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Extension of the Lagrange multiplier test for error cross-section independence to large panels with non normal errors
Consider the fixed effects panel data model
where $i$ indexes the cross-sectional (individual) units, and $t$ the time series observations. The dependent variable is $y_{it}$ and $x_{it}$ denotes the exogenous regressors of dimension $k\times 1$, $\beta$ is the corresponding $k\times 1$ vector of parameters, $\mu_i$ denotes the time-invariant individual effect which could be correlated with the regressors $x_{it}$. Throughout the paper the number of covariates $k$ is fixed while the two dimensions $n$ and $T$ may grow to infinity. A more general model is the following heterogeneous panel model
where the slope parameters, $\beta_i$, are allowed to vary across $i$.
Our focus is to test the following cross-section independence hypothesis, that is
Testing such cross-section independence is important because it is a preliminary step for many existing inference procedures for the panel model. These procedures become biased or even inconsistent when cross-sectional units are correlated, see for example chudik2013large for the case of commonly used panel unit root tests.
Several procedures exist in the literature for this independence test. A popular procedure is the Lagrange multiplier (LM) test proposed by breusch1980lagrange based on the sum of squared pair-wise sample correlation coefficients of the OLS residuals. Recent efforts have concentrated in large panel models where the panel size $n$ is large compared to the sample size $T$. It has been observed that the LM test is particularly biased in such large panels. pesaran2008bias found an approximation for the mean and variance of the LM statistic and established its asymptotic normality for their modified $LM_{adj}$ test under the following {\em sequential limit scheme} \[ \text{SEQ-L:}\qquad T \to \infty \text{~~first, followed by}~~ n\to \infty. \] pesaran2004general also proposed a cross-sectional dependence (CD) test using sum of (non-squared) sample correlations of the OLS residuals. pesaran2015testing showed that the null of the CD test is rather weak cross-sectional dependence as defined in chudik2011. Again the asymptotic normality for the test statistics is derived under the sequential limit scheme.
When the panel size $n$ and the sample size $T$ are large while having comparable magnitude, it has been argued in recent high-dimensional statistic literature that more reliable asymptotic results can be found by considering a {\em simultaneous limit scheme} YZB15-book \[ \text{SIM-L:}\qquad T \to \infty, n=n(T) \quad \text{such that } \lim_{T\to\infty} \frac{n}{T} = c>0. \] Technically, the derivation of asymptotic normality for LM-type statistics under the SIM-L scheme is challenging. Actually, baltagi2012lagrange succeeded for the fixed effects homogeneous panel data model (ref) by precisely identifying a mean shift in the asymptotic distribution of the LM statistic due to the SIM-L scheme. This bias-corrected test, $LM_{bc}$, however needs to assume the normality of the errors $(\nu_{it})$ as its development relies on a previous result established in schott2005testing on sample correlations of normal-distributed errors. As for the heterogeneous model (ref), a recent work jiangworking2020 establishes the asymptotic normality of the LM test under the SIM-L scheme while also requiring normally distributed errors $(\nu_{it})$ as well as normally distributed regressors $(x_{it})$. This test, $LM_{RMT}$, is extensively compared to the CD and $LM_{adj}$ tests in various dimension and panel size combinations via simulation. The $LM_{RMT}$ has been shown to be comparable to $LM_{adj}$ in terms of size and power, with a slight preference for $LM_{RMT}$ when the sample size $T$ is not very large (or when the panel size $n$ is relatively large). The CD test is universally correctly sized but generally lacks power under local alternatives.
An important aim of this paper to extend the LM test to large panel models under the SIM-L scheme {\em and} without the normality assumption on the errors $(\nu_{it})$. Because the normality assumption is fundamental to the analytic methods used in both the $LM_{bc}$ and $LM_{RMT}$ tests, we achieve our goal by employing complete different tools and derive a new asymptotic normal distribution for the LM statistic in Theorem (ref). A corresponding test $LM_e$ is thus derived for cross-section independence for the panel model (ref). We also establish a surprising fact that this asymptotic distribution for the $LM_e$ coincide with those of both the $LM_{bc}$ and $LM_{RMT}$ tests although the technical derivations of the three tests are all different. This universality of the asymptotic distribution for the LM statistic under the SIM-L scheme ensures a welcomed robustness of the LM test, thus extending its application scope to various large panel models.
A second contribution from the paper is to go beyond the LM statistic by considering fourth powers of residual correlations (instead of their squares in the LM statistic). The idea here is that fourth powers weight more heavily large correlations than smaller ones. This magnification effect enhances the power of the test, particularly under an alternative where the cross-section dependence is weak or sparse. Following the same setting and tools as for the LM statistic, we establish the asymptotic normality for this power-enhanced test (PET).
Next we designed several simulation experiments following commonly used settings in the large panel model literature. These experiments confirm the excellent empirical performance of the new $LM_e$ test at a level comparable to the tests $LM_{RMT}$ and $LM_{adj}$. When it comes to compare the powers of the tests, the power-enhanced test PET indeed dominates all the existing tests discussed above. As expected, the advantage of the PET test is particularly significant under weak or sparse dependence alternatives.
We have also conducted a real data analysis to assess the properties of these tests in real-life large panels. This findings basically confirm the comparison results found through simulation experiments.
The rest of the paper is organised as follows. Section (ref) presents OLS regression residuals in the panel model (ref) and the existing tests discussed above. Sections (ref) and (ref) introduce, respectively, the extended LM test $LM_e$ and the power-enhanced test PET, and establish their asymptotic normality under the null and the SIM-L scheme without assuming normality of the errors. Section (ref) presents a detailed simulation study for comparison of finite-sample performance of the various tests. A real data analysis is carried out in Section (ref). Some discussions are offered in the last conclusion section. All technical proofs are grouped to the Appendix.
Consider the population correlation matrix $R=[\mathop{\text{diag}}(\Sigma)]^{-1/2} \Sigma [\mathop{\text{diag}}(\Sigma)]^{-1/2}$ of the error vectors $\nu_t$. Clearly, under the independence hypothesis (ref), one has $R=I$. It is thus natural to deign test statistics based on estimates for these error correlations. Natural estimates for these correlations are obtained using the residuals from some fitted panel regression model. Precisely, consider the centralised variables \[ \tilde{y}_{it}=y_{it}-\frac{1}{T}\sum_{t=1}^T y_{it}, \quad \tilde{x}_{it}=x_{it}-\frac{1}{T}\sum_{t=1}^Tx_{it}, \quad \text{and}\quad \tilde{\nu}_{it}=\nu_{it}-\frac{1}{T}\sum_{t=1}^T \nu_{it}. \] The model (ref) takes a simpler form with these centralised variables, \[ \tilde y_{it} = \tilde x_{it}^\prime \beta +\tilde\nu_{it}, \quad \textrm{for} \ 1\le i\le n;\ 1\le t\le T. \] The OLS estimator of the regression parameter $\beta$ in model (ref) is
Introduce also the design matrices \[ X_i=
_{k\times T},\quad X=
_{k\times nT}, \] and the staked observation vectors, \[ Y_i=
^{\prime},\quad Y=
^{\prime}. \] Then $\hat{\beta}=\left(XX^{\prime}\right)^{-1}XY$. It is well known that under fairly general assumptions on the panel model and the independence hypothesis $H_0$, $\hat{\beta}$ is a consistent estimator of $\beta$ no matter whether $n$ is fixed or tends to infinity jointly with $T$ baltagi2011testing,baltagi2017asymptotic. The regression residuals $\hat{\nu}_{it}$ are
Let $\hat{\nu}_t = (\hat{\nu}_{1t},\ldots,\hat{\nu}_{nt})^\prime$ for $t=1,\ldots, T$. The {\em sample residual covariance matrix} $\hat{S}_T$ and the {\em sample residual correlation matrix} $\hat{R}_T$ are respectively,
The entries of $\hat{R}_T $ are denoted as $\hat{R}_T=\{\hat\rho_{rs}, ~1\le r,s\le n\}$.
Introduce also the {\em sample error covariance and correlation matrix} defined analogously but with the centralised errors $\{\tilde{\nu}_t\}$, that is
This very first test for cross-section independence is proposed by breusch1980lagrange and uses the statistic
Note that $\sum_{r\ne s} \hat\rho_{rs}^2=||\hat{R}_T-I||_F^2 $ is the squared Frobenius distance between ${\hat R}_T$ and $I$, the $LM$ statistic is indeed a scaled estimator for the population distance $||{R}-I||_F^2$ which is zero under the null hypothesis.
Consider the following assumptions:
Under a {\em large sample limit scheme} where $n$ is fixed while $T\to\infty$ and Assumptions (A1) and (A2), breusch1980lagrange established that under the null, the $LM$ statistic has an asymptotic $\chi^2_{n(n-1)/2}$ distribution. It has been well discussed in the literature that this LM test based on its large sample limiting chi-squared distribution suffers from severe size distortions for large panels where $n$ is large compared to time series size $T$.
To accommodate large panels, pesaran2004general proposed the following CD statistic
Consider the following additional assumptions:
Under the Assumptions (A3)-(A4)-(A5), pesaran2004general established that the CD statistic is asymptotically standard normal under the sequential limit scheme. It is widely reported that the CD test enjoys a very accurate size in general while suffering from certain loss of power when the panel-wise correlations have varying signs. This loss of power can be understood by the fact that the CD statistic is averaging the sample residual correlations $\{\hat{\rho}_{rs}\}$; so if they carry varying signs across panel units, this averaging may lead to certain cancellation of correlations of opposite signs, thus neutralising the statistic and its power. This fact is confirmed in one setting of the simulation experiments where the cross-sectional dependence comes from a factor model with mean zero factor loadings, see Table (ref).
In order to adapt the LM test to large panels, pesaran2008bias proposed the following bias-adjusted version of the LM statistic:
where
Under the assumptions (A1)-(A2)-(A4) (which imply the null hypothesis), the authors proved that under the SEQ-L scheme, $LM_{adj}$ is asymptotically standard normal. This bias-adjusted test indeed perform much better in large panels than the original LM test. The only known issue on this test is that because of the employed sequential limiting scheme, the test may be over-conservative in “micro-panels” where $T$ is quite limited compared to panel size $n$.
pesaran2004general also suggested the following scaled version of the LM test for large panels: \[ LM_P=CD_{lm}=\sqrt{\frac{1}{4n(n-1)}} \sum_{r\neq s}\left(T \hat{\rho}_{rs}^{2}-1\right). \] Under the Assumptions (A1)-(A2) and the SEQ-L scheme, $LM_P$ is shown to have a standard normal limiting distribution. An issue with the test statistic as mentioned in pesaran2004general is that when $T$ is not large, the scale adjustment made in $LM_P$ may not be accurate, thus exhibiting substantial size distortions. Consequently, baltagi2012lagrange considered the SEQ-L scheme where a bias-corrected version was found for the fixed effects model (ref). Precisely, the authors established that under assumptions (A1)-(A2)-(A4) and the SEQ-L scheme,
It is remarkable that the asymptotic bias in $LM_P$ is exactly identified as $n/2(T-1)$, and the authors demonstrated that this bias is caused by the fact that the sample correlations are calculated using OLS residuals, instead of the (unobserved) model errors. Note that this bias disappears in the traditional large sample scheme where the ratio $n/T\to 0$, thus emphasising the particular large panel effect. The derivation of (ref) uses the asymptotic results on sample error correlation matrix in schott2005testing, thus requiring the normality assumption of the errors, see Assumption (A2).
Simulation experiments in baltagi2012lagrange show that the bias-corrected $LM_{bc}$ test has an excellent finite sample performance in comparison to the CD test and the $LM_{adj}$ test. It is particularly recommendable for micro-panels where $T$ is relatively small.
Based on asymptotic results from random matrix theory literature, jiangworking2020 introduced another limiting distribution for the LM statistic under the SIM-L scheme and for the heterogeneous panel data (ref). Precisely, they considered the statistic,
where, setting $c_T=n/T$ and $\kappa = \frac{3T(T-k+2)}{(T+2)(T-k)}$,
Consider the following assumptions.
Under the assumptions (A1)-(A2) and (A6)-(A7), the null and the SIM-L scheme, jiangworking2020 established
The simulation experiments in this reference showed that $LM_{RMT}$ is generally comparable to $LM_{adj}$ in terms of size and power, with however a slight preference for $LM_{RMT}$ when the sample size $T$ is relatively small. This result requires normality for for both the model errors and the regressors, see Assumption (A2) and (A6).
As a main result of the present paper, we extend the test $LM_{RMT}$ to large panels where the errors are not necessarily normal distributed. Because in large panels, most of the previous results on the Lagrange multiplier rely on the normality assumption, the derivation of the asymptotic distribution of the LM statistic here requires new tools. This is achieved via recent results on sample correlation matrix from random matrix theory literature.
To introduce the conditions on the centralised design matrices $X_i$, we need to consider $E_k$, the subspace of $R^k$ which is orthogonal to the constant vector ${\mathbf 1}_k$ (with all coordinates equal to 1), that is, \[ E_k =\{ u=(u_1,\ldots,u_k)\in R^k:~~ u_1+ \ldots+u_k=0.\} \]
We will use the following assumptions on the panel model.
The Assumption (B1) is a standard moment condition on the errors $\{\nu_{it} \}$ which are serially uncorrelated over time with a constant cross-section covariance matrix $\Sigma=\mathop{\text{cov}}(\nu_t)$. The Assumption (B2) ensures the regularity of the design matrices with the centralised regression variables $\{\tilde x_{it}\}$. The independence between regressors and errors required in Assumption (B2) is slightly stronger than the strict exogeneousity in Assumption (A4).
\begin{theorem2} Suppose Assumptions (B1) and (B2) hold for the panel data model ((ref)). Then under the SIM-L scheme and the null hypothesis,
with
\end{theorem2}
This result is established in two steps. Using recent results from random matrix theory, we first find that
Next, we show that
The first step result (ref) is justified in Appendix (ref). The justification of the second step result (ref) requires more calculations; they are detailed in Appendix (ref), see Proposition (ref).
We now compare the new test $LM_e$ to two existing tests also developed in the SIM-L scheme. With respect to the bias-corrected test $LM_{bc}$, note that the $LM_p$ statistic (Section (ref)) can be rewritten as \[ LM_p = \frac{ tr(\hat{R}_T^2)-\mu_{LM_p} }{ \sigma_{LM_p}} \] with \[ \mu_{LM_p}= n(1+c_T)-c_T+o(1), \quad \text{and}\quad \sigma_{LM_p}^2=4c_T^2=\sigma_{LM_e}^2. \] We have \[ \frac{tr(\hat{R}_T^2)-\mu_{LM_e}}{2c_T}=\frac{tr(\hat{R}_T^2)-\mu_{LM_p}-c_T^2}{2c_T} = LM_P -\frac{c_T}{2}. \] Thus by Theorem (ref), \[LM_P -\frac{c_T}{2} \stackrel{D}{\longrightarrow} N(0,1), \] under the SIM-L scheme. This coincide with the asymptotic normality of $LM_{bc}$ given in ((ref)). Therefore, our $LM_e$ test can be seen as an extension of the $LM_{bc}$ test to panels with non normal-distributed errors.
Next we compare the $LM_{e}$ test to the result (ref) for the Gaussian $LM_{RMT}$ test. We find that $\mu_{rmt}=\mu_{LM_e}$, and $\sigma_{rmt}^2\sim \sigma_{LM_e}^2\sim 4c^2$ for large $T$ (and $n$) since as $\kappa\to 3$ when $k$ is fixed and $T\to \infty$. This means that under the SIM-L scheme, the asymptotic distribution of $tr(\hat{R}_{T}^2)$ derived in Theorem (ref) for the extended LM test $LM_e$ is also valid for the heterogeneous model ((ref)) assuming normality of the errors (and the regression variables).
In conclusion of these comparisons, the asymptotic distribution of $tr(\hat{R}_{T}^2)$ derived in Theorem (ref) has a welcomed universality, being valid for the three tests $LM_{e}$, $LM_{adj}$ and $LM_{RMT}$ which cover quite different large panel models. Such distributional robustness enlarges the application scope of the LM statistic to various large panel models .
Anticipating the simulation results shown in Section (ref), the various large-panel versions of the LM test, namely $LM_{adj}$, $LM_{bc}$, $LM_{RMT}$ as well as the new test $LM_e$ may suffer from the problem of low power against large panels where the units are weakly dependent. Such weak dependence arises for example when the cross-sectional correlation matrix $R$ is sparse. Recent literature in high-dimensional statistics indicates that an efficient way for detection of sparse correlations is to weight those relatively significant sample correlations $\hat\rho_{rs}$ more heavily than those small sample correlations. An extreme method in this regard is for example to take the overall maximum $\max\limits_{r\ne s}|\hat\rho_{rs}|$ as a test statistic. It is however unclear in the current SIM-L large panel setting how to derive a limiting distribution for such maximum type statistic. Here we propose a manageable compromise by considering the sum of fourth powers of the sample residuals correlations, that is, to consider \[ tr(\hat{R}_T^4)=\sum\limits_{r\ne s} {\hat\rho}^4_{rs}, \] as the new test statistic. The rational here is that compared to the sum of the squares $\{ {\hat\rho}^4_{rs}\}$ used in the LM statistic, $tr(\hat{R}^4)$ is weighting more heavily larger sample correlations than smaller ones. This will enhance the power of the test when either very few sample correlations are significantly non zero, or they are many but with relatively small amplitudes. Such situations arise under an alternative with sparse cross-section dependence, or with globally weak cross-dependence.
We derive the asymptotic normality of $tr(\hat{R}_T^4)$ in the following theorem.
\begin{theorem2} Suppose Assumptions (B1) and (B2) hold for the panel data model ((ref)). Then under the SIM-L scheme and under the null $H_0$ in ((ref)),
with
\end{theorem2}
The limiting normality of $tr(\hat{R}^4)$ in ((ref)) under the null allows us to perform a level-$\alpha$ test for the null hypothesis $H_0$. This test is hereafter referred as the {\em power enhanced test (PET)} for cross-section independence in large panels.
The proof of Theorem (ref) follows the strategy for that of Theorem (ref) and also proceeds in two steps. In the first step, using recent results from random matrix theory, we find that
Next, we show that
The first step result (ref) is justified in Appendix (ref). The justification of the second step result (ref) also requires more calculations; they are detailed in Appendix (ref), see Proposition (ref).
We conduct a simulation study to investigate the finite sample performance of the proposed tests $LM_e$ and $PET$. Comparisons are made with the bias-adjusted LM test ($LM_{adj}$) in ((ref)) and the cross-section dependence test (CD) in ((ref)). The original Lagrange Multiplier test (LM) in ((ref)) is excluded due to its well-known non applicability to large panels. The Gaussian high-dimensional Lagrange Multiplier test $LM_{RMT}$ is also excluded in comparison as it is equivalent to our proposed general Lagrange multiplier test ($LM_e$), see comments after Theorem (ref).
We consider the following data generating process proposed in pesaran2008bias:
where $a$ and $\beta_l$ are set arbitrarily to 1 and $l$, respectively, $\mu_i\stackrel{i.i.d.}{\sim}N(1,1)$ and here $k$ is the number of regressors including the intercept $a$. The regressors are generated as
with $x_{li,-50}=0$, $\sigma_{li}^2=\tau_{li}^2/(1-0.6^2)$, $\tau_{li}^2 \stackrel{i.i.d.}{\sim} \chi^2(6)/6$ and $u_{lit}\stackrel{i.i.d.}{\sim} N(0,1)$. The first 50 observations in ((ref)) are disregarded. Under the null, the errors $\nu_{it}$ are assumed to be i.i.d. across individuals and over time, that take the form \[\nu_{it}=\sigma_i \varepsilon_{it}, \quad 1\leq i\leq n; ~1\leq t\leq T, \] where $\sigma_i^2\stackrel{i.i.d.}{\sim}\chi^2(2)/2$ and the $\varepsilon_{it}$'s are {also i.i.d.} and generated from different populations: (i) normal, $N(0,1)$, (ii) Student-t, $t_7/\sqrt{7/5}$, and (iii) chi-square, $(\chi^2(5)-5)/\sqrt{10}$. The normalisation's in (ii) and (iii) are such that these variable have mean zero and unit variance.
We explore the performance of different tests using various combination of $(n,T)$ with $n\in\{50, 100, 200\}$ and $T\in\{50, 100\}$. Empirical sizes and powers of the tests are evaluated from 2,000 independent replications. The nominal test level is 5%.
Table (ref) presents the empirical sizes of $LM_e$, PET, $LM_{adj}$ and CD tests (values close to 5% are better) under three different error distributions. The proposed PET test and the CD test have a similar performance a little better than $LM_e$, which is in trun slightly better than $LM_{adj}$. However, the $LM_{adj}$ test is noticeably computationally more demanding than the other tests. It also has a large downside size distortion under the cases with $T=50$, $n=200$ and $k=4$.
To evaluate the power of the tests considered, the disturbances are generated according to the following single-factor model, \[\nu_{it} = \lambda_{i}f_{t} +\epsilon_{it},\quad 1\leq i\leq n; ~1\leq t\leq T, \] where $\lambda_{i}$ is the factor loading of individual $i$ for the common factor $f_{t}$ in period $t$, with $f_t\stackrel{i.i.d.}{\sim}N(0,1)$. The factor loadings are constructed under three different scenarios:
In the dense case, all cross-sectional units are correlated. The correlation between units $i$ and $i'$ is $\lambda_i\lambda_{i'}$ and the overall strength of correlation is controlled by $h$. We study the empirical powers of the tests when $h$ varies while remaining bounded, that is, $h/n\to 0$. This thus corresponds to the setting of weak factor alternative used in baltagi2017asymptotic. In the sparse case, only a few, about $n^{0.3}$, cross-sectional units are correlated while other units are uncorrelated. When this number of correlated cross-sectional units increases to $n^{0.5}$, we call it a less-sparse case.
Table (ref) shows the empirical powers of the tests in the dense case. The dimensions $n$, $T$ and $k$ are set to be (50, 100), 100 and 2, respectively. The cross-sectional correlation strength $h$ varies from 1 to 7. As expected, the empirical powers of all tests increase with the strength $h$. The PET test largely outperforms the others. Compared to $LM_e$ and $LM_{adj}$, PET indeed boosts the power by up to 36% for cases with small value of $h$. $LM_{adj}$ performs a little better than $LM_e$. The CD test has very low powers, confirming the fact that its implicit null is rather weak cross-sectional dependence pesaran2004general. Plots at the bottom of Table (ref) illustrate the evolution of these powers in function of the varying strength $h$ and for the cases with chi-square distributed errors.
Tables (ref) and (ref) show the empirical powers for the sparse and the less-sparse cases, respectively. As expected, all the tests have higher power in the less-sparse case than in the sparse cases. The proposed PET test again performs best. It boosts the power by up to 16% under these sparse cases as compared to $LM_e$ and $LM_{adj}$. The proposed $LM_e$ test performs better than $LM_{adj}$. The latter has low powers in the cases with $T=50$ and $k=4$, which is consistent with the conservative sizes observed in Table (ref). The CD test has no powers for the sparse case, but has some powers for the less-sparse case; it has however an overall poor performance compared to the other tests.
The two proposed tests $LM_e$ and $PET$ are established for the fixed effects panel data model (ref). Additional simulations are conducted to show the finite sample performance of the proposed tests for the heterogeneous panel data model (ref) although this model is not covered by the developed theory. We consider the following data generating process of pesaran2008bias: \[y_{it}=a +\sum_{l=2}^k x_{lit}\beta_{li}+\mu_i+\nu_{it}, \quad 1\leq i\leq n; ~1\leq t\leq T, \] with $\beta_{li}\stackrel{i.i.d.}{\sim} N(1,0.04)$. All other settings are kept the same as for the fixed effect model. It is striking and satisfactory to observe that for the heterogeneous panel model, conclusions from the simulation experiments are in general very similar to those reported in the previous section for the fixed-effect panel model. The following tables report some empirical results for the case of $k=2$ (the other results with $k=4$ are very similar and thus omitted). Table (ref) presents the empirical sizes of all tests under three different error distributions. The empirical sizes of the considered tests are all close to the nominal level. The proposed tests $LM_e$ and PET perform slightly better than $LM_{adj}$. Tables (ref), (ref) and (ref) show the empirical powers of the tests for the dense case, the sparse case and the less-sparse case, respectively. As expected, $LM_e$, PET and $LM_{adj}$ all show higher powers when the correlation matrix of errors becomes denser. Again the proposed PET has consistently the highest power. In conclusion, the proposed two tests seem also valid for the heterogeneous panel data model ((ref)) according to these experiments.
We apply our two new tests $LM_e$ and PET to the public health data sets in liworking2020 for the investigation of association between air pollution, hypertension and blood pressure of people. They used a unique sample of elderly people in Nanjing (China) containing 21 individuals with 441 medical records in total with precise examination dates, that is, $n=21, T=21$. Fixed effects panel data models are constructed to evaluate the effect of the special particulate $PM_{2.5}$ (diameter $<2.5~ \mu$m) on the hypertension (HY), systolic blood pressure (SBP) and diastolic blood pressure (DBP), respectively, by controlling the individual fixed effects $\delta_i, i=1,\ldots, n$. Temperature (temp) is also considered as another control variable due to its positive effect on blood pressure. Therefore, three panel models are considered
To investigate whether the cross-sectional uncorrelation assumption in three models is justified, we applied the $LM_{adj}$ test, the $LM_e$ test, the PET test and the CD test to each regression model. The values of the corresponding test statistics are reported in Table (ref).
Among the other four tests and considering a 5% nominal level, none of them can reject the null hypothesis of cross-sectional uncorrelation under Model 1. But they all reject the null under both Model 2 and Model 3. That means, individuals are uncorrelated in terms of hypertension, but correlated in terms of SBP and DBP. Meanwhile, the values of PET are much larger than others in Models 2 and 3 which confirm its power-enhancement property. Overall, the statistical results drawn from Model 1 seem reliable and consistent with each other, while more control variables need to be investigated to study the association of air pollution with SBP or DBP.
As illustrated by this example, the results of this paper seem to have potential application in the important area of panel data modelling.
For large fixed effects panel data model, we propose two new and efficient tests to detect the existence of cross-sectional correlation (dependence). The asymptotic normalities of test statistics are constructed under a simultaneous limit scheme (SIM-L) where the cross-sectional units dimension $n$ and the time series dimension $T$ are both large with comparable magnitude. Meanwhile, these results do not need the normality assumption on the errors and/or on the random design, while such normality assumptions are essential for the theoretical justification of most of the existing tests for large panels. Extensive Monte-Carlo experiments demonstrates the superiority of our proposed tests over some popular existing methods in terms of size and power. Especially, the power enhanced high-dimensional test PET consistently outperforms all the other methods considered.
There are still several avenues for future research. The tests proposed here are based on the fixed effects panel data model. It is highly valuable to investigate their validity in other large panel models. For example, our simulation experiments have shown the applicability of these proposed tests for the heterogeneous panel data model. A thorough theoretical investigation of such observation is missing though.