Vasilis Sarafidis

Takashi Yamagata

Donald Robertson

Vasilis Sara…dis

A test of cross section dependence for a linear dynamic panel model with regressors

Keywords: cross section dependence, generalised method of moments, dynamic panel data, overidentifying restrictions test. JEL Classi…cation: C12, C13, C15, C33

This paper proposes a new testing procedure for detecting error cross section dependence after estimating a linear dynamic panel data model with regressors using the generalised method of moments (GMM). The test is valid when the crosssectional dimension of the panel is large relative to the time series dimension. Importantly, our approach allows one to examine whether any error cross section dependence remains after including time dummies (or after transforming the data in terms of deviations from time-speci…c averages), which will be the case under heterogeneous error cross section dependence. Finite sample simulation-based results suggest that our tests perform well, particularly the version based on the Blundell and Bond (1998) system GMM estimator. In addition, it is shown that the system GMM estimator, based only on partial instruments consisting of the regressors, can be a reliable alternative to the standard GMM estimators under heterogeneous error cross section dependence. The proposed tests are applied to employment equations using UK …rm data and the results show little evidence of heterogeneous error cross section dependence.

Introduction

During the past decade a substantial literature has been developed analysing the e¤ects of cross section dependence as well as advancing ways of dealing with it in panel data models. Cross section dependence may arise for several reasons often, due to spatial correlations, economic distance and common unobserved shocks. In the case of spatial attributes, where a natural immutable distance measure is available, the dependence may be captured through spatial lags using techniques that are familiar from the time series literature [START_REF] Anselin | Spatial Econometrics: Method and Models[END_REF][START_REF] Anselin | Spatial Econometrics[END_REF]. In economic applications, spatial techniques are often adapted using alternative measures of economic distance (see e.g. [START_REF] Conley | GMM estimation with cross-sectional dependence[END_REF], Kapoor, Kelejian and Prucha, 2004, [START_REF] Lee | Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models[END_REF][START_REF] Lee | GMM and 2SLS estimation of mixed regressive spatial autoregressive models[END_REF], and others). There are several contributions in the literature that allow for time-varying individual e¤ects (Holtz-Eakin, Newey and Rosen, 1988, Ahn, Lee and[START_REF] Ahn | GMM estimation of linear panel data models with time-varying individual e¤ects[END_REF][START_REF] Han | Estimation of a panel data model with parametric temporal variation in individual e¤ects[END_REF]Schmidt, 2005). Recently, a number of researchers have modelled cross section dependence by restricting the covariance matrix of the errors using a common factor speci…cation with a …xed number of unobserved factors and individual-speci…c factor loadings that give rise to heterogenous cross section dependence (see [START_REF] Forni | Let's Get Real: A Factor Analytical Approach to Disaggregated Business Cycle Dynamics[END_REF][START_REF] Robertson | Factor residuals in SUR regressions: estimating panels allowing for cross-sectional correlation[END_REF][START_REF] Phillips | Dynamic panel estimation and homogeneity testing under cross section dependence[END_REF][START_REF] Stock | Macroeconomic Forecasting Using Di¤usion Indexes[END_REF][START_REF] Bai | A PANIC attack on unit roots and cointegration[END_REF][START_REF] Moon | Testing for A Unit Root in Panels with Dynamic Factors[END_REF][START_REF] Pesaran | Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure[END_REF], among others). The factor structure approach is widely used because it can approximate a wide variety of error cross section dependence. For example, in a panel data set of …rms we may think of the factors as capturing ‡uctuations in economic activity or changes in regulatory policy for the industry as a whole, and so on. The impact of these factors will vary across …rms, due to di¤erences in size, liquidity constraints, market share etc. In a macro panel data model, the factors may represent a general demand shock or an oil price shock with the factor loadings re ‡ecting the relative openness of the economies, di¤erences in technological constraints, and so on. 1In the literature of estimating linear dynamic panel data models with a large number of cross-sectional units (N ) and a moderately small number of time series observations (T ), generalised method of moments (GMM) estimators are widely used, such as those proposed by [START_REF] Arellano | Some tests of speci…cation for panel data: Monte Carlo evidence and an application to employment equations[END_REF], [START_REF] Ahn | E¢ cient Estimations of Models for Dynamic Panel Data[END_REF], [START_REF] Arellano | Another look at the instrumental-variable estimation of error components models[END_REF] and [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]. These methods typically assume that the disturbances are cross-sectionally independent. On the other hand, in empirical applications it is common practice to include time dummies, or, equivalently, to transform the observations in terms of deviations from time-speci…c averages (i.e. to cross-sectionally demean the data) in order to eliminate any common time-varying shocks; see, for example, [START_REF] Arellano | Some tests of speci…cation for panel data: Monte Carlo evidence and an application to employment equations[END_REF] and [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]. This transformation will marginal out these common e¤ects, unless their impact di¤ers across cross-sectional units (heterogeneous cross section dependence). In this case, the standard GMM estimators used in the literature will not be consistent, as shown by Sara…dis and Robertson (2007) and in the current paper.

Several tests for cross section dependence have been proposed in the econometric literature. The most widely used test is perhaps the Lagrange Multiplier (LM) test proposed by [START_REF] Breusch | The Lagrange Multiplier test and its application to model speci…cations in econometrics[END_REF], which is based on the squared pair-wise Pearson's correlation coe¢ cients of the residuals. This test requires T being much larger than N . [START_REF] Frees | Assessing cross-sectional correlation in panel data[END_REF] proposed a cross section dependence test that is based on the squared Spearman rank correlation coe¢ cients and allows N to be larger than T . Recently, [START_REF] Ng | Testing cross-section correlation in panel data using spacings[END_REF] has developed tools for guiding practitioners as to how much residual cross section correlation is in the data and which cross-sectional units are responsible for this in particular, through tests that are based on probability integral transformations of the ordered residual correlations. However, the proposed procedures are valid only in panels for which p T -consistent estimates are available. Pesaran, Ullah and Yamagata (2006) developed bias-adjusted normal approximation versions of the LM test of [START_REF] Breusch | The Lagrange Multiplier test and its application to model speci…cations in econometrics[END_REF], which are valid for large-N panel data models but with strictly exogenous regressors only. [START_REF] Pesaran | General diagnostic tests for cross section dependence in panels[END_REF] proposed another test for cross section dependence, called the CD test, which is closely related to Friedman's (1937) test statistic. Pesaran showed that the CD test can also be applied to a wide variety of models, including heterogeneous dynamic models with multiple breaks and non-stationary dynamic models with small/large N and T . However, as Frees (1995) implied and [START_REF] Pesaran | General diagnostic tests for cross section dependence in panels[END_REF] pointed out, the problem of the CD test is that in a stationary dynamic panel data model it will fail to reject the null of error cross section independence when the factor loadings have zero mean in the cross-sectional dimension. It follows that the CD test will have poor power properties when it is applied to a regression with time dummies or on cross-sectionally demeaned data.

This paper proposes a new testing procedure for error cross section dependence after estimating a linear dynamic panel data model with covariates by the generalised method of moments. This is valid when N is large relative to T . Importantly, unlike the CD test, our approach allows one to examine whether any error cross section dependence remains after including time dummies, or after transforming the data in terms of deviations from time-speci…c averages, which will be the case under heterogeneous error cross section dependence.

The small sample performance of our proposed tests is investigated by means of Monte Carlo experiments and we show that they have correct size and satisfactory power for a wide variety of simulation designs. Furthermore, the paper suggests a consistent GMM estimator under heterogeneous error cross section dependence. Results on the …nite sample properties of the estimator are reported and discussed.

Our proposed tests and estimators are applied to employment equations using UK …rm data, and it is found that there is little evidence of heterogeneous cross section dependence in this data set.

The remainder of the paper proceeds as follows. Section 2 reviews some relevant existing tests for error cross section dependence. Section 3 proposes a new test for cross section dependence and a consistent GMM estimator under these circumstances. Section 4 reports the results of our Monte Carlo experiments. Section 5 illustrates an empirical application of our approach. Finally, Section 6 contains concluding remarks.

Consider a panel data model y it = i + 0 x it + u it , i = 1; 2; : : : ; N , t = 1; 2; : : : ; T ,

where the u it may exhibit cross section dependence. The hypothesis of interest is

H 0 : E(u it u jt ) = 0 8 t for all i 6 = j; (2)
vs H 1 : E(u it u jt ) 6 = 0 for some t and some i 6 = j;

where the number of possible pairings (u it ; u jt ) rises with N . In the literature several tests for error cross section dependence have been proposed, and some relevant ones are discussed in this section.

Breusch-Pagan (1980) Lagrange Multiplier Test

Breusch and Pagan (1980) proposed a Lagrange multiplier (LM) statistic for testing the null of zero cross-equation error correlations, which is de…ned as

LM = T N 1 X i=1 N X j=i+1 ^ 2 ij ; (4)
where ^ ij is the sample estimate of the pair-wise Pearson correlation coe¢ cient of the residuals

^ ij = ^ ji = P T t=1 e it e jt P T t=1 e 2 it 1=2 P T t=1 e 2 jt 1=2 ; (5)
where e it is the Ordinary Least Squares (OLS) estimate of u it in [START_REF] Ahn | GMM estimation of linear panel data models with time-varying individual e¤ects[END_REF]. LM is asymptotically distributed as chi-squared with N (N 1)=2 degrees of freedom under the null hypothesis, as T ! 1 with N …xed.

Pesaran' s (2004) CD Test

Recently [START_REF] Pesaran | General diagnostic tests for cross section dependence in panels[END_REF] proposed another test for cross section dependence, called CD test, which allows for a ‡exible model structure, including fairly general heterogeneous dynamic models and nonstationary models. The test statistic is de…ned as

CD = s 2T N (N 1) 0 @ N 1 X i=1 N X j=i+1 ^ ij 1 A . ( 6
)
For su¢ ciently large N and T , the CD test statistic tends to a standard normal variate under the null of cross section independence. 2 The …nite sample evidence in [START_REF] Pesaran | General diagnostic tests for cross section dependence in panels[END_REF] shows that the estimated size of the test is very close to the nominal level for any combinations of N and T considered. As [START_REF] Pesaran | General diagnostic tests for cross section dependence in panels[END_REF] notes, there are two important cases in which the CD test can be unreliable. Firstly, when the distribution of the errors is not symmetric, the CD test becomes invalid and it may not have correct size. 3Secondly, the CD test may lack power towards some directions of alternatives. To see this, consider the following single-factor structure for the error process

u it = i f t + " it ,
where i is a factor loading that is …xed and bounded, f t is an unobserved common factor such that f t i:i:d:(0; 1), " it i:i:d:(0; 2 ) and E(f t " it ) = 0 for all i and t. The common factor f t generates error cross section dependence because of the fact that cov(u it ; u jt ) = i j , and the power of the CD test hinges on this non-zero covariance. Now suppose that i i:i:d:(0; 2 ) and i is uncorrelated with f t and " it . In this case, cov(u it ; u jt ) = E( i )E( j ) = 0, even if there does exist (potentially large) error cross section dependence.

In the next section, we elaborate on the stochastic properties of the factors and factor loadings, and develop a new cross section dependence test.

Sargan' s Di¤erence Tests for Heterogeneous Error Cross

Section Dependence in a Linear Dynamic Model with Regressors

Model Speci…cation

Consider the following model y it = i + y i;t 1 + 0 x it + u it , i = 1; 2; :::; N , t = 1; 2; :::; T , [START_REF] Arellano | Dynamic panel data estimation using DPD98 for Gauss: A guide for users[END_REF] where j j < 1, is a (K 1) parameter vector that is bounded and non-zero, x it is a (K 1) vector of regressors with x it = (x 1it ; x 2it ; :::; x Kit ) 0 , i is a random e¤ect with …nite mean and …nite variance, and u it has a multi-factor structure such that

u it = 0 i f t + " it ; (8)
where i = ( 1i ; 2i ; :::; M i ) 0 is a (M 1) vector of factor loadings that is assumed to be i:i:d:( ; ) with being a positive semi-de…nite matrix, f t = (f 1t ; f 2t ; :::; f M t ) 0 is a (M 1) vector of time-varying common factors that are assumed to be non-stochastic and bounded, and " it is an independently distributed random variable over i with zero-mean and …nite variance 2 i . The error (multi-) factor structure has been employed extensively in the economic literature. 4 For example, in a macro panel, a common factor could be an unobserved technological shock, and the factor loadings can be thought of as capturing a crosssectionally heterogeneous response to such shock. Note that since our asymptotic is N ! 1 with T …xed, f t is treated as non-stochastic here. In this paper we explicitly employ a random coe¢ cient assumption for the factor loadings.

The process for x it is de…ned as

x it = i + x i;t 1 + xi f t + " i;t 1 + v it , i =
1; 2; :::; N , t = 1; 2; :::; T ,

where i = ( i1 ; i2 ; :::; iK ) 0 is a (K 1) vector of random e¤ects with …nite mean and …nite variance, signi…es the Hadamard product, = ( 1 ; 2 ; :::; K ) 0 such that j k j < 1 for k = 1; 2; :::; K, xi = ( 1i ; 2i ; :::; Ki ) 0 with ki = ( ki1 ; ki2 ; :::; kiM ) 0 , such that xi i:i:d:( x ; x ), = ( 1 ; 2 ; :::; K ) 0 , and v it is a vector of independently distributed random variables over i with mean vector zero and a …nite variance matrix vi = diag( 2 vki ), k = 1; 2; :::; K. The model de…ned by [START_REF] Arellano | Dynamic panel data estimation using DPD98 for Gauss: A guide for users[END_REF], [START_REF] Arellano | Another look at the instrumental-variable estimation of error components models[END_REF], and ( 9) is general enough to allow for a large variety of plausible speci…cations that are widely used in the economic literature. Furthermore, this model accommodates more simple processes for x it ; such as those where x it is strictly exogenous, or exogenous with respect to f t :

The null hypothesis of interest is then

H 0 : var( i ) = = 0 (10)
against the alternative

H 1 : 6 = 0, (11)
as opposed to (2) and (3).

Remark 1 Observe that error cross section dependence may occur under the null hypothesis when i = for all i, since E(u it u jt ) = 0 f t f 0 t , which is not zero unless = 0. However, such error cross section dependence can be eliminated simply by including time dummies, or equivalently by cross-sectionally demeaning the data. This implies that the null hypothesis in [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] can be interpreted as saying that the cross section dependence is homogeneous across pairs of cross-sectional units, against the alternative hypothesis (11) of heterogeneous error cross section dependence.

We make the following assumptions: Assumption 1: E( i " it ) = 0, E( i " it ) = 0, and E( i v it ) = 0, for all i, t. Also E (v it " is ) = 0 for all t and s.

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

Assumption 3: E(y i0 " it ) = 0 and E(x i1 " it ) = 0 for all i and t = 1; 2; :::; T .

Assumption 4: Time-varying common factors, f m;t , m = 1; 2; :::; M ; t = 1; 2; :::; T are non-stochastic and bounded.

Assumption 5:

(a) E(" it i ) = 0, E(v it 0 i ) = 0, E(" it xi ) = 0, cov( i ; i ) = 0, cov( i ; xi ) = 0, cov( i ; 0 i ) = 0, cov( 0 xi ; i ) = 0, for all i and t. (b) cov( xi ; i ) = 0 and cov( i ; x 0 i1 ) = 0.
Assumption 6: 6 = 0.

Assumptions 1-3 are standard in the GMM literature; see, for example, [START_REF] Ahn | E¢ cient Estimations of Models for Dynamic Panel Data[END_REF]. Assumption 4 ensures that the initial observations of y and x are bounded.

Assumption 5(a), a random coe¢ cient type assumption on the factor loadings, allows cross-sectionally heterogeneous inter-dependence among both level and …rst-di¤erenced variables (y it ; x 1it ; x 2it ; :::; x Kit ), as well as cross section dependence of each variable through the common factor f t . This contrasts to a simple time e¤ects assumption, namely i = and xi = x for all i, which is stronger than ours. On the other hand, Assumption 5(a) is stronger than assuming i and xi to be merely bounded, in that the cross-sectionally demeaned variables of (y it ; x 1it ; x 2it ; :::; x Kit ) become cross-sectionally uncorrelated to each other, asymptotically.

Assumption 5(b) implies that the cross-sectionally demeaned x i;t 1 and (u it u i;t 1 ) or the cross-sectionally demeaned (x i;t 1 x i;t 2 ) and u it are uncorrelated under the alternative hypothesis [START_REF] Blundell | GMM estimation with persistent panel data: an application to production functions[END_REF]. This will obviously hold true if the common factor components have no impact on x it namely, when xi = 0 for all i in equation ( 9). This restriction can be relaxed provided that the direct e¤ect of the factors on y it is uncorrelated with the e¤ect of the factors on x it , which satis…es Assumption 5(b). 5Assumption 6, 6 = 0, is also necessary in order to obtain a consistent estimator under the alternative of heterogeneous error cross section dependence, although this is probably not a restrictive assumption in many applications.

For further discussion, stacking [START_REF] Arellano | Dynamic panel data estimation using DPD98 for Gauss: A guide for users[END_REF] for each i yields

y i = i T + y i; 1 + X i + u i , i = 1; 2; :::; N , ( 12
)
u i = F i + " i , (13)
where y i = (y i1 ; y i2 ; :::; y iT ) 0 , g is a (g 1) vector of unity, y i; 1 = (y i0 ; y i1 ; :::; y iT 1 ) 0 , X i = (x i1 ; x i2 ; :::; x iT ) 0 , u i = (u i1 ; u i2 ; :::; u iT ) 0 , F = (f 1 ; f 2 ; :::; f T ) 0 , " i = (" i1 ; " i2 ; :::; " iT ) 0 , and the cross-sectionally demeaned and di¤erenced equation is de…ned by

y i = y i; 1 + X i + u i , i = 1; 2; :::; N , ( 14
)
u i = F i + " i , (15)
or

y i = W i + u i , (16)
where the underline signi…es that the variables are cross-sectionally demeaned, and " " denotes the …rst-di¤erenced operator. For example, y i = ( y i2 ; y i3 ; :::; y iT ) 0 , y it = (y it y it 1 ), y it = (y it y t ) with y t = N 1 P N i=1 y it , W i = ( y i; 1 ; X i ), = ( ; 0 ) 0 , with obvious notation.

Remark 2 Pesaran's CD test statistic as de…ned in ( 6) can fail to reject the null hypothesis of homogeneous error cross section dependence given in [START_REF] Blundell | GMM estimation with persistent panel data: an application to production functions[END_REF], when E( i ) = 0. This arises when time dummies are introduced or the data are cross-sectionally demeaned, since E( i ) = 0. On the other hand, the Breusch and Pagan (1980) LM test de…ned as in (4) will have power, although its use is justi…ed only when T is much larger relative to N .

Sargan' s Di¤erence Tests based on the First-di¤erenced GMM Estimator

De…ne the matrices of instruments

Z Y i = 2
where h y = T (T 1)=2, and

Z Xi = 2 6 6 6 6 6 4
x 0 i1 0 0 0 0 0 x 0 i1 x 0 i2 0 0 0 0 0 x 0 i1 0 . . . . . . . . . . . .
. . . ::: ::: . . .

0 0 0 0 x 0 i1 x 0 iT 1 3 7 7 7 7 7 5
, (T 1 h x ) (18)
where

h x = KT (T 1)=2. If x it is strictly exogenous that is, = 0 in (9), Z Xi = T 1 vec(X i ) 0 . 6 A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT Proposition 3 Consider two sets of moment conditions Moment Conditions I: E[Z 0 Y i u i ] = 0 (19)
and Moment Conditions II:

E[Z 0 Xi u i ] = 0, (20)
where Z Y i , Z Xi , and u i are de…ned as in ( 17), [START_REF] Frees | Assessing cross-sectional correlation in panel data[END_REF], and ( 15), respectively. Under Assumptions 1-4, 5(a)-(b), 6 and model ( 7), both sets of moment conditions given in ( 19) and ( 20) hold under the null hypothesis [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]. However, under the alternative hypothesis given in [START_REF] Blundell | GMM estimation with persistent panel data: an application to production functions[END_REF], [START_REF] Hahn | Long di¤erence instrumental variables estimation for dynamic panel models with …xed e¤ects[END_REF] holds but [START_REF] Friedman | The use of ranks to avoid the assumption of normality implicit in the analysis of variance[END_REF] does not.

Proof. See Appendix A. Now de…ne the full set of moment conditions

E[Z 0 i u i ] = 0, (21)
where

Z i = Z Y i Z Xi (T 1 h); (22)
and h = h y + h x .

Based on Proposition 3, the hypothesis testing for (10) reduces to a test for the validity of the subset of the moment conditions in [START_REF] Han | Estimation of a panel data model with parametric temporal variation in individual e¤ects[END_REF], as given in [START_REF] Friedman | The use of ranks to avoid the assumption of normality implicit in the analysis of variance[END_REF]. There are at least three possible approaches. The …rst one is to take a Hausman-type (1978) approach. [START_REF] Newey | Generalized method of moments speci…cation testing[END_REF] showed that the Hausman test may be inconsistent against some local alternatives. In our application, this may happen when h y = T (T 1)=2 > K + 1, which is the most likely empirical situation. 7 In addition, Arellano and Bond (1991) examined a Hausman-type test for serial correlation and found that it has poor …nite sample performance, in that it rejects the null hypothesis far too often. The second approach is to adopt Newey's (1985) optimal test. The third approach is to use [START_REF] Sargan | Testing for misspeci…cation after estimating using instrumental variables[END_REF] di¤erence test, which we are adopting, since it is relatively easier to compute than the Newey's test statistic, and furthermore, they are asymptotically equivalent. We consider two versions of Sargan's di¤erence test. One is based on the Arellano and Bond (1991) two-step …rst-di¤erenced GMM estimator hereafter DIF and the other is based on the Blundell and Bond (1998) two-step system GMM estimator hereafter, SYS.

Initially, the test based on DIF is considered. This is de…ned as

e • DIF 2 = N X i=1 W 0 i Z i e _ 1 N X i=1 Z 0 i W i ! 1 N X i=1 W 0 i Z 1 i e _ 1 N X i=1 Z 0 i y i ; (23)
where

e _ = N 1 P N i=1 Z 0 i e _ u i e _ u 0 i Z i with e _ u i = y i W i e _ DIF 1
, which is the residual vector based on the one-step DIF estimator (denoted by DIF 1),

e _ DIF 1 = N X i=1 W 0 i Z 1 i e 1 N X i=1 Z 0 i W i ! 1 N X i=1 W 0 i Z i e 1 N X i=1 Z 0 i y i ; (24)
where e = N 1 P N i=1 Z 0 i HZ i , and H is the square matrix of order T 1, with 2's on the main diagonal, -1's on the …rst o¤-diagonals, and zeros elsewhere. Now, the two-step DIF estimator based only on the restricted set of moment conditions in [START_REF] Hahn | Long di¤erence instrumental variables estimation for dynamic panel models with …xed e¤ects[END_REF], which is consistent both under the null and the alternative, is de…ned as

b • DIF X2 = N X i=1 W 0 i Z Xi b _ 1 X N X i=1 Z 0 Xi W i ! 1 N X i=1 W 0 i Z Xi b _ 1 X N X i=1 Z 0 Xi y i ; (25) where b _ X = N 1 P N i=1 Z 0 Xi b _ u i b _ u 0 i Z Xi with b _ u i = y i W i b _ DIF X1
, where b _ u i is the residual vector based on the one-step DIF estimator that exploits [START_REF] Hahn | Long di¤erence instrumental variables estimation for dynamic panel models with …xed e¤ects[END_REF], denoted by

DIF X1, b _ DIF X1 = N X i=1 W 0 i Z Xi b 1 X N X i=1 Z 0 Xi W i ! 1 N X i=1 W 0 i Z Xi b 1 X N X i=1 Z 0 Xi y i ; ( 26
)
where b X = N 1 P N i=1 Z 0 Xi HZ Xi . Sargan's (1958), or [START_REF] Hansen | Large sample properties of generalized method of moments estimators[END_REF] test statistic of overidentifying restrictions for the full set of moment conditions in ( 21) is based on DIF 2 and is given by

S DIF 2 = N 1 N X i=1 e • u 0 i Z i ! e _ 1 N X i=1 Z 0 i e • u i ! , (27)
where e

• u i = y i W i e • DIF 2 .
Sargan's statistic for [START_REF] Hahn | Long di¤erence instrumental variables estimation for dynamic panel models with …xed e¤ects[END_REF] that is based on DIF X2 is given by

S DIF X2 = N 1 N X i=1 b • u 0 i Z Xi ! b _ 1 X N X i=1 Z 0 Xi b • u i ! ; ( 28
)
where b

• u i = y i W i b • DIF X2
. We are now ready to state a proposition on Sargan's di¤erence test of heterogeneous cross section dependence: hx (K+1) under the null hypothesis and the alternative hypothesis, and

D DIF 2 = (S DIF 2 S DIF X2 ) d ! 2 hy (29)
under the null hypothesis, where S DIF 2 , S DIF X2 are as de…ned in ( 27) and ( 28) respectively, while h = h y + h x with h y and h x de…ned as in ( 17) and ( 18) respectively.

Proof. See Appendix B.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Remark 5 The cross section correlation of the sample moment conditions Z 0 i e • u i ; which is induced as a result of the cross-sectional demeaning of the series, is asymptotically negligible, and standard central limit theorems can be applied to the asymptotic expansion of Arellano and Bover (1995) proposed the use of lagged di¤erences as possible instruments for the equations in levels, E (u it w it ) = 0 with w it = (y t 1 ; x 0 t ) 0 for t = 3; 4; :::; T , which is valid under the null hypothesis and Assumptions 1-3. In addition, Blundell andBond (1998, 2000) proposed using an additional condition E (u i2 w i2 ) = 0 under mild conditions upon the initial observations, which would follow from joint stationarity of the y and x processes. Accordingly, we add Assumption 5(c):

N 1=2 P N i=1 Z 0 i e • u i ; See
Assumption 5(c): cov( i ; y i1 ) = 0 and cov( i ; x i2 ) = 0.

De…ne

Z + Y i = Z Y i 0 0 Z L Y i
(2(T 1) h ys );

where Z L Y i = diag( y i;t 1 ) for t = 2; 3; :::; T , h ys = h y + (T 1); and

Z + Xi = Z Xi 0 0 Z L Xi (2(T 1) h xs ); (31)
where

Z L Xi is a (T 1 K(T 1
)) matrix whose s th diagonal raw vector is x 0 is , s = 2; 3; :::; T , otherwise zeros, and h xs = h x + K(T 1).

Proposition 8 Consider two sets of moment conditions

Moment Conditions I :

E[Z +0 Y i i ] = 0 (32)
and Moment Conditions II :

E[Z +0 Xi i ] = 0, (33)
where Z + Y i and Z + Xi are de…ned as in ( 30) and ( 31) respectively, i = ( u 0 i ; u +0 i ) 0 ; and u + i = (u i2 ; u i3 ; :::; u iT ; ) 0 .

Under Assumptions 1-4, 5(a)-(c), 6 and model [START_REF] Arellano | Dynamic panel data estimation using DPD98 for Gauss: A guide for users[END_REF], both ( 32) and ( 33) hold under the null hypothesis in [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]. However, under the alternative hypothesis given in ( 11), [START_REF] Pesaran | General diagnostic tests for cross section dependence in panels[END_REF] holds but [START_REF] Ng | Testing cross-section correlation in panel data using spacings[END_REF] does not.

Proof. See Appendix C.

In the same manner as D DIF 2 , we de…ne the Sargan's di¤erence test statistic based on SYS estimator

D SY S2 = (S SY S2 S SY SX2 ) (35)
where S SY S2 is Sargan's test statistic of overidentifying restrictions based on the twostep SYS estimator that makes use of both sets of moment conditions I and II in Proposition 8 (denoted by SY S2), namely

E[Z +0 i i ] = 0, (36)
where

Z + i = Z + Y i Z + Xi (2(T 1) h s ); (37)
h s = h sy +h sx , and S SY SX2 is Sargan's test statistic based on the two-step SYS estimator that exploits only Moment Conditions II , as de…ned in [START_REF] Pesaran | General diagnostic tests for cross section dependence in panels[END_REF], denoted by SY SX2. 9 One-step and two-step SYS estimators based on the moment conditions [START_REF] Pesaran | A Bias-Adjusted LM Test of Error Cross Section Independence[END_REF]

Discussion

The overidentifying restrictions test can be regarded as a misspeci…cation test, in a sense that it is designed to detect violations of moment conditions, which are the heart of GMM methods. Thus, it will have power under the alternative hypothesis of heterogenous error cross section dependence. Nonetheless, the proposed Sargan's di¤erence test is expected to have higher power than the overidentifying restrictions test, so long as Assumption 6, 6 = 0, holds, since the former exploits extra information about the validity of the moment conditions under the alternative hypothesis, which the latter does not use. This 9 With small samples DDIF 2 may not be positive, but it can be patched easily. See, for example, Hayashi (2000, p.220). However, we did not adopt this modi…cation here since one of our aims is to show the properties of a consistent estimator based only on orthogonality conditions

E[Z 0 Xi u i ] = 0 or E[Z +0
Xi i ] = 0. 1 0 The initial weighting matrix for one-step GMM-SYS estimator is de…ned as a block diagnoal matrix of order 2(T 1), whose diagnoal blocks are H and IT 1. also implies that, when = 0, the overidentifying restrictions test should replace our approach. 11 Now consider a violation of Assumption 2, E(" it " is ) = 0 for t 6 = s, no error serial correlation. Under the alternative of error cross section dependence, the composite error u it = 0 i f t + " it will be serially correlated, since E(u it u is ) = f 0 t E( i 0 i )f s 6 = 0 for all i. This means that the second-order serial correlation test based on DIF2 or SYS2, the m 2 test, proposed by [START_REF] Arellano | Some tests of speci…cation for panel data: Monte Carlo evidence and an application to employment equations[END_REF], is likely to reject the hypothesis of no error serial correlation, under the alternative of heterogeneous error cross section dependence. 12 Then, a question that may arise is how to distinguish between error cross section dependence and serial correlation in the idiosyncratic errors. To answer this question, consider two scenarios. First, suppose that there is …rst-order autoregressive serial correlation but no heterogeneous error cross section dependence, such that " it = " " it 1 + it with j " j < 1 and it i:i:d:(0;2 ). In this case, the problem can be solved in a straightforward manner by adding a further lag of the dependent variable on the right hand side of (7) and using (up to) y it 3 as instruments for y it 1 and y it 2 . Second, suppose there is both …rst-order autoregressive error serial correlation as above and heterogeneous error cross section dependence. Clearly, the m 2 test based on DIFX2 or SYSX2 is likely to reject the null even when E(" it " is ) = 0 for t 6 = s. Meanwhile, the probability of rejecting the null by the overidentifying restrictions test for the restrictions based only on the subset of X i (de…ned by [START_REF] Hahn | Long di¤erence instrumental variables estimation for dynamic panel models with …xed e¤ects[END_REF] or [START_REF] Pesaran | General diagnostic tests for cross section dependence in panels[END_REF]), tends to its signi…cance level when E(" it " is ) = 0 for t 6 = s, but such a probability goes to one when E(" it " is ) 6 = 0 for t 6 = s. Therefore, the solution given in the …rst case applies, but the test statistic to employ is the overidentifying restrictions test based only on the subset of X i , not the m 2 test based on DIFX2 or SYSX2.

Finally, we have shown that the moment conditions ( 20) and ( 33) hold under the alternative of error cross section dependence, therefore, the DIFX2 and SYSX2 estimators are consistent. However, in …nite samples there could be a trade-o¤ between e¢ ciency and bias. If the degree of heterogeneity of the error cross section dependence is relatively small, then the bias of the standard GMM estimators exploiting moment conditions including [START_REF] Friedman | The use of ranks to avoid the assumption of normality implicit in the analysis of variance[END_REF] or [START_REF] Hahn | Long di¤erence instrumental variables estimation for dynamic panel models with …xed e¤ects[END_REF] which are invalid, may be small enough so that these estimators are preferred (in root mean square errors terms) to a consistent estimator based only on the valid moment conditions [START_REF] Hahn | Long di¤erence instrumental variables estimation for dynamic panel models with …xed e¤ects[END_REF] or [START_REF] Pesaran | General diagnostic tests for cross section dependence in panels[END_REF]. We will investigate the …nite sample performance of these estimators in the next section. 13

Tests

This section investigates by means of Monte Carlo experiments the …nite sample performance of our tests, the Breusch and Pagan (1980) LM test and Pesaran's (2004) CD test, all based on cross-sectionally demeaned variables. Our main focus is on the e¤ects of (i) the degree and heterogeneity of error cross section dependence, (ii) the relative importance of the variance of the factor loadings and the idiosyncratic errors, and (iii) di¤erent values of and . In order to make the results comparable across experiments, we control the population signal-to-noise ratio and the impact of the ratio between the variance of the individual-speci…c time-invariant e¤ects and the variance of the idiosyncratic errors and the common factor on y it . To this end, we extend the Monte Carlo design of [START_REF] Kiviet | On bias, inconsistency, and e¢ ciency of various estimators in dynamic panel data models[END_REF] and [START_REF] Bun | The e¤ects of dynamic feedbacks on LS and MM estimator accuracy in panel data models[END_REF] to accommodate a factor structure in the error process.

Recently [START_REF] Bowsher | On testing overidentifying restrictions in dynamic panel data models[END_REF] reports …nite sample evidence that Sargan's overidentifying restrictions test exploiting all moment conditions available can reject the null hypothesis too infrequently in linear dynamic panel models. Thus, we only make use of y it 2 and y it 3 as instruments for y it 1 and we use x it 2 and x it 3 as instruments for x it 1 .

Design

The data generating process (DGP) we consider is given by

y it = i + y i;t 1 + x it + u it ; (38)
u it = i f t + " it , i = 1; 2; :::; N ; t = 48; 47; :::; T , where i i:i:d:N (1; 2 ) and f t i:i:d:N (0; 2 f ). 14 " it is drawn from (i) i:i:d:N (0; 2 " ) and (ii) i:i:d:( 21 1)= p 2, in order to investigate the e¤ect of non-normal errors. y i; 49 = 0 and the …rst 49 observations are discarded. 15 To control the degree and heterogeneity of cross section dependence three speci…cations for the distribution of i are considered: Also, as we change the value of = 0:2; 0:5; 0:8, is equated to 1 in order to keep the long run e¤ect of x on y constant.

The DGP of x it considered here is given by

x it = x i;t 1 + " i;t 1 + i f t + v it , i =
1; 2; :::; N ; t = 48; 47; :::; T , [START_REF] Sara…dis | On The Impact of Cross-sectional Dependence in Short Dynamic Panel Estimation[END_REF] where = 0:5, i i:i:d:U [ 1; 2], v it i:i:d:N (0; 2 v ). is set to 0:5. 16 x i; 49 = 0 and the …rst 50 observations are discarded.

Since our focus is on the performance of the tests and estimators, we pay careful attention to the main factors that a¤ect it namely, (i) the signal-to-noise ratio, (ii) the relative importance of the variance of the factor loadings and the idiosyncratic errors, and (iii) the impact of the ratio between the variance of the individual-speci…c e¤ects and the variance of the idiosyncractic error and factor loadings on y it . To illustrate, we de…ne the signal as 2 s = var (y it u it ), where y it = y it i =( 1). Then, denoting the variance of the composite error by 2 u = var(u it ) =

(2) 2

f + 2 " with (2) = E( 2 i )
, we de…ne the signal-to-noise ratio as = 2 s = 2 u . We set = 3. The relative importance in terms of the magnitude of the variance of i f t and " it , as measured by

(2) 2 f = 2
" , is thought in the literature to be an important factor to control for and we achieve this by changing (2) and applying the normalisation

2 f = 2 " = 1.
As it has been discussed by [START_REF] Kiviet | On bias, inconsistency, and e¢ ciency of various estimators in dynamic panel data models[END_REF], [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF], and [START_REF] Bun | The e¤ects of dynamic feedbacks on LS and MM estimator accuracy in panel data models[END_REF], in order to compare the performance of estimators across di¤erent experimental designs it is important to control the relative importance of i and (" it , f t ). We choose 2 such that the ratio of the impact on var(y it ) of the two variance components i and (" it , f t ) is constant across designs. 17 We consider all combinations of N = 50; 100; 200; 400, and T = 5; 9. All experiments are based on 2,000 replications.

Results

Tables 1 reports the size and power of the tests for T = 5. 18 LM denotes Breusch and Pagan's LM test, as de…ned in (4), and CD denotes Pesaran's CD test, de…ned in [START_REF] Arellano | Some tests of speci…cation for panel data: Monte Carlo evidence and an application to employment equations[END_REF], both of which are based on the …xed e¤ects estimator. D DIF 2 is Sargan's di¤erence test based on the two-step DIF estimator de…ned in [START_REF] Lee | GMM and 2SLS estimation of mixed regressive spatial autoregressive models[END_REF], and D SY S2 is Sargan's di¤erence test based on the two-step SYS estimator de…ned in [START_REF] Pesaran | Estimating long-run relationships from dynamic heterogeneous panels[END_REF]. The size of the LM test is always indistinguishable from 100% and therefore it is not recommended. The CD test does not reject the null in all experiments, and has no power across experiments. On the other hand, although the size of D DIF 2 and D SY S2 is below the nominal level for N = 50 (especially for the latter), as N becomes larger the size quickly approaches its nominal size. In addition, our proposed tests have satisfactory power. D SY S2 has more power than D DIF 2 in general, unless D SY S2 rejects the null too infrequently. Di¤erent values of seem to have very little e¤ect on the performance of D DIF 2 and D SY S2 .

We now turn our attention to the performance of the estimators. Table 2 reports the bias of the estimators for . 19 DIF 1 and DIF 2 are the one-step and two-step DIF estimators respectively, de…ned by ( 24) and ( 23), and they are based on the full set of moment conditions I and II in Proposition 3. DIF X1 and DIF X2 denote the one-step and two-step DIF estimators de…ned by ( 26) and ( 25), and they are based only on the subset of the moment conditions II. SY S1 and SY S2 are the one-step and two-step SYS estimators respectively, and they are based on the full set of moment conditions I and II in Proposition 8, and SY SX1 and SY SX2 denote the one-step and two-step SYS estimators based only on the subset of moment conditions II . The bias of all GMM estimators under low cross section dependence is not noticeably di¤erent from that under zero cross section dependence. As the degree of error cross section dependence rises, the bias of the GMM estimators based on the full set of moment conditions increases, which is expected as only those estimators based on Moment Conditions II or II are consistent. As a result, the relative bias between those estimators that use the full set of moment conditions and those that use only Moment Conditions II or II increases. Table 3 reports root mean square errors of the estimators for . Under no error cross section dependence and low cross section dependence, DIF 2 and SY S2 outperform DIF X2 and SY SX2 respectively in terms of root mean square error. However, under moderate and high cross section dependence, DIF X2 and SY SX2 have a smaller root mean square error compared to DIF 2 and SY S2 respectively, in most cases.

An Empirical Example: Employment Equations of U.K. Firms

In this section we examine the homogeneity of error cross section dependence of the employment equations using (unbalanced) panel data for a sample of UK companies, which is an updated version of that used by [START_REF] Arellano | Some tests of speci…cation for panel data: Monte Carlo evidence and an application to employment equations[END_REF], and it is contained in the DPD-Ox package. 20 Brie ‡y, these authors select a sample of 140 companies that operate mainly in the UK with at least 7 continuous observations during the period 1976-1984. We apply our test to the model speci…cations of [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]. The model we estimated is given by

y it = i + 1 y it 1 + 0 w it + 1 w it 1 + ' 0 it + ' 1 it 1 + u it ,
where y it is log of the number of employees of company i, w it is log of real product wage, it is the log of gross capital stock.

Table 4 presents estimation and test results. 21 Observe that year dummies are included to remove possible time e¤ects, therefore no cross-sectional demeaning of the series is implemented. Our estimation results based on the full sets of instruments, Z i and Z + i , as de…ned in ( 22) and ( 37) but without cross-sectional demeaning, resemble those reported in the last two columns of Table 4 in [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF], although the values do not match exactly due to di¤erences in computations and the data set used. First, all m 2 tests suggest that there is no evidence of error serial correlation and this implies possibly no heterogeneous error cross section dependence. This is con…rmed by the fact that both Sargan's di¤erence tests based on DIF and SYS for heterogeneous error cross section dependence safely fail to reject the null hypothesis of homogeneous error cross section dependence. The estimation results based only on partial instruments consisting of the covariates, Z Xi and Z + Xi , as de…ned by ( 18) and ( 31) but without crosssectional demeaning, are largely downward biased for the DIF estimator and less so for the SYS estimator. This indicates that the e¢ ciency loss of SYS that does not contain Z + Y i in the instrument set is much smaller compared to the e¢ ciency loss of DIF. This feature seems to have some e¤ect on the testing results. For example, the p-value of the …rst-order serial correlation test, m 1 , for DIF with the full set of instruments, Z i , is zero up to three decimal points, but it goes up to 0.028 with the subset of instruments Z Xi . On the other hand, the p-value of m 1 for SYS is zero up to three decimal points in both cases.

Concluding Remarks

This paper has proposed a new testing procedure for error cross section dependence after estimating a linear dynamic panel data model with regressors by the generalised method of moments (GMM). The procedure is valid when the cross-sectional dimension is large and the time series dimension of the panel is small. Importantly, our approach allows one to examine whether any error cross section dependence remains after including time dummies, or after transforming the data in terms of deviations from time-speci…c averages, which will be the case under heterogeneous error cross section dependence. The …nite sample simulation-based results suggest that our tests perform well, particularly the version based on the Blundell and Bond (1988) system GMM estimator. On the other hand, the LM test of [START_REF] Breusch | The Lagrange Multiplier test and its application to model speci…cations in econometrics[END_REF] overrejects the null hypothesis substantially and Pesaran's (2004) CD test lacks power. Also it is shown that the system GMM estimator, based only on partial instruments consisting of the regressors, can be a reliable alternative to the standard GMM estimators under heterogeneous error cross section dependence. The proposed tests are applied to employment equations using UK …rm data, and the results show little evidence of heterogeneous error cross section dependence.

1 p N N X i=1 Z o0 Xi u o i d ! N (0; I hx ).

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Noting that rank(MB) = hx (K + 1) we have

SDIF X2 = 1 N N X i=1 u o0 i Z o Xi ! 1=2 X MB 1=2 X N X i=1 Z o0 Xi u o i ! + op(1) (52) d ! 2 hx (K+1) ,
under the alternative hypothesis of H1 : 6 = 0, as required. Under the null hypothesis of H0 : = 0, (52) follows immediately. Also it is straightforward to establish that SDIF

2 d ! 2 h (K+1)
, where SDIF 2 is de…ned as in [START_REF] Kiviet | On bias, inconsistency, and e¢ ciency of various estimators in dynamic panel data models[END_REF], in line with the proof provided for (52). Now we provide the asymptotic distribution of SDIF 2 SDIF X2. Consider the local alternative

HN : i = + i N 1=4 ,
where 0 < jj i jj < 1 for all i, which are assumed to be non-stochastic for expositional convenience. Here the analysis is based on the instruments in terms of deviations from their true mean, rather than from the cross-sectional average, since we have already shown that the e¤ect of such replacement is asymptotically negligible. Without loss of generality, consider Z o Y i = diag(y o i;t 2 ), t = 2; 3; :::; T , hy = T 1. Also de…ne

Z o i = Z o Y i Z o Xi , = p lim N !1 N 1 N X i=1 E Z o0 i u o i u o0 Z o i , (53) where =
Y Y X XY X , (54)
with block elements that are conformable with Z o Y i and Z o Xi . By using [START_REF] Stock | Macroeconomic Forecasting Using Di¤usion Indexes[END_REF] we have

E N 1=2 N X i=1 Z o0 Y i u o i ! = N ,
where N = O(1) is a (T 1 1) vector whose …rst element is N 1 P N i=1 E(y i0 i 0 ) f2 and the (t 1) th elements are N 1 P N i=1 t E y i0 i 0 ft+2 + f 0 t+2 i i 0 P t 1 j=0 j ft j , for t = 2; 3; :::; T 1. De…ne

Z o i = Z o i L 0 , with L = Ih y Y X 1 X 0 Ih x
where L is non-singular, so that

1 p N N X i=1 Z o 0 i u o i N 0 d ! N (0;
) , where

= L L 0 = Y 0 0 X ; with Y = Y Y X 1 X XY . It follows that SDIF 2 = 1 N N X i=1 u o0 i Z o i ! 1=2 M B 1=2 N X i=1 Z o 0 i u o i ! + op(1)
with

M B = I B (B 0 B ) 1 B 0 , B = B Y B = 1=2 N 1 N X i=1 Z o 0 i W o i , so that SDIF 2 SDIF X2 = 1 N N X i=1 u o0 i Z o i ! 1=2 M B 1=2 N X i=1 Z o 0 i u o i ! (55) 1 N N X i=1 u o0 i Z o Xi ! 1=2 X MB 1=2 X N X i=1 Z o0 Xi u o i ! + op(1) = 1 N N X i=1 u o0 i Z o i ! 1=2 M 1=2 N X i=1 Z o 0 i u o i ! + op(1)
where

M = M B 0 0 0 MB = M11 M12 M21 M22 , ( 56
)
which is a symmetric and idempotent matrix of rank hy. Finally, we have

SDIF 2 SDIF X2 = 1 N N X i=1 u o0 i Z o 0 i ! 1=2 M 1=2 N X i=1 Z o 0 i u o i ! + op(1) d ! 2 (hy; )
which is a non-central chi-squared distribution with hy degrees of freedom and non-centrality parameter

= p limN!1 0 N 1=2 Y M11 1=2 Y N > 0, so long as N 6 = 0. Therefore, the result SDIF 2 SDIF X2 d ! 2
hy readily follows under the null hypothesis, as required.

Furthermore, using (55) and (56), it is easily seen that SDIF 2 d ! 2 (h (K + 1); ) under the local alternatives, where is the same non-centrality parameter of the asymptotic distribution of (SDIF 2 SDIF X2). Therefore, when h (K + 1) > hy, or subtracting hy from both sides hx > (K + 1), the Sargan's di¤erence test is locally more powerful than the overidentifying restrictions test.

C Proof of Proposition 8

In line with the proof of Proposition 3 in Appendix A, consider E( x it ( i + u it )). For t 3 and using [START_REF] Sargan | The estimation of economic relationships using instrumental variables[END_REF] together with u it = 0 i ft + " it , we have under the alternative hypothesis of i 6 = , E( x it ( i + u it )) = 0, under Assumptions 1-4, 5(a)-(c), 6 and model [START_REF] Arellano | Dynamic panel data estimation using DPD98 for Gauss: A guide for users[END_REF]. A similar line of argument proves that E( x i2 ( i + u i2 )) = 0. However, for t 2 and using [START_REF] Sargan | The estimation of economic relationships using instrumental variables[END_REF] and [START_REF] Sargan | Testing for misspeci…cation after estimating using instrumental variables[END_REF] we have under the alternative

E( y it ( i + u it+1 )) = t E y i1 0 i ft+1 + f 0 t+1 E i 0 i t 1 X j=0 j ft j 6 = 0;
under Assumptions 1-4, 5(a)-(c), 6 and model [START_REF] Arellano | Dynamic panel data estimation using DPD98 for Gauss: A guide for users[END_REF]. A similar line of argument will prove that E( y i1 ( i + u i2 )) 6 = 0 under the alternative. Finally, under the null hypothesis, it is also easily seen that E( y it ( i + u it+1 )) = 0 for t 1 and E( x it ( i + u it )) = 0 for t 2, which completes the proof. Furthermore, it is straightforward to show that E( x i;t 1 x it ) 6 = 0, E( x i;t 1 y i;t 1 ) 6 = 0, E( y i;t 1 x it ) 6 = 0, E( y i;t 1 y i;t 1 ) 6 = 0.

D Derivations of Parameters in Monte Carlo Experiments

Using the lag operator, L, we can write yit and xit as

yit = i 1 + 1 L xit + i 1 L ft + 1 1 L "it (57) xit = L 1 L "it + i 1 L ft + 1 1 L vit (58)
and thereby substituting (58) into (57) yields

yit = i 1 + vit (1 L) (1 L) + i + i (1 L) (1 L) (1 L) ft + 1 + ( )L (1 L) (1 L) "it. (59)
De…ne y it = yit i=( 1), such that (57) can be rewritten as

y it = y it 1 + xit + uit (60)
and let the signal-to-noise ratio be denoted by 2 s = 2 u ; where 2 s is the variance of the signal,

2 s = var (y it uit) = var (y it ) + var (uit) 2cov (y it ; uit) . ( 61
)
2 s varies across designs, with the aim being to keep the signal-to-noise ratio constant over changes in and the distribution of i , so that the explanatory power of the model does not change. In particular, we set 2 s = 2 u = = 3, where 2 u = var (uit). We normalise 2 f = 2 " = 1 and we keep the total signal-to-noise ratio …xed by modifying 2 s accordingly through changes in 2 v . It can be shown that

2 s = 2 2 v + 2 f b1 + 2 " b2 a1 2 2 f 2 u ,
where a1 = (1+ ) ( 12 )( 12 )( 1)

, b1 = + 2 + 2 2( )( + ) 1+ and b2 = 1 + ( ) 2 + 2( )( + ) 1+ . Applying the normalisation 2 f = 2 " = 1, substituting 2 s = 2 u , and solving for 2 v yields 2 v = 2 2 u (1 + ) + 2 =a1 (b1 + b2) .
In line with the simulation design of [START_REF] Bun | The e¤ects of dynamic feedbacks on LS and MM estimator accuracy in panel data models[END_REF], we choose 2 such that the ratio of the impacts on var(yit) of the two variance components i and ("it, ft) is 2 . By ( 59)

var(yit) = var 1 1 i + var (1 L) (1 L) vit +var i + i (1 L) (1 L) (1 L) ft + var 1 + ( ) L (1 L) (1 L) "it = 2 (1 ) 2 + ( 2 2 v + 2 f b1 + 2 " b2)a1. Now de…ne 2 such that 2 ( 1
) 2 = 2 ( 2 f b1 + 2 " b2)a1. By applying the normalisation 2 " = 2 f = 1, we set 2 = 2 (1 ) 2 (b1 + b2)a1: We choose = 1.
[0.502] --Notes: The estimated model is yit = i + t + 1 yit 1 + 0wit + 1wit 1 + ' 0 it + ' 1 it 1 + uit, where yit is the log of the number of employees of company i, wit is the log of real product wage and it is the log of gross capital stock. Year dummies are included in all speci…cations. The standard errors reported are those of the robust one-step GMM estimator. The …rst row of the test results reports Sargan's statistic for overidentifying restrictions. m1 and m2 are the …rst-order and secondorder serial correlation tests in the …rst-di¤erenced residuals. DDIF 2 denotes Sargan's Di¤erence test for heterogeneous error cross section dependence based on the two-step Arellano and Bond (1991) DIF GMM estimator. DSY S2 denotes Sargan's Di¤erence test based on the two-step Blundell and Bond (1998) SYS GMM estimator. Sargan test and Sargan's di¤erence test are distributed as 2 under the null with degrees of freedom reported in parentheses. Instruments used in each equation are for DIF: yi;t 2; yi;t 3; :::; yi;0;wi;t 2; wi;t 3; :::; wi;0; i;t 2; i;t 3; :::; i;0, and for SYS: yi;t 1, wi;t 1, i;t 1.

Zi, ZXi, Z + i , Z + Xi are de…ned by ( 22), ( 18), ( 37), [START_REF] Newey | Generalized method of moments speci…cation testing[END_REF] but without cross-sectional demeaning, respectively.

,

  (T 1 h y )

Proposition 4 2 h

 42 Under Assumptions 1-4, 5(a)-(b), 6 and model (7), S DIF 2 d ! (K+1) under the null hypothesis, S DIF X2 d ! 2

:

  Low cross section dependence:i i:i:d:U [ 0:3; 0:7] Medium cross section dependence: i i:i:d:U [ 1; 2] High cross section dependence: i i:i:d:U [ 1; 4]:

  proof in Appendix B. Proposition 4 holds with unbalanced panel data, so long as min N t ! 1, where N t is the number of cross-sectional units for a given time t. In this case, the crosssectional demeaning for y t , say, is now de…ned as y t = y t y t , where y t = N 1 Sargan' s Di¤erence Tests based on the System GMM Estimator It has been well documented (see, for example,[START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF]) that the DIF estimator can su¤er from a weak instruments problem when is close to unity and/or the variance of the individual e¤ects is large relative to that of the idiosyncratic errors. Thus, we also consider another version of Sargan's di¤erence test based on the Blundell and Bond (1998) SYS estimator, which is known to be more robust to the problem of weak instruments under certain conditions.

	Remark 6 Sargan's di¤ erence test has non-trivial asymptotic local power; See proof in
	Appendix B.
	Remark 7 t	P Nt i=1 y it . 8
	3.3

Table 2 :

 2 Bias ( 1000) of Fixed E¤ects and GMM estimators for ; in the Case with Predetermined Regressors with T = 5 .91 -29.04 -26.18 -29.00 -83.57 -77.34 -72.34 -64.51 -459.56 -434.21 -438.01 -414.08 DIF2 -30.25 -22.18 -20.73 -22.36 -76.68 -68.73 -62.60 -54.55 -466.73 -441.78 -447.79 -427.26 See notes to Table1. FE is the …xed e¤ects estimator, DIF1 and DIF2 are the Arellano and Bond (1991) one-step and two-step …rst di¤erenced GMM (DIF) estimators, respectively. DIFX1 and DIFX2 are the one-step and two-step DIF estimators, respectively, which are based on the instruments consisting of subsets of Xi only. SYS1 and SYS2 are the Blundell and Bond (1998) one-step and two-step system GMM (SYS) estimators, respectively. SYSX1 and SYSX2 are the one-step and two-step SYS estimators, respectively, which are based on the instruments consisting of subsets of Xi only.

			= 0:2			= 0:5			= 0:8
	Te st,N	50	100	200	400	50	100	200	400	50	100	200	400
					No Cross Section Dependence: i = = 0
	FE	-115.62 -114.70 -116.66 -115.46 -187.03 -183.89 -185.06 -184.63 -392.65 -389.05 -388.91 -388.59
	DIF1	-23.68 -10.76	-7.76	-2.46 -46.94 -24.22 -10.82	-5.29 -205.25 -105.79 -59.39 -26.46
	DIF2	-22.83 -10.11	-7.65	-1.98 -48.67 -24.47 -10.48	-5.25 -223.06 -115.40 -61.99 -27.85
	DIFX1 -20.58 -10.07	-7.70	-2.30 -65.99 -32.32 -15.45	-7.80 -438.04 -326.91 -209.79 -116.42
	DIFX2 -19.93 -10.18	-7.62	-2.11 -66.16 -33.28 -14.50	-7.79 -457.19 -339.03 -214.87 -118.03
	SYS1	13.43	7.31	1.92	2.04	6.89	2.96	1.59	1.42 -12.35	-3.91	-5.17	-1.46
	SYS2	11.58	6.06	0.34	1.51	7.24	4.18	2.20	1.42	-4.91	2.04	-0.20	0.54
	SYSX1	9.57	4.71	0.54	1.47	5.67	5.55	3.01	2.37 -50.06 -42.01 -30.40 -13.39
	SYSX2	7.94	3.37	-0.59	1.45 10.07	8.00	6.87	4.32 -37.91 -28.31 -15.51	-2.08
				Low Cross Section Dependence: i i:i:d:U [ 0:3; 0:7]
	FE	-107.82 -106.72 -108.15 -107.89 -171.34 -168.69 -169.18 -168.50 -325.46 -321.39 -321.04 -319.85
	DIF1	-22.29 -10.66	-7.80	-3.68 -42.94 -23.58 -11.97	-6.50 -140.41 -75.11 -45.60 -24.33
	DIF2	-20.85	-9.55	-7.44	-2.92 -43.64 -23.37 -11.12	-6.08 -147.90 -77.45 -46.13 -24.42
	DIFX1 -18.53	-8.89	-7.33	-2.29 -60.43 -29.10 -13.46	-7.00 -353.16 -251.67 -159.44 -87.67
	DIFX2 -17.71	-8.66	-7.26	-2.05 -60.16 -29.79 -12.58	-6.89 -365.60 -254.65 -158.36 -85.55
	SYS1	12.24	6.24	1.06	0.53	2.80	-0.67	-1.42	-0.59 -18.63 -11.68 -10.26	-6.72
	SYS2	10.12	4.93	-0.41	0.12	3.05	1.22	-0.17	-0.37 -12.05	-5.93	-5.40	-3.88
	SYSX1	9.36	4.72	0.38	1.27	2.25	3.82	2.44	1.75 -53.49 -43.00 -30.35 -15.23
	SYSX2	7.18	3.42	-0.89	1.29	6.42	6.01	6.09	3.68 -43.81 -31.09 -18.55	-6.48
				Moderate Cross Section Dependence: i i:i:d:U [ 1; 2]
	FE	-87.17 -85.41 -85.48 -87.94 -143.05 -141.46 -140.35 -138.30 -281.73 -278.38 -277.75 -274.77
	DIF1	-30.36 -21.66 -18.81 -19.59 -60.73 -50.04 -42.90 -35.16 -202.24 -165.42 -151.49 -136.35
	DIF2	-27.14 -16.75 -15.14 -14.83 -57.18 -44.98 -36.37 -29.22 -204.32 -163.72 -150.64 -135.37
	DIFX1 -14.25	-6.81	-5.99	-1.91 -51.31 -24.03 -11.00	-5.72 -289.33 -199.89 -124.44 -68.86
	DIFX2 -12.92	-5.71	-5.65	-1.29 -46.85 -21.95	-8.43	-4.37 -281.22 -184.10 -106.67 -55.71
	SYS1	-0.15	-2.22	-6.09	-9.19 -15.90 -19.55 -20.10 -15.63 -50.84 -48.19 -46.00 -48.00
	SYS2	-1.04	-1.85	-6.40	-8.35 -13.94 -16.42 -17.16 -13.89 -45.48 -43.31 -41.70 -42.90
	SYSX1	6.00	3.00	-0.28	0.67	-3.87	1.04	1.15	0.76 -56.04 -40.74 -28.56 -16.86
	SYSX2	3.96	2.14	-1.32	1.02	0.77	3.29	4.38	2.87 -43.59 -25.70 -16.02	-5.77
				High Cross Section Dependence: i i:i:d:U [ 1; 4]
	F E	-63.49 -61.52 -60.98 -64.34 -119.29 -118.17 -116.32 -113.69 -374.01 -371.82 -372.45 -366.96
	DIFX1	-9.91	-4.75	-4.25	-1.22 -41.29 -19.10	-8.88	-4.72 -453.74 -344.77 -244.99 -148.10
	DIFX2	-8.17	-3.38	-3.57	-0.52 -33.56 -14.40	-4.84	-2.42 -436.91 -313.41 -198.66 -106.04
	SYS1	-0.25	-3.56	-8.64 -13.43 -17.11 -26.10 -29.33 -25.59 -79.86 -95.60 -100.75 -114.51
	SYS2	-1.63	-3.71	-8.97 -12.56 -15.61 -24.64 -27.43 -24.05 -77.14 -94.71 -100.97 -113.50
	SYSX1	9.23	4.76	1.29	1.22 11.12	9.72	5.95	3.11	-5.01	2.03	0.03	1.38
	SYSX2	4.93	2.10	-0.78	0.87 12.32	8.56	6.71	3.56	4.83 18.18 17.69 22.97
	Notes:

DIF1

-34

Table 3 :

 3 Root Mean Square Errors ( 1000) of Fixed E¤ects and GMM estimators for ; in the Case with Predetermined Regressors with T = 5 FE 15.45 14.24 14.27 13.75 37.79 35.41 35.12 34.69 159.49 153.85 152.75 151.86 DIF1

		= 0:2	= 0:5		= 0:8
	Test,N	50 100 200 400	50 100 200 400	50	100	200	400
		No Cross Section Dependence: i = = 0

Table 4 :

 4 Homogeneity Error Cross Section Dependence Tests and Estimates of Employment Equation, 140 Firms with 9-Year Observations

	A: Two-Step DIF Estimator, 1976-84
	Estimation Results
		Based on Zi	Based on ZXi
		Coef. Std.Err. Coef. Std.Err.
	yi;t 1	0.679	(0.084)	0.401	(0.124)
	wit	-0.720	(0.117)	-0.551	(0.130)
	wit 1	0.463	(0.111)	0.347	(0.112)
	it	0.454	(0.101)	0.447	(0.110)
	it 1	-0.191	(0.086)	-0.079	(0.105)
	cons	0.005	(0.017)	0.003	(0.014)
	Test Results
		statistics p-values statistics p-values
	Sargan	88:8 (79)	[0.211]	62:0 (51)	[0.140]
	m1	-4.46	[0.000]	-2.19	[0.028]
	m2	-0.17	[0.866]	-0.47	[0.641]
	DDIF 2	26:84 (28)	[0.527]	-	-
	B: Two-Step SYS Estimator, 1976-84
	Estimation Results
		Based on Z +
	yi;t 1	0.873	(0.044)	0.825	(0.071)
	wit	-0.780	(0.116)	-0.717	(0.105)
	wit 1	0.527	(0.168)	0.560	(0.149)
	it	0.470	(0.071)	0.395	(0.088)
	it 1	-0.358	(0.072)	-0.253	(0.092)
	cons	0.948	(0.390)	0.720	(0.402)
	Test Results
		statistics p-values statistics p-values
	Sargan	111:6 (100)	[0.201]	77:3 (65)	[0.142]
	m1	-5.81	[0.000]	-5.19	[0.000]
	m2	-0.15	[0.883]	-0.12	[0.906]
	DSY S2	34:30

i Based on Z + Xi Coef. Std.Err. Coef. Std.Err.

Other examples are provided by Ahn, Lee and Schmidt (2001).

[START_REF] Frees | Assessing cross-sectional correlation in panel data[END_REF] pointed out, test statistics similar to the CD test of Pesaran were proposed by[START_REF] Friedman | The use of ranks to avoid the assumption of normality implicit in the analysis of variance[END_REF], based on the Spearman rank correlation coe¢ cient (which is expected to be robust against non-normality). Although we do not consider the Friedman test in this paper, results that are similar to the CD test would apply for this test.

However, the experimental results of[START_REF] Pesaran | General diagnostic tests for cross section dependence in panels[END_REF] illustrate that the CD test is robust to skewed errors.

See e.g.[START_REF] Robertson | Factor residuals in SUR regressions: estimating panels allowing for cross-sectional correlation[END_REF],[START_REF] Phillips | Dynamic panel estimation and homogeneity testing under cross section dependence[END_REF],[START_REF] Bai | A PANIC attack on unit roots and cointegration[END_REF], Moon and Perron (2004), and Pesaran (2006) among others. Notice that these methods are only justi…ed when T is large. For related work that allows for time-varying individual e¤ects in the …xed T , large N case, see Holtz-Eakin, Newey and Rosen (1988), Ahn, Lee, and Schmidt (2001) and Han, Orea and Schmidt (2005).

For su¢ ciently large N , the overidentifying restrictions test (de…ned by equation (28) below) is expected to have enough power to detect the violation of Assumption 5(b).

One can construct ZXi such that both strictly exogenous and predetermined regressors are present. See, for example,[START_REF] Arellano | Panel Data Econometrics[END_REF].

See Proposition 5 and its proof in[START_REF] Newey | Generalized method of moments speci…cation testing[END_REF], under the assymption of i.i.d. errors.

For more details on the computations of the[START_REF] Arellano | Some tests of speci…cation for panel data: Monte Carlo evidence and an application to employment equations[END_REF] and Blundell and Bond (1998) estimators with unbalanced panels, see[START_REF] Arellano | Dynamic panel data estimation using DPD98 for Gauss: A guide for users[END_REF].

[START_REF] Ahn | GMM estimation of linear panel data models with time-varying individual e¤ects[END_REF] In Appendix B, it is formally shown that when hx > (K + 1), the DDIF

test is asymptotically more powerful than the SDIF 2 test under the local alternatives.1 2 Heterogeneity of would also render the error term serially correlated, as discussed in Pesaran and Smith (1995).1

Other solutions have been proposed in the literature, such as a panel feasible generalized median unbiased estimator, proposed by[START_REF] Phillips | Dynamic panel estimation and homogeneity testing under cross section dependence[END_REF], or the common correlated e¤ects (CCE) estimator proposed by[START_REF] Pesaran | Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure[END_REF]. However, both estimators require a larger value for T than that considered in this paper.

[START_REF] Anselin | Spatial Econometrics[END_REF] Note that the unobserved common factor, ft, is randomly drawn to control the signal-to-noise ratio without loss of generality.1 5 We do not report the results based on non-normal errors in this paper, since the results were very similar. They are available from the authors upon request.

[START_REF] Arellano | Some tests of speci…cation for panel data: Monte Carlo evidence and an application to employment equations[END_REF] xit has zero mean without loss of generality, since we cross section-demean all data before computing the statistics.1 7 See Appendix D for the details of the way of controlling these parameters.1 8 We do not report the results for T = 9 in this paper, since these were similar to those for T = 5. They are available from the authors upon request.1 9 We do not report the performance of the estimators for , since it has a similar pattern to that for ; although it is not as much a¤ected by error cross section dependence.

0 The data set used is available at http://www.doornik.com/download/dpdox121.zip 2 1 The GMM estimates of the parameters have been obtained using the xtabond2 command in Stata; see Roodman, D., (2005). xtabond2: Stata module to extend xtabond dynamic panel data estimator. Center for Global Development, Washington. http://econpapers.repec.org/software/bocbocode/s435901.htm

Tom Flavin, Hashem Pesaran and Neville Weber for helpful discussions. We have also bene…ted from the constructive and most helpful comments of Cheng Hsiao and two anonymous referees. Sara…dis gratefully acknowledges full …nancial support from the ESRC during his PhD studies at Cambridge University (PTA-030-2002-00328). Yamagata gratefully acknowledges the …nancial support from the ESRC (Grant No. RES-000-23-0135).

Appendices

A Proof of Proposition 3

To simplify the analysis without loss of generality we consider the case where K = 1. For t 2 + j, given j such that 0 j t 1, it can be shown that

`"i;t j 1 `+ t j 2 X `=0 `vi;t j `; [START_REF] Sargan | The estimation of economic relationships using instrumental variables[END_REF] and for t 1

Firstly we consider E(y i;t s u it ) for s t T , given 2 s T , under the alternative hypothesis of i 6 = . But this is equivalent to considering E(y it u i;t+s ) for 0 t T s, given 2 s T . Initially we focus on the case of s = 2. When t = 0, E(y i0 u i2 ) = E(y i0 0 i ) f2 by Assumption 3, which is not necessarily zero under the alternative. When t 1, using ( 40) and ( 41), together with

under Assumptions 1-4, 5(a)-(b), 6 and model [START_REF] Arellano | Dynamic panel data estimation using DPD98 for Gauss: A guide for users[END_REF]. A similar approach for the case of s > 2 leads to the conclusion that E(y it u i;t+s ) 6 = 0 for t 0, s 2 under the alternative, as required. Under the null of i = , it follows immediately that E(y it u i;t+s ) = 0 for t 0, s 2. Now we consider E(x i;t s u it ) for s < t T , given 1 s T 1, under the alternative, which is equivalent to considering E(x it u it+s ) for 1 t < T s, given 1 s T 1. Initially we focus on the case of s = 1. When t = 1, E(x i1 u i2 ) = 0 due to Assumption 3 and 5(b). For t 2 and using [START_REF] Sargan | The estimation of economic relationships using instrumental variables[END_REF], together with u it = 0 i ft + " it , we have E(x it u it+1 ) = 0, under Assumptions 1-4, 5(a)-(b), 6 and model [START_REF] Arellano | Dynamic panel data estimation using DPD98 for Gauss: A guide for users[END_REF], under the alternative. A similar approach for the case where s > 1 leads to the conclusion that E(x it u i;t+s ) 6 = 0 for t 0, s 1 under the alternative, as required. Under the null of i = , it follows immediately that E(y it u i;t+s ) = 0 for t 0, s 1, which completes the proof. In addition, it is straightforward to show that E(x i;t 1 y i;t 1 ) 6 = 0.

B Proof of Proposition 4

Firstly we establish that SDIF X2 d ! 2 hx (K+1) under the alternative of H1 : 6 = 0, where SDIF X2 is de…ned in [START_REF] Lee | Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models[END_REF]. Rewriting 1

where

since MB B = 0.

We can express the instruments as deviations from their cross-sectional averages:

where ZXi is de…ned similarly to Z Xi but all x 0 it s are replaced with xit, and ZX is de…ned similarly but all non-zero elements are replaced with their cross-sectional averages, xt. Also de…ne the instruments in terms of deviations from their mean as

and

where mZ X = E(ZXi), = E( i ). Using ( 46)-( 48)

It is easily seen that the second term of ( 49) is asymptotically negligible. Consider the …rst term of (49).

Reminding ourselves that ui = F i + "i, we have

By the assumptions i i:i:d:( ; ), "it i:i:d:(0; 2 i ) above, = Op(N 1=2 ) and " = Op(N 1=2 ) as well as N 1=2 P N i=1 Z o Xi = Op(1). Then it follows that

with obvious notations, together with (50), (45) can be written as

As Z o0 Xi u o i are independent across i, a suitable Central Limit Theorem ensures that , ft i:i:d:N (0; 1), "it i:i:d:N (0; 1); xit = xi;t 1 + "i;t 1 + i ft + vit, i = 1; 2; :::; N , t = 48; 47; :::; T with xi; 49 = 0 and the initial 50 observations being discarded. = = 0:5, = 0:5; i i:i:d:U [ 1; 2], vit i:i:d:N (0; 2 v ), 2 v is chosen such that the signal-to-noise ratio equals 3. 2 is chosen such that the impact of the two variance components i and (ft, "it) on var(yit) is constant. All variables are cross-sectionally demeaned before computing statistics. All experiments are based on 2,000 replications.


E Malicka

H Duda

A W Pacyna

T Mydlarz

R Sitko

] \n

Spin-glass-like behavior in Zn x Cr y Al z Se 4

Keywords: A. Chalcogenides, B. Magnetic properties, C. Electrical conductivity ________________________________________________________________

The magnetic and electrical properties of the Al-doped polycrystalline spinels Zn x Cr y Al z Se 4 (0.13 d z d 0.55) with the antiferromagnetic order and semiconducting behavior were investigated. A complex antiferromagnetic structure below a Néel temperature T N | 23 K for the samples with z up to 0.4 contrasting with the strong ferromagnetic interactions evidenced by a large positive Curie-Weiss temperature T CW decreasing from 62.2 K for z = 0.13 to 37.5 K for z = 0.55 was observed. Detailed investigations revealed a divergence between the zero-field-cooling (ZFC) and field-cooling (FC) susceptibilities at A c c e p t e d m a n u s c r i p t 2 temperature smaller than T N suggesting bond frustration due to competing ferromagnetic and antiferromagnetic exchange interactions in the compositional range 0.13 d z d 0.4. Meanwhile, for z = 0.55 a spin-glass-like behavior of cluster type with randomly oriented magnetic moments is observed as the ZFC-FC splitting goes up to the freezing temperature T f = 11.5 K and the critical fields connected both with a transformation of the antiferromagnetic spin spiral via conical magnetic structure into ferromagnetic phase disappear.

Introduction

Recently, the single crystals of the ZnCr 2-x Al x Se 4 spinel system in the compositional range 0.0 d x d 0.23 came into the focus because of the sharp first-order anomalies at the antiferromagnetic transition [START_REF] Hemberger | DF]RURZVNL[END_REF]2]. Previous neutrondiffraction investigations of the pure ZnCr 2 Se 4 spinel revealed a helical antiferromagnetic (AFM) spin structure below T N | 20 K with a strong ferromagnetic (FM) component evidenced by a large positive Curie-Weiss (CW) temperature of 115 K [START_REF] Lotgering | Proceedings of the International Conference on Magnetism[END_REF][START_REF] Plumier | [END_REF]. The helical structure has a FM arrangement in the (001) planes with a turning angle of 42º between the spins in adjacent (001) planes. The transition to the AFM state at T N is accompanied by structural transformation from cubic Fd 3 m to tetragonal I4 1 /amd symmetry with a small A c c e p t e d m a n u s c r i p t 3 contraction along the c axis [5], although the latest structural investigations using synchrotron radiation defined this latter symmetry as orthorhombic Fddd [6]. It is well known from literature that many magnetically ordered chromium spinels manifest structural instabilities which do not originate from a conventional Jahn-Teller instability, as the half-filled t 2g ground state of the Cr 3+

ions is orbitally non-degenerate [START_REF] Hemberger | DF]RURZVNL[END_REF]. Another source of the structural instability is a competition between FM and AFM exchange of nearly equal strength yielding strong bond frustration as it was identified in AFM ZnCr 2 S 4 [7]. Next, in the isostructural ZnCr 2x Al 2-2x S 4 spinel system in a very large concentration range a spin-glass behavior has been observed [8].

A detailed latest investigation of ZnCr 2 Se 4 single crystal [START_REF] Hemberger | DF]RURZVNL[END_REF] revealed the AMF order at T N = 21 K with a positive CW temperature T CW = 90 K, a metamagnetic transition at a critical field H c1 of about 10 kOe for T = 2 K connected with a simple spiral spin arrangement, the breakdown of the conical spin arrangement at the critical field H c2 of about 65 kOe for T = 2 K, the full saturation magnetisation of about 3 P B per Cr ion above H c2 for the FM spin arrangement, a negative thermal expansion resulting from the high frustration of the lattice degrees of freedom, a large magnetostriction comparable to giant magnetostrictive materials, and an extremely strong suppression of the anomalies in the specific heat and thermal expansion by external magnetic fields suggesting a spin-driven origin of the structural transformation. Next, the structural and magnetic investigations of ZnCr and AFM interactions from one side and the randomly oriented magnetic moments for high diluted magnetic system from the other.

Experimental

Powder samples (polycrystals) of the Zn x Cr y Al z Se 4 spinel series were obtained in the range of 0.13 d z d 0.55 using a ceramic method [START_REF] 2nrĕvnd ; R]árzvnd | URN = DQRUJ DOOJ &KHP[END_REF]. The The mass susceptibilities F V (T) and F ac (T), magnetic moment P(H), electrical resistivity V(T) and thermopower S were measured using: 1) a Lake Shore 7225 ac susceptometer/dc magnetometer up to 60 kOe as well as a Cahn automatic magnetic electrobalance of the Faraday type in the temperature range 4.2 -300 K, 2) a step-magnetometer at 4.2 K and in applied external fields up to 140 kOe, 3) a 4-point dc method in the temperature range 300-500 K, and 4) a differential method with the temperature gradient 'T of about 5 K, respectively. For electrical measurements the powder samples were compacted in disc form (10 mm in diameter and 1-2 mm thick) using a pressure of 1.5 GPa and they were next sintered through 2 h at 473 K.

Results

The X-ray diffraction carried out on the polycrystalline Zn x Cr y Al z Se 4 samples revealed a single-phase material with the cubic spinel structure (Fd3m).

The lattice and positional anion parameters do not significantly depend on the Al content (Table 1). A Rietveld refinement [START_REF]FullProf.2k (Version 4.00 -May, 2007 -ILL JRC). >@ 7 *URĔ + 'XGD -:DUF[END_REF] showed that the zinc ions have a preference to be located in the tetrahedral sites and both the chromium and aluminium ions prefer to be located in the octahedral sites of the spinel structure. The analysis of the chemical composition (see The temperature dependences of dc mass susceptibility F V (T) depicted in for z = 0.13 to P sat = 3.81 P B for z = 0.4 (see inset of Fig. 3 and Table 2). This feature confirms, additionally, that the Al ions are octahedrally coordinated. For the spin-glass sample with z = 0.55 the magnetic moment does not reach the full saturation up to 140 kOe (Fig. 2) and the peaks at the critical fields H c1 and H c2 are not observed in the measured range of magnetic field up to 60 kOe (Fig. 3).

In Fig. 4, one can observe that the zero-field-cooling (ZFC) and field-cooling Really, in the spinels with essentially ferromagnetic near-neighbors interaction, the J 1 is thought to be large and positive and J 2 negative, however, for antiferromagnetic ZnCr 2 Se 4 , J 1 is negative and J 2 positive (cf. Fig. 1 in Ref. 13).

From the comparison with the experiment we determined the exchange integrals J 1 and J 2 of interactions between the first two coordination spheres for the antiferromagnetic system by the Holland and Brown equations [13]:

60 9 1 N CW T J T , (1) 120 3
2

N CW T J T , (2)
and the exchange integral J sg of the spin-glass system using a random energy model [14,15]. The J sg reads as [15]:

2 4ln T J f sg . ( 3
)
The results of the exchange integral calculations of the Zn x Cr y Al z Se 4 spinels are presented in Table 3. For the antiferromagnetically ordered samples with Al content up to z = 0.4 the integral J 1 is negative and the integral J 2 is positive and their absolute values do not exceed 3 K, whereas for the usual spin-glass-like sample with z = 0.55 the exchange integral J sg = 19.1 K and it is over seven times larger in comparison with the J 1 of the antiferromagnetic ones. For pure ZnCr 2 Se 4 spinel the FM coupling in the (001) planes is stronger than the AFM coupling between them [13], while for the Al-doped samples an opposite

A c c e p t e d m a n u s c r i p t

behavior is observed (Table 3). The lower values of J 1 are characteristic for the frustration state while the higher values of J sg are characteristic for the usual SG state. The latter feature is related to the experimentally observed in Figs. 2 and3 a hard magnetization, a small hysteresis below T f and a lack of the critical fields at H c1 and at H c2 where the simple spin spiral and the conical spin arrangement breakdown, respectively.

Basing on the magnetic data above described, we have proposed in Fig. 8

Conclusions

We investigated an influence of spin defects on magnetic and electrical Note: R F2 , R F2w and F 2 -criteria of fit [START_REF]FullProf.2k (Version 4.00 -May, 2007 -ILL JRC). >@ 7 *URĔ + 'XGD -:DUF[END_REF]. a and u are the lattice and anion parameters, respectively. Atomic positions are given in standard setting for space group Fd3m (No. 227):

Zn 8a (1/8, 1/8, 1/8), Cr/Al 16d (½ ½, ½), Se 32e (u, u, u).

A c c e p t e d m a n u s c r i p t

Table 2 Magnetic and electrical parameters for the polycrystalline Zn x Cr y Al z Se 4 spinels.

T N , T CW and T f are the Néel, Curie-Weiss and freezing temperatures, respectively, C V is the Curie constant, P eff is the effective magnetic moment, P sat is the saturation magnetic moment at 4.2 K and at 140 kOe, E A and S are the activation energy and thermopower at 300 K, respectively. Experimental data for z = 0.0 were taken from Refs.

A c c e p t e d m a n u s c r i p t

A c c e p t e d m a n u s c r i p t

  with CW temperatures of 108 and 114 K, respectively, a metamagnetic transition at the critical field H c1 of about 15 kOe for T = 1.72 K, the breakdown of the conical spin arrangement at the critical field H c2 about 70 kOe for T = 4.2 K, and a decreasing saturation magnetic moment from 2.5 to 2.15 P B per Cr atom with increasing Al content.The present contribution reports the magnetic and electrical properties for ZnCr 2 Se 4 compensated spin array harboring mixed couplings doped with the octahedrally coordinated the non-magnetic Al ions up to 55 %. It was thus natural to expect the spin frustration connected with a competition of the FM

A c c e p t e d m a n u s c r i p t 5 structure

 5 chemical composition of polycrystals under study was determined by wavelength-dispersive X-ray fluorescence spectrometry (PW1410, Philips, Almelo, The Netherlands). The phase analysis of the polycrystalline samples was done by the X-ray diffraction methods using a SIEMENS D5000 diffractometer (CuK D radiation) over an angular range of 2T: 20 y 116°. The refinements were done using a Rietveld method and a FullProf.2k (Version 4.00) computer program[START_REF]FullProf.2k (Version 4.00 -May, 2007 -ILL JRC). >@ 7 *URĔ + 'XGD -:DUF[END_REF].

Fig. 1 A c c e p t e d m a n u s c r i p t 7 field

 17 Fig.1and the magnetic parameters collected in Table2 reveal typical

(

  FC) susceptibilities, F V (T), start diverging below the ordering temperature T N , in the compositional range 0.13 d z d 0.4. For z = 0.55 the ZFC-FC splitting reaches a minimum at T f = 11.5 K and simultaneously the FC curve remains temperature independent below T f . These two slightly differing spin-glass-like states in the compositional range 0.13 d z d 0.4 and for z = 0.55 are indicated by SG1 and SG2, respectively, to make easier further discussion.The second harmonic of ac susceptibility was used for interpretation of spin arrangement in the spinels under study, too. Figures 5 and 6 show the inphase ) ( ' 2 T F and out-of-phase ) ( ' ' 2 T F second harmonics of ac susceptibilities in A c c e p t e d m a n u s c r i p t 8 the temperature range 4.2 -140 K for the samples with z = 0.13 and 0.55, have one feature in common: they vanish both below T N and T f in accordance with the simple molecular field theory [12]. The ) ( ' 2 T F in Fig.5 for a sample with z = 0.13 has a complex magnetic structure above T N . We observe a negative value in the temperature range T N -90 K indicating a parallel spin arrangement and a positive value above 90 K indicating an antiprallel spin arrangement. The out-of-phase second harmonic ) ( ' ' 2 T F in Fig.5 is positive and has a high maximum also close to 90 K relating to the energy dissipation in the temperature range T N -130 K. A bond frustration of competing AFM and FM exchange interactions of nearly equal strength may lead to the SG1 state in the compositional range 0.0 d z d 0.4, all the more so as for z = 0.0 the ZFC and FC splitting below T N is also observed [in Fig. 6 for a sample with z = 0.55 have completely different behavior as considered above. They vanish above T f indicating both a lack of the long-range magnetic interactions and energy losses. This behavior in connection with the ZFC and FC splitting below T f is characteristic for the usual SG-like behavior with randomly oriented magnetic moments, indicated earlier by SG2.

Figure 7 andA c c e p t e d m a n u s c r i p t 4 .

 74 Figure 7 and Table 2 illustrate a high temperature electrical conductivity,

A c c e p t e d m a n u s c r i p t 10 strong

 10 bond frustration. For that the exchange integrals J 1 and J 2 have been estimated according to the Holland and Brown procedure described in Ref.13.

A c c e p t e d m a n u s c r i p t 12 only

 12 a generalized phase diagram for the Zn x Cr y Al z Se 4 spinel system. Below the freezing temperature T f one observes a metamagnetic behavior, where the FC and ZFC curves describe a fractal decomposition of the spin state. A SG1 state would necessarily define a region between T f and T N , connected both with the structural instability and a strong competition between AFM and FM interactions. Above T N we have a paramagnetic state. For higher Al content the usual spin-glass-like state, SG2, occurs at low temperatures and above T f a direct transition to paramagnetic state is observed. It is probably connected both with a lack of structural instability and a competition of magnetic interactions as the critical fields are not observed. The SG1 state occupies a very large domain of concentrations whereas the SG2 state only exists in the high concentration range. The SG1 state is usually accompanied with long-range AFM and shortrange FM interactions and the antiferromagnetic transition line remains almost constant in the compositional range 0.0 d z d 0.4. In the case of the FM and AFM exchange of nearly equal strength the SG1 state is usually called the reentrant spin-glass state (RSG)[16]. For z = 0.55 the SG2 state is accompanied with short-range ferromagnetic interaction indicating its cluster character.The overall shape of the phase diagram shown in Fig.8for the Zn x Cr y Al z Se 4 spinel system differs from the ZnCr 2x Al 2-2x S 4 diagram[8] where the antiferromagnetic transition line decreases very quickly as the Al concentration increases. It depends on the electronic configurations of the Cr 3+ ions with a half filled t 2g level, and their mutual nearest neighbour Cr-Cr distance due to larger radius of the selenium ion as compared to the sulphur one, giving stronger ferromagnetic interactions via the selenium ions in the spinels under study.

  properties in the polycrystalline Zn x Cr y Al z Se 4 spinel system and found a frustration of RSG type in the compositional range 0.13 d z d 0.4 and the usual SG state of cluster type for z = 0.55. It was also observed that with increasing Al content z a change of spin arrangement from antiparallel into spin-glass-like behavior caused the lowering of the electrical conductivity and a change of sign of thermopower from negative into positive.

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Mass susceptibility F V vs. temperature T recorded at H dc = 572 Oe of the

Fig. 4 .

 4 Fig. 4. ZFC and FC dc mass susceptibility F V vs. temperature T recorded at H dc

Fig. 5 .Fig. 6 .

 56 Fig. 5. Temperature dependence of the in-phase ) ( ' 2 T F and out-of-phase ) ( ' ' 2 T F

Fig. 7 .

 7 Fig. 7. Electrical conductivity lnV vs. reciprocal temperature 10 3 /T of the

Fig. 8 .

 8 Fig. 8. Phase diagram of temperature T vs. Al content z for the Zn x Cr y Al z Se 4

Figure

  Figure(s)

Figure(s)

  Figure(s)

  1.85 Al 0.15 Se 4 and ZnCr 1.77 Al 0.23 Se 4

Table 1 ) showed that A c c e p t e d m a n u s c r i p t 6 the

 1 polycrystals of the Zn x Cr y Al z Se 4 spinel system are almost stoichiometric. The total content of the Zn, Cr and Al cations equals approximately 3 per one molecule. However, the structural refinements revealed that the values of the

	anion parameter presented in Table 1 are higher than 0.375 (which is typical for
	an ideal spinel structure), testifying that the spinel unit cell is distorted. The
	sample composition taken from chemical analysis was a basis of the refined site
	occupation for the cation involved giving the best fit. All these features suggest
	a very likely presence of the anion vacancies, which act as donors, in the

compositional range 0.13 d z d 0.4, while for the highest Al-content z = 0.55 the non-magnetic Al ions (spin defects) in the octahedral coordination seem to be responsible for a p-type conduction.

  Table 2) are observed. From Table2it follows that the effective magnetic moment P eff | 6.07 P B /f.u. in the compositional range 0.13 d z d 0.4 is higher than the spinonly value (p eff = 5.48 P B /f.u.) of the pure ZnCr 2 Se 4 spinel suggesting an appearance of the Cr 2+ ions in the high spin configuration 3d4

	t	3 2	1 g g e	. Thus, the n-
	type conduction here may be connected with the hopping process involving a
	transfer of electrons from Cr 2+ to Cr 3+ ions in the extremely narrow mixed
	valence bands. A nature of the n-type conductivity in this case may originate
	from selenium deficiencies, too, as a slight non-stoichiometry and the
	distortions of the unit cell are observed. For a sample with the highest Al-
	content (z = 0.55) the lowest Cr 3+ Mott-Hubbard sub-band of 3d 3	t	3 2	g e	0 g	narrowed
	band plays a role of the acceptor level giving p-type conduction.

In general, the spin-glass-like state is discussed in a framework of a competition between FM and AFM exchange of nearly equal strength yielding

Table 1

 1 Chemical composition and structural parameters obtained from the Rietveld refinement for Zn x Cr y Al z Se 4 spinels.

	Chemical composition
	x(Zn)	0.99	1.03	1.08	1.02	1.02
	y(Cr)	1.87	1.78	1.63	1.58	1.43
	z(Al)	0.13	0.22	0.33	0.4	0.55
	Confidence interval: 'x(Zn) = r0.003, 'y(Cr) = r0.006, 'z(Al) = r0.0015, '(Se) = r0.012
	Crystal data and refinement
	a (pm)	1049.862(9) 1047.854(7) 1049.761(10) 1049.745(1) 1050.851(8)
	u	0.3845(14) -	0.38584(19)	-	0.38536(13)
	R F2 (%)	4.06	-	9.52	-	10.9
	R F2w (%)	3.44	-	7.24	-	8.93
	F 2	0.16510 4	-	0.54410 4	-	0.10210 5
	Selected interatomic distances and angles
	Zn-Se (pm)	244.58(7)	-	246.99(10)	-	246.37(7)
	Cr/Al-Se (pm)	252.89(6)	-	251.58(8)	-	252.30(6)
	Se-Cr/Al-Se (deg) 180.00(11)	-	180.00(14)	-	180.00(10)
		85.39(8)	-	84.70(11)	-	84.95(7)
		94.61(8)	-	95.30(11)	-	95.05(8)
	Se-Zn-Se (deg)	109.47(9)	-	109.47(13)	-	109.47(9)

  1 and 11.

	z	T N	T CW	T f	C V	P eff	P sat	E A	S
		(K)	(K)	(K)	(Kcm 3 /g)	(P B /f.u.)	(P B /f.u.)	(eV)	(PV/K)
	0.0	21	90	12	8.36110 -3	5.70	6	0.2	450
	0.13 22.6 62.2	18	9.46410 -3	6.04	5.44	0.23	-358
	0.22 23.4 49.8	15	9.68710 -3	6.10	4.80	0.23	-279
	0.33 23.1 49.3	18	9.15310 -3	5.93	4.49	0.17	-239
	0.4	22.9 46.6	17	1.0110 -2	6.20	3.81	0.19	-123
	0.55 -	37.5	11.5 8.63310 -3	5.71	-	0.43

Table 3

 3 Exchange integrals for the polycrystalline Zn x Cr y Al z Se 4 spinels. J 1 and J 2 are the exchange integrals for the first and second coordination spheres, respectively, °J1 /J 2 °is the ratio of the exchange integrals, J sg is the exchange integral of the spin-glass system. The values of J 1 , J 2 and °J1 /J 2 °for ZnCr 2 Se 4 are taken from Ref.13.

	z	J 1	J 2	°J1 /J 2	°0.0
		(K)	(K)
		-1.08	1.46	0.74
	0.13	-2.35	1.08	2.18
	0.22	-2.68	1.00	2.68
	0.33	-2.64	0.99	2.67
	0.4	-2.66	0.96	2.77
	0.55	J sg =19.1 K		-

Acknowledgments

A c c e p t e d m a n u s c r i p t 13 This work is partly supported by Ministry of Scientific Research and Information Technology (Poland) and funded from science resources for years 2007-2009 as a research project (project No. N N204 178433).

A c c e p t e d m a n u s c r i p t