β) =E( ˆβ 1 β) A( ˆβ 1 β) = σ 2 trv(ra 1 R ) 1 (28) R A ( ˆβ 2,β) =E( ˆβ 2 β) A( ˆβ 2 β) = σ 2 trv(ra 1 R ) 1 + σ 2 nβ [ β Aβ + σ 2 trv(ra 1 R ) 1 = σ 2 trv(ra 1 R ) 1 + σ 2 nβ σ 2 ] σ 2 + nβ β Mβ [ β Aβ + σ 2 trv(ra 1 R ) 1] σ 4 β Mβ n 2 (β ) 2 + O(n 3 ) (29) R A ( β 1,β) =E( β 1 β) A( β 1 β) = σ 2 trv(ra 1 R ) 1 + σ 2 n trmq 1 (30) R A ( β 2,β) =E( β 2 β) A( β 2 β) = σ 2 trv(ra 1 R ) 1 + σ 2 σ 4 n 2 β [ 2trMQ 1 5 n trmq 1 ( β Mβ β )] + O(n 3 ). (31) Looking at the above expressions, it is interesting to note that the reative performance of one estimator over the other is same as observed under the criterion of the conditiona risk given r. 4 Simuation Study We conducted a simuation experiment to study the performance of the estimators β 1 and β 2 with respect to the ordinary east squares estimator b. The sampe size was fixed at n = 30. The design matrix X contained an intercept term and six covariates which were generated from mutivariate norma distribution with variance 1 and equa correation of 0.4. The mean vector of the covariates was ( 2, 2, 2,2,2,2). The true response vector (without the error term ε) was then cacuated as ỹ = Xβ with the 7 1 true parameter vector β =(10,10,10,10, 1, 1, 1). The restriction matrix R was generated as a 3 7 matrix containing uniform random numbers. The true restriction vector (without the error term φ) was cacuated as r = Rβ. Then in a oop with 5,000 repications, in every repication, new error terms ε and φ were added in ỹ and r to get y and r respectivey. The errors were generated

13 194 H. Toutenburg et a. independenty from norma random variabes with variances σ 2 = 40 for ε i, i = 1,...,n and σ 2 /c for φ j, j = 1,2,3. The factor c contros the accuracy of the prior information compared to the noise in the data. If c is high, the prior information is more accurate than the case when c is ow. Note that c < 1 means that the prior information is more noisy than the data which indicates that it is probaby useess in practice. In fact we ony expect the proposed estimators to be better than b if c is consideraby arger than 1. For comparison of the estimators, we cacuated the measure MRMSE = k=1 1 7 ( ˆβ β) ( ˆβ β), (mean of root mean squared errors) where ˆβ stands for one of the estimators b, β1 or β2. Figure 1 shows the distribution of the root mean squared errors 1 7 ( ˆβ β) ( ˆβ β) for each estimator based on 5,000 repications with c = 100. This means that the prior information was not perfect but very reiabe (σ 2 /c = 40/100 = 0.4). A considerabe gain can be observed by using one of the new proposed estimators whie there is no noticeabe difference between β 1 and β 2.The MRMSEs in that run were 2.64 for b and 1.85 for β 1 and β 2. The picture changes when we decrease c. For exampe when c = 4 (which means that the standard error of φ j is haf of the standard error of the noise in the data), then the MRMSE were 3.09 for b and 2.65 for β 1 and β 2. Figure 2 shows the corresponding boxpots. But a genera concusion is not possibe since the resuts ceary aso depend on the matrices X, R and vector β itsef. OLS beta1 beta Fig. 1 Boxpot of root mean squared errors of the three estimators with c = 100

14 Optima Estimation in a Linear Regression Mode 195 OLS beta1 beta Fig. 2 Boxpot of root mean squared errors of the three estimators with c = 4 5 Some Summarizing Remarks We have considered the minimum risk approach for the estimation of coefficients in a inear regression mode when incompete prior information specifying a set of inear stochastic restrictions with unknown variance covariance matrix is avaiabe. In the inear and weaky unbiased heterogeneous and homogeneous casses of estimators, the optima estimators obtained by Toutenburg et a. [4] as we as their feasibe versions are presented. Properties of these four estimators are then discussed. Anayzing the effect of operationaizing the optima estimators, we have observed that the property of inearity is retained ony in case of heterogeneous estimation. So far as the property of weaky unbiasedness is concerned, the process of operationaization has no infuence. But when the traditiona unbiasedness is considered, it is seen that the optima heterogeneous estimator remains unbiased whie the optima homogeneous estimator is generay biased. This remains true when their feasibe versions are considered. In other words, the process of operationaizations does not bring any change in the performance of estimators. Looking at the direction and magnitude of bias, we have found that the optima estimator and its feasibe version in the case of homogeneous estimation have identica bias vectors to order O(n 1 ) impying that the operationaization process has no effect on the bias vector in arge sampes. But when the sampe size is not arge enough and the term of order O(n 2 ) is no more negigibe, the effect of operationaization appears. If we compare the optima estimator and its feasibe version with respect to the criterion of the ength of the bias vector to order O(n 3 ),itis seen that the operationaization improves the performance provided that there are two or more expanatory variabes in the mode. This resut remains true in the case of one expanatory variabe aso under a certain condition. Examining the risk functions, it is observed that the reative performance of one estimator over the other remains unatered whether the criterion is conditiona risk given r or the unconditiona risk.

15 196 H. Toutenburg et a. When we compare the risk functions of the optima heterogeneous estimator and its feasibe version, it is found that the process of operationaization invariaby increases the risk. Such is not the case when we compare the optima homogeneous estimator and its feasibe version. Here the operationaization may ead to a reduction in risk at times; see the condition (23). Next, it is observed that the optima heterogeneous estimator has aways smaer risk in comparison to the optima homogeneous estimator. When they are operationaized in a bid to obtain feasibe estimators, the property of uniform superiority is ost. We have therefore obtained sufficient conditions for the superiority of one feasibe estimator over the other. An important aspect of these conditions is that they are simpe and easy to check in practice. Further, we have observed the magnitude of change in the risk of one optima estimator over the other optima estimator is generay different when their feasibe versions are considered. In case of optima estimators, the change occurs at the eve of order O(n 1 ) but when the feasibe estimators are compared, this eve is of order O(n 2 ). This brings out the impact of operationaization process. Finay, it may be remarked that if we consider the asymptotic distribution of the estimation error, i.e., the difference between the estimator and the coefficient vector, both the optima estimators as we as their feasibe versions have same asymptotic distribution. Thus the process of operationaization does not show any impact on the asymptotic properties of estimators. It may ater the performance of estimators when the number of observations is not sufficienty arge. The difference in the performance of estimators is cear in finite sampes through simuation experiment. Appendix If we define we can write z = 1 n 1/2 X W 1 ε, 1 u = σ 2 n 1/2 ε W 1 ε n 1/2, v = 1 σ 2 ε W 1 XS 1 X W 1 ε, b Sb = β Sβ + 2β X W 1 ε + ε W 1 XS 1 X W 1 ε = nβ + 2n 1/2 β z + σ 2 v (32) s 2 1 = (n p) (y Xb) W 1 (y Xb) [ = σ u n 1/2 v ] + O p (n 3/2 ). (33) n

16 Optima Estimation in a Linear Regression Mode 197 Using these, we can express s 2 s 2 + b Sb = σ 2 nβ [ 1 + 2β z [ 1 + u n 1/2 v ] n + O p(n 3/2 ) n 1/2 β + σ 2 (1 + v) nβ + O p(n 3/2 ) = σ 2 [ nβ 1 + u n 1/2 v ] n + O p(n 3/2 ) [ 1 2β ( z n 1/2 β σ v 4β zz ) β σ 2 β = σ 2 nβ + σ 2 ( u 2β ) z β σ 2 n 2 β Utiizing these resuts, we can express ] 1 n 3/2 β ( v + 2uβ z + σ 2 + σ 2 v β 4β zz β (β ) 2 ] + O p (n 3/2 ) ) + O p (n 5/2 ). (34) where ( β 2 β) =( β 1 β) s 2 s 2 + b Sb A 1 Mb = ξ n 1/2 ξ 1/2 + 1 n ξ n 3/2 ξ 3/2 + 1 n 2 ξ 2 + O p (n 5/2 ), (35) ξ 0 = A 1 R (R A 1 R ) 1 (r Rβ) ξ 1/2 = A 1 MQ 1 z ξ 1 = σ 2 β A 1 Mβ ξ 3/2 = σ 2 [( β u 2β ) z β ξ 2 = σ 2 β [( v + 2uβ z + σ 2 + σ 2 v β ) ] ( u 2β z β A 1 MQ 1 z By virtue of normaity of ε, it is easy to see that ] A 1 Mβ + A 1 MQ 1 z E(ξ 0 r) =ξ 0, E(ξ 0 )=0, E(ξ 1/2 r) =E(ξ 1/2 )=0, E(ξ 1 r) =E(ξ 1 )=ξ 1,. 4β zz ) β (β ) 2 A 1 Mβ

17 198 H. Toutenburg et a. E(ξ 3/2 r) =E(ξ 3/2 )=0, E(ξ 2 r) =E(ξ 2 )= σ 2 [p β +(p 1) σ 2 ] β A 1 Mβ. Using these resuts, we obtain from (35) the expression (15) of Theorem I. Next, we observe from (35) that the conditiona risk function of β 2 to order O(n 2 ) is given by R A ( β 2,β r) =E[( β 2 β) A( β 2 β) r] = ξ 0Aξ n 1/2 E(ξ 0Aξ 1/2 ) Now it can be easiy seen that + 1 n E(ξ 1/2 Aξ 1/2 + 2ξ 0Aξ 1 )+ 2 n 3/2 E[ξ 0Aξ 3/2 + ξ 1/2 Aξ 1] + 1 n 2 E(ξ 1Aξ 1 + 2ξ 0Aξ 2 + 2ξ 1/2 Aξ 3/2)+O p (n 5/2 ). (36) E(ξ 0Aξ 1/2 r) =0, E(ξ 1/2 Aξ 1/2 r) =σ 2 trmq 1, E(ξ 0Aξ 1 r) =0, E(ξ 0Aξ 3/2 r) =0, E(ξ 1/2 Aξ 1 r) =0, E(ξ 1Aξ 1 r) = E(ξ 0Aξ 2 r) =0, E(ξ 1/2 Aξ 3/2 r) = σ 4 β σ 4 (β ) 2 β Mβ, [ trmq ( β )] Mβ β, where repeated use has been made of the resuts RA 1 M = 0 and MA 1 M = M. Substituting these resuts in (36), we obtain the resut stated in Theorem II. References [1] Rao, C.R., Toutenburg, H., Shaabh, Heumann, C.: Linear Modes and Generaizations: Least Squares and Aternatives (3rd ed.). Springer, New York (2008) [2] Srivastava, V.K.: Estimation of inear singe-equation and simutaneous equation modes under stochastic inear constraints: An annotated bibiography. Int. Stat. Rev. 48, (1980)

18 Optima Estimation in a Linear Regression Mode 199 [3] Thei, H., Godberger, A.S.: On pure and mixed estimation in econometrics. Int. Econ. Rev. 2, (1961) [4] Toutenburg, H., Trenker, G., Liski, E.P.: Optima estimation methods under weakened inear restrictions. Comput. Stat. Data An. 14, (1992)