a note on estimating linear trend in a regression model with serially correlated error components

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A NOTE ON ESTIMATING LINEAR TREND IN AREGRESSION MODEL WITH SERIALLY CORRELATEDERROR COMPONENTSSeuck Heun Song a , Dietmar Stemann b & Byoung Cheol Jung ca Department of Statistics , Korea University , Seoul, 136-701, Koreab Department of Economics , University of Hagen , Hagen, 58084, Germanyc Department of Economics , Korea University , Seoul, 136-701, KoreaPublished online: 02 Sep 2006.

To cite this article: Seuck Heun Song , Dietmar Stemann & Byoung Cheol Jung (2002) A NOTE ON ESTIMATING LINEAR TREND INA REGRESSION MODEL WITH SERIALLY CORRELATED ERROR COMPONENTS, Communications in Statistics - Theory and Methods,31:8, 1385-1398, DOI: 10.1081/STA-120006075

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©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.

MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016

REGRESSION ANALYSIS

A NOTE ON ESTIMATING LINEAR TREND

IN A REGRESSION MODEL WITH

SERIALLY CORRELATED ERROR

COMPONENTS

Seuck Heun Song,1,* Dietmar Stemann,3

and Byoung Cheol Jung2

1Department of Statistics, and 2Department ofEconomics, Korea University, Seoul 136-701, Korea3Department of Economics, University of Hagen,

Hagen 58084, Germany

ABSTRACT

We consider a linear trend regression model when the distur-bances follow a serially correlated one-way error componentmodel. In this model, we investigate the performance of theOrdinary Least Squares Esitmator (OLSE), First DifferenceEstimator (FDE), Generalized Least Squares Estimator(GLSE) and the Cochrane-Orcutt-Transformation Estimator(COTE) of slope coefficient in terms of efficiency. The mainfindings are as follows: ð1Þ when the autocorrelation is close tounity, then the FDE is approximately the GLSE; ð2Þ theOLSE is better than the COTE; and ð3Þ when the value ofthe autocorrelation is kept constant and T ! 1, the OLSE,

1385

Copyright & 2002 by Marcel Dekker, Inc. www.dekker.com

COMMUN. STATIST.—THEORY METH., 31(8), 1385–1398 (2002)

*E-mail: [email protected]

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COTE and GLSE are asymptotically equivalent whereas theFDE is worse than the other estimators in terms of efficiency.

Key Words: Panel data regression; OLSE; FDE; GLSE;COTE; Autocorrelation

1. INTRODUCTION

For the linear regression model with autocorrelated disturbances, avariety of estimators for the regression coefficients have been proposed inthe literature. In earlier applied studies, research workers frequentlyattempted to deal with the problem of autoregression in disturbances byusing the method of first differences (Refs. [1–4]). In the linear regressionmodel with AR(1) error process[5] has shown that the FDE is asymptoticallyequivalent to the GLSE for fixed sample size as the correlation increases, forboth homogeneous and inhomogeneous regressions.

In this paper we extend Kramer’s (Refs. [4] and [5]) results to the panellinear regression model with a serially correlated one-way error component.We will obtain a close relationship between the FDE and the GLSE whenthe autocorrelation is close to unity. Moreover, we will show that, in whichcases, an estimator is worse or better than its competitors in terms of effi-ciency which is defined here as the ratio of there variances.

This paper is organized as follows: In Section 2, we consider a lineartrend regression model when the disturbances follow a serially correlatedone-way error components. In Section 3, we derive the variances of theGLSE, FDE, OLSE and the COTE of a slope coefficient. In Section 4, wediscuss the relative performance of estimators.

2. THE MODEL

We consider the following simple linear trend regression model:

yit ¼ �þ �tþ uit, i ¼ 1, . . . ,N, t ¼ 1, . . . ,T , ð2:1Þ

where yit is an observation on a dependent variable for the ith cross sec-tional unit (firms, individuals or countries) for the t-th time period, � is theintercept and � is the slope parameter. The model (2.1) can be written inmatrix notation as

y ¼ X�þ u, ð2:2Þ

1386 SONG, STEMANN, AND JUNG

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where y is an NT � 1 observation vector, X ¼ ð_��NT , _��N � lT Þ with _��NT isNT � 1 vector of ones and lT ¼ ð1, 2, . . . ,TÞ

0, � ¼ ð�,�Þ0, and u is anNT � 1 disturbance vector.

A popular specification of the disturbances is the error componentmodel (see, Refs. [6] and [7])

uit ¼ �i þ �it, i ¼ 1, . . . ,N, t ¼ 1, . . . ,T , ð2:3Þ

where the �i denote the unobservable individual specific effects which areassumed to be i:i:d: ð0, �2

�Þ and �it are the remaining disturbances which areassumed to be i:i:d: ð0, �2

� Þ. The �i’s and the �it’s are independent of eachother. This may be a restrictive assumption for economic relationships, likeinvestment or consumption, where an observed shock this period will affectthe behavioral relationship for the next few period. In this paper, we focuson the error component model with individual effects and a serially corre-lated remainder term, i.e., the �it are are assumed to be generated by a first-order autoregressive process ðARð1ÞÞ (see Refs. [8] and [9] study on earningsof American scientists over the decade 1960–1970):

�it ¼ �it1 þ "it, jj < 1, i ¼ 1, . . . ,N, t ¼ 1, . . . ,T , ð2:4Þ

where the "it are i:i:d: ð0, �2" Þ and �2

� ¼ �2" =ð1 2Þ and �2

" is held constant inwhat follows. Under these assumptions, the NT �NT disturbance covari-ance matrix can be written as

Eðuu0Þ ¼ O ¼ IN � ð�2�JT þ �2

�VÞ, ð2:5Þ

where IN is an N �N identity matrix, JT ¼ _��T _��0T with _��T is a T � 1 vector of

ones, and V is the T � T ARð1Þ-correlation matrix:

V ¼

1 2 � � � T1

1 � � � T2

2 1 � � � T3

..

. . .. ..

.

T1 T2 T3� � � 1

2666664

3777775: ð2:6Þ

3. THE OLSE, GLSE, COTE, AND FDE

To compare the relative efficiency of the GLSE with regard to theOLSE, COTE and FDE of �, we derive the variances of estimators.

ESTIMATING LINEAR TREND IN PANEL MODEL 1387

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In the context of the simple model (2.1) the OLSE of � is given by

��OLS ¼ X 0X� �1

X 0y ð3:1Þ

with covariance matrix

Covð��OLSÞ ¼ ðX 0XÞ1X 0OXðX 0XÞ

1, ð3:2Þ

Where

X 0OX ¼_��0N � _��0T

_��0N � l0T

�2�ðIN � JT Þ þ �2

� ðIN � VÞ� �

_��N � _��T ,0_��N � lT

� �

¼ �2�

NT2 NT2ðT þ 1Þ

2

NT2ðT þ 1Þ

2

NT2ðT þ 1Þ2

4

0BB@

1CCAþ �2

�N_��0TV _��T _��0TVlT

l0TV _��T l0TVlT

ð3:3Þ

and

ðX 0XÞ1

¼1

NTðT2 1Þ

2ðT þ 1Þð2T þ 1Þ 6ðT þ 1Þ6ðT þ 1Þ 12

: ð3:4Þ

Varð��OLSÞ is presented in the southeast corner of matrix of Eq. (3.2) andreduces to

Varð��OLSÞ ¼ 12�2"

n6Tþ1

½ðT 1Þ ðT þ 1Þ�2 ðT3TÞ4

þ 2ðT2 1ÞðT 3Þ3 þ 12ðT2

þ 1Þ2 2ðT2 1ÞðT þ 3Þ

þ ðT3TÞ

o.ð1 2Þð1 Þ4NðT3

TÞ2: ð3:5Þ

To obtain the GLSE for �, ��GLS ¼ ðX 0O1XÞ1XO1y, we use the

Prais–Winsten transformation:[10]

RT ¼

ð1 2Þ1=2 0 0 � � � 0 0 1 0 � � � 0 00 1 � � � 0 0

..

. ... ..

. . .. ..

. ...

0 0 0 � � � 1 00 0 0 � � � 1

266666664

377777775

ð3:6Þ


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in the model (2.1). Then, the transformed regression model is

y� ¼ X��þ u�, ð3:7Þ

where y� ¼ ðIN � RT Þy, X�¼ ðIN � RT ÞX and u� ¼ ðIN � RT Þu.

The covariance matrix of the transformed disturbances is

O�¼ Eðu�u�

0

Þ ¼ ðIN � RT Þ �2�ðIN � JT Þ þ �2

� ðIN � VÞ� �

ðIN � R0T Þ

¼ �2�ð1 Þ2ðIN � _��T _��

�0

T Þ þ �2� ð1 2ÞðIN � IT Þ, ð3:8Þ

where _��0

T ¼ ð�, 1, . . . , 1Þ, � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ Þ=ð1 Þ

p.

Alternatively Eq. (3.8) can be written as

O�¼ d2�2

�ð1 Þ2ðIN � �JJ�T Þ þ �2

" ðIN � IT Þ

¼ �2" ðIN � E�

T Þ þ1

1ðIN � �JJ�

T Þ

� �, ð3:9Þ

where �JJ�T ¼ _��T _��

�0

T =d2, d2

¼ �2þ T 1, E�

T ¼ IT �JJ�T and 1 ¼ �2

" =½d2�

ð1 Þ2�2� þ �2

" �:Using the spectral decomposition of O�, we obtain

O�1¼

1

�2"

½ðIN � E�T Þ þ 1ðIN � �JJ�

T Þ� ð3:10Þ

Since

X�¼ ðIN � RT ÞX ¼ ðIN � RT Þð_��N � _��T , _��N � lT Þ

¼

n_��N � ð1 Þ_��T , _��N � ð1 Þl�T þ _��0T

� �o, ð3:11Þ

where l�T ¼ ð�, 1, . . . ,T 1Þ0, _��0T ¼ ð0, 1, . . . , 1Þ0,we have

X�0

O�1X�¼

1

�2"

_��0N � ð1 Þ_��0

T

_��0N � ð1 Þl�T þ _��0T� �0

0@

1A IN � E�

T þ 1ðIN � �JJ�T Þ

� �

� _��N � ð1 Þ_��T , _��N � ð1 Þl�T þ _��0T� ��

: ð3:12Þ


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Making use of _��0

TE�T _��

�T ¼ 0, _��

0

T�JJ�T _��

�T ¼ d2 1,

_��0

TE�T ð1 Þl�T þ _��0T� �

¼ 0

_��0

T�JJ�T ð1 Þl�T þ _��0T� �

¼ _��0

T ð1 Þl�T þ _��0T� �

ð1 Þl�T þ _��0T� �0 �JJ�

T _��0

T ¼T þ 1

2½T ðT 2Þ�

_��0

T ð1 Þl�T þ _��0T� �

¼T þ 1

2½T ðT 2Þ�

and

ð1 Þl�T þ _��0T� �0

ð1 Þl�T þ _��0T� �

¼TðT 1Þð2T 1Þ

6ð1 Þ2 þ TðT 1Þð1 Þ þ T 1þ ð1 2Þ:

The covariance matrix of the GLSE for � is given by

Covð��GLSÞ ¼ ðX�0

O�1X�Þ1

¼ �2"

Nd2ð1Þ2 1

NðT þ 1Þð1Þ 12

ðT ðT 2ÞÞ

NðT þ1Þð1Þ 12

ðT ðT 2ÞÞ Q

0B@

1CA1

,

where

Q ¼ N

"TðT 1Þð2T 1Þ

6ð1 Þ2 þ TðT 1Þð1 Þ

þ T 1þ ð1 2Þ

#NðT þ 1Þ2ð1 1Þ

4d2½T ðT 2Þ�2:

Using the following fact

detðX�0

O�1X�Þ ¼ N2d2

ð1 Þ2 1

(T 1

12½ðT 3ÞðT 2Þ2

2ðT 3ÞðT þ 1Þþ TðT þ 1Þ�

)


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Varð��GLSÞ is given in the southeast corner of ðX�0

O�1X�Þ1

Varð��GLSÞ

¼12�2

"

NðT 1Þ ðT 3ÞðT 2Þ2 2ðT 3ÞðT þ 1Þþ TðT þ 1Þ� � :

ð3:13Þ

Let R2 be the ðT 1Þ � T matrix obtained from RT in Eq. (3.6) bydeleting its top row. Reference [11] suggested the model

y�� ¼ X��þ u��, ð3:14Þ

where y�� ¼ ðIN � R2Þy, X��¼ ðIN � R2ÞX and u�� ¼ ðIN � R2Þu, where R2

amounts to dropping the first observation from the transformed modely� ¼ X��þ u�. Then the covariance matrix of CO-transformed disturbancesu�� is

O��¼ Eðu��u��

0

Þ ¼ ðIN � R2Þ �2�ðIN � JT Þ þ �2

� ðIN � VÞ� �

ðIN � R02Þ

¼ �2�ðIN � R2JTR

02Þ þ �2

� ð1 2ÞðIN � IT1Þ

¼ �2�½IN � ð1 Þ2_��T1_��

0T1� þ �2

� ð1 2ÞðIN � IT1Þ

¼ ðT 1Þ�2�ð1 Þ2ðIN � �JJT1Þ þ �2

" ðIN � IT1Þ

¼ �2" IN � ET1 þ

1

2ðIN � �JJT1Þ

� �, ð3:15Þ

where �JJT1 ¼ _��T1_��0T1=ðT 1Þ,ET1 ¼ IT1

�JJT1 and 2 ¼ �2" =½ðT 1Þ �

ð1 Þ2�2� þ �2

" �:Using the spectral decomposition of O��, we obtain

O��1¼

1

�2"

½ðIN � ET1Þ þ 2ðIN � �JJT1Þ� ð3:16Þ

The covariance of the COTE for � is given by

Covð��COT Þ ¼ ðX��0

O��1X��Þ1: ð3:17Þ

where

X��¼ ðIN � R2ÞX ¼ ðIN � R2Þ _��N � _��T , _��N � lTð Þ

¼ _��N � ð1 Þ_��T1, _��N � ½ð1 ÞlT1 þ _��T1�ð Þ


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and

X��0

O��1X��¼

1

�2"

_��0N � ð1 Þ_��0T1

_��0N � ð1 ÞlT1 þ _��T1½ �0

� IN � ET1 þ 2ðIN � �JJT1Þ

� �� _��N � ð1 Þ_��T1, _��N � ð1 ÞlT1 þ _��T1½ �ð Þ:

Using the following results

_��0T1 ð1 ÞlT1 þ _��T1½ � ¼T 1

2ðT þ 2 TÞ,

ð1 ÞlT1 þ _��T1½ �0ð1 ÞlT1 þ _��T1½ �

¼TðT 1Þð2T 1Þ

6ð1 Þ2 þ 2ð1 Þ

TðT 1Þ

2þ T 1:

we obtain

X��0

O��1X��

¼

NðT 1Þð1Þ2 2NðT 1Þð1Þ 2

2ðT þ2TÞ

NðT 1Þð1Þ 22

ðT þ2TÞ P

0B@

1CA,

where

P ¼ NTðT 1Þð2T 1Þ

6ð1 Þ2 þ TðT 1Þð1 Þ þ T þ 1

� �

Nð1 2Þ

T 1

T 1

2ðT þ 2 TÞ

� �2:

Using the following fact

detðX��0

O��1X��Þ ¼ N2

ðT 1Þ2ðT 2Þð1 Þ4 2=12,

we have

Varð��COT Þ ¼ 12�2" =ð1 Þ2NðT3

3T2þ 2TÞ ð3:18Þ

Finally, the FDE of � is obtained by applying OLS to the differencedmodel

Dy ¼ DX�þDu, ð3:19Þ


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where D ¼ IN �D1 and D1 is the ðT 1Þ � T first-difference matrix

D1 ¼

1 1 0 � � � 0 00 1 1 � � � 0 00 0 1 � � � 0 0... ..

. ... . .

. ... ..

.

0 0 0 � � � 1 1

266664

377775: ð3:20Þ

Therefore the FDE for � is

��FD ¼ ½ðDXÞ0ðDXÞ�

1ðDXÞ

0ðDyÞ ð3:21Þ

with

Covð��FDÞ ¼ ½ðDXÞ0ðDXÞ�

1ðDXÞ

0DOD0ðDXÞ½ðDXÞ

0ðDXÞ�

1: ð3:22Þ

In Eq. (3.22)

DX ¼ ðIN �D1Þ _��N � _��T , _��N � lTð Þ ¼ ð0, _��N � _��T1Þ,

ðDXÞ0ðDXÞ ¼ 1=NðT 1Þ

and

DOD0¼ ðIN �D1Þ½IN � ð�2

�V þ �2�_��T _��

0T Þ�ðIN �D0

1Þ

¼ IN � �2�D1VD

01:

This implies

Varð��FDÞ ¼1

N2ðT 1Þ2ð_��0N � _��0T1ÞðIN � �2

�D1VD01Þð_��N � _��T1Þ

¼2

NðT 1Þ2�2� ð1 T1

Þ ¼2�2

" ð1 T1Þ

NðT 1Þ2ð1 2Þ: ð3:23Þ

4. DISCUSSION

To compare the relative efficiency of estimators, we summarize againthe preceding variances of estimators,

Varð��OLSÞ ¼ 12�2"

n6Tþ1

½ðT 1Þ ðT þ 1Þ�2 ðT3 TÞ4

þ 2ðT2 1ÞðT 3Þ3 þ 12ðT2

þ 1Þ2 2ðT2 1ÞðT þ 3Þ

þ ðT3 TÞ

o.ð1 2Þð1 Þ4NðT3

TÞ2, ð4:1Þ


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Varð��GLSÞ

¼12�2

"

NðT 1Þ½ðT 3ÞðT 2Þ2 2ðT 3ÞðT þ 1Þþ TðT þ 1Þ�, ð4:2Þ

Varð��COT Þ ¼ 12�2" =ð1 Þ2NðT3

3T2þ 2TÞ, ð4:3Þ

Varð��FDÞ ¼2�2

" ð1 T1Þ

NðT 1Þ2ð1 2Þ: ð4:4Þ

As the measure of the relative efficiency of estimators we used the ratioof their variances. Since the relative efficiency is a function involving only Tand , the relative efficiency is plotted for 2 ½1:1� and selectedT ¼ 10, 20, 50, 100, in Figures 1–6.

In Figure 1, we show that the efficiency of the GLSE with respective tothe OLSE. For all value of , the GLSE is more efficient than the OLSE.From Eqs. (4.1) and (4.2) we have lim!1 Varð��GLSÞ=Varð��OLSÞ ¼

5ðT2þ TÞ= 6ðT2

þ 1Þ < 1 (provided T > 3, see Ref. [12], P. 120). Keeping�2" constant, it is also verified from Eqs. (4.2) and (4.4) that

lim!1

Varð��FDÞ ¼ lim!1

Varð��GLSÞ ¼�2"

NðT 1Þ2< 1, ð4:5Þ

Figure 1. Relative efficiency of GLSE with regard to OLSE.


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Figure 2. Relative efficiency of GLSE with regard to FDE.

Figure 3. Relative efficiency of OLSE with regard to COTE.


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Figure 4. Relative efficiency of OLSE with regard to FDE.

Figure 5. Relative efficiency of GLS with regard to COTE.


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which implies that lim!1 Varð��FDÞ=Varð��GLSÞ ¼ 1, i.e., the FDE is approxi-mately BLUE for high value of (see Figure 2). It is further seen fromEqs. (4.2)–(4.4) that lim!1 Varð��GLSÞ=Varð��COT Þ ¼ lim!1 Varð��FDÞ=Varð��COT Þ ¼ 0 (cf. Figures 5 and 6). This means that when the FDE ismore efficient than the COTE for ! 1. In Figures 3 and 4, it can beobserved that for high negative value of , the COTE considerably increasesthe efficiency as compare to the OLSE and in particular as compare to theFDE. However, for positive value of , the OLSE is more efficient thanthe COTE.

Up to now we have been concerned with the limiting behavior ofestimates for rising values of , given T . Completely different propositionsare obtained when is kept constant and T ! 1. It can be obtain from thepreceding variance formulas that Varð��FDÞ ¼ OðN1T2

Þ, whereas the vari-ance of the remaining estimators is OðN1T3

Þ. Thus it is verified thatasymptotic relative efficiency of the FDE with respect to the other estimatorsof �. Varð��FDÞ=Varð��Þ, where � denotes OLS, COT , or GLS, respectively,tends to 1 as T ! 1 (cf. Figures 2, 4 and 6). Thus one has the strikingresult that, for instance, limT!1 Varð��FDÞ ¼ Varð��COT Þ ¼ 1 for any given, and lim!1 Varð��FDÞ ¼ Varð��COT Þ ¼ 0 for any finite and constant T . Thismeans that the value for which the FDE is overtaking the COTE in termsof efficiency is an increasing function of T .

Figure 6. Relative efficiency of FDE with regard to COTE.


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ACKNOWLEDGEMENT

The authors would like to thank the editor, William B. Smith and twoanonymous referees for their valuable comments and suggestions.

REFERENCES

1. Kadiyala, K.R. A Transformation Used to Circumvent the Problem ofAutocorrelation. Econometrica 1968, 36, 93–96.

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4. Kramer, W. Note on Estimating Linear Trend when Residuals areAutocorrelated. Econometrica 1982, 50, 1065–1067.

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a note on estimating linear trend in a regression model with serially correlated error components

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