A Lagrange Multiplier Test for Cross-Sectional Dependence in a Fixed Effects Panel Data Model

It is well known that the standard Breusch and Pagan (1980) LM test for cross-equation correlation in a SUR model is not appropriate for testing cross-sectional dependence in panel data models when the number of cross-sectional units (n) is large and the number of time periods (T) is small. In fact, a scaled version of this LM test was proposed by Pesaran (2004) and its finite sample bias was corrected by Pesaran, Ullah and Yamagata (2008). This was done in the context of a heterogeneous panel data model. This paper derives the asymptotic bias of this scaled version of the LM test in the context of a fixed effects homogeneous panel data model. This asymptotic bias is found to be a constant related to n and T, which suggests a simple bias corrected LM test for the null hypothesis. Additionally, the paper carries out some Monte Carlo experiments to compare the finite sample properties of this proposed test with existing tests for cross-sectional dependence.


Introduction
Cross-sectional dependence, described as the interaction between cross-sectional units (e.g., households, …rms and states etc.), has been well discussed in the spatial literature. Intuitively, dependence across "space", can be regarded as the counterpart of serial correlation in time series. It could arise from the behavioral interaction between individuals, e.g., imitation and learning among consumers in a community, or …rms in the same industry. This has been widely studied in game theory and industrial organization. It could also be due to unobservable common factors or common shocks popular in macroeconomics.
As is the case under serial correlation in time series, cross-sectional dependence leads to e¢ciency loss for least squares and invalidates conventional t-tests and F -tests which use standard variance-covariance estimators. In some cases, it could potentially result in inconsistent estimators (Lee, 2002;Andrews, 2005). Several estimators have been proposed to deal with cross-sectional dependence, including the popular spatial methods (Anselin, 1988;Anselin and Bera, 1998; Kelejian and Prucha, 1999; Kapoor, Kelejian and Prucha, 2007;Lee, 2007;Lee and Yu, 2010), and factor models in panel data (Pesaran, 2006, Kapetanios, Pesaran andYamagata, 2011;Bai, 2009).
However, before imposing any structure on the disturbances of our model, it may be wise to test the existence of cross-sectional dependence.
There has been a lot of work on testing for cross-sectional dependence in the spatial econometrics literature, see Anselin and Bera (1998) for cross-sectional data and Baltagi, Song and Koh (2003) for panel data, to mention a few. The latter derives a joint Lagrange Multiplier (LM ) test for the existence of spatial error correlation as well as random region e¤ects in a panel data regression model. Panel data provide richer information on the covariance matrix of the errors than crosssectional data. This is especially relevant for the o¤-diagonal elements which are of particular importance in determining cross-sectional dependence. With panel data one can test for crosssectional dependence without imposing ad hoc speci…cations on the error structure generating the covariance matrix, e.g., the spatial autoregressive model in the spatial literature, or the single or multiple factor structures imposed on the errors in the macro literature. Ng (2006) and Pesaran (2004) propose two test procedures based on the sample covariance matrix in panel data. Ng (2006) develops a test tool using spacing method in a panel model. Pesaran (2004) proposes a crosssectional dependence (CD) test using the pairwise average of the o¤-diagonal sample correlation coe¢ cients in a seemingly unrelated regressions model. The CD test is closely related to the R AV E 1 test statistic advanced by Frees (1995). Unlike the traditional Breusch-Pagan (1980 John (1972) and Ledoit and Wolf (2002) in the statistics literature. Sphericity means that the variance-covariance matrix is proportional to the identity matrix. However, rejection of the null could be due to cross-sectional dependence or heteroskedasticity or both. For a recent survey of some cross-sectional dependence tests in panels, see Moscone and Tosetti (2009).
In the …xed n case and as T ! 1, the Breusch and Pagan's (1980) LM test can be applied to test for the cross-sectional dependence in panels. Under the null hypothesis, the test statistic is asymptotically Chi-square distributed with n(n 1)=2 degrees of freedom. However, this test is not applicable when n ! 1. Therefore, Pesaran (2004) proposed a scaled version of this LM test, denoted by CD lm which has a N (0; 1) distribution as T ! 1 …rst, followed by n ! 1. As pointed out by Pesaran (2004), the CD lm test is not correctly centered at zero for …nite T and is likely to exhibit large size distortions as n increases. To solve this problem, Pesaran (2004) proposes a diagnostic test based on the average of the sample correlations, which he denotes by the CD test, and this is valid for large n. Additionally, Pesaran, Ullah and Yamagata (2008) develop a bias-adjusted LM test using …nite sample approximations in the context of a heterogeneous panel model. This paper derives the asymptotic bias of this scaled version of the LM test in the context of a …xed e¤ects homogeneous panel data model. Because it is based on the …xed e¤ects residuals, we denote it by LM P to distinguish it from CD lm . The asymptotic bias of LM P is found to be a constant related to n and T which suggests a simple bias corrected LM test for the null hypothesis. This paper di¤ers from the bias-adjusted LM test of Pesaran, Ullah and Yamagata (2008) in that the latter assumes a heterogeneous panel data model, whereas this paper assumes a …xed e¤ects homogeneous panel data model. Also, the bias correction derived in this paper is based on asymptotic results as (n; T ) ! 1; while the bias adjustment in Pesaran, Ullah and Yamagata (2008) is obtained using …nite sample approximation. Phillips and Moon (1999) provide regression limit theory for panels with (n; T ) ! 1. Here, we adopt the asymptotics used in the statistics literature for high dimensional inference, see Ledoit and Wolf (2002) and Schott (2005), to mention a few. This literature usually deals with multivariate normal distributed variables where the number of variables (in our case n) is comparably as large as the sample size (T ). Our paper …nds that under this joint asymptotics framework with (n; T ) ! 1 simultaneously, the limiting distribution of the LM P statistic is not standard normal under the assumption of a …xed e¤ects model. Consequently, it can su¤er from large size distortions.
The organization of the paper is as follows. The next section discusses the LM tests for crosssectional dependence in the context of a …xed e¤ects panel data model. Section 3 derives the limiting distribution of the LM P test in the raw data case. Section 4 derives the corresponding limiting distribution of the LM P test in the context of a …xed e¤ects model. Section 5 compares the size and power of the proposed test as well as other tests for cross-sectional dependence using Monte Carlo experiments. In section 6, we show that the proposed bias-corrected LM test can be extended to the dynamic panel data model. Section 7 concludes. The appendix contains all the proofs and the technical details.

LM Tests for Cross-sectional Dependence
Consider the heterogeneous panel data model: it i + u it ; for i = 1; :::; n; t = 1; :::; T; where i indexes the cross-sectional units and t the time series observations. y it is the dependent variable and x it denotes the exogenous regressors of dimension k 1 with slope parameters i that are allowed to vary across i. u it is allowed to be cross-sectionally dependent but is uncorrelated with x it . Let U t = (u 1t ; ; u nt ) 0 . The n 1 vectors U 1 ; U 2; ; U T are assumed iid N (0; u ) over time. Let ij be the (i; j)th element of the n n matrix u . The errors u it (i = 1; :::; n; t = 1; :::; T ) are cross-sectionally dependent if u is non-diagonal, i.e., ij 6 = 0 for i 6 = j. The null hypothesis of cross-sectional independence can be written as: or equivalently as where ij is the correlation coe¢ cient of the errors with ij = ij q 2 i 2 j : Under the alternative hypothesis, there is at least one non-zero correlation coe¢ cient ij , i.e., H a : ij 6 = 0 for some i 6 = j: 3 The OLS estimator of y it on x it for each i, denoted by^ i , is consistent. The corresponding OLS residualsû it de…ned byû it = y it x 0 it^ i are used to compute the sample correlation ij as follows: In the …xed n case and as T ! 1, the Breusch and Pagan's (1980) LM test can be applied to test for the cross-sectional dependence in heterogeneous panels. In this case it is given by: This is asymptotically distributed under the null as a 2 with n(n 1)=2 degrees of freedom.
However, this Breusch-Pagan LM test statistic is not applicable when n ! 1. In this case, Pesaran (2004) proposes a scaled version of the LM BP test given by: Pesaran (2004) shows that CD lm is asymptotically distributed as N (0; 1); under the null, with T ! 1 …rst, and then n ! 1. However, as pointed out by Pesaran (2004), for …nite T , E[T 2 ij 1] is not correctly centered at zero, and with large n, the incorrect centering of the CD lm statistic is likely to be accentuated. Thus, the standard normal distribution may be a bad approximation of the null distribution of the CD lm statistic in …nite samples, and using the critical values of a standard normal may lead to big size distortion. Using …nite sample approximations, Pesaran, Ullah and Yamagata (2008) rescale and recenter the CD lm test. The new LM test, denoted as PUY's LM test, is given by where is the exact mean of (T k) 2 ij and is its exact variance. Here ; ; x iT ) 0 contains T observations on the k regressors for the i-th individual regression. PUY's LM is asymptotically distributed as N (0; 1); under the null, with T ! 1 …rst, and then n ! 1.
This paper considers the …xed e¤ects homogeneous panel data model for i = 1; :::; n; t = 1; :::; T where i denotes the time-invariant individual e¤ect. The k 1 regressors x it could be correlated with i , but are uncorrelated with the idiosyncratic error v it . This is a standard model in the applied panel data literature and di¤ers from (1) in that the 0 i s are the same, and heterogeneity is introduced through the 0 i s. The intercept appears explicitly so that the regressor vector x it includes only time-variant variables. Throughout our derivations for the …xed e¤ects model, we require the following assumptions: c is a non-zero bounded constant. This assumption approximates the case where the dimension n is comparably as large as T .
For a static panel data model, we assume: are stochastic bounded for all i = 1; ; n and j = 1; ; n, and lim (n;T )!1 exists and is nonsingular.
The normality assumption 2.i) above may be strict but it is a standard assumption in the statistical literature and is also assumed by Pesaran, Ullah and Yamagata (2008). Other distributions will be examined for robustness checks in the Monte Carlo experiments. Assumptions 2.ii) is standard. Assumption 2.iii) excludes nonstationary or trending regressors. Under these assumptions, the within estimator~ is p nT -consistent. This estimator is obtained by regressing The corresponding within residuals given by b v it =ỹ it x 0 it~ are used 1 Vt and v form triangular arrays as both n and T increase. Strictly speaking, Vt ( v ) should be written as Vt;n ( v;n). To avoid index cluttering, we suppress the subscript n.

5
to compute the sample correlation^ ij as follows: For a dynamic panel data model with the lagged dependent variable as a regressor, more assumptions are needed. We will discuss this case in Section 6.
The scaled version of the LM BP test suggested by Pesaran (2004) but now applied to the …xed e¤ects model is given by: This replaces ij with^ ij and it now tests the null given in (2) only applied to the remainder disturbance v it : In order to see this, let u it = i + v it denote the disturbances in (6). The …xed e¤ects estimator wipes out the individual e¤ects, and that is why it does not matter whether the 0 i s are correlated with the regressors or not. The test for no cross-sectional dependence of the disturbances given in (2) becomes a test for no cross-sectional dependence of the v it : This LM P test, for the …xed e¤ects model (8), su¤ers from the same problems discussed by Pesaran (2004) for the corresponding CD lm statistic (4) for the heterogeneous panel model. We show that it will exhibit substantial size distortions due to incorrect centering when n is large. Unlike the …nite sample adjustment in Pesaran, Ullah and Yamagata (2008), this paper derives the asymptotic distribution of the LM P statistic under the null as (n; T ) ! 1; and proposes a bias corrected LM test. The asymptotics are done using the high dimensional inference in the statistics literature, see Ledoit and Wolf (2002) and Schott (2005), to mention a few. Our derivation begins with the raw data case and then extends it to a …xed e¤ects regression model. We …nd that in a …xed e¤ects panel data model, after subtracting a constant that is a function of n and T , the LM P test is asymptotically distributed, under the null, as a standard normal. Therefore, a bias-corrected LM test is proposed.

LM P Test in the Raw Data Case
In the raw data case, the n 1 vectors Z 1 ; Z 2; ; Z T are a random sample drawn from N (0; z ): The t th observation Z t has n components, Z t = (z 1t ; ; z nt ) 0 . The null hypothesis of independence among these n components is the same as (2) but now pertaining to z rather than u : For …xed n, and as T ! 1, the traditional LM test statistic is T n 1 P i=1 n P j=i+1 r 2 ij , which converges in distribution to 2 n(n 1)=2 under the null of independence. The sample correlation r ij is de…ned as However, as the dimension n becomes as comparably large as T , this traditional LM test becomes invalid. A scaled LM test statistic is thus considered. This LM z statistic (10) is closely related to the test statistic proposed by Schott For high-dimensional data, as n=T ! c 2 (0; 1 ), Schott (Theorem 1, 2005) shows that under the null of independence, or, equivalently, that s T 2 (T + 2) n(n 1)(T 1) Using Schott's (2005) result and the fact that s T 2 (T + 2) n(n 1)(T 1) it is straightforward to infer that the limiting distribution of LM z is N (0; 1) under the null. 2

LM P Test in a Fixed E¤ects Panel Data Model
This section derives the limiting distribution of the LM P test de…ned in (8). This tests the null of no cross-sectional dependence in the …xed e¤ects regression model given in (6). The null hypothesis of no cross-sectional dependence is the same as (2) but now pertaining to rather than u .
Theorem 1 Under Assumptions 1, 2 and the null hypothesis of no cross-section dependence The proof of the theorem is provided in the Appendix. The asymptotics are derived under the joint asymptotics of (n; T ) ! 1 with n=T ! c 2 (0; 1).
Based on this result, this paper proposes a bias-corrected LM test statistic given by: Hence, the accuracy of the within residuals depends on T . For small T , the within residuals are inaccurate, and so are the sample correlations^ ij 's computed using the within residuals. For large T , the terms involved with odd power of 1 T P T s=1 v is vanish due to the law of large numbers. However, the sum of a large number of squared terms of 1 T P T s=1 v is can not be ignored. The inaccuracy due to the within transformation accumulates in the sum of squared terms of the statistic with comparably large n and n=T ! c 2 (0; 1), consequently, resulting in asymptotic bias.

Monte Carlo Simulations
This section employs Monte Carlo simulations to examine the empirical size and power of our biascorrected LM test de…ned in (11) in a static panel data model. We compare its performance with that of Pesaran's (2004) CD test given by and PUY's LM test given in (5). The sample correlations ij are computed using OLS residuals, see (3). We also include the John test for the null of sphericity discussed by Baltagi, Feng and Kao (2011). Sphericity means that is proportional to the identity matrix. The John test statistic is given by t is the n n sample covariance matrix computed using the within residualŝ V t = (v 1t ; :::;v nt ) 0 . trŜ is the trace of the matrixŜ. Under normality and homoskedasticity, the John test can be used to test for cross-sectional dependence. However, John's test is not robust to heteroskedasticity, and rejection of the null hypothesis using the John test could be due to heteroskedasticity or cross-sectional dependence. For this reason we include the John test in our experiments only under the homoskedastic case. The John test is not recommended for testing cross-section dependence when heteroskedasticity is present. 3

Experiment Design
The experiments use the following data generating process: Following Im, Ahn, Schmidt and Wooldridge (1999) x it in (13) is correlated with the i , but not To calculate the power of the tests considered, two di¤erent models of the cross-sectional dependence are used: a factor model and a spatial model. In the former, it is assumed that where f t (t = 1; ; T ) are the factors and i (i = 1; ; n) are the loadings. In a spatial model, we consider a …rst-order spatial autocorrelation (SAR(1) in (15)) and a spatial moving average (SMA(1) in (16)) model as follows: v it = (0:5" i 1;t + 0:5" i+1;t ) + " it : Cross-sectional dependence can also be modelled by including a spatially lagged dependent variable, denoted as the mixed regressive, spatial autoregressive (MRSAR) model: where, similar to the SAR(1) model in (15), the term (0:5y i 1;t + 0:5y i+1;t ) represents the spatial interaction in the dependent variable. The null can be regarded as a special case of i = 0 in the factor model (14) and = 0 in the spatial models (15), (16) and (17).
v it (under the null) and " it (under the alternative) are from iidN (0; 2 i ). To model the heteroskedasticity, we follow Baltagi, Song and Kwon (2009) and Roy (2002) and assume that where x i is the individual mean of x it . Here is assigned values 0, 0:5 with = 0 denoting the homoskedastic case. For non-zero , we …x the average value of 2 i across i as 0:5 in our experiments. We obtain the value of 2 = 0:5= 1 (18)

Results
Tables 1 and 2 present the empirical size of these tests under the null of cross-sectional independence with homoskedasticity ( = 0) and heteroskedasticity ( = 0:5), respectively. The size of the biascorrected LM test is close to 5%, even for micro panels with small T and large n. has the correct size for all combinations of n and T . 4 The size of the John test is also reported in Table 1 for comparison purposes. It performs well except for micro panels, in which case the John test is oversized under homoskedasticity. Table 3 shows the size adjusted power of these tests under the alternative speci…ed by a factor model. The bias-corrected LM test has bigger size adjusted power than PUY's LM test for small T .
However, both tests have size adjusted power that is almost 1 when n and T are larger than 20. By contrast, the power of Pesaran's CD test is much smaller than those of the two LM tests. While the power of the LM tests becomes one for large n and T; the power of the CD test reaches a maximum of 36:5% for n = 200 and T = 50 when i is drawn from U ( 0:5; 0:55). This is expected under the current design. As pointed out by Pesaran, Ullah and Yamagata (2008), in the factor model above in (14),  (1), SMA(1) and MRSAR, respectively. In these cases, the size adjusted power of Pesaran's CD test performs much better than in the case of a factor model, increasing to 1 with T . 5 4 Pesaran's CD test is designed for hetergeneous panels and is based on the sample correlation of the residuals of individual heterogeneous OLS regressions. We performed the experiments again using the CD test but computed with …xed e¤ects residuals. Pesaran's CD test always has correct size for all combinations of n and T under the homoskedastic case. However, it is a little oversized under heteroskedasticity for large n and small T . 5 Tables 4 and 5 show that in the SAR(1) and SMA(1) models, the size-adjusted power of tests is low when n is relatively large and T is small. For example, in the SAR (1) model, when T = 10; n = 200, the size-adjusted power is 73:6%; 45:6% and 52:9% for the proposed bias-corrected LM, PUY's LM and Pesaran's CD tests, respectively. However, when T gets large, the size-adjusted power of these tests increases to 1. By contrast, Table 6 shows that in the MRSAR model, the size-adjusted power of these two LM tests is large and increases to 1 with n no matter whether T is small or large.
The power of the tests under the spatial model depends upon the spatial autocorrelation parameter . For example, for = 0:8, in the SAR (1) model, when T = 10, n = 100, the size-adjusted power is 100%, 100% and 93:9% for the proposed bias-corrected LM, PUY's LM and Pesaran's CD tests, respectively. Pesaran (2004) discusses the local power of the CD test in factor model and SAR(1) model. Similarly, one can investigate the asymptotic power of the proposed bias-corrected LM under di¤erent alternatives. Since our proposed test statistic is based on the sum of Table 7 provides the results of robustness check on the size of the tests with some non-normal or asymmetric distributions on the errors. We ran experiments with uniform distribution U [1; 2], Chi-square distribution with 1 degree of freedom, 2 1 , and t-distribution with 5 degrees of freedom, t(5), and we compare these results with those of Gaussian case N (0; 0:5). By and large, these experiments show that the size of the bias-corrected LM, PUY's LM and Pesaran's CD tests are not that sensitive to the normality assumption on the errors. The same results obtain although the magnitude are di¤erent. PUY's LM test is still oversized around 8% for large n = 100, small T = 10 no matter what distribution is used. The bias-corrected LM test has size close to 5% for the uniform and t distributions and is a little oversized for T 10 when using the 2 1 distribution. 6

Dynamic Panel Data Models
In a dynamic panel data model 1 T P T s=1 y i;s 1 , and the corresponding sample correlations^ ij and the bias-corrected LM test statistic (LM BC ). 7 We show that as long as b b is p nT -consistent, squared sample correlations constructed from within residuals, these derivations will be quite involved and are not pursued in this paper. 6 We performed Monte Carlo simulations where the true DGP is a heterogeneous panel. When T is large (T = 50) and n is small (n = 10), the size of the proposed bias-corrected LM , PUY's LM and Pesaran's CD test is 5:6; 5:4 and 5:5 respectively. When n is relatively larger than T , our simulations show that the proposed bias-corrected LM test is not robust to slope heterogeneity. For example, the size of our proposed bias-corrected LM test is 13:4% for T = 30 and n = 50: The proposed test used in the heterogeneous panels is actually CD lm minus n 2(T 1) . When T is much larger than n, the CD lm test has size close to the nominal level. Since the term n 2(T 1) is small in this case, the estimated size of the proposed test is also close to the nominal level. 7 Theorem 1 of Hahn and Kuersteiner (2002) shows that the limiting distribution of p nT (~ ), where~ denotes the within estimator, is not centered at zero when both n and T are large. Due to this noncentrality, we …nd in Monte the proposed LM BC test in the dynamic panel data model still has standard normal limiting distribution under the null. However, stronger assumptions are needed than the static panel data model. In particular,   in micro panels with large n and small T , and this fact is also observed in Table 6

Appendix
This appendix includes the proofs of the main results in the text.
In the …xed-e¤ects model y it = + x 0 it + i + v it ,~ is the within estimator and the within residuals are given by Using this notation, the sample correlation r ij in the raw data case can be written as and its sample counterpart using within residuals in the …xed e¤ects model is given bŷ Dividingv it by i , we obtainv As shown below, the terms involving (x it i ) 0 (~ ) have no e¤ect on the test statistic asymptotically. Without loss of generality, i is assumed to be 1 in the derivations below. Under Assumption 2, ). In addition, we need the following lemma in the proofs below.
Lemma 1 Under Assumptions 1, 2 and the null, Proof. To calculate the order of magnitude of a random variable, we can use Lemma 1 in Baltagi, Feng and Kao (2011). Speci…cally, for a random sequence fZ n g, if EZ 2 n = O(n ) and EZ n = 0, where is a constant, then Z n = O p (n =2 ). Using this result, we can prove this lemma by calculating the order of magnitude of the second moments of random variables. 1) 8) Similar to 6), Lemma 2 Under Assumptions 1, 2 and the null, Proof. 1) Using (20), we have By Lemma 1, iV j can be written as: Lemma 3 Under Assumptions 1, 2 and the null, Proof. 1) Using Lemma 2, we obtain, By denoting G the sum of expressions (25) and (26), and denoting H the sum of expressions (27) and (28), we obtain and Note that the term H contains terms involved with F , E i and E j . We will show that this term vanishes asymptotically.
2) As in the proof of Lemma 1, By Lemma 2, it follows that Lemma 4 Under the Assumptions 1, 2 and the null, Proof. a) c) Similar to the proof of b), Lemma 5 Under Assumptions 1, 2 and the null, 2). q 1 n(n 1) .
The proof of part 1) is given below. Part 2) through part 6) can be shown in the same way. The proofs are included in the Supplementary Appendix which is available upon request from the authors.
There are 5 cases of (t; s; ): (1) t = s = ; (2) (t = s) 6 = ; (3) (t = ) 6 = s; (4) t 6 = (s = ); (5) t 6 = s 6 = : We can write Using Lemma 4, we get Now we are in good position to prove Theorem 1. Proof of Theorem 1. It is equivalent to show that for large n and T , By (21), (22) and Lemma 3, Using H = O p (n 1=2 ) from (29), the second term above can be written as follows: . Thus, it is straightforward to calculate the order of magnitude of the third term, Now we consider the …rst term, By Lemma 5,

Proof of Theorem 2
For the dynamic panel data model (19), the sample correlations^ ij used in the bias-corrected LM test statistic LM BC are computed using the within residuals b v it =ỹ it (ỹ i;t 1 ;x 0 it ) b b where b b is the bias-corrected estimator proposed by Hahn and Kuersteiner (2002). Denote the regressors in (19) by w it = (y i;t 1 ; x 0 it ) 0 and the demeaned regressors byw it = (ỹ i;t 1 ;x 0 it ) 0 . LetW i = (w i1 ; ;w iT ) 0 .
Under Assumption 3.i) p nT ( b b ) = O p (1). Replacing~ andX i with b b andW i , the proof of Theorem 2 follows along the same lines as above. We need to verify that Lemmas 1.5, 1.6, 1.7, 1.8 and Lemma 3.6 still hold for the dynamic panel data model. Lemma 5. Under Assumptions 1, 2, 3 and the null, Proof. In (19), the within residual is given by b Proof of 1): Without loss of generality, assume k = 1; Under Assumption 2, as in the proof of Lemma 1.5 in the static model above, 1 Since v it is uncorrelated with y i;s 1 for s < t and v is is uncorrelated with y i;t 1 for s > t; the …rst term is 1 T P T t=1 y i;t 1 v it = O p (T 1=2 ). Consider the second term, we obtain The proofs of 3) and 4) are similar. The proof of 5) can be found in the supplementary appendix which is available upon request from the authors.