Consistent Estimation of Dynamic Panel Data ModelsWith Cross-sectional Dependence

Vasilis Sara�dis�

March 2007

Abstract

This paper proposes new moment estimators for autoregressive panels with cross-sectional dependence. In particular, for each unit i the proposed estimators makeuse of instruments with respect to another cross-section, unit j, rather than withrespect to lagged values of the endogenous regressor (either in levels or in �rst-di¤erences) of unit i itself. This is a rather di¤erent approach compared to theconventional one applied in the dynamic panel literature. The resulting estimatorsare asymptotically valid when the time dimension of the panel is �xed. The �nite-sample evidence of the Monte Carlo experiments suggests that the new momentestimators are by far superior in terms of bias and RMSE compared to the existingones, under cross-sectional dependence in the error process.Key Words: dynamic panel data, cross-sectional dependence, generalised method

of moments, Sargan�s test.JEL Classi�cation: C12; C13; C15; C33.

1 Introduction

In dynamic panel data models, it is well known that the �xed e¤ects estimator� hereafterFE � is biased for �nite T (the number of time series observations), and this resultholds regardless of the number of cross-sections observed in the sample, N .1 To tacklethis problem, applied economic research usually follows the practice of �rst-di¤erencingthe data and then using an estimator based either on instrumental variables (such as theAnderson-Hsiao estimator, 1981), or on the generalised method of moments (GMM),proposed by Arellano and Bond (1991), Arellano and Bover (1995), Ahn and (1995)and Blundell and Bond (1998). These estimators are consistent for �xed T , however,this property hinges crucially upon the assumption that the cross-sectional units areindependently drawn, such that the only characteristic giving merit to the panel structureof the sample is a set of common coe¢ cients shared by all individuals. Recently, this

�Discipline of Econometrics and Business Statistics, University of Sydney, NSW 2006, Australia. Tel:+61-2-9036 9120; e-mail: v.sara�dis@econ.usyd.edu.au.

1See Nickell (1981).

1

assumption has come into question by a large body of the panel literature, which �ndsthat panel data sets often exhibit substantial cross-sectional dependence, which may beattributable to spatial correlations speci�ed on the basis of economic distance or relativelocation, as well as unobserved common factors that ultimately become part of the errorterm. See, for example, Anselin (2001), Bai (2005) and Pesaran (2006).

The impact of cross-sectional dependence in dynamic panels has been investigated byPhillips and Sul (2003) in the context of the FE estimator, and Sara�dis and Robertson(2006) in GMM estimation. The latter demonstrated that the conventional estimatorsbased on instrumental variables and the generalised method of moments are asymptoti-cally biased for �xed T , when factor structure dependence is present in the error process.Intuitively, this result holds because for a �nite time dimension the common unobservedfactor that is present in the disturbances is not averaged away to zero as N !1, even ifit is zero-mean distributed. Therefore, plimN!1

n1N

PNi (uituit�s)

o6= 0 8 s (where uit

is the disturbance term that includes the unobserved shock), which implies that there isno valid instrument to be used with respect to a lagged value of the dependent variable,regardless of how large the di¤erence apart in time between the instrument and theendogenous regressor is.2

Sara�dis, Yamagata and Robertson (2006) showed that in dynamic panel modelsthat include covariates, rather than only lags of the dependent variable, one may cir-cumvent this problem by using appropriate lagged values of xit as instruments for theendogenous regressors. The proposed solution is valid provided that certain regularityconditions hold � namely, that the factor loadings of the yit process and the xit processare uncorrelated. This paper o¤ers an alternative procedure, which is valid even inpure autoregressive models with no covariates, as well as in more general models wherethe above uncorrelatedness assumption on the factor loadings cannot be warranted. Inparticular, a solution is proposed based on instrumenting the endogenous regressors forindividual i by appropriate lags of the dependent variable (either in levels or in �rst-di¤erences) of a di¤erent cross-section, individual j. We demonstrate the validity ofthese moment conditions and analyse the properties of the resulting GMM estimators,including cases where the problem of weak instruments applies. The superiority of thenew estimators is shown using Monte Carlo experiments.

The remaining of the paper is organised as follows. The next section describesthe speci�cation of the model that we have in mind and section (3) analyses the newmoment estimators developed in this paper. Section (4) investigates the performanceof the estimators using simulated data and a �nal section concludes.

2See the reference above for more details.

2

2 Model Speci�cation

We consider dynamic panel data models of the following form:

yit =PXp=1

�pyit�p + �0xit + vit; i = 1; :::; N and t = 1; :::; T

vit = �i + uit; uit =

MXm=1

�mifmt + "it +

�X�=1

"i(modN)+�;t (1)

where yit is the dependent variable of unit i at time t and vit is the composite error termthat consists of the individual unobserved e¤ect, a multiple-factor structure and a spatialmoving average process on the random component "it. Thus, uit gives rise to a generalform of cross-sectional dependence that may arise in some applications due to unobservedcommon shocks and omitted variables that are spatially correlated. i (modN) is themodulo operator, which is de�ned as the remainder after numerical division of i by Nto obtain integer values. Thus, for i = 1; :::; N �1, and � = 1; i (modN)+ � = i+1 andfor i = N , N (modN)+1 = 1. Without loss of generality and for easiness of expositionof the asymptotics we are going to assume that P = 1 and � = 1, such that the impliedN �N spatial weighting matrix, W , collapses to

W =

26666664

0 1 0 : : 00 0 1 : : 00 : : 1 : 0: : : : : :: : : : : 11 : : : 0 0

37777775 (2)

Moreover, to simplify things without changing the main results of the paper, wewill impose the restriction � = 0, which also emphasises that our proposed momentestimators are valid even in pure autoregressive panel data processes.

Hence, the resulting model may be written as

yit = �yit�1 + vit; i = 1; :::; N and t = 1; :::; T

vit = �i + uit; uit = �0ift + "it + "i(modN)+1;t (3)

where j�j < 1 and j j � 1.3 �i is a random e¤ect with zero mean and �nite variance,equal to �2�, and uit is a composite term that contains a multiple-factor structure and aspatial moving average component of �rst order. In particular, �i = (�1i; �2i; :::; �Mi)

0

is a (M � 1) vector of factor loadings which is assumed to be iid(�;��) with �� beinga positive semi-de�nite matrix and ft = (f1t; f2t; :::; fMt)

0 is a (M � 1) time-varyingcommon factor which is iid(0;�f ). Notice that since our asymptotic is N !1, �xed

3The restriction on is not necessary for analytic purposes, however, it makes sense to impose in thedata applications we have in mind.

3

T , �i may justi�ably be treated as stochastic here. "it is a randomly distributed termthat represents purely idiosyncratic e¤ects, such that "it � iid(0; �2).

Furthermore, we make the following standard assumptions:

Assumption 1: cov(�i"it) = 0, for all i, t:

Assumption 2: E("it"is) = 0 for all i and t 6= s.

Assumption 3: E(yi1"it) = 0 for all i and t = 2; 3; :::; T .

Assumption 4:

E("it"i0t) =

8<:�2" for i = i0; t = s �2" for i0 = i (modN) + 1; t = s0 otherwise

Assumption 5: (a) yit and ft are covariance stationary processes.

Assumption 6: E("it�i) = 0, cov(�i;�i) = 0, E("itft) = 0; cov(�i; ft) = 0, cov(�i; ft) =0 for all i and t.

Assumptions 1-2 are standard in the literature and they are required for the consis-tency of the Arellano and Bond (1991) �rst-di¤erenced GMM estimator (together withassumption 5). Assumption 3 is an extra assumption, necessary for the consistency ofthe Blundell and Bond (1998) system GMM estimator. Assumption 4 re�ects the spa-tial nature of the "it process. Assumption 6 is a random coe¢ cient type of assumptionon the unobserved factors and factor loadings, which is similar in nature to Assumption1.

Stacking (3) for each i yields

yi = �yi;�1 + vi, i = 1; 2; :::; N , (4)

vi = �i�T�1 + F�i + "i + "i(modN)+1;t (5)

where yi = (yi2; yi3; :::; yiT )0, � � is a (� � 1) vector of ones, yi;�1 = (yi1; yi2; :::; yiT�1)0,vi = (vi2; vi3; :::; viT )

0, F = (f2; f3; :::; fT )0, and "i = ("i2; "i3; :::; "iT )0.

It is useful to express the model above in terms of deviations from time-speci�ccross-sectional averages. E¤ectively, this is usual practice in short, dynamic panels andresults in removing the mean impact of the unobserved factors from the disturbanceterm � as such, it results in alleviating the distortions induced in estimation from thefactor structure.4 Hence, the demeaned and �rst-di¤erenced equation is de�ned as

�yi= ��y

i;�1 +�vi, i = 1; 2; :::; N ,

�vi = �F�i+�"i + �"i(modN)+1 (6)

4For instance, if the factor structure was homogeneous such that �i = �, this transformation wouldremove the factor-structure dependence completely from the error process.

4

where the underline signi�es that the variables are expressed in terms of deviations fromtime-speci�c averages, and ��� signi�es that the variables are �rst-di¤erenced. For

example, �yi=��y

i3;�y

i4; :::;�y

iT

�0, where �y

it=�yit� y

it�1

�, y

it= (yit � �yt)

and �yt = N�1PNi=1 yit and so on.

Also, in order to save some space regarding the notation we de�ne

j = i (modN) + 1 (7)

3 Estimation Using Moment Estimators

3.1 Equations in First-di¤erences

The presence of cross-sectional dependence in the error process may result in largeasymptotic biases when the standard IV and GMM estimators are employed, see Sara�disand Robertson (2006). Intuitively, this is because the e¤ect of each factor is not averagedout within a �nite time dimension, even if it has mean at zero. A solution to thisproblem for a dynamic model that is not a strict autoregressive process was proposedby Sara�dis, Yamagata and Robertson (2006), who suggested using instruments withrespect to appropriate lags of the covariates, as opposed to the conventional ones thatmake use of lags of the endogenous regressors (either in levels or in �rst-di¤erences).This approach was shown to yield consistent estimates of the parameters, provided thatcertain regularity conditions on the factor loadings are satis�ed � namely, that the factorloadings of the yit process and those of the covariates are independently drawn. Whetherthis is a tenable assumption or not depends on the particular application of course. Ina pure autoregressive process, there are no covariates that can be used as instruments,and therefore the method proposed above is certainly not appropriate. However, itturns out that in a set up like (3) there are other moments available, as the followingproposition demonstrates:

Proposition 1 Under Assumptions 1-6, model (3) can be estimated consistently usingmoment estimators that rely on the following moment conditions:

plimN!11

N

NXi=1

y0j;�2�vi

!= 0 (8)

and

plimN!11

N

NXi=1

y0j;�2�yi;�1

!= � (T � 2)

1 + ��2" (9)

where yj;�2 = (y

j;1; yj;2; :::; y

j;T�2)0, �y

i;�1 = (�yi;2;�y

i;3; :::;�y

i;T�1)0 and �vi =

(�vi3;�vi4; :::;�viT )0.

Proof. See Appendix A.

5

The above implies that model (3) can be estimated consistently using a simple IVestimator that employs y

j;t�2 as an instrument for �yi;t�1, or a �rst-di¤erenced GMM-type estimator that makes use of y

j;t�s for s = 2; 3; ::: to instrument �yi;t�1; since thecorrelation between y

j;t�sand �yi;t�1 is non-zero and the correlation between yj;t�sand

�vi;t�1 remains zero. In particular, de�ning ZMM =�ZMM1 ;ZMM

2 ; :::;ZMMN

�0with

ZMMi =

�yj;1; yj;2; :::; y

j;T�2

�0and the matrix of instruments

Zyi =

266664yj;1

0 0 0 � � � � � � � � � 0

0 yj;1

yj;2

0 � � � � � � � � � 0...

......

.... . . ::: :::

...0 0 0 0 � � � y

j;1� � � y

j;T�2

377775 ; (T � 2� hy) (10)

such that Zy =�Zy1;Z

y2; :::;Z

yN

�0, where hy = [(T � 1)� (T � 2) =2], proposition (1)

implies that the following moment estimators are appropriate:

b�MM =�Z0MM�y�1

��1 �Z0MM�y

�(11)

and

b::�GMM�FD =

"�y0�1Z

y�b::y��1

Zy0�y�1

#�1�y0�1Z

y�b::y��1

Zy0�y (12)

where b�MM denotes the simple IV estimator, similarly to Anderson and Hsiao, andb::�GMM�FD denotes the �rst-di¤erenced two-step GMM estimator with�y =

��y

1;�y

2; :::;�y

N

�;

and �y�1 =��y

1;�1;�y2;�1; :::;�yN;�1

�0.b::yis the weighting matrix of the two-step

GMM estimator, which is estimated by

b::y= Zy0

h��b_u�b_u0�HiiZy (13)

where Hi is a (T � 2)� (T � 2) matrix that takes the value of 2 on the main diagonal,�1 on the �rst o¤-diagonals and zeros otherwise and �b_u is the N � (T � 2) matrix ofresiduals, obtained from the following �rst-step GMM estimator:

b:�GMM�FD =

"�y0�1Z

y�b:y��1

Zy0�y�1

#�1�y0�1Z

y�b:y��1

Zy0�y (14)

b:�GMM�FD is identical to (12) except that it uses the following weighting matrix:

b:y= Zy0HyZy (15)

6

whereHy = IN Hi (16)

and IN is an N �N identity matrix. Hy essentially assumes homoscedastic and purelyidiosyncratic disturbances in the levels equations, with the �rst component of the kro-necker product in (16) re�ecting cross-sectional independence in the error structure andthe latter component re�ecting �rst-order serial correlation, induced as a result of the�rst-di¤erencing of the variables.

Notice that the least-squares estimate of � _u� _u0 in (13) is rank de�cient because itis an N�N matrix and has rank T �2. The matrix inside the square brackets of (13) isalso rank de�cient because it is of order N (T � 2)�N (T � 2) and has rank (T � 2)2 :However, the covariance matrix of the moment conditions is a square hy � hy matrix,

which has rank equal to min�hy; (T � 2)2

�. Therefore, provided that we do not use

too many di¤erent cross-sections from which to obtain instruments for the endogenousregressor of unit i , i.e. hy < (T � 2)2 ; (13) will be of full rank and the weighting matrixwill exist.

To investigate the properties of these moment estimators it is useful to write explicitlythe instrumental variable equation implied by using a single instrument for �y

i;t�1:

�yi;t�1 = �dy

j;t�2 + wit�1 (17)

The ordinary least-squares estimate of �d equals

b�d =

PNi=1

PTt=3 yj;t�2�yi;t�1PN

i=1

PTt=3 y

2j;t�2

(18)

=(�� 1)

PNi=1

PTt=3 yj;t�2yi;t�2 +

Pyj;t�2

�yi;t�1 � �yi;t�2

�PNi=1

PTt=3 y

2j;t�2

7

Using the assumptions listed above it is straightforward to show that the plim of b�dequals

plimN!1�b�d�

=plimN!1

1N

PNi=1

PTt=3 yj;t�2yi;t�2 + plimN!1

1N

Pyj;t�2

�yi;t�1 � �yi;t�2

�plimN!1

1N

PNi=1

PTt=3 y

2j;t�2

= (�� 1) �2"1� �2

�"

�2�

(1� �)2+ trace

(�� �

TXt=3

W0t�2Wt�2

)+

�2"1� �2

�1 + 2

�#�1

= ��1� �1 + �

� �"

�2��2"

�1

1� �

�+ (1� �) � trace

(���2"

�TXt=3

W0t�2Wt�2

)+

�1

1 + �

��1 + 2

�#�1(19)

whereWt�2 =P1s=0 �

sft�s�2.

Thus, we can see that the plim of b�d depends on a number of parameters, namely�; ; �2�; �� and �

2": For example, as � ! 1; the plim of the estimator converges to

zero, implying that the correlation between yj;t�2 and �yi;t�1 becomes weaker. The

intuition behind this is illustrated in the following �gure:

1−∆ ity

2−ity

Case 1: 1 & = 0

1−∆ ity

2−ity

1−∆ ity

2−ity

1−∆ ity

2−ity

1−∆ jty

2−jty 2−jty

2−jty 2−jty

1−∆ jty

1−∆ jty

1−∆ jty

Case 2: 1 & = 0

Case 3: 1 & = 0 Case 4: 1 & = 0

1−∆ ity

2−ity

Case 1: 1 & = 0

1−∆ ity

2−ity

1−∆ ity

2−ity

1−∆ ity

2−ity

1−∆ jty 1−∆ jty

2−jty 2−jty 2−jty 2−jty

2−jty 2−jty 2−jty 2−jty

1−∆ jty 1−∆ jty

1−∆ jty 1−∆ jty

1−∆ jty 1−∆ jty

Case 2: 1 & = 0

Case 3: 1 & = 0 Case 4: 1 & = 0

GMM instruments with cross-sectional dependence.

As we can see, the �gure distinguishes among four possibilities of combining weakinstruments with cross-sectional dependence in the error process, while holding �2�; ��

8

and �2" �xed. The �rst one concerns the case where � 9 1 and = 0. This situationhas been discussed by Sara�dis and Robertson (2006), who showed that the standardIV and GMM-type estimators used in the literature are asymptotically biased, mainlydue to the non-zero correlation between y

i;t�2 and lagged values of uit for �nite T . Inthis case, the use of y

j;t�2 as an instrument for �yi;t�1 is not appropriate because thereis no correlation between these variables. The second possibility refers to the situationwhere �9 1 and 6= 0. In this case, while the standard instrument, y

i;t�2, used in theliterature for �y

i;t�1 is not valid given that �� 6= 0, yj;t�2 is correlated with �yi;t�1 andtherefore it may be used as an instrument. The third possibility re�ects the situationwhere � ! 1 and = 0, such that there is no spatial dependence in the error process.Here the same consequences apply as in Case 1. Finally, there is also a possibility that� ! 1 and 6= 0. In this case, there is a problem of weak instruments and the linkbetween �y

j;t�1 and �yi;t�1 is not e¤ective anymore since yj;t�2 is poorly correlatedwith �y

j;t�1; while the link between yj;t�2 and yi;t�2 does not help either because yi;t�2is poorly correlated with �y

i;t�1. This paper confronts Cases 2 and 4. In particular,we are mainly interested in �nding out what gains can be achieved in GMM estimationby exploiting instruments with respect to a cross-section that is di¤erent from unit i, incases where either �! 1 or �9 1 with 6= 0.

When there is no heterogeneous factor structure dependence, such that �� = 0,the plim of b�d remains non-zero but at the same time the standard instruments withrespect to lagged values of yit are also valid. On the other hand, for a given non-zerovalue of and � less than one in absolute terms, the plim of the estimator converges tozero as either

��2�=�

2"

�! 1 or

���=�

2"

�! 1. The former is similar to the result by

Blundell and Bond (1998) that applies in standard GMM estimation of dynamic panelsusing the equations in �rst-di¤erences. Interestingly, the same appears to apply for theratio between �� and �2". Intuitively, this is because the contribution of the spatialcomponent of the error process in vi � and thereby the correlation between �yit�1 andyjt�2 � diminishes with high values of ��; and increases with high values of �2".

3.2 Equations in Levels

To circumvent the problem of weak instruments in GMM estimation with no cross-sectional dependence, Blundell and Bond (1998) proposed using extra moment conditionswith respect to the equations in levels. The authors showed that these moments remainrelevant even in those cases where the standard �rst-di¤erenced GMM estimator breaksdown, namely when � ! 1, or

��2�=�

2"

�! 1. Similarly, in our set-up of error cross-

sectional dependence there are additional moments to be exploited in the equations givenin levels.

In particular, in the regression yi;t= �y

i;t�1 + vi;t; yi;t�1 is correlated with vi;t asboth terms involve �i, while and �yi;t�1 and yj;t�1 are correlated with vi;t due to thefactor structure. However, it turns out that we can now instrument y

i;t�1 by �yj;t�1as the following proposition demonstrates:

9

Proposition 2 Under Assumptions 1-6 of model (3), the following set of moment con-ditions becomes relevant in the equations given in levels

plimN!11

N

NXi=1

�y0j;�2vi

!= 0 (20)

and

plimN!11

N

NXi=1

�y0j;�1yi;�1

!=

(T � 2)1 + �

�2" (21)

where�yj;�1 = (�yj;2; yj;3; :::; yj;T�1)

0, yi;�1 = (yi;2; yi;3; :::; yi;T�1)

0 and vi = (vi3; vi4; :::; viT )0.

Proof. See Appendix B.

Thus, de�ning

Zzi =

"Zyi 0

0 ZyLi

#(2(T � 2)� hz); (22)

where hz = hy + (T � 2); and

ZyLi = diag(�yj;t�1), t = 3; 4; :::; T , (23)

such that Zz =�Zz1;Z

z2; :::;Z

zN

�0, proposition (2) implies that the following system GMM

estimator is appropriate:

b::�GMM�sys =

"Y0�1Z

z�b::z��1

Zz0Y�1

#�1Y0�1Z

z�b::z��1

Zz0Y (24)

whereb::�GMM�sys denotes the system two-step GMM estimator, Y = (Y1;Y2; :::;YN )

0,Yi = (�yi3; :::;�yiT ; yi3; :::; yiT ), Y�1 =

�Y1;�1;Y2;�1; :::;YN;�1

�0, and �nally Yi;�1 =

(�yi2; :::;�yiT�1; yi2; :::; yiT�1).b::zis the estimated weighting matrix of the two-step

system GMM estimator, given by

b::z= Zz0

b:QZz (25)

where Zz is de�ned above.b:Q equals

b:Q =

24 ��e_u�e_u0� (Hi) 0

0�e_ue_u0� (IT�2)

35 (26)

where e_u is the N�(T � 2) matrix of residuals, obtained from the �rst-step system GMMestimator:

b:�GMM�sys =

"Y0�1Z

z�b:z��1

Zz0Y�1

#�1Y0�1Z

z�b:z��1

Zz0Y (27)

10

b:�GMM�sys is identical to (24), except that it uses the following weighting matrix:

b:z= Zz0HzZz (28)

where

Hz =

�Hy 00 Hsys

�(29)

and Hy has been de�ned above with Hsys being equal to Hsys = IN IT�2.Similarly to the analysis in section (3.1), we can obtain a useful insight into the

properties of the system GMM estimator by writing out explicitly the expression for theordinary least-squares estimator in the instrumental variable regression for y

i;t�1

b�l =

PNi=1

PTt=3�yj;t�1yi;t�1PN

i=1

PTt=3

��y

j;t�1

�2=

PNi=1

PTt=3 yj;t�1yi;t�1 � �

PNi=1

PTt=3 yj;t�2yi;t�2 �

PNi=1

PTt=3 yj;t�2vi;t�1PN

i=1

PTt=3

��y

j;t�1

�2(30)

Using assumptions 1-6, it is straightforward to show that the plim of b�l equalsplimN!1

�b�d�=

plimN!11N

PNi=1

PTt=3 yj;t�1yi;t�1 � �plimN!1

1N

Pyj;t�2yi;t�2

plimN!11N

PNi=1

PTt=3

��y

j;t�1

�2=

"(1� �) �2"�

1� �2�# � �2� 1� �

1� �2

��1 + 2

��2"

+trace

(��

TXt=3

W0t�1Wt�1 + (1� 2�)

TXt=3

W0t�2Wt�2

!)#�1(31)

As we can see, as �! 1 the above expression converges to

plimN!1�b�d� = 1

2

"�1 + 2

�+ trace

(���=�

2"

� TXt=3

��W0

t�1Wt�1�)#�1

(32)

and so �yj;t�1 remains informative as an instrument for yi;t�1, provided that 6= 0.

Hence, the problem of weak instruments does not apply as far as the value of � isconcerned. In addition, without heterogeneous factor structure dependence in the error

11

process, �� = 0 and therefore transforming the data in terms of deviations from time-speci�c averages has removed the impact of the factors completely. As a result, therandom element in (32) disappears and the expression becomes equal to a constant

number � speci�cally, plimN!1�b�d� = =

�2�1 + 2

��. Similarly to (19) ; the plim of

the ordinary least-squares estimator converges to zero as���=�

2"

�! 1 for the same

reason discussed previously � that is, because the contribution of the spatial componentof the error process in uit diminishes.

3.3 Testing for Factor Structure Dependence

Notice that under Assumptions 1-6 of model (3), a test for factor structure dependencecan be constructed using the same rationale as in Sara�dis, Yamagata and Robertson(2006). Speci�cally, in a pure autoregressive process, while � for each unit i � thereare no valid instruments with respect to lagged values of yit itself, instruments withrespect to a di¤erent cross-section, unit j, are valid, as propositions (1) and (2) havedemonstrated. This conclusion suggests that the hypothesis testing procedure for factorstructure dependence can be implemented as a test for the validity of the subset of themoment conditions � that with respect to lagged values of yit itself � using Sargan�s(1958, 1988) di¤erence test.

Thus, we can de�ne the matrix of instruments

Zyyi =

26666664yj;1

yi1

0 0 0 0 � � � � � � � � � 0 � � � � � � 0

0 0 yj;1

yj;2

yi;1

yi;2

0 � � � � � � 0 � � � � � � 0

0 0 0 0 � � � � � � � � � � � � � � � 0 � � � � � � 0...

......

......

. . . ::: :::...

0 0 0 0 0 � � � yj;1

� � � yj;T�2 y

j;1� � � y

j;T�2

37777775, (T � 2� hyy) (33)

where hyy = (T � 1)(T � 2), such that Zyy =�Zyy1 ;Z

yy2 ; :::;Z

yyN

�0.

Also, we can de�ne

Zzzi =

"Zyyi 0

0 ZzLi

#(2(T � 2)� hzz); (34)

with hzz = hyy + 2(T � 2); and

ZzLi =hdiag(�y

j;t�1) diag(�yi;t�1)

i, t = 3; 4; :::; T (35)

such that Zzz =�Zzz1 ;Z

zz2 ; :::;Z

zzN

�0.

We may then test the hypothesis of factor structure dependence in a pure autore-gressive process with �nite T using the following Sargan�s di¤erence test statistics

Dfd2 = (Sfd2 � eSfd2) (36)d! �2(hy�hyy)

12

and

Dsys2 = (Ssys2 � eSsys2) (37)d! �2(hz�hzz)

where eSfd2 is Sargan�s test statistic of overidenti�ed restrictions based on the �rst-di¤erenced two-step GMM estimator that is given in (12), Sfd2 is Sargan�s test statisticof overidenti�ed restrictions based on the �rst-di¤erenced two-step GMM estimator thatmakes use of the �grant�set of instruments, Zyy, and similarly for eSsys2 (which makesuse of Sargan�s test based on (24)) and Ssys2.

The �nite-sample performance of these tests has not been investigated in this currentversion of the paper.

4 Small Sample Properties of Moment Estimators

This section investigates the �nite-sample performance of the various estimators pro-posed in this paper using Monte Carlo experiments. The main focus of the analysisis on the e¤ects of the relative importance of the unobserved factors in the total er-ror process, as well as on the e¤ects of di¤erent values of N and � on the estimators.To make the results comparable across experiments we control the impact of the ratiobetween �i and uit on yit using an approach similar to Bun and Kiviet (2006). The �rst-di¤erenced two-step GMM estimators make use of either yit�s or yjt�s as instrumentsfor �yit�1, setting s = 2; 3.

4.1 Monte Carlo Design

The underlying data generating process is given by

yit = �yit�1 + �i + uit;

uit = �ift + "it + "jt, i = 1; 2; :::; N ; t = �48;�47; :::; T . (38)

where �i � iidN�0; �2�

�; "it � iidN

�0; �2"

�, ft � iidN

�0; �2f

�and j has already been

de�ned in (7) :5 Also, the factor loadings are drawn from

�i � iidU [0; 0:5] (39)

The signal-to-noise ratio of a pure autoregressive process like (38) is not controllableand depends solely on �. To see this, we may de�ne y�it = yit � 1

1���i and let thesignal-to-noise ratio be denoted by �2s=�

2u; where �

2s is the variance of the signal and �

2u

is the total error variance. In this case, the model can be rewritten as

y�it = �y�it�1 + uit (40)

5Adding covariates in (38) would not make any di¤erence to the main results of the experiment.

13

with �2s being equal to

�2s = var (y�it � uit) = var (y�it) + var (uit)� 2cov (y�it; uit) (41)

and

var (y�it) = var

1Xs=0

�suit�s

!=

1

1� �2�2u

cov (y�it; uit) = E

" 1Xs=0

�suit�s

!uit

#= �2u (42)

Combining these terms produces

�2s =�2

1� �2�2u

�2s=�2u =

�2

1� �2(43)

which con�rms the statement above.The performance of GMM estimation depends crucially upon the ratio of the two

variance components, ai and uit; on var(yit) as shown in (19). This implies that as thevalue of � falls, or as the amount of cross-sectional dependence increases, the impact of �ion var (yit) will tend to decrease, in which case the GMM estimators will become worsein RMSE performance and thereby the comparisons across experiments with di¤erentlevels of cross-sectional dependence will not be valid. To control this ratio we use thefollowing simple result

var (yit) = var

"�i1� � +

1Xs=0

�suit�s

!#=

�2�

(1� �)2+

�2u1� �2

(44)

and we set �2� = [(1� �) = (1 + �)]�2u with = 0:5.In addition to �2�=�

2u, the performance of the estimators will depend on the proportion

of �2u attributed to the factor structure in uit� hereafter this proportion is denoted by�(d), d = 1; :::; 4.

6 Therefore, noticing that

�2u =����2 � �2f + �2� � �2f + �2" �1 + 2� (45)

and normalising �2f = 1, we can produce the following result

�2" =

�1� �(d)

��(d) (1 +

2)

h����2+ �2�

i(46)

6This implies that the proportion attributed to random noise, "it + "jt, equals�1� �(d)

�.

14

Since the values of����2 and �2� are determined solely by (39) and so they are �xed,

normalising = 0:5 implies that �2" will change only according to �(d). As this ratioincreases, the impact of the factor structure in the error process will rise. We choosethe following values for �(d):8>><>>:

Low impact of factor structure on uit: �(1) = 1=3

Medium impact of factor structure on uit: �(2) = 1=2

Medium-to-high impact of factor structure on uit: �(3) = 2=3

High impact of factor structure on uit: �(4) = 3=4

We consider N = 400; 800 and T = 6, 10, since our focus is T �xed, N ! 1. �alternates between 0:5; 0:7 and 0:9: The initial value of yit has been set equal to zerobut the �rst 50 observations have been discarded before choosing the sample, so as toensure that the initial zero values do not have an impact on the results. All experimentsare based on 2,000 replications.

4.2 Results

Tables 1-2 report the simulation results in terms of the mean value of b� and RMSEfor each of the estimators used in the experiment. FE is the �xed e¤ects estimator, IVis the simple instrumental variable estimator by Anderson and Hsiao (1981) and FDand SYS denote the �rst-di¤erenced and system two-step GMM estimators respectively,proposed by Arellano Bond (1991) and Blundell and Bond (1998). The superscript ���indicates that the corresponding estimator uses instrument(s) with respect to anothercross-section, unit j.

As expected, the performance of all estimators depends on �(d), the value of �, andthe size of T and N . Speci�cally, as the value of � increases for a given value of �,T and N , the estimators su¤er a rise in bias and in RMSE. This is natural becauseas the relative impact of the factor structure in the total error process increases, theinvalidity of the instruments used with respect to unit i itself (such as in IV, FD andSYS) is magni�ed. For the estimators using instruments with respect to unit j, the risein bias and RMSE is also intuitive because as � increases, the contribution of the spatialcomponent in the error process � and thereby the correlation between �yit�1 and yjt�s� diminishes.

Having said that, two things are clear from these results; �rstly, that IV�, FD�

and SYS� outperform IV, FD and SYS respectively under all circumstances. Secondly,that the relative performance of IV�, FD� and SYS� improves with larger values of �.This is also intuitive � ultimately, as � ! 0 the factor structure in the error processdiminishes and all estimators become consistent, with the conventional estimators beingmore e¢ cient, unless = 1.7 Notice also that in RMSE terms, SYS� performs betterthan FD�, which performs better than IV�, with the relative di¤erence in performance

7Of course, in this case one may also use a GMM estimator that makes use of instruments both withrespect to unit i and unit j. For instance, instead of using two lags of yit�1 as instruments for �yit�1,it is possible to use one lag of yit�1 and one lag of yjt�1.

15

being increased according to the value of �. As T rises, the performance of the estimatorsimproves without exception.

Finally, it is important to emphasise that as the size of N increases, the bias andRMSE of IV�, FD� and SYS� decreases considerably. This is not the case for theconventional estimators, IV, FD and SYS, the performance of which � if anything �deteriorates with larger values of N .

5 Concluding Remarks

This paper has proposed new moment estimators for estimating short dynamic panel datamodels with cross-sectional dependence. For each unit i, the proposed estimators makeuse of instruments not with respect to unit i itself, but with respect to another cross-section, unit j. The validity of the implied moment conditions has been demonstratedand the properties of the resulting GMM estimators have been analysed under di¤erentcircumstances. The simulated experiments have shown that the newly proposed momentestimators are by far superior compared to the conventional ones under cross-sectionaldependence. This result is magni�ed as the impact of the factor structure in the totalerror process increases. Finally, larger values of N are accompanied by a considerabledecrease in bias and RMSE for the estimators put forward in this paper. This is not thecase with the conventional estimators, the performance of which is naturally not a¤ectedby the size of N .

16

References

[1] Anderson, T.W. and C. Hsiao (1981) �Estimation of Dynamic Models with ErrorComponents�, Journal of the American Statistical Association, 76, 598-606.

[2] Ahn, S.C and P. Schmidt (1995) �E¢ cient Estimations of Models for Dynamic PanelData�, Journal of Econometrics, 68, 5-28.

[3] Anselin, L. (2001) �Spatial Econometrics�, Chapter 14 in B. Baltagi, ed., A Com-panion to Theoretical Econometrics, Blackwell Publishers, Massachuttes.

[4] Arellano, M. and S. Bond (1991) �Some Tests of Speci�cation for Panel Data:Monte Carlo Evidence and an Application to Employment Equations�, Review ofEconomic Studies, 58, 277-297.

[5] Arellano, M. and O. Bover (1995) �Another Look at the Instrumental VariableEstimation of Error-Component Models�, Journal of Econometrics, 68, 29-51.

[6] Bai, J. (2005) �Panel Data Models with Interactive Fixed E¤ects�, mimeo.

[7] Blundell, R. and S. Bond (1998) �Initial Conditions and Moment Restrictions inDynamic Panel Data Models�, Journal of Econometrics, 87, 115-143.

[8] Bun, M. and J.F. Kiviet (2006) �The E¤ects of Dynamic Feedbacks on LS andMM Estimator Accuracy in Panel Data Models�, Journal of Econometrics, 127(2),409-444

[9] Nickell, S. (1981) �Biases in Dynamic Models with Fixed E¤ects�, Econometrica,49, 1417-1426.

[10] Pesaran, H. (2005) �Estimation and Inference in Large Heterogeneous Panels witha Multifactor Error Structure�, Econometrica, 74(4), 967-1012.

[11] Phillips, P. and D. Sul (2003) �Dynamic panel estimation and homogeneity testingunder cross section dependence�, The Econometrics Journal, Vol. 6, 217-259.

[12] Sara�dis, V. and D. Robertson (2006) �On The Impact of Cross Section Dependencein Short Dynamic Panel Data Estimation�, mimeo.

[13] Sara�dis, V., Yamagata, T., and D. Robertson (2006) �A Test of Cross SectionDependence for a Linear Dynamic Panel Model with Regressors�, mimeo.

[14] Sargan, J.D. (1958) �The Estimation of Economic Relationships Using InstrumentalVariables�, Econometrica, 26, 393-495.

[15] Sargan, J.D. (1988) �Testing for misspeci�cation after estimating using instrumen-tal variables�, in Maasoumi, E. (ed.), Contributions to Econometrics: John DenisSargan, Vol. 1. Cambridge: Cambridge University Press.

17

Appendices

A Proof of Proposition 1

Assuming that the yit process has started a long time ago, it can be shown that

yj;t�2 =

�j1� � + �

0j

1Xs=0

�sft�s�2 +

1Xs=0

�s"j;t�s�2 +

1Xs=0

�s"j0;t�s�2 (47)

wherej0 = j (modN) + 1 (48)

and j is de�ned in (7).Hence, we have

plimN!11

N

"NXi=1

TXt=3

yj;t�2�uit

#

= plimN!11

N

"NXi=1

TXt=3

�j1� � + �

0j

1Xs=0

�sft�s�2 +

1Xs=0

�s"j;t�s�2 +

1Xs=0

�s"j0;t�s�2

!��0i�ft +�"it + �"i+1;t

�i=

TXt=3

"�f 0t � plimN!1

1

N

NXi=1

�j�0i

! 1Xs=0

�sft�s�2

#= 0; (49)

under Assumptions 1-6. On the other hand, the covariance between yj;t�2 and �yit�1 is di¤erent from

zero:

plimN!11

N

"NXi=1

TXt=3

yj;t�2�yit�1

#

= plimN!11

N

(NXi=1

TXt=3

yj;t�2

h(�� 1) y

i;t�2 + vi;t�1

i)

= (�� 1) plimN!11

N

"NXi=1

TXt=3

yj;t�2yi;t�2

#+ plimN!1

1

N

"NXi=1

TXt=3

yj;t�2vi;t�1

#

= (�� 1) plimN!11

N

"NXi=1

TXt=3

�j1� � + �

0j

1Xs=0

�sft�s�2 +

1Xs=0

�s"j;t�s�2 +

1Xs=0

�s"j0;t�s�2

!

�i1� � + �

0i

1Xs=0

�sft�s�2 +

1Xs=0

�s"i;t�s�2 +

1Xs=0

�s"j;t�s�2

!#

+plimN!11

N

"NXi=1

TXt=3

�j1� � + �

0j

1Xs=0

�sft�s�2 +

1Xs=0

�s"j;t�s�2 +

1Xs=0

�s"j0;t�s�2

!�

�i1� � + �

0ift�1 + "it�1 + "i+1;t�1

��

= (�� 1) � plimN!11

N

"

NXi=1

TXt=3

1Xs=0

�s"j;t�s�2

!2#= (�� 1) (T � 2)�

2"

1� �2

= � (T � 2)1 + �

�2" 6= 0 (50)

18

B Proof of Proposition 2Assuming that the yit process has been started long time ago it can be shown that

�yj;t�1 = �

0j

1Xs=0

�s�ft�s�1 +

1Xs=0

�s�"j;t�s�1 +

1Xs=0

�s�"j0;t�s�1 (51)

Therefore, we have

plimN!11

N

"NXi=1

TXt=3

�yj;t�1vi;t

#

= plimN!11

N

"NXi=1

TXt=3

�0j

1Xs=0

�s�ft�s�1 +

1Xs=0

�s�"j;t�s�1 +

1Xs=0

�s�"j0;t�s�1

!��i + �

0ift + "it

�i=

TXt=3

f 0t � plimN!11

N

NXi=1

�j�0i

! 1Xs=0

�s�ft�1�s

= 0; (52)

under Assumptions 1-6. In addition, the covariance between �yj;t�1 and yit�1 is di¤erent from zero:

plimN!11

N

"NXi=1

TXt=3

�yj;t�1yi;t�1

#

= plimN!11

N

"NXi=1

TXt=3

�yj;t�1yi;t�1 �

��y

i;t�2 + vi;t�1

�yj;t�2

�#

= plimN!11

N

"NXi=1

TXt=3

�j1� � + �

0j

1Xs=0

�sft�s�1 +

1Xs=0

�s"j;t�s�1 +

1Xs=0

�s"j0;t�s�1

!

�i1� � + �

0i

1Xs=0

�sft�s�1 +

1Xs=0

�s"i;t�s�1 +

1Xs=0

�s"j;t�s�1

!#

�� � plimN!11

N

"NXi=1

TXt=3

�j1� � + �

0j

1Xs=0

�sft�s�2 +

1Xs=0

�s"j;t�s�2 +

1Xs=0

�s"j0;t�s�2

!

�i1� � + �

0i

1Xs=0

�sft�s�2 +

1Xs=0

�s"i;t�s�2 +

1Xs=0

�s"j;t�s�2

!#

�plimN!11

N

"NXi=1

TXt=3

�j1� � + �

0j

1Xs=0

�sft�s�2 +

1Xs=0

�s"j;t�s�2 +

1Xs=0

�s"j0;t�s�2

!��i + �

0ift�1 + "it�1

�i= plimN!1

1

N

(NXi=1

TXt=3

"

1Xs=0

�s"j;t�s�1

!2� �

1Xs=0

�s"j;t�s�2

!2#)

= (T � 2)1 + �

�2" 6= 0 (53)

19

SIMULATIONRESULTS

Table1.MonteCarloresults,b �(R

MSE)

N=400

�=0:5

�=0:7

�=0:9

FE

IVIV

?FD

FD?

SYSSYS?

FE

IVIV

?FD

FD?

SYSSYS?

FE

IVIV

?FD

FD?

SYSSYS?

z=1=3;T=6

:161

1:09

:507

:416

:501

:476

:490

:296

:721

:713

:561

:683

:661

:684

:424

7:88

1:01

:560

:788

:844

:876

(:364)(25:6)(:175)(:226)(:190)(:159)(:109)(:426)(1:14)(:222)(:309)(:219)(:164)(:112)(:495)(307)(2:92)(:557)(:348)(:165)(:112)

z=1=2;T=6

:153

:497

:512

:344

:491

:457

:487

:285

:588

:697

:453

:659

:635

:679

:409

1:20

:058

:430

:726

:821

:869

(:397)(2:33)(:300)(:339)(:218)(:219)(:125)(:458)(8:94)(:971)(:450)(:257)(:223)(:129)(:528)(14:9)(3:47)(:693)(:422)(:216)(:127)

z=2=3;T=6

:140

:483

:570

:258

:467

:436

:481

:269

7:43

:739

:342

:616

:611

:669

:389

:134

:736

:351

:649

:797

:858

(:442)(3:34)(2:22)(:453)(:251)(:276)(:151)(:504)(261)(2:00)(:574)(:310)(:277)(:155)(:575)(21:0)(7:59)(:757)(:487)(:262)(:152)

z=3=4;T=6

:132

:565

:574

:212

:404

:423

:475

:258

-3:80

:511

:293

:578

:599

:661

:375

:612

:591

:312

:598

:788

:850

(:470)(1:84)(1:77)(:507)(:276)(:304)(:171)(:533)(215)(15:3)(:621)(:348)(:303)(:175)(:606)(6:57)(5:42)(:785)(:526)(:283)(:171)

z=1=3;T=10:312

:506

:498

:439

:503

:484

:491

:471

:707

:699

:609

:696

:672

:688

:622

:915

:903

:697

:850

:856

:882

(:213)(:207)(:113)(:145)(:107)(:115)(:072)(:247)(:250)(:129)(:176)(:113)(:115)(:072)(:293)(2:55)(:210)(:320)(:162)(:113)(:070)

z=1=2;T=10:303

:518

:501

:385

:494

:472

:489

:459

:723

:701

:531

:680

:653

:694

:608

1:23

:913

:591

:814

:835

:878

(:244)(:313)(:160)(:224)(:118)(:165)(:083)(:276)(:663)(:191)(:279)(:130)(:166)(:083)(:318)(15:2)(:457)(:429)(:207)(:153)(:081)

z=2=3;T=10:291

:533

:534

:320

:477

:457

:485

:443

:718

:705

:447

:652

:634

:678

:589

:570

:935

:515

:759

:816

:870

(:283)(:560)(:524)(:306)(:134)(:212)(:102)(:315)(1:29)(2:89)(:374)(:158)(:206)(:102)(:355)(16:4)(13:7)(:501)(:267)(:191)(:097)

z=3=4;T=10:283

:832

:497

:287

:464

:449

:481

:433

:773

:560

:409

:631

:625

:673

:576

:970

1:13

:489

:719

:808

:863

(:307)(10:4)(1:58)(:345)(:146)(:235)(:118)(:340)(1:43)(4:23)(:414)(:179)(:228)(:118)(:379)(13:8)(15:1)(:527)(:305)(:208)(:110)

Notes:FEisthe�xede¤ectsestimator,IVistheAnderson-HsiaoestimatorandFDandSYSarethe�rst-di¤erencedandsystem

two-step

GMMestimators,proposedby

Arellano

andBond(1991)andBlundellandBond(1998)respectively.

Thesuperscript���indicatesthatthe

correspondingestimatorusesinstrument(s)withrespecttoanothercross-section,unitj.

FDandFD�makeuseofyit�sandyjt�srespectivelyas

instrumentsfor�yit�1,settings=2;3.Thedatageneratingprocess(DGP)isyit=�i+�yi;t�1+�if t+" it+ " jt,i=1;2;:::;N,t=�48;�47;:::;T

withyi;�49=0,withtheinitial50observationsbeingdiscarded.

�i�iidN(0;�

2 �)," it�iidN(0;�

2 "),f t�iidN(0;�

2 f)and�i�iidU[0;0:5].�2 �

ischosentoensurethattheimpactofthetwovariancecomponents,�ianduit,onvar(yit)isconstant.�2 fisnormalisedtothevalueof1and�2 "

issetaccordingto(46),suchthatitchangesaccordingtotheproportionof�2 uattributedtothefactorstructure.

�alternatesbetween0.5,0.7,

0.9,while isnormalisedto0:5.Allvariablesaretime-demeanedbeforecomputingstatistics.Allexperimentsarebasedon2,000replications.

20

Table2.MonteCarloresults,b �(R

MSE)

N=800

�=0:5

�=0:7

�=0:9

FE

IVIV

?FD

FD?

SYSSYS?

FE

IVIV

?FD

FD?

SYSSYS?

FE

IVIV

?FD

FD?

SYSSYS?

z=1=3;T=6

:162

:534

:501

:427

:508

:478

:496

:295

:750

:703

:569

:701

:663

:693

:424

:968

:885

:548

:867

:846

:888

(:365)(:402)(:120)(:231)(:144)(:159)(:076)(:427)(:795)(:146)(:312)(:164)(:162)(:080)(:495)(6:07)(1:94)(:569)(:250)(:163)(:085)

z=1=2;T=6

:153

:563

:517

:355

:506

:459

:495

:283

:011

:720

:452

:693

:637

:690

:409

:185

:937

:407

:823

:828

:883

(:401)(:717)(:164)(:353)(:162)(:226)(:088)(:462)(21:2)(:204)(:474)(:187)(:228)(:093)(:527)(37:8)(:527)(:717)(:303)(:217)(:098)

z=2=3;T=6

:140

:116

:520

:266

:499

:440

:492

:266

:323

:709

:335

:674

:612

:684

:388

:656

:786

:322

:768

:801

:875

(:450)(20:1)(:498)(:472)(:190)(:291)(:108)(:512)(18:9)(:709)(:584)(:232)(:292)(:114)(:578)(6:69)(2:90)(:788)(:375)(:276)(:120)

z=3=4;T=6

:131

:555

:546

:219

:489

:430

:488

:254

5:75

:771

:281

:653

:600

:679

:374

1:16

:953

:296

:716

:792

:867

(:481)(6:64)(1:55)(:523)(:211)(:323)(:124)(:544)(7:53)(4:23)(:638)(:263)(:323)(:131)(:611)(13:0)(4:51)(:810)(:430)(:293)(:138)

z=1=3;T=10:313

:505

:501

:442

:505

:486

:499

:468

:741

:706

:614

:700

:674

:697

:618

1:05

:898

:693

:885

:851

:897

(:216)(:199)(:089)(:148)(:067)(:119)(:054)(:252)(:531)(:094)(:179)(:076)(:119)(:047)(:299)(2:91)(:137)(:326)(:109)(:117)(:049)

z=1=2;T=10:305

:532

:501

:391

:501

:474

:498

:455

:719

:714

:534

:699

:654

:698

:601

1:03

:921

:582

:880

:837

:891

(:249)(:472)(:123)(:229)(:078)(:169)(:056)(:279)(:610)(:153)(:281)(:072)(:169)(:057)(:323)(12:1)(:236)(:436)(:154)(:159)(:059)

z=2=3;T=10:290

:542

:512

:325

:501

:460

:497

:441

:771

:703

:449

:681

:636

:692

:585

:649

:891

:502

:829

:819

:886

(:289)(1:26)(:211)(:311)(:091)(:219)(:071)(:319)(:994)(:451)(:380)(:101)(:213)(:073)(:358)(10:5)(:879)(:521)(:211)(:201)(:069)

z=3=4;T=10:280

:741

:525

:290

:493

:451

:492

:430

:815

:731

:404

:672

:629

:689

:572

:961

:939

:473

:765

:810

:878

(:313)(3:41)(:576)(:365)(:101)(:243)(:081)(:347)(2:39)(1:74)(:423)(:124)(:232)(:082)(:384)(11:2)(2:22)(:541)(:252)(:219)(:079)

Notes:FEisthe�xede¤ectsestimator,IVistheAnderson-HsiaoestimatorandFDandSYSarethe�rst-di¤erencedandsystem

two-step

GMMestimators,proposedby

Arellano

andBond(1991)andBlundellandBond(1998)respectively.

Thesuperscript���indicatesthatthe

correspondingestimatorusesinstrument(s)withrespecttoanothercross-section,unitj.

FDandFD�makeuseofyit�sandyjt�srespectivelyas

instrumentsfor�yit�1,settings=2;3.Thedatageneratingprocess(DGP)isyit=�i+�yi;t�1+�if t+" it+ " jt,i=1;2;:::;N,t=�48;�47;:::;T

withyi;�49=0,withtheinitial50observationsbeingdiscarded.

�i�iidN(0;�

2 �)," it�iidN(0;�

2 "),f t�iidN(0;�

2 f)and�i�iidU[0;0:5].�2 �

ischosentoensurethattheimpactofthetwovariancecomponents,�ianduit,onvar(yit)isconstant.�2 fisnormalisedtothevalueof1and�2 "

issetaccordingto(46),suchthatitchangesaccordingtotheproportionof�2 uattributedtothefactorstructure.

�alternatesbetween0.5,0.7,

0.9,while isnormalisedto0:5.Allvariablesaretime-demeanedbeforecomputingstatistics.Allexperimentsarebasedon2,000replications.

21