Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
38,290 characters · 8 sections · 34 citation commands
Detection Boundaries for Panel Slope Homogeneity Tests Under Small-Group Heterogeneity
\pagenumbering{arabic} \thispagestyle{empty}
JEL classification: C13, C23, C33
Keywords: Panel Data; Detection Boundary; Slope Heterogeneity; Local Power; Group Structures.
Slope homogeneity is a convenient and widely used restriction in panel data analysis. If slopes are common across units, pooling information over the cross section improves precision and yields estimators with a simple interpretation. If slopes are heterogeneous, common-slope estimands may be difficult to interpret or even misleading (see, e.g., BreitungSalish2021). Empirical researchers therefore use slope-homogeneity tests to assess whether slopes are common and to choose between pooled and heterogeneous specifications. A key difficulty is that heterogeneity may be concentrated in a small number of units, so that non-rejection may reflect limited power against such departures rather than genuine homogeneity. This paper quantifies the local power of standard slope-homogeneity tests against departures confined to small groups. Our results characterize, in terms of panel dimensions and group size, the conditions under which standard tests have power against such localized departures.
The literature on testing slope homogeneity is long-standing. A seminal starting point is swamy1970efficient, based on the dispersion of individual-slope estimates around a pooled estimator; refinements such as pesaran2008testing remain among the most widely used tools for linear panel models. In parallel, several LM-type procedures have been developed in random-coefficient models, where the null of slope homogeneity is equivalent to a zero-variance restriction on the random coefficients. Building on breusch1979simple (see also hsiao2022analysis), juhl2014test propose LM statistics for zero variance, and breitung2016lagrange further extend the methodology to empirically relevant settings. Other LM tests assess whether the regressors have predictive power for the restricted residuals under the null of slope homogeneity: su2013testing in models with interactive fixed effects, and lu2017determining, whose test for the number of latent groups nests slope homogeneity as a special case.
We analyze the power of established test statistics—pesaran2008testing, su2013testing\footnote{In the setup considered here, the test of su2013testing coincides with that of lu2017determining when specialized to the single-group null; our results therefore cover both simultaneously.}, and the LM-type test of breitung2016lagrange—under a sequence of doubly local alternatives designed to capture small-group heterogeneity. These alternatives are doubly local in the sense that $(i)$ only the slopes of small groups of individuals deviate from the dominant common slope, and $(ii)$ the slope differences across groups shrink at a controlled (Pitman-type) rate. This design characterizes how power depends jointly on the group-specific slope differences and the size of the groups. We develop the analysis for a benchmark setting with two groups (one dominant majority, one heterogeneous remainder), and for extensions with multiple groups, with either a fixed or growing number of groups. The analysis covers both large-$N$, large-$T$ panels and large-$N$, fixed-$T$ panels, where $T$ denotes the time dimension.
Our main results deliver a detection boundary by linking the rate at which slope differences vanish, $\gamma$, to the size of the heterogeneous group(s), $M$. In large-$N$, large-$T$ panels, this boundary is given by \[ \gamma \asymp \frac{M^{1/2}T^{1/2}}{N^{1/4}}, \] which nests the classical $N^{1/4}\sqrt{T}$ rate—obtained when $M/N \to c>0$—reported in pesaran2008testing, su2013testing, and breitung2016lagrange under analogous Pitman-drift sequences. Importantly, this boundary shows that the tests remain informative even if $M/N \to 0$, provided $MT \to \infty$ and the boundary condition above holds, thereby enabling detection of small-groups alternatives. When $T$ is fixed, detectability requires $M$ to be of order $\sqrt{N}$ or larger; below this threshold, heterogeneity is undetectable. Finally, when the number of heterogeneous groups increases with $N$, the tests retain nontrivial power and detectability depends on the total number of units outside the dominant majority rather than on the number of groups per se.
The results also speak to the literature on latent group structure in panel models, where such departures arise naturally—see, e.g., bonhomme2015grouped, bester2016grouped, su2016identifying, lu2017determining, and dzemski2024confidence for identification, estimation, and inference for group membership and group-specific parameters. Our contribution is complementary, as we characterize when standard slope-homogeneity tests can detect departures consistent with latent group structure. This perspective also connects to recent work on inference when latent groups are not well separated; see AkgunOkui2025.
The remainder of the paper is organized as follows. Section 2 introduces the model and notation. Section 3 reviews the test statistics. Section 4 derives the local-power results under doubly local alternatives, and Section 5 reports a Monte Carlo study. Proofs are relegated to the appendix.
Consider the linear panel data model
where $y_i=(y_{i1},\ldots,y_{iT})'$ is the $T$-dimensional vector of dependent variables for unit $i$; $x_i=(x_{i1},\ldots,x_{iT})'$ is the $T\times K$ design matrix of regressors for unit $i$; and $\epsilon_i=(\epsilon_{i1},\ldots,\epsilon_{iT})'$ is the corresponding vector of idiosyncratic errors. The slope parameter, $\beta_{g(i)}$, is group-specific, with unknown group membership determined by the mapping $g:\mathcal{N} \to \{G_1, \ldots, G_P\}$, where $\mathcal{N} = \{1, \ldots, N\}$. The function $g$ assigns unit $i$ to one of the $P$ groups partitioning the sample; that is, $G_p \cap G_q = \emptyset$ for all $p \neq q$, and $\bigcup_{p=1}^P G_p = \mathcal{N}$.
We are interested in testing the presence of group heterogeneity in the slopes by means of existing slope homogeneity tests. The null hypothesis of identical slopes is
versus the alternative that there exists a group of individuals with different slopes,
As discussed in the introduction, a variety of well-established tests for slope homogeneity are available in the literature. To analyze their asymptotic behavior within a unified framework, we adopt a common set of assumptions sufficiently broad to capture their essential features. Throughout, $C$ denotes a generic constant that may vary across expressions, and $\#G$ denotes the cardinality of a generic set $G\subseteq \mathcal{N}$.
In particular, taking $G=\mathcal{N}$, $S=S^{(\mathcal{N})}$ is a positive definite non-stochastic matrix.
Assumption (ref)(a), corresponding to Assumption 1(i)--(ii) in pesaran2008testing, rules out serial correlation, and cross-sectional dependence. Together with model (ref), which excludes endogenous components in the error term, they provide a simplified framework for presenting our main results in a unified manner. We emphasize, however, that the tests considered here have been developed or extended to accommodate more general error structures (see original papers as well as blomquist2013testing; ando2015simple). In all such cases, the main conclusions of our analysis remain valid.
We focus here on three representative and widely used slope homogeneity tests: (i) the dispersion statistic $\Delta$ from pesaran2008testing (hereafter, PY); (ii) the residual-based LM test $\mathcal{J}$ from su2013testing (SC); and (iii) the LM-type variance test $LM$ from breitung2016lagrange (BRS). The next section provides the necessary details for each of the test statistics.
The $\Delta$ test of PY refines the swamy1970efficient test, originally designed for testing slope homogeneity in large $T$, fixed $N$, scenarios. Both tests compare individual slope estimates, $\hat{\beta}_i = (x_i'x_i)^{-1}x_i'y_i$, with the weighted least squares (WLS) estimator $\hat{\beta}_{WLS}(\bar{\sigma}_i^2) = \bigl(\sum_{i=1}^N x_i'x_i/\bar{\sigma}_i^2\bigr)^{-1}\bigl(\sum_{i=1}^N x_i'y_i/\bar{\sigma}_i^2\bigr)$, using some estimator $\bar{\sigma}^2_i$ of $\sigma^2_i$. The original Swamy statistic is $\mathcal{S}_{SW} = \mathcal{S}(\hat{\sigma}^2_i)$, where \[ \mathcal{S}(\bar{\sigma}^2_i) = \sum_{i=1}^N \left(\hat{\beta}_i-\hat{\beta}_{WLS}(\hat{\sigma}^2_i)\right)' \left(\frac{x_i'x_i}{\bar{\sigma}^2_i}\right) \left(\hat{\beta}_i-\hat{\beta}_{WLS}(\hat{\sigma}^2_i)\right), \] and $\hat{\sigma}^2_i = (T-K-1)^{-1}(y_{i} - x_{i}\hat{\beta}_i)'(y_{i} - x_{i}\hat{\beta}_i)$. This statistic has a $\chi^2$ null distribution with $K(N-1)$ degrees of freedom when $T\to\infty$ and $N$ is fixed. For the large-$N$, large-$T$ framework, PY propose to replace $\hat{\sigma}^2_i$ with
where $\hat{\epsilon}_i = (\hat{\epsilon}_{i1},...,\hat{\epsilon}_{iT})'$, $\hat{\epsilon}_{it} = y_{it} - x_{it}'\hat{\beta}_{LS}$, and $\hat{\beta}_{LS}=(\sum_{i=1}^N x_i'x_i)^{-1}\sum_{i=1}^N x_i'y_i$ - the pooled least square estimator. The resulting dispersion statistic, $\mathcal{S}_{PY}=\mathcal{S}(\tilde{\sigma}^2_i)$, after centering and standardization, converges to the standard normal distribution under $H_0$,
\noindentThe residual-based $\mathcal{J}$ test. An alternative approach is the LM test of SC,\footnote{With additive fixed effects (or none, as in the present case), $\mathcal{J}$ is numerically equivalent to the test of lu2017determining when applied to a single-group case under the null.} based on the idea that, under $H_0$ and with predetermined regressors, $x_{it}$ should not have explanatory power for the residuals in (ref). This motivates a test of $H_0^*: \phi_i=0\;\text{for all } i=1,\ldots,N$ in the auxiliary regression \[ \hat{\epsilon}_{it} = \phi_i'x_{it} + \eta_{it}, \qquad \text{with} \quad i=1,\ldots \,N, t=1,\ldots,T, \] where $\eta_{it}$ is a mean-zero error term.
Let $h_i=x_i(x_i'x_i)^{-1}x_i'$ be the projection matrix with diagonal elements $h_{i,tt}$, and define $\hat{b}_{it}=\hat{S}_i^{-1/2}x_{it}$, where $\hat{S}_i = T^{-1}x_i'x_i$. The SC test statistic is given by the following centered and standardized $LM$ test statistic, which has a limiting normal distribution under $H_0^*$:
where
and $ \hat{B}_{NT} = N^{-1/2}\sum_{i=1}^N\sum_{t=1}^T\hat{\epsilon}_{it}^2h_{i,tt}$ and $\hat{V}_{NT} = 4T^{-2}N^{-1}\sum_{i=1}^N\sum_{t=2}^T\left[\hat{\epsilon}_{it}\hat{b}_{it}'\sum_{s=1}^{t-1}\hat{b}_{is}\hat{\epsilon}_{is}\right]^2$ estimate the mean and variance of $LM_{SC}$ under the null, respectively.
\noindentThe LM variance test. A third approach reformulates the null hypothesis within the random coefficients model. Specifically, assuming $\beta_i=\beta+v_i$ with $\mathbb{E}[v_iv_i'|x_i]=\text{diag}(\sigma^2_{v,1},\dots,\sigma^2_{v,K})$, slope homogeneity is equivalent to $H_0^{**}: \sigma^2_{v,k}=0$, for all $k=1,\dots,K$, which reduces the $N(K-1)$ linear restrictions tested by $\Delta$ and $\mathcal{J}$ to $K$.
Building on this idea, BRS develop an LM test valid for panels with large $N$ and either fixed $T$ or large $T$. The statistic is based on the score vector $\hat{s}=(\hat{s}_1,\ldots,\hat{s}_K)'$, where
The test statistic is then given by
where $\hat{V}$ has $(k,l)$-th element
Under $H_0^{**}$ and $N\to\infty$, with $T$ either fixed or $T\to\infty$, the test has a $\chi^2_K$ limiting distribution.
The three tests, $\Delta$, $\mathcal{J}$, and $LM$, are consistent for $H_0$ and have well-established local power properties. However, none has been explicitly studied for local power against small-group heterogeneity, which is the focus of the next section.
We study the local power of $\Delta$, $\mathcal{J}$, and $LM$ tests under grouped alternatives, focusing on (i) how large slope differences must be to be detected and (ii) how large the groups must be for detection. To this end, we introduce a sequence of doubly local alternatives that extends the classical Pitman framework and integrates these two aspects. We first present the results for the case where the sample is partitioned into two groups, $\{G_1,G_2\}$, with $N,T\to\infty$. Group $G_1$ is treated as the “dominant” group and $G_2$ as the “alternative” one, containing the remaining individuals. We then extend these results to panels with $N \to \infty$ and fixed $T$, as well as to settings with more than two groups.
Formally, define
where $\lambda_2\neq0$ is a $K\times 1$ vector, and $\{G_1,G_2\}$ form a partition of $\mathcal{N}$. The number of individuals in each group is
where $M\to\infty$ as $N\to\infty$ and $\gamma=\gamma(N,T,M)$ diverges with $N$, $T$, and $M$.
Allowing the scaling factor $\gamma$ to depend on $M$ as well as on $N$ and $T$ reflects that our proposed local alternatives control for both the slope difference and the size of the heterogeneous group. This extension is essential: the relative magnitude of $M$ with respect to $N$ and $T$ determines when slope homogeneity tests can detect group-based heterogeneity. Theorem (ref) formalizes this statement.
\setcounter{theorem}{0}
Theorem (ref) highlights a trade-off between the rate at which slope differences vanish and the size of the alternative group. The critical rate,
marks the boundary between trivial and nontrivial power. If $\gamma$ grows faster than this rate, all three tests have trivial power. If it grows at or below this rate, they have power to detect the grouped alternative.
Two cases deserve attention. If the fraction of alternative units remains bounded away from zero ($m_0 > 0$, so $M \asymp N$), the rate (ref) reduces to the classical order $O(\sqrt{T}N^{1/4})$, consistent with the local power results of PY, SC, and BRS. When the alternative group is small ($m_0=0$), the rate is slower than $O(\sqrt{T}N^{1/4})$, yet the tests retain nontrivial power as long as $M$ diverges with $N$ — even as the fraction of units with distinct slopes vanishes. This contrasts with the prevailing assumption in the grouped-patterns literature, which requires heterogeneous slopes to affect a non-negligible share of the sample bonhomme2015grouped,su2016identifying,lu2017determining.
\paragraph{On the noncentrality parameters.} Theorem (ref) shows that all three tests share the same asymptotic detection boundary (ref). Their local power need not coincide, however, since the noncentrality parameters differ. A general comparison is intractable, as the parameters depend on both model and design in complex ways. We therefore consider the benchmark case of a single regressor, $K=1$, to build intuition.
Let $x_{it}$ be scalar and i.i.d.\ across $i$ and $t$, with $\mathbb{E}\left[x_{it}\right]=0$, $\mathbb{E}\left[x_{it}^2\right]=\sigma_{i,x}^2$, and uniformly bounded fourth moments. The regressor may thus exhibit unit-specific variance, constant over time. For any subset $G\subseteq \mathcal N$, define \[ \mu_{2,x}^{(G)}=\lim_{\#G\to\infty}\frac{1}{\#G}\sum_{i\in G}\sigma_{i,x}^2,\quad \mu_{4,x}^{(G)}=\lim_{\#G\to\infty}\frac{1}{\#G}\sum_{i\in G}\bigl(\sigma_{i,x}^2\bigr)^2, \quad q_x^{(G)}=\lim_{\#G\to\infty}\frac{1}{\#G}\sum_{i\in G}\frac{\sigma_{i,x}^2}{\sigma_i^2}, \] with $\mu_{2,x}=\mu_{2,x}^{(\mathcal{N})}$, $\mu_{4,x}=\mu_{4,x}^{(\mathcal{N})}$, and $q_x=q_x^{(\mathcal{N})}$. Now suppose $m_0=0$, so $M/N\to 0$ and the heterogeneous subset is asymptotically negligible. The second term in each noncentrality parameter vanishes, and Theorem (ref) reduces to \[ \delta_{\Delta}=\frac{c^2\lambda_2^2}{\sqrt{2}}\,q_x^{(G_2)}, \qquad \delta_{\mathcal J}=\frac{c^2\lambda_2^2}{\sqrt{V_0}}\,\mu_{2,x}^{(G_2)}, \] and \[ \xi_{LM}=\delta_{LM}'\Psi^{-1}\delta_{LM} =\frac{c^4\lambda_2^4}{4\Psi}\,\left(\mu_{4,x}^{(G_2)}\right)^2, \] where $ V_0 = 2\lim_{N\to\infty}N^{-1}\sum_{i=1}^N \sigma_i^4$ and $\Psi = \lim_{N\to\infty}(2N)^{-1}\sum_{i=1}^N \sigma_i^4(\sigma_{i,x}^2)^2$. Even in the scalar case, PY and SC depend on different weighted second moments, while BRS depends on fourth-order cross-sectional moments of the regressor.
To compare PY and SC directly, note that \[ \frac{\delta_{\Delta}}{\delta_{\mathcal{J}}} = \lim_{N\to\infty} \mathrm{RMS}(\sigma^2_i) \sum_{i\in G_2}\frac{\omega_{x,i}}{\sigma^2_i}\; \] where $\omega_{x,i}=\sigma^2_{x,i}/\sum_{i\in G_2}\sigma^2_{x,i}$, and $\mathrm{RMS}(\sigma^2_i)=\sqrt{N^{-1}\sum_{i=1}^{n}\sigma^4_i}$. It follows that, \[ \lim_{N\to\infty}\frac{\mathrm{RMS}(\sigma^2_i)}{\max_{i\in G_2}\sigma^2_i} \leq \frac{\delta_{\Delta}}{\delta_{\mathcal{J}}} \leq \lim_{N\to\infty} \frac{\mathrm{RMS}(\sigma^2_i)}{\min_{i\in G_2}\sigma^2_i}. \] When $G_2$ concentrates on units with small $\sigma^2_i$, the $\Delta$ test dominates; otherwise the $\mathcal{J}$ test has higher local power. Under homoskedasticity ($\sigma_i^2=\sigma^2$ for all $i$), $\delta_{\Delta} = \delta_{\mathcal J}$ and the two tests are locally equivalent.
Comparing PY and SC with BRS is more complex. Under homoskedasticity and constant regressor variance ($\sigma_{i,x}^2=\sigma_x^2$ for all $i$), all three procedures are locally equivalent: $\xi_{LM}/\delta_{\Delta}^2 = \xi_{LM}/\delta_{\mathcal J}^2 =1$. If errors are homoskedastic and $\{\sigma_{x,i}^2\}$ unrestricted with $\mu_{2,x}^{(G_2)}=\mu_{2,x}$, and $\mu_{4,x}^{(G_2)}=\mu_{4,x}$, then \[ \frac{\xi_{LM}}{\delta_{\Delta}^2} = \frac{\mu_{4,x}}{\mu_{2,x}^2} \geq 1. \] Hence, BRS has a larger local noncentrality parameter than PY and SC, with the ratio $\xi_{LM}/\delta_\Delta^2$ governed by the cross-sectional dispersion in regressor variance.
A complementary comparison is obtained when regressor variance is constant across units but errors are heteroskedastic. Then, $\delta_{\mathcal J}^2 = \xi_{LM}$, so SC and BRS tests are locally equivalent. If $G_2$ is also asymptotically representative with respect to error variances — that is $\lim_{M\to\infty}M^{-1}\sum_{i\in G_2}\sigma_i^{-2} = \lim_{N\to\infty}N^{-1}\sum_{i=1}^n\sigma_i^{-2}$ — then \[ \frac{\xi_{LM}}{\delta_{\Delta}^2} = \left(\frac{\delta_{\mathcal J}}{\delta_{\Delta}}\right)^2 = \lim_{N\to\infty} \frac{1} {\left(N^{-1}\sum_{i=1}^N \sigma_i^4\right) \left(N^{-1}\sum_{i=1}^N \sigma_i^{-2}\right)^2} \leq 1, \] with equality under homoskedasticity. When regressor variance is constant, error heteroskedasticity thus favors PY over SC and BRS
When $m_0>0$ (i.e., $M \asymp N$), the comparison is more complex as the second term in each expression of noncentrality parameters is no longer asymptotically negligible. However, the analysis of the same scalar benchmark points in the same direction. In particular, under homoskedasticity, $\delta_{\Delta} = \delta_{\mathcal J}$; if, in addition, the regressor variance is constant, all three tests are locally equivalent; if, instead, regressor variance is unrestricted and $\mu_{2,x}^{(G_2)}=\mu_{2,x}$, and $\mu_{4,x}^{(G_2)}=\mu_{4,x}$, \[ \frac{\xi_{LM}}{\delta_{\Delta}^2} = \frac{\mu_{4,x}}{\mu_{2,x}^2} \geq 1. \] Outside these benchmarks, the comparison depends on the joint distribution of $\{\sigma^2_i\}$ and $\{\sigma^2_{x,i}\}$ across the full sample and the remainder group $G_2$.
When $T$ is fixed and $N$ grows, the LM-type test of BRS is the only one available. The restrictions on the regressors can be relaxed, as in Assumption (ref) in the Appendix, which also contains the corresponding version of Theorem (ref). In this scenario, the boundary between trivial and nontrivial power is governed by the rate
the analogue of (ref) in this setting.
When the alternative group is proportional to the total cross-section ($m_0 > 0$, or $M \!\asymp\! N$), the rate in (ref) reduces to $\gamma \!\asymp\! N^{1/4}$, matching the local-power behavior established in BRS. More interestingly, this rate marks the boundary at which the test begins to exhibit power. If $M$ diverges at least as fast as $N^{1/2}$, the $LM$ test retains nontrivial local power even when $M/N \!\to\! 0$. Essentially, the minimal detectable group size is of order $N^{1/2}$: below this threshold, grouped heterogeneity is undetectable.
The main message of our analysis remain valid even if the number of groups is allowed to grow with $N$. Consider
where $\{G_1,...,G_P\}$ forms a partition of $\mathcal{N}$, with $\lambda_p\neq\lambda_q$ for any $p,q\in\{1,...,P\}$ such that $p\neq q$. We let $M_{p} = \#G_p$ and $\mathbb{M} = \sum_{p=2}^P M_{p}$.
Analyzing the local power when $P \to \infty$ as $N \to \infty$ is complicated because the size of each group $\{G_2, \dots, G_P\}$ can also depend on $N$. Both the number of groups and the number of individuals per group determine the expansion rates under which the test statistics exhibit nontrivial power.
\setcounter{theorem}{0}
Corollary (ref) generalizes Theorem (ref). If $P$ is fixed and $M^\ast = \max_{2 \le p \le P} M_p \to \infty$ as $N \to \infty$, replacing $M$ with $M^\ast$ yields Theorem (ref). If $P \to \infty$ while $\sup_{2 \le p \le P} M_p = \kappa < \infty$, the expansion rate satisfies $PT / (\sqrt{N} \gamma^2) \to c > 0$ as $N, T \to \infty$. More generally, when both $P \to \infty$ and $M_p \to \infty$ for at least one $p$, the expansion rate is determined by the total “volume” of individuals in $\{G_2, \dots, G_P\}$, denoted as $\mathbb{M}$.
We evaluate the finite-sample performance of the tests by comparing their power and assessing the theoretical results presented in the main text. We consider the three test statistics ($\Delta$, $\mathcal{J}$, and $LM$) under a data-generating process adapted from PY to accommodate grouped slope patterns; i.e., for each $i=1,\ldots,N$ and $t=1,\ldots,T$:
where $x_{it}=(x_{it,1},\ldots,x_{it,K})'$, $\epsilon_{it}\overset{iid}{\sim}\mathcal{N}(0,1)$, $v_{it}\overset{iid}{\sim}\mathcal{N}(0,\sigma_{ix}^2)$, $\alpha_i\overset{iid}{\sim}\mathcal{N}(1,1)$, and $\rho_i\overset{iid}{\sim}\mathcal{U}(0.05,0.95)$, with $\sigma_{ix}^2\sim \chi^2_1$. The first 50 observations of $x_{it}$ ($t=-49,\ldots,0$) are discarded to mitigate the effect of initial values. Group-specific slopes, $\beta_{g(i)}=(\beta_{g(i),1},...,\beta_{g(i),K})'$, are generated as: \[ \beta_{g(i),k} =
\] for $k=1,\ldots,K$, where $p=2,\ldots,P$ and $e_p\overset{iid}{\sim}\mathcal{U}(-h,h)$. The dominant group $G_1$ contains $N-\sum_{p=2}^{P}M_p$ individuals, while the remaining $P-1$ groups $\{G_2,\ldots,G_P\}$ represent alternative groups of size $M_p$. The parameters $\lambda$ and $h$ control, respectively, the degree of heterogeneity relative to the dominant group and the dispersion across alternative groups.
We consider two simulation designs to examine how the size of the alternative group and the magnitude of slope differences affect power:
Each design is run at $(N,T) = (100,10)$ and $(100,100)$, corresponding to short and long panels, and at $K=1$ and $K=4$, corresponding to single and multiple regression.
In both designs, the $\mathcal{J}$ test is severely biased when $T$ is small, especially for $K=4$, and its rejection rate far exceeds that of the other two tests. To compare the three tests on equal footing, we use size-adjusted critical values obtained by simulating the null distribution of each test with $R_2=10000$ replications.
The data-generating process includes individual effects, which are removed through appropriate demeaning procedures for each test statistic. For the $LM$ test, we employ orthogonal deviations as in arellano1995another, following Remark 5.1 in BRS. For the $\Delta$ and $\mathcal{J}$ tests, we apply the standard within transformation, pre-multiplying $y_i$ and $x_i$ by $M_0=I_T-T^{-1}\mathbf{i}_T\mathbf{i}_T'$, where $\mathbf{i}_T$ is a $T\times1$ vector of ones, as suggested in PY and lu2017determining.
All simulations report rejection frequencies at the 5% significance level based on 1{,}000 Monte Carlo replications. Results for the two-group design are shown in Figure (ref), and those for the multiple-group design in Figure (ref).
The two-group results closely mirror our theoretical findings in Theorem (ref). Rejection rates increase monotonically with the alternative group size $M$ and with the slope deviation parameter $\lambda$. All tests tend to have a power close to trivial when $M$ is smaller than $\sqrt{N}$, but the power increases significantly after this threshold and as $T$ grows.
In the multiple-group experiment, the decisive role is played by the number of individuals in the alternative groups, $\mathbb{M}$. Tests exhibit substantial power when $\mathbb{M}$ grows at least proportionally to $\sqrt{N}$, consistent with Corollary (ref). Power declines slightly as $P$ increases, due to greater dispersion in group-specific slopes, but this effect is secondary once $\mathbb{M}$ or $T$ is sufficiently large.
In most cases, all three tests show comparable power. The $J$ and $LM$ tests, however, tend to gain power somewhat more rapidly than the $\Delta$ test, especially in the short-panel designs with $T=10$ and $K=4$. This divergence is much less visible once $T$ increases. When $T=100$, the three procedures behave in a way that is broadly consistent with the asymptotic comparison in Section 4.
We derive the detection boundary of standard slope-homogeneity tests under small-group heterogeneity. The power of the tests is evaluated under a sequence of doubly local alternatives, linking the degree of heterogeneity across groups and each group size to its impact on test performance. Our analysis shows that slope-homogeneity tests are effective in detecting such structures, including sparse groups whose cardinality is negligible relative to the cross-sectional size. Monte Carlo simulations support similar conclusions in finite samples.
\phantomsection \addcontentsline{toc}{section}{References} \makeatletter \makeatother