Effect of Misspecifying the Disturbance Covariance Matrix on a Family of Shrinkage Estimators

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Effect of Misspecifying the Disturbance Covariance Matrix on a Family ofShrinkage EstimatorsAnoop Chaturvedia; Suchita Kesarwania

a Department of Statistics, University of Allahabad, Allahabad, India

Online publication date: 22 September 2010

To cite this Article Chaturvedi, Anoop and Kesarwani, Suchita(2011) 'Effect of Misspecifying the Disturbance CovarianceMatrix on a Family of Shrinkage Estimators', Communications in Statistics - Theory and Methods, 40: 1, 53 — 67To link to this Article: DOI: 10.1080/03610920903353691URL: http://dx.doi.org/10.1080/03610920903353691

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Communications in Statistics—Theory and Methods, 40: 53–67, 2011Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920903353691

Effect ofMisspecifying the Disturbance CovarianceMatrix on a Family of Shrinkage Estimators

ANOOP CHATURVEDI AND SUCHITA KESARWANI

Department of Statistics, University of Allahabad,Allahabad, India

The present article deals with the problem of misspecifying the disturbance-covariance matrix as scalar, when it is locally non scalar. We consider a familyof shrinkage estimators based on OLS estimator and compare its asymptoticproperties with the properties of OLS estimator. We proposed a similar familyof estimators based on FGLS and compared its asymptotic properties with theshrinkage estimator based on OLS under a Pitman’s drift process. The effect ofmisspecifying the disturbances covariance matrix was analyzed with the help ofa numerical simulation.

Keywords Asymptotic distribution; Linear model; Non spherical disturbances;Pitman’s drift process; Risk under quadratic loss; Shrinkage estimator; Steinrule estimators.

Mathematics Subject Classification 62J05; 62J07.

1. Introduction

The regression coefficients of a regression model are generally estimated underthe specification that the covariance matrix is spherical. The ordinary least square(OLS) estimator yields the best linear unbiased estimator in such cases. Theproblem of misspecification, in the context of linear model, arises when the nonspherical disturbances in the model are considered as spherical and, accordingly,OLS estimator is used for estimating the parameter vector. In case the disturbancesare non spherical, OLS remains unbiased and consistent but not efficient. In thiscase, the generalized least squares (GLS) estimator is more efficient than theOLS estimator provided the disturbances covariance matrix in known. When itis unknown, a feasible GLE (FGLS) estimator can be obtained by replacing theunknown parameter vector in the covariance matrix by its consistent estimator.However, it is often observed that, in finite samples, the estimation error ofcovariance matrix parameter leads to less efficient FGLS estimator as compared toOLS estimator.

Received July 18, 2008; Accepted September 21, 2009Address correspondence to Anoop Chaturvedi, Department of Statistics, University of

Allahabad, Allahabad, U.P. 211002, India; E-mail: [email protected]

53

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54 Chaturvedi and Kesarwani

Banerjee and Magnus (1999) proposed sensitivity statistics to decide whetherthe OLS estimators of the coefficients and the disturbance variance are sensitive todeviations from the assumption of spherical error covariance matrix. Banerjee andMagnus (2000) further showed the sensitivity of conventional F and t statistics tocovariance misspecification. Wan et al. (2007) extended their work by investigatingthe sensitivity of the restricted least squares estimator to covariance misspecificationwhere the restrictions may or may not be correct. They also provided analyticaljustification to some of the numerical findings reported in Banerjee and Magnus(1999). Grubb and Magee (1988) considered the distribution of OLS and FGLSestimators when the covariance parameter follows a Pitman drift process andcompare the variance of the OLS estimator with that of the FGLS estimator whendisturbances covariance matrix is locally non-scalar. Chaturvedi and Shukla (1990)considered a family of Stein-rule estimators for the coefficient vector obtainedthrough shrinking the OLS estimator towards the null vector and derived itsasymptotic distribution when the disturbances covariance matrix is non-scalar.They also compared the risk of this estimator with that of OLS estimator undera quadratic loss function. Chaturvedi et al. (1993) investigated the effects ofmisspecifying disturbances on the efficiency properties of Stein-rule estimator.Chaturvedi et al. (1996) proposed a family of shrinkage estimators for the restrictedregression model when regression coefficients are subject to linear restrictions andinvestigated the risk properties of proposed estimator.

In this article, we consider a general family of improved shrinkage estimatorsbased on the OLS estimator and compare the asymptotic properties of this estimatorwith the usual OLS estimator under a quadratic loss function when disturbances arenon spherical. We also compare the asymptotic properties of this class of estimatorswith that of shrinkage estimator based on the FGLS estimator under a Pitman’sdrift process. The results of a numerical simulation have been reported to observethe effect of misspecifying disturbance covariance matrix on different estimators infinite sample.

2. Family of Estimators and Asymptotic Distribution

Consider the following general linear model with non spherical disturbances

y = X� + u� (2.1)

where y�n×1� is a vector of n observations on the dependent variable, X�n×p�

is a matrix of observations on p weakly exogenous variables with full columnrank, ��p×1� is a vector of unknown regression coefficients, and u�n×1� is a vectorof disturbances assumed to follow normal distribution with E�u� = 0� E�uu′� =�2�−1��.The elements of� ≡ �� are assumed to be functions of (q × 1) parametervector � ∈ �; � being an open subset of the q-dimensional Euclidean space.

The ordinary least squares (OLS) estimator of � is given by

b = X′X−1X′y (2.2)

for which

E�b� = �

E�b − ��b − ��′ = �2�X′X�−1X′�−1��X�X′X�−1�

Clearly, the OLS estimator is still unbiased but no longer remains efficient.


https://www.researchgate.net/publication/237482549_A_NOTE_ON_THE_STEIN_RULE_ESTIMATION_IN_LINEAR_MODELS_WITH_NONSCALAR_ERROR_COVARIANCE_MATRIX?el=1_x_8&enrichId=rgreq-8cb4fdbd-60bf-4217-acb5-bdff305d9389&enrichSource=Y292ZXJQYWdlOzIzMzI4NDkwMTtBUzoxMDEzNjI5ODk3OTczODVAMTQwMTE3ODIyODY2Mw==

https://www.researchgate.net/publication/227391536_A_variance_comparison_of_OLS_and_feasible_GLS_estimators?el=1_x_8&enrichId=rgreq-8cb4fdbd-60bf-4217-acb5-bdff305d9389&enrichSource=Y292ZXJQYWdlOzIzMzI4NDkwMTtBUzoxMDEzNjI5ODk3OTczODVAMTQwMTE3ODIyODY2Mw==

https://www.researchgate.net/publication/4902208_On_the_sensitivity_of_the_restricted_least_squares_estimators_to_covariance_misspecification?el=1_x_8&enrichId=rgreq-8cb4fdbd-60bf-4217-acb5-bdff305d9389&enrichSource=Y292ZXJQYWdlOzIzMzI4NDkwMTtBUzoxMDEzNjI5ODk3OTczODVAMTQwMTE3ODIyODY2Mw==

https://www.researchgate.net/publication/222481483_The_sensitivity_of_OLS_when_the_variance_matrix_is_partially_known?el=1_x_8&enrichId=rgreq-8cb4fdbd-60bf-4217-acb5-bdff305d9389&enrichSource=Y292ZXJQYWdlOzIzMzI4NDkwMTtBUzoxMDEzNjI5ODk3OTczODVAMTQwMTE3ODIyODY2Mw==

Effect of Misspecifying 55

If � is known, the generalized least squares (GLS) estimator of � is given by

�∗ = X′��X−1X′��y� (2.3)

If � is unknown, then we replace � by its consistent estimator, say �, and get thefollowing FGLS estimator of �

� = X′��X−1X′��y� (2.4)

We define, for all j� k = 1� 2� � � � � q,

�j =�

��j�� jk =

�2

��j��k��

A = 1nX′�X Aj =

1nX′�jX Ajk =

1nX′�jkX

= B′A� j = �′Aj� jk = �′Ajk�

� = 1√nX′�u �j =

1√nX′�ju �jk =

1√nX′�jku�

We denote the class of all matrices or vectors having the same number of indices byboldface letters subscripted in bracket by that number. For example, A�2� denotesthe set of all matrices �Ajk� j� k = 1� 2� � � � � q�.

The following regularity conditions are required for the validity of Edgeworthexpansion of the distribution of proposed estimator (see Chaturvedi and Shukla,1990; Rothenberg, 1984):

C1: As n → �, matrix A approaches to a non-singular finite matrix.C2: Each matrix in the set A�1�, A�2�� A�5� and covariance matrix of each vector

in ��1�� 2�� 5� converges to a finite matrix as n → �C3: n−1�X′C2X� is bounded and tends to a finite matrix for all C in �6.C4: � has a stochastic expansion of the following form

√n��− �� = e+ Op�T

−1��

where e is asymptotically distributed as normal with mean vector of orderO�n− 1

2 � and covariance matrix �+ O�n− 12 �. Further, the third-order cumulants of

(�11� �22� � � � � �pp) are of orderO�n− 1

2 � and higher-order cumulants are of orderO�n−1�.For estimating the coefficients vector �, we consider a general family of

shrinkage estimators

�S =[1− k

n− p+ 2r��

�

]�� (2.5)

where k �>0� is a characterizing scalar, � ≡ ��.

� = �′X�X�

�� = �y − X��′��y − X��

and r�� is a real valued function of �.



Let us consider the following family of shrinkage estimator for �, whichis obtained by shrinking the OLS estimator towards the null vector, under theassumption of disturbances being homoscedastic:

bS =[1− k

n− p+ 2r��

�

]b� (2.6)

where

� = b′X′Xb�y − Xb�′�y − Xb�

and r�� is a function of �. If we substitute k = 0 in (2.6), we get the OLSestimator b.

Let us introduce the following notations:

D = 1nX′X D∗ = 1

nX′�−1��X � = 1

ntr[�−1��

]� = 1√

nX′u �0 =

�

�2��−1/2 =

2

�2�√n�′� �−1 =

1n��2�

�′D−1�

� = �′D��

Further, we write

rB =√n

�D1/2bS − � �B = − kv�√

n�r��0�D

1/2�

VB = D−1/2D∗D−1/2 − 2kv�2

n�

[r��0�

(D−1/2D∗D−1/2 − 2

�D1/2��′D∗D−1/2

)+ 2�2v�

r ′��0�D1/2��′D∗D−1/2

]�

The following theorem gives the asymptotic distribution of rs when n is large.

Theorem 2.1. The asymptotic distribution of rB , to order Op�n−1�, is normal with mean

vector �B and dispersion matrix VB.

Using Theorem 2.1, we obtain the following expression for the bias vector ofbS , to order O�n−1�:

EbS − � = −kv�2

n�r��0�� (2.7)

Further, MSE matrix, to order O�n−2�, is given by

E�bS − ��bS − ��′

= �2

n

[D−1D∗D−1 − kv�2

n�

{r��0�

(2D−1D∗D−1 − 4

��′D∗D−1 − kv

�r��0��

′)

+ 4�2v�

r ′��0��′D∗D−1

}]� (2.8)



3. Comparison of Estimators

3.1. Comparison Under Quadratic Loss Function

The quadratic loss function associated with an estimator � is defined as

L�� = �� − ��′Qn�� − �� (3.1)

where Qn is a p× p positive definite symmetric matrix of the loss function, theelements of which may depend on n. Further, we assume that Qn converges to apositive definite symmetric matrix Q as n → �.

The risk under quadratic loss function of an arbitrary estimator � is defined as

R� = E� − �′Qn� − �� (3.2)

Using (3.2), we observe that under the quadratic loss function criterion, thedifference between the risks of the estimators bS and b is given by

Rb− RbS =kv�4

n2�

{r��0�

(2tr�D−1D∗D−1Qn�−

4��′D∗D−1Qn� − kv

�r��0��

′Qn�

)+ 4

�2v�r ′��0��

′D∗D−1Qn�

}� (3.3)

Let �l�·� and �S�·� be, respectively, the maximum characteristic root and theminimum characteristic root of the matrix inside the bracket. Now we observe that

tr�D−1D∗D−1Qn� ≥ �S�−1��tr�D−1Qn�

�′C��

≤ �l�D−1Qn�

�′D∗D−1Qn�

�≤ ��′D∗��1/2��′QnD

−1D∗D−1Qn��1/2

�

≤ �l�−1��

��′QnD−1Qn��

1/2

��′D��1/2

≤ �l�−1��lD

−1Qn

v = 1ntr�−1�� ≤ �l�

−1��

Hence, we have:

Rb− RbS ≥kv�4

n2�

{r��0�

(2�S�

−1��tr�D−1Qn�− 4�l�−1��l�D

−1Qn�

− kr��0��l�−1��l�D

−1Qn�)+ 4�

�2vr ′��0��l�

−1��l�D−1Qn�

}�

Let us define

g = �S�−1��tr�D−1Qn�

�l�−1��l�D

−1Qn��



Then a sufficient condition that bS will dominate b is given by

0 ≤ kr��0� ≤ 2�g − 2�+ 4��2v

r ′��0�

r��0�� g > 2� (3.4)

If r ′��0� ≥ 0, a sufficient dominance condition is

0 ≤ kr��0� ≤ 2�g − 2� g > 2� (3.5)

3.2. Comparison Under a Pitman’s Drift Process for �

Let �0 be a value of �0 such that ��0� = In. Further, we assume that � ≡ �n followsthe process

�n = �0 + n−1/2�� (3.6)

where �n×1 is a non stochastic vector of order Op�1�. Thus,

�� → In as n → �

Let

�l��0� =�

��j��

∣∣∣∣�=�0

�jk��0� =�2

��j��k��

∣∣∣∣�=�0

∀j� k = 1� 2� 3 � � � � q�

We assume that n− 12X′�∗

t X and n− 12X′�tX are of order O�1� for all integers t > 3,

where �∗t is any product of the matrix subscripts in q and ��t� is any tth order

derivative of �� evaluated at � = �0. Further, we write

Aj��0� =1nX′�j��0�X

Ajk��0� =1nX′�jk��0�X

A∗jk��0� =

1nX′�j��0��k��0�X�

The following theorem states the MSE of estimators bS and �S .

Theorem 3.1. If �n follows the process (3.4), the expression for the MSE matrices ofthe estimators bS and �S , to order O�n−2�, are given by:

E[�bS − ��bS − ��′

]= �2

n

[D−1 − 1√

nD−1

∑Aj��0��jD

−1 + 1nD−1

∑A∗

jk��0��j�kD−1

− 12n

D−1∑

Ajk��0��j�kD−1

− k�2

n�

{r��0�

(2D−1 − 4+ kr��0�

��′)+ 4

�2r ′��0��

′}]

(3.7)



E[��S − ��S − ��′

]= �2

n

[D−1 − 1√

nD−1


−1 + 1nD−1

∑Aj��0�D

−1Ak��0��j�kD−1

− 12n

D−1∑

Ajk��0��j�kD−1 − 1

nD−1

∑Aj��0�D

−1Ak��0��jkD−1

+ 1nD−1

∑A∗

jk��0��jkD−1

− k�2

n�

{r��0�

(2D−1 − 4+ kr��0�

��′)+ 4

�2r ′��0��

′}]

� (3.8)

See the Appendix for the proof of above theorem.Further, the difference between the MSE matrices of bS and �S , to order O�n−2�

is given by

E[�bS − ��bS − ��′

]− E[��S − ��S − ��′

]= �2

n2D−1

[∑j�k

Pjk��0��j�k − �jk�

]D−1�

where

Pj�k��0� = A∗jk��0�− Aj��0�D

−1Ak��0�

= 1nX′�j��0�

[I − 1

nXD−1X′

]�k��0X

Thus, a sufficient condition for �S to dominate bS is that the matrix

D−1

[∑j�k

Pjk��0��j�k − �jk�

]D−1

is positive definite.

4. Monte-Carlo Study

We employed large sample asymptotic theory to derive the efficiency propertiesof the estimators. In order to judge the performance of these estimators in finitesample, we conducted a Monte-Carlo simulation experiment. We assume that thedisturbance term in the linear model

y = X� + u

follows a first-order autoregressive process

ut = �ut−1 + �t t = 1� 2� � � � � n�



where � is the autoregressive parameter with �� < 1 and �t’s are iid disturbancesfrom N�0� �2

�� so that Var�ut� = �2�1−�2

. We have taken our results for the followingparameter settings:

n = 20� 50� 100� p = 4� 10�

� = −0�3�−0�25�−0�2�−0�15�−0�10�−0�05� 0�0� 0�05� 0�10� 0�15�

0�20� 0�25� 0�30� 0�35� 0�4�

�′� = 1� 5�28� 10�56� 15�84� 21�12� 26�4� 31�68� 42�24� 47�52� 52�8�

where �′� is the length of the coefficient vector. The bias and MSE are empiricallyevaluated using 2,500 replications for each set of parameter values. Eor completeoutcomes, see Tables 1–5.

For each experimental setting, we observe that there is a range of values for �where �� is close to zero and there OLS based estimators uniformly dominate thecorresponding FGLS-based estimators. Similarly, there is a range of values � whereOLS-based estimators are uniformly dominated by the corresponding FGLS basedestimators.

We observe that as the number of observations n increases, the range whereOLS-based shrinkage estimators perform better than corresponding FGLS-basedshrinkage estimators usually shrinks. The relative risk of FGLS based shrinkageestimators as compared to that of OLS based shrinkage estimators is maximumfor �� < 0�05 and it have a tendency to decrease as the value of �� increases.Table 1 provides the range of values of � for which the two type of estimatorsuniformly dominate each other. Tables 2–5 provide the relative risk of FGLS-basedshrinkage estimators as compared to OLS-based shrinkage estimators for differentparameter settings. Since the change in value of �′� does not have much effect onthe relative performance of various estimators, we have provided tables only for n =20, 50 and �′� = 1, 10.56. The possible reason for relative risk not much effectedby changes in �′� is that the relative risk involves �′� in higher order terms andeffect of changes in �′� is not clearly visible from the values. We compared the

Table 1Range of � for which the two types of estimators perform uniformly

better than each other

n p

Range where OLS-basedshrinkage estimators

dominate theFGLS-based shrinkage

estimators

Range whereFGLS-based shrinkageestimators dominate theOLS-based shrinkage

estimators

20 4 −0�15 ≤ � ≤ 0�2 � < −0�2; � > 0�220 10 −0�15 ≤ � ≤ 0�25 � < −0�2; � > 0�350 4 −0�1 ≤ � ≤ 0�1 � < −0�1; � > 0�150 l0 −0�1 ≤ � ≤ 0�1 � < −0�1; � > 0�15100 4 −0�1 ≤ � ≤ 0�05 � < −0�1; � > 0�15100 10 −0�05 ≤ � ≤ 0�1 � < −0�1; � > 0�15



Table 2Relative risks of the different versions of shrinkage estimators based on FGLS as

compared to that of based on OLS (n = 20, �′� = 1)

k = 4 k = 10

� SR KK MSE AMSE SR KK MMSE AMSE

−0�3 0.9405 0.9405 0.9403 0.9405 0.9540 0.9540 0.9537 0.9540−0�25 0.9641 0.9641 0.9640 0.9640 0.9740 0.9740 0.9739 0.9740−0�2 1.0087 1.0087 1.0086 1.0087 0.9982 0.9982 0.9980 0.9982−0�15 1.0238 1.0238 1.0238 1.0238 1.0114 1.0114 1.0112 1.0115−0�1 1.0253 1.0253 1.0252 1.0253 1.0283 1.0283 1.0282 1.0283−0�05 1.0436 1.0436 1.0435 1.0436 1.0313 1.0313 1.0312 1.03130 1.0476 1.0476 1.0475 1.0476 1.0410 1.0410 1.0409 1.04090�05 1.0419 1.0419 1.0419 1.0420 1.0371 1.0371 1.0369 1.03730�1 1.0368 1.0368 1.0368 1.0368 1.0383 1.0383 1.0382 1.03830�15 1.0257 1.0257 1.0257 1.0257 1.0364 1.0364 1.0363 1.03640�2 1.0117 1.0117 1.0117 1.0117 1.0249 1.0249 1.0249 1.02490�25 0.9985 0.9985 0.9986 0.9985 1.0172 1.0172 1.0171 1.01730�3 0.9601 0.9601 0.9601 0.9601 1.0072 1.0072 1.0071 1.00720�35 0.9462 0.9462 0.9462 0.9462 0.9974 0.9974 0.9973 0.99730�4 0.9184 0.9184 0.9185 0.9184 0.9717 0.9717 0.9716 0.9717


compared to that of based on OLS (n = 20, �′� = 10�56)

k = 4 k = 10


−0�25 0.9709 0.9709 0.9708 0.9709 0.9708 0.9708 0.9708 0.9708−0�2 1.0006 1.0006 1.0006 1.0006 0.9943 0.9943 0.9943 0.9943−0�15 1.0257 1.0257 1.0257 1.0257 1.0110 1.0110 1.0110 1.0110−0�1 1.0362 1.0362 1.0362 1.0362 1.0236 1.0236 1.0236 1.0236−0�05 1.0405 1.0405 1.0405 1.0405 1.0322 1.0322 1.0322 1.03220 1.0499 1.0499 1.0499 1.0499 1.0383 1.0383 1.0383 1.03830�05 1.0433 1.0433 1.0434 1.0433 1.0469 1.0469 1.0469 1.04690�1 1.0361 1.0361 1.0361 1.0361 1.0356 1.0356 1.0356 1.03560�15 1.0369 1.0369 1.0369 1.0369 1.0382 1.0382 1.0382 1.03820�2 1.0084 1.0084 1.0084 1.0084 1.0243 1.0243 1.0243 1.02430�25 0.9960 0.9960 0.9960 0.9960 1.0174 1.0174 1.0173 1.01740�3 0.9736 0.9736 0.9736 0.9736 1.0055 1.0055 1.0055 1.00550�35 0.9423 0.9423 0.9422 0.9423 0.9919 0.9919 0.9918 0.99190�4 0.9199 0.9199 0.9199 0.9199 0.9763 0.9763 0.9763 0.9762




compared to that of based on OLS (n = 50, �′� = 1)

k = 4 k = 10


−0�3 0.8643 0.8643 0.8642 0.8644 0.8974 0.8973 0.8973 0.8974−0�25 0.9245 0.9245 0.9245 0.9244 0.9366 0.9366 0.9365 0.9367−0�2 0.9493 0.9493 0.9492 0.9493 0.9631 0.9631 0.9631 0.9631−0�15 0.9893 0.9893 0.9892 0.9893 0.9953 0.9953 0.9953 0.9954−0�1 1.0148 1.0148 1.0148 1.0147 1.0103 1.0103 1.0102 1.0103−0�05 1.0256 1.0256 1.0256 1.0256 1.0207 1.0207 1.0207 1.02060 1.0303 1.0303 1.0303 1.0303 1.0269 1.0269 1.0269 1.02690�05 1.0274 1.0274 1.0273 1.0274 1.0234 1.0234 1.0233 1.02350�1 1.0161 1.0161 1.0161 1.0161 1.0183 1.0184 1.0183 1.01830�15 0.9909 0.9909 0.9909 0.9910 0.9976 0.9976 0.9976 0.99750�2 0.9641 0.9641 0.9640 0.9642 0.9739 0.9739 0.9739 0.97400�25 0.9366 0.9366 0.9366 0.9366 0.9471 0.9471 0.9471 0.94720�3 0.8969 0.8969 0.8969 0.8968 0.9137 0.9137 0.9137 0.91370�35 0.8824 0.8824 0.8824 0.8824 0.8824 0.8824 0.8824 0.88230�4 0.8309 0.8309 0.8308 0.8309 0.8492 0.8492 0.8492 0.8491


compared to that of based on OLS (n = 50, �′� = 10�56)

k = 4 k = 10


−0�3 0.8638 0.8638 0.8638 0.8638 0.8967 0.8967 0.8967 0.8967−0�25 0.9097 0.9097 0.9097 0.9097 0.9310 0.9310 0.9310 0.9310−0�2 0.9622 0.9622 0.9623 0.9622 0.9584 0.9584 0.9584 0.9584−0�15 0.9876 0.9876 0.9876 0.9876 0.9923 0.9923 0.9923 0.9923−0�1 1.0108 1.0108 1.0108 1.0108 1.0121 1.0121 1.0121 1.0121−0�05 1.0194 1.0194 1.0194 1.0194 1.0194 1.0194 1.0194 1.01940 1.0229 1.0229 1.0229 1.0229 1.0263 1.0263 1.0263 1.02630�05 1.0277 1.0277 1.0277 1.0277 1.0221 1.0221 1.0221 1.02210�1 1.0122 1.0122 1.0122 1.0122 1.0120 1.0120 1.0120 1.01200�15 0.9907 0.9907 0.9907 0.9907 0.9995 0.9995 0.9995 0.99950�2 0.9727 0.9727 0.9727 0.9727 0.9779 0.9779 0.9779 0.97800�25 0.9303 0.9303 0.9303 0.9304 0.9515 0.9515 0.9515 0.95160�3 0.9087 0.9087 0.9086 0.9087 0.9201 0.9201 0.9201 0.92020�35 0.8671 0.8671 0.8671 0.8670 0.8768 0.8768 0.8768 0.87680�4 0.8365 0.8365 0.8365 0.8365 0.8409 0.8409 0.8409 0.8409



risk properties of the four very common forms of shrinkage estimators, Stein-ruleestimators (SR), double-K class of estimator (KK), Minimum Mean Square Errorestimator (MMSE), and Admissible Feasible MMSE (AFMMSE, denoted as AMSEin the tables).

Our findings clearly support the theoretical results that for the values of �,which are very close to zero, OLS-based shrinkage estimators usually perform betterthan corresponding FGLS-based shrinkage estimators.

Appendix

Proof of Theorem 2.1. We observe that:

1n�y − Xb�′�y − Xb� = 1

nu′In − X�X′X�−1X′u

= 1ntr�In − X�X′X�−1X′�uu′

= �2

ntr�In − X�X′X�−1X′��−1+ �Op�n

−1�

= �2v+ Op�n−1�

1b′X′Xb

= 1

n(� + 1√

nD−1�

)′D(� + 1√

nD−1�

)= 1

n�

[1+ 2

�√n�′� + 1

n��′D−1�

]= 1

n�

[1− 2

�√n�′�]+ Op�n

−2��

Up to order O�n−1�, we have:

rB =√n

�D1/2

[b − � − kr��

�n− p+ 2��b

]�

Further,

1�n− p+ 2��

= 1b′X′Xb

1�n− p+ 2�

�y − Xb�′�y − Xb�

= �2v

n�

[1− 2

�√n�′�]

so that

� = �

�2v

[1+ 2

�√n�′� + 1

n��′D−1�

]= �0 + �−1/2 + �−1�

Thus,

r�� = �0 + �−1/2 + �−1

= r��0�+ �−1/2r′��0�+ O�n−1��



Hence,

rB = 1�D1/2

[D−1� − kv�2

�√n

(1− 2

�√n�′�)(

� + 1√nD−1�

)�r��0�+ �−1/2r

′��0��

]= 1

�D1/2

[D−1� − kv�2

�√n

(� − 2

�√n�′�� + 1√

n�−1�

)r��0�

− kv�2

�√nr ′��0��−1/2

]= 1

�D−1/2� − kv�

�√nr��0�D

1/2� − kv�

n�

[r��0��D

−1/2� − 2D1/2�′��

+ 2�2v

r ′��0�D1/2��′�

](A.1)

Using (A.1), we get the cumulant generating function of rB as

KB�h� = lnE exp�ih′rB�

= −ikv�

�√nr��0�h

′D1/2� + ln[exp

(i1�h′D−1/2�

)×{1− i

kv�

n�h′(r��0��D

−1/2�− 2D1/2��′��+ 1�2v

r ′��0�D1/2��′�

)}]� (A.2)

Further,

E

[exp

(i1�h′D1/2�

)]= exp

[−12h′D−1/2D∗D−1/2h

](A.3)

E

[S exp

(i1�h′D1/2�

)]= i�D∗D−1/2h� exp

[−12h′D−1/2D∗D−1/2h

]Using (A.3) in (A.2), we get

KB�h� = −ikv�

�√nr��0�h

′D1/2� + ln[exp

(−12h′D−1/2D∗D−1/2

)×{1− i

kv�

n�h′(r��0��D

−1/2 − 2D1/2��′�+ 2�2v

r ′��0�D1/2��′

)}i�D∗D−1/2h

]= −i

kv�

n�r��0�h

′D1/2� +−12h′D−1/2D∗D−1/2h− i

kv�2

n�h′[r��0��D

−1/2D∗D−1/2

− 2D1/2��′D∗D−1/2�+ 2�2v

r ′��0�D1/2��′D∗D−1/2

]h

= ih′�B −12h′VBh (A.4)

which is the cumulant generating function of normal distribution N��B� VB�. Thisleads to the result of the theorem.



Proof of Theorem 3.1. When �0 follows the process (3.6), we observe

�� = In +1√n

∑i

�j��0��j +12n

∑j�k

�jk��0��j�k + O�n−3/2�

�−1�� = In −1√n

∑i

�j��0��j +1n

∑j�k

�j��0��k��0��j�k

− 12n

∑j�k

�jk��0��j�k + O�n−3/2��

Hence, we have:

D∗ = D − 1√n

∑j

Aj��0��j +1n

∑j�k

A∗jk��0��j�k

− 12n

∑j�k

Ajk��0��j�k + O�n−3/2� (A.5)

v = 1ntr�−1��

= 1+ O�n−1/2�� (A.6)

Substituting the approximate value of D∗ and v from (A.5) and (A.6) into expression(2.8), we get

MSE�bS� = E�bS − ��bS − ��′

= �2

n

[D−1 − 1√

nD−1

∑i

Aj��0�njD−1 + 1

nD−1

∑j�k

A∗jk��0��j�kD

−1

− 12n

D−1∑j�k

Ajk��0��j�kD−1

− k�2

n�

{r��0�

(2D−1 − 4+ kr��0�

��′)+ 4

�2r ′��0��

′}]

which gives (3.7). Again, we have:

E[��S − ��S − ��′ = �2

n

[A−1 + 1

nA−1

(∑j�k

Qjk�jk

)A−1 − k�2

n r ��

×{(

2A−1 − 4+ kr ��

��′)

+ 4�2

r ′ �� ′}]

� (A.7)

where

� =

�2= �′A�

�2A = 1

nX′��X�



We observe that

A = 1nX′��X

= D + 1√n

∑j

Aj��0��j +12n

∑j�k

Ajk��0��j�k + O�n−3/2�

A−1 = D−1 − 1√nD−1

∑j

Aj��0��jD−1 + 1

nD−1

∑j�k

Aj��0�D−1Ak��0��j�kD

−1

− 12n

D−1∑j�k

Ajk��0��j�kD−1 + O�n−3/2�

Qjk =1nX′�j�

−1��kX − AjA−1Ak

= 1nX′�j��0��k��0�X − Aj��0�D

−1Ak��0�+ O�n−1/2��

Substituting the above expression for A−1 and Qjk in (A.7), we have:

E[(

�S − �) (

�S − �)′]

= �2

n

[D−1 − 1√

nD−1


−1 + 1nD−1

∑Aj��0�D

−1Ak��0��j�kD−1

− 12n

D−1∑

Ajk��0��j�kD−1 − 1

nD−1

∑Aj��0�D

−1Ak��0��jkD−1

+ 1nD−1

∑A∗

jk��0��jkD−1

− k�2

n�

{r��0�

(2D−1 − 4+ kr��0�

��′)+ 4

�2r ′��0��

′}]

which leads to (3.8).

Acknowledgments

The authors are thankful to Prof. Shalabh for his helpful comments on an earlierdraft. The second author gratefully acknowledges the financial support from CSIRto carry out the present work.

References

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Chaturvedi, A., Van Hoa, T., Shukla, G. (1993). Performance of the Stein-rule estimatorswhen disturbances are misspecified as spherical. Econ. Stud. Quart. 44:97–107.



Chaturvedi, A., Van Hoa, T., Shukla, G. (1996). Improved estimation in the restrictedregression model with non spherical disturbances. J. Quant. Econ. 12:115–123.

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Rothenberg, T. J. (1984). Approximate normality of generalized least squares estimates.Econometrica 52(4):811–825.

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Effect of Misspecifying the Disturbance Covariance Matrix on a Family of Shrinkage Estimators

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