Unified and robust Lagrange multiplier type tests for cross-sectional independence in large panel data models

Unified and robust Lagrange multiplier type tests for cross-sectional independence in large panel data models

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{\it Keywords:} Cross-sectional dependence, Large panels, Coefficient heterogeneity, Weak exogenous, Random Matrix Theory

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Introduction

In panel data analysis, the problem of error cross-sectional dependence has attracted substantial attention in recent years. This cross-sectional dependence can arise for various reasons, such as omitted common or spatial effects. Ignoring error cross-sectional dependence can have dramatic effects on conventional panel estimators, e.g., the least squares, fixed and random effect estimators, and yield invalid inferential procedures such as commonly used panel unit root tests, where these tests assume cross-sectional independence. Therefore, designing efficacious tests for cross-sectional dependence is essential in panel data analysis.

There has been much work on testing for cross-sectional dependence in the literature. breusch1980lagrange proposed a Lagrange multiplier ($LM$) test based on the squared pair-wise Pearson correlation coefficients of the residuals. Under the null hypothesis of no cross-sectional dependence, the $LM$ test is asymptotically chi-squared distributed with $T\rightarrow\infty$ and $n$ fixed. However, it is not applicable for large $n$, which renders its popularity considering recent researches have focused on the large panels where both $T$ and $n$ can be large. In such high dimensional setting, there are two mainstream schemes considered by statisticians and econometricians, being known as sequential limit scheme and simultaneous limit scheme defined as following: $$\text{SEQ-L:} \; T\rightarrow\infty, \; \text{followed by} \; n\rightarrow\infty,$$ and $$\text{SIM-L:} \; (n, T) \rightarrow\infty \; \text{such that} \; \frac{n}{T}\rightarrow c \in (0,\infty), $$ respectively. Under the SIM-L scheme, frees1995assessing proposed a distribution free $LM$ type test allowing for large $n$, $R^2_{AVE}$, based on the squared pair-wise Spearman rank correlation coefficients, which is asymptotically distributed as Chi-squared. The test has however imposed limitations on the number of regressors and could be oversized for small $T$. pesaran2004general suggested a scaled version of the $LM$ test, denoted by $CD_{LM}$, and showed its asymptotic property. The author however pointed out that the $CD_{LM}$ test is not correctly centered at zero for small $T$, and is likely to exhibit large size distortions as $n$ increases. pesaran2004general also proposed an alternative approach, the $CD_P$ test, which employs pair-wise Pearson correlation coefficients of the residuals as well, but without squaring them. This test has universally correct size under a broad course of panel data model designs, but it lacks power when correlation coefficients within panel units have variable signs leading to certain cancellation effect. Particularly, this happens when the errors are generated from a factor model where the loadings average to zero. pesaran2015testing extended the $CD_P$ test to the scenario of weak cross-sectional dependence. pesaran2008bias put forward another approach, $LM_{adj}$, by deriving the exact expected values and variances of the squared correlation coefficients under the assumption of normally distributed errors and strictly exogenous regressors. By applying the classical central limit theory, it is shown that the $LM_{adj}$ test converges to standard normal distribution under the SEQ-L scheme. bailey2021lagrange proposed a new $LM$ type test, $LM_{RMT}$, and proved its asymptotic normality under the SIM-L scheme using Random Matrix Theory. However, they require the assumptions of normal regressors and normal errors. In a slope homogeneity setting, baltagi2012lagrange analyzed the performance of the $CD_{LM}$ test in the fixed effects panel data model, then presented a bias corrected test, $LM_{bc}$, and established its asymptotic normality under the SIM-L scheme assuming normal errors and strictly exogenous regressors. baltagi2012lagrange also showed that the $LM_{bc}$ test can be applied to dynamic panel data model with fixed effects, using the within estimator proposed by hahn2002asymptotically. However, the slope homogeneity restriction has often been rejected in empirical analyses, see a detailed survey by baltagi2008pool. Meanwhile, there are limited tests designed for the dynamic panel data model. Examples include the GMM approach of sarafidis2009test applied to panels with homogeneous coefficients under factor model representation, and a heteroskedasticity robust $LM$ test, $LM_{HOY}$, of halunga2017heteroskedasticity with $n^2/T\rightarrow 0$ required.

We make two distinct contributions in this paper. First, we propose a $LM$ type test statistic that can be applied to a wide class of linear panel models. Specifically, we show that our test is robust to both static and dynamic heterogeneous panel data models with weakly exogenous regressors and non-normal errors. With tools from Random Matrix Theory, we treat sample correlation matrix of residuals directly as a random matrix to establish the asymptotic normality of the test statistics under the SIM-L scheme, which has been suggested to be a more reliable strategy when dealing with high-dimensional statistical problems, see yao2015sample. We also show that the proposed test is mathematically equivalent to both the $LM_{RMT}$ and the $LM_{bc}$ test. This finding theoretically enriches the $LM_{RMT}$ and $LM_{bc}$ test by relaxing the restrictive assumptions they need. It is worth mentioning that the existing literature on testing for cross-sectional dependence has mostly focused on the case of strictly exogenous regressors, including the $LM_{adj}$, $LM_{bc}$ and $LM_{HOY}$ tests. Though pesaran2015testing showed that the $CD_P$ test is also applicable to autoregressive panel data models so long as the errors are symmetrically distributed, the properties of the $CD_P$ test for dynamic panels that include weakly exogenous regressors have not yet been investigated. The new proposed test fills this gap for the weakly exogenous regressors case, and to the best of our knowledge, there is no such a unified test so far.

Second, weak cross-sectional dependence is common in empirical applications, see, for example, bailey2016exponent, ertur2017weak. This leads to a sparse correlation structure with few nonzero off-diagonal entries. Therefore, it is important to test the existence of such weak cross-sectional dependence, which corresponds to sparse alternatives in the high-dimensional statistics literature. The mainstream of more powerful tests for sparse alternatives are based on the maxima absolute value of sample correlations, see cai2011limiting and hall2010innovated. However, these tests require stringent conditions that are `unfeasible' in econometric applications and often suffer from size distortions due to slow rates of convergence. In view of this, we propose a novel and easy-implemented test statistic for cross-sectional dependence based on the fourth moment of sample correlation to boost the power of detecting sparse alternatives. Again, this test can be used in any aforementioned panel setting under suitable conditions on higher moments of the errors.

The remainder of the paper is organized as follows. Section (ref) discusses the existing tests for no cross-sectional dependence. Section (ref) introduces the new test statistics and establishes the limiting distributions under the SIM-L scheme. Sections (ref) and (ref) demonstrate that the proposed tests can be extended to both the dynamic and fixed effects panel data models. Section (ref) reports the results of Monte Carlo simulations. Section (ref) provides concluding remarks and further discussions.

Notations. Throughout the paper, for a matrix $\mathbf{A} \in \mathcal{R}^{n\times n}$, $tr(\mathbf{A})$ represents the trace $\mathbf{A}$. We use $\lambda_1(\mathbf{A}), \dots, \lambda_n(\mathbf{A})$ to denote $n$ eigenvalues of $\mathbf{A}$. Further, for vectors $\mathbf{a}, \mathbf{b} \in \mathcal{R}^{n\times 1}$, we write $\langle \mathbf{a}, \mathbf{b} \rangle=\mathbf{a}^{T}\mathbf{b}$ for their scalar product. In addition. $\|\cdot\|$ represents the Euclidean norm for a vector and the induced operator norm for a matrix.

Existing tests for cross-sectional dependence based on sample correlation

Consider the heterogeneous panel data model:

where $i$ indexes the cross-sectional units and $t$ the time series observations, $y_{it}$ is the response variable and $\mathbf{x}_{it}$ is a $k \times 1$ vector of regressors with unity on the first row with coefficients $\boldsymbol{\beta}_i$ allowed to vary cross the cross-sectional units. For each $i$, the error term, $v_{it}$, are assumed to be serially independent with zero mean and finite variance. The null hypothesis of interest in the literature is

When $T$ is sufficiently large, a natural way to test $H_0$ is based on some reliable estimates ($\hat{\rho}_{ij}$) for the pair-wise error sample correlations ($\rho_{ij}$). Specifically,

where $\hat{v}_{it}$ is the Ordinary Least Squares (OLS) estimate of $v_{it}$ in ((ref)) defined by

with $\hat{\boldsymbol{\beta}}_i$ being the OLS estimate of $\boldsymbol{\beta}_i$ by regressing the $T$ sample observations $y_{it}$ on $\mathbf{x}_{it}$ for each $i$. In the seemingly unrelated regression equations (SURE) context with fixed $n$ and as $T\rightarrow\infty$, breusch1980lagrange proposed a Lagrange multiplier ($LM$) test for testing $H_0$ given by

where $\hat{\mathbf{R}}=(\hat{\rho}_{ij})_{1\leq i, j\leq n}$ is the sample correlation matrix of residuals. It has been shown that $LM\stackrel{d}{\longrightarrow} \chi^2_d$, where $d=\frac{1}{2}n(n-1)$ under $H_0$ and normal errors assumption. However, it is well known that the $LM$ test is severely oversized when $n$ is relatively large compared to $T$. To amend this size distortion, pesaran2004general put forward a scaled version of the $LM$ test given by

which is asymptotically distributed as $N(0,1)$ under the SEQ-L scheme. There are two important cases where the $CD_{LM}$ test is not reliable. Firstly, as baltagi2012lagrange noted, it will exhibit substantial size distortions in the homogeneous panel data model. Secondly, pesaran2004general pointed out that, in finite $T$ case, the $CD_{LM}$ test tends to over-reject the null due to the fact that $E(T\hat{\rho}_{ij}^2-1)$ is not correctly centered at zero. This kind of bias even accumulates as $n$ becomes larger. In this case, dufour2002exact suggested to apply bootstrap method to ((ref)). pesaran2004general and pesaran2015testing proposed an alternative adjustment based on the raw, non-squared, sample correlation coefficients given by

The test is asymptotically distributed as standard normal under both SEQ-L and SIM-L schemes. However, it is widely reported that the $CD_P$ test suffers from a specific loss of power when the loadings have zero mean in the cross-sectional dimension under factor representation. baltagi2012lagrange proposed another modified version of the $CD_{LM}$ test for fixed effect panel data model, $LM_{bc}$, given by

The test is asymptotically standard normal with normal errors and strictly exogenous regressors under the SIM-L scheme. pesaran2008bias proposed an alternative finite sample adjustment to the $LM$ test by deriving the exact moments of the squared sample correlation coefficients under normal errors and strictly exogenous regressors assumptions. Their $LM_{adj}$ test statistic is given by

where $$\mu_{T,i,j}=\frac{1}{T-k}tr(\mathbf{M}_i\mathbf{M}_j), \; \sigma_{T,i,j}^2=\left[tr(\mathbf{M}_i\mathbf{M}_j) \right]^2a_{1T}+ tr(\mathbf{M}_i\mathbf{M}_j)^2a_{2T},$$ and $$a_{1T}=a_{2T}-\frac{1}{(T-k)^2}, \; a_{2T}=\frac{3}{(T-k+2)^2},$$ $\mathbf{M}_i=\mathbf{I}_T-\mathbf{X}_i'(\mathbf{X}_i\mathbf{X}_i')^{-1}\mathbf{X}_i$ is the projection matrix, where $\mathbf{X}_i=(\mathbf{x}_{i1}, \cdots, \mathbf{x}_{iT})$ contains $T$ samples on the $k$ regressors for the $i$-th individual regression. Under $H_0$ and the SEQ-L scheme, $LM_{adj}$ was shown to be asymptotically distributed as $N(0,1).$ However, as pointed out by pesaran2008bias, the $LM_{adj}$ test is not robust in panel data models with weakly exogenous regressors. bailey2021lagrange proposed another modified $LM$ test for heterogenous panel data models based on Random Matrix Theory, $LM_{RMT}$, given by

where

with $\kappa=\frac{3T(T-k-2)}{(T+2)(T-k)}$. Under the assumptions of normal regressors and normal errors, the authors showed that $LM_{RMT}$ is asymptotically distributed as $N(0,1)$ under the SIM-L scheme. The application scopes of the discussed tests are summarized in Table (ref). (The table also contains the new tests proposed in this paper in the last two columns so-called $RLM$ and $RLM_{PE}$, which are developed later.)

The RLM test and its power enhancement

The RLM test

Motivated by the existing well-known tests based on the sum of squared sample correlation coefficients in ((ref)), ((ref)), ((ref)), ((ref)) and ((ref)), it is natural to consider the limiting behavior of $tr(\hat{\mathbf{R}}^2)$ under the SIM-L scheme. Throughout the paper, we consider the following assumptions.

For a static heterogeneous panel data model, we further assume

Assumption (ref) is standard allowing for heteroskedastic errors across units. Assumption (ref) requires suitable moments of the errors for the two proposed test procedures, respectively. It helps relax the often-met normal error assumption by Random Matrix Theory. Assumption (ref)(i) only requires the regressors to be weakly exogenous. Assumption (ref)(ii) and (iii) impose mild conditions on the design matrix. We note that Assumption (ref) does not impose the dependence structure between errors and regressors, which allows for the regressors to be weakly exogenous. Under these assumptions, $\sqrt{T}(\hat{\boldsymbol{\beta}}_i-\boldsymbol{\beta}_i)$ is asymptotically normal according to lai1982least.

For a dynamic heterogeneous panel data model with lagged dependent variable included in regressors, more assumptions are needed which will be discussed in Section (ref).

Now we are in the position of introducing the $RLM$ test and establishing its asymptotic property in the following theorem.

The proof of Theorem (ref) is provided in the Appendix, and the method is in two stages. In the first stage, Lemma 1 establishes the Central Limit Theorem of $tr(\mathbf{R}^2)$ with tools from Random Matrix Theory, where $\mathbf{R}=(\rho_{ij})_{1\leq i, j \leq n}$. In the second stage, Lemma 2 shows that the asymptotic bias of $tr(\hat{\mathbf{R}}^2)$ disappears under the SIM-L scheme with $c_T \rightarrow c \in (0, \infty)$.

Relationship between the $RLM$ and $LM_{RMT}$ tests

By the respective definitions of the $LM_{RMT}$ and $RLM$ in ((ref)) and Theorem (ref), we have

and

as $\kappa=3+o(1)$. It follows that

which indicates that $LM_{RMT}$ is asymptotically equivalent to $RLM$ regardless of model specifications and assumptions. Note that the proof the asymptotic normality of $LM_{RMT}$ in bailey2021lagrange heavily relies on the assumptions of normal regressors and normal errors as it is used to ensure the residuals have desirable properties, and then transform the sample correlation matrix of residuals to the sample correlation matrix of a nomarlized population with unit covariance matrix (see details in Section 3.1 in bailey2021lagrange). From ((ref)), we conclude that the $LM_{RMT}$ is also valid without the restrictive assumptions of normality. Besides, as we will show later, $RLM$ is also valid in both dynamic and fixed effects panel data models, which theoretically extends the application scope of $LM_{RMT}$. This finding is consistent with the simulation findings that show such robustness of $LM_{RMT}$ in bailey2021lagrange.

Relationship of the $RLM$, $LM_{bc}$ and $CD_{LM}$ tests

By the respective definitions of the $CD_{LM}$, $LM_{bc}$ and $RLM$ tests in ((ref)), ((ref)) and Theorem (ref), we have the following identities

and

Note that the factor $\sqrt{\frac{n}{n-1}}\rightarrow 1$ and the remainder $\frac{\sqrt{n}}{2(T-1)(\sqrt{n}+\sqrt{n-1})}\rightarrow 0$. It follows that the two tests, $RLM$ and $LM_{bc}$, are always asymptotically equivalent, while the $CD_{LM}$ statistic has always a positive mean shift of value $\frac{n}{2(T-1)}$.

In particular, Theorem (ref) is also valid for the $LM_{bc}$ statistic. Moreover, anticipating Theorems (ref) and (ref) in Sections (ref) and (ref), for dynamic and fixed effects panel data model, respectively, these asymptotic normality are also valid for the $LM_{bc}$ statistic. In this sense, the results from the paper can also be considered as new extension of the $LM_{bc}$ test, originally developed for homogeneous fixed effects panel data model in baltagi2012lagrange, to various large panel models with coefficient heterogeneity.

The $RLM_{PE}$ test

In the high dimensional setting, for testing the identity hypothesis $H_0: \mathrm{corr}(\mathbf{v}_t)=\mathbf{I}_n$, where $\mathbf{v}_t=(v_{1t},\dots,v_{nt})'$, there are mainly two types of test statistics. The majority of existing tests are based on the squared Frobenius norm $\|\mathbf{R}-\mathbf{I}_n\|_F^2=tr( \mathbf{R}^2)-n=\sum_{i\neq j}\rho_{ij}^2.$ However, this quadratic statistic lacks power if $ \mathrm{corr}(\mathbf{v}_t)$ is a sparse matrix, see fan2015power. Considering this, tests based on the maxima of absolute values, $\max_{i<j}|\rho_{ij}|$, which share a asymptotic type I extreme value distribution, are generally powerful under sparse alternatives. This approach has however a main drawback that such test can suffer from size distortions, which is common for statistics of the maximum type, see liu2008asymptotic. Besides, this way is not as appropriate as the Frobenius norm (sum) type in some cases. For example, consider the alternative

where $\mathbf{P}_n$ is a perturbation matrix with diagonal entries being zero and $s$ non-zero off diagonal entries, where $s \in \{1, \dots, n^2-n \}$. Intuitively, $\mathbf{P}_n$ can be designed as a dense matrix but with weak coefficients such that $\max_{i<j}|\rho_{ij}|<z_\alpha$ for any $s$. Consequently, the extreme value type tests will fail to detect such a matrix. In such instances, the sum type tests are more suitable in the light of the fact that the eigenvalues of $\mathbf{R}$ could vary from $H_0$ to $H_1$, which results in larger quadratic statistic value by $tr( \mathbf{R}^2)=\sum_{i=1}^n\lambda_i^2({\mathbf{R}})$. In order to realize an interpolation of the two types of statistics above, namely the maximum type and the sum type, we propose a new test statistic based on $tr(\mathbf{R}^4)=\sum_{i=1}^n\lambda_i^4({\mathbf{R}})$. The reason is that large empirical correlations, $\rho_{ij}^4$, would be more emphasized in $tr(\mathbf{R}^4)$ than in $tr(\mathbf{R}^2)$. To see this, consider increasingly large powers of the sample correlations, $\rho_{ij}^m$, where $m$ is a positive integer. Let $E=\text{argmax}_{(i,j):\;i<j}|\rho_{ij}|$, then

where $\mathrm{card}(E)$ denotes the cardinality of the set $E$. Therefore, the new statistic with $m=4$ can mimic some properties of the maximum type, while remaining a sum type smoothing statistic. The resulting power is expected to be higher than $tr(\mathbf{R}^2)$ when very few sample correlations are significantly non-zero under sparse alternatives, and higher than maximum type statistics when there are many but relatively small correlations.

Test based on $\sum_{i\neq j}\hat{\rho}_{ij}^4$

On the ground of analyses above, we propose a new test statistic based on the fourth power of $\hat{\rho}_{ij}$ in the following theorem.

The proof of the Theorem (ref) is similar to that of Theorem (ref), which requires the two lemmas given in Appendix.

Dynamic panel data model

In this section, we show that the $RLM$ and $RLM_{PE}$ tests are asymptotically valid in a dynamic panel data model, which is specified as following:

for $i=1,\dots,n; t=1,\dots,T$, where $y_{i,t-1}$ is the lagged dependent variable. Let $\mathbf{z}_{it}=(y_{i,t-1}, \mathbf{x}_{it}')$, $\boldsymbol{\phi}_i=(\boldsymbol{\beta}_i,\alpha_i)'$, then ((ref)) can be rewritten as $y_{it}=\mathbf{z}_{it}'\boldsymbol{\phi}_i+v_{it}$. We show that the proposed $RLM$ and $RLM_{PE}$ tests still have standard normal limiting distribution under the null hypothesis in the dynamic panel data model. To establish the asymptotic normality, we need additional assumptions as following,

We establish the limiting distributions of the proposed tests in the following theorems

Under Assumption (ref), the proofs of Theorems (ref) and (ref) follow along the same lines as that of static panel data model. See the Appendix.

Fixed effects panel data model

In this section, we establish the asymptotic normality of the $RLM$ and $RLM_{PE}$ tests in a fixed effects panel data model. We find that as long as the coefficient estimator is $\sqrt{T}$-consistent, the proposed tests still have standard normal limiting distribution under the null. To allow for weakly exogenous regressors, various consistent estimators have been proposed in the literature including chudik2015common, chudik2018half etc. However, these estimators require stronger assumptions than the static panel data model. For simplicity of illustration, we focus on residuals obtained by the within estimator. The strictly exogenous assumption is then necessary for the consistency of the within estimator. One can relax this assumption to the weakly exogenous one by applying a $\sqrt{T}$-consistent estimator.

Consider a fixed effects panel data model:

for $i=1,\dots,n; t=1,\dots,T$, where $\mu_i$ denotes the time-invariant individual effect. The within estimator in ((ref)) is specified by

where $\tilde{\mathbf{x}}_{it}=\mathbf{x}_{it}-\frac{1}{T}\sum_{t=1}^T\mathbf{x}_{it}$ and $\tilde{y}_{it}=y_{it}-\frac{1}{T}\sum_{t=1}^Ty_{it}$.

Under the Assumptions (ref), (ref), (ref) and (ref), $\hat{\boldsymbol{\beta}}$ is $\sqrt{nT}-$consistent. We establish the validity of our proposed tests in the following theorems.

The proofs of Theorems (ref) and (ref) are given in the Appendix.

Monte Carlo simulations

In this section, we conduct Monte Carlo simulations to examine the empirical sizes and powers of our $RLM$ and $RLM_{PE}$ tests, which are defined by ((ref)) and ((ref)), respectively, and compare their performances to that of the $CD_p$ test and the $LM_{adj}$ test defined by ((ref)) and ((ref)), respectively. We consider four data generating processes (DGPs): heterogeneous panel data model with either strictly or weakly exogenous regressors, fixed effects panel data model and pure dynamic panel data model.

Before looking at the simulation results, we consider the estimated rejection frequencies within range from 3.6% to 6.5% to provide evidence consistent with the robustness of the tests, following the arguments in halunga2017heteroskedasticity. Besides, we don't include the $LM_{RMT}$ and the $LM_{bc}$ tests since they are almost identical to the $RLM$ test by ((ref)) and ((ref)).

Monte Carlo design

DGP1: Heterogeneous panel data model with strictly exogenous regressors

We first consider the DGP used in pesaran2008bias, which is specified by

where $\alpha_i\sim IIDN(1,1)$, $\beta_{li}\sim IIDN(1,0.04)$. The regressors are generated as

with $x_{li,-51}=0$ where $u_{lit}\sim IIDN(0, \tau_{li}^2/(1-0.6^2))$, $\tau_{li}^2\sim IID \chi^2(6)/6$. The first 50 observations are discarded to lessen the effects of initial values. Now we generate the disturbances under the null $H_0$ as $v_{it}=\sigma_i\epsilon_{it}$, where $\sigma_i\sim\chi^2(2)/2 $ and $\{\epsilon_{it}\}$ are generated from three different distributions: (i) normal, $N(0,1)$, (ii) chi-squared, $(\chi^2(5)-5)/\sqrt{10}$ and (iii) student-t, $t_{10}/\sqrt{10/8}$. The normalizations in (ii) and (iii) are such that errors have mean one and variance one. To investigate the effects of the number of regressors, $k=2, 4$ are considered.

To examine the powers of the proposed tests, the disturbances are generated by a factor model as following:

where $f_t (t=1,\dots,T)$ are the factors with $f_t\sim IIDN(0,1)$ and $\lambda_i (i=1,\dots,n)$ are the loadings. We consider the following three cases of loading construction:

In the dense case, $h$ measures the degree of cross-sectional dependence. The sparse case and the less-sparse case follow the design used in bailey2016exponent to model the weak and strong cross-sectional dependence, respectively. The Monte Carlo experiments are conducted for $T=50, 100 ,200$, and three different choices of ratio $n/T=0.5, 1, 2$ basing on 2000 replications. To obtain the empirical size, the proposed $RLM$ test, $RLM_{PE}$ test and $LM_{adj}$ test are implemented at the one-sided 5% nominal significance level, while $CD_P$ test is conducted at the two-sided 5% nominal significance level.

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DGP2: Heterogeneous panel data model with weakly exogenous regressors

To investigate the performances of the $RLM$ and $RLM_{PE}$ tests in panel data models with weakly exogenous regressors, we consider the following DGP:

where $\alpha_i\sim IIDN(1,1)$, $\beta_{li}\sim IIDN(1,0.04)$, and

with $y_{i,-51}=x_{1,i,-51}=0$ where $u_{1it}, u_{2it}\sim IIDN(0, \tau_{li}^2/(1-0.6^2))$, $\tau_{li}^2\sim IID \chi^2(6)/6$. This set up allows for feedback from $y_{it-1}$ to the regressors, thus rendering weakly exogenous. The errors, $\{v_{it}\}$, are generated in the same way as DGP1.

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DGP3: Fixed effects panel model

The third DGP considered is a fixed effects panel data model with homogeneous coefficients, which is specified as

where $\alpha$ and $\beta_{l}$ are set arbitrarily to 1 and $l$, respectively, $\mu_i\sim IIDN(1,1)$. The regressors and errors are generated in the same way as DGP1.

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DGP4: Dynamic panel data model

To examine the properties of the $RLM$ and $RLM_{PE}$ tests in a dynamic panel data model, we follow the design of pesaran2008bias:

with $y_{i,-51}=0$, where $\beta_{i}\sim IIDN(1,0.04)$, and the fixed effects, $\xi_i$, are drawn as $v_{i0}+\eta_i$, with $\eta_i \sim IIDN(1,2)$. The errors, $\{v_{it}\}$, are generated in the same way as DGP1.

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Simulation results

Table (ref) reports the empirical size of these tests for the DGP1. The proposed $RLM$ and $RLM_{PE}$ tests successfully control the size under almost all settings, irrespective of number of regressors included in the panel data model\footnote{However, for small sample size with more regressors, i.e. $T=50$ and $k=4$, the $RLM$ and $RLM_{PE}$ tests would be slightly oversized. For example, the empirical sizes of $RLM$ and $RLM_{PE}$ are 7.65 and 7.4 under normal errors, respectively. }. For a fixed ratio $n/T$, the empirical size of the $RLM$ and $RLM_{PE}$ tests converge to the nominal size of $0.05$ as $T\rightarrow\infty$, that authenticates the asymptotic normality of the tests under the SIM-L scheme. Besides, the performance of $RLM$ are almost identical to $LM_{adj}$. The $CD_P$ test has correct size in all cases.

Table (ref) demonstrates the empirical power of these tests under the alternative with dense factors. The $RLM$ test has comparable power to $LM_{adj}$ regardless of $(n,T)$ combinations and error distributions. In contrast, the $CD_P$ test suffers from little power by construction, where mean of factor loading is close to zero as mentioned by pesaran2008bias. The power enhancement version of $RLM$, the $RLM_{PE}$ test, outperforms others across the board, especially when $n/T=2$. For example, the power of the $RLM_{PE}$ test is 78.25% for $T=100, n=200, k=4, h=3$ and student-t errors, whereas the power of the $RLM$ and $LM_{adj}$ tests are 60.1% and 56.75%, respectively. It improves the power by up to around 30%. Besides, the power of the $RLM_{PE}$ test is 69.2% for $T=100, n=200, k=2, h=3$ and chi-square errors, and the power of the $RLM$ and $LM_{adj}$ tests are 51.2% and 51.1%, respectively. These results indicate that the $RLM_{PE}$ test successfully boost the power.

The empirical power of these tests under the alternative with sparse and less sparse factors are summarized in Tables (ref) and (ref), respectively. The $RLM$ test again has similar performance to $LM_{adj}$. The empirical power of $RLM_{PE}$ show that it performs the best among those tests. The power of the $CD_P$ test floats around 5% throughout as in pesaran2015testing.

For the heterogeneous panel data model with weakly exogenous regressors, Table (ref) shows that $LM_{adj}$ becomes considerably oversized, especially for the case $n/T=2$, where it has size around 28%. This shows that the $LM_{adj}$ test is not robust to weakly exogenous regressors, which is also observed in bailey2021lagrange. However, our proposed $RLM$ test and $RLM_{PE}$ tests control the size well, which are not sensitive to the strictly exogenous assumptions on regressors. Therefore, though it is widely reported that the $LM_{adj}$ test has generally satisfying empirical performances regardless of restrictive strictly exogenous regressors and normal errors assumptions, the present DGP2 is indeed a rare situation where the $LM_{adj}$ test is outperformed by others.

Table (ref) reports the size of the tests for the fixed effects panel data model. It shows that the proposed $RLM$ and $RLM_{PE}$ tests have the correct size, close to the 5% nominal significance level, for example, $RLM$ has 5.1% and 5% size results, respectively for $T=50,n=25$ under normal errors and for $T=100,n=200$ under chi-squared errors. Similar results for $RLM_{PE}$ can be also observed in this table. Pesaran's $LM_{adj}$ and $CD_P$ tests have correct size in this setting as in pesaran2004general and pesaran2008bias.

Finally, Table (ref) gives the empirical size of these tests for dynamic panel data model. It shows that the proposed $RLM$ and $RLM_{PE}$ tests have the correct size, e.g. 5.15% for $n=100, T=100$ with chi-squared error, which is comparable to the $LM_{adj}$ test. The $CD_P$ always has correct size as in pesaran2004general. The results of empirical power for DGP2, DGP3 and DGP4 are similar to those of DGP1, so we omit it here.

Based on these findings, the $RLM_{PE}$ test is strongly recommended for practitioners if it is not clear whether weakly exogenous regressors are present or not, given its universally correct size and better power performances. Instead, when regressors are believed to be strictly exogenous, then the $RLM_{PE}$ test, a easily implemented and computationally cheap procedure, is preferred for large panels ($T\geq 50$). For $T\leq 50$, $RLM_{PE}$ is still applicable though it might be slightly oversized, or the $LM_{adj}$ test is a suggested method at the risk of intensive computation.

Conclusion

This paper has developed a Lagrange multiplier type test for the null hypothesis of no cross-sectional dependence in large panel models. The procedure can be applied to a wide class of linear panel data models and shows robustness to quite general forms of non-normality in the disturbance distribution. We further proposed a power enhancement version of the $LM$ type test based on the fourth moment of the sample correlations obtained from residuals to boost power under sparse alternatives, which only requires existence of higher moment but still shares such robustness. The simulations illustrate that this test has satisfactory power under the sparse alternatives of weak cross-sectional dependence, and both of the tests successfully control the size in different data generating processes.

For future work, it is interesting to explore theoretically the power properties of $RLM_{PE}$ and the optimal $m$ that would maximise power. Also, it would be of interest to investigate the performance of the $RLM$ and $RLM_{PE}$ tests in the weakly cross-sectional dependence framework. In addition, testing the null hypothesis with no cross-sectional dependence when errors are serial dependent will also be studied.