Panel Data Estimation and Inference: Homogeneity versus Heterogeneity

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Introduction

Panel data analysis has seen its popularity in the past thirty years or so. Comprehensive reviews have been conducted at different stages, while the literature evolves. See, for example, ARELLANO20013229, Petersen2009, CP2015, hsiao2022analysis, etc. Among all challenges raised in different surveys, this article aims to offer a unified framework and a set of toolkit to

In what follows, we review the relevant literature, point out the challenges, and then highlight our contributions.

We start with the hypothesis testing about homogeneity against heterogeneity, which has always been a central topic in empirical studies. Without loss of generality, we consider a simple setup as follows:

where $x_{it}$, $\mu_i$, and $\epsilon_{it}$ are all scalars, $(i,t)\in [N]\times [T]$, $[L]\coloneqq \{1,\ldots,L \}$ for any given positive integer $L$, $N$ stands for the number of individuals, and $T$ stands for the total number of periods. More often than not, the key hypotheses are

Sometimes, the hypotheses are even more straightforward, e.g.,

The model (ref) extends the location model of Lazarus to panel data settings. To settle (ref), a large literature (PY2008,GAO2020329; and references therein) adopts the quadratic test statistics. However, as explained in fan2015power, the tests based on quadratic forms often suffer from low powers, and fail to detect the sparse alternatives. Therefore, fan2015power and YU2024105458 provide power enhanced test statistics to tackle this issue.

To the best of our knowledge, the above literature largely (if not all) ignores the dependence along both dimensions. As time series autocorrelation (TSA) has been well discussed in the literature (see FanYao; Gao2007), we justify the necessity of accounting for the cross-sectional dependence (CD) here. As surveyed by CP2015, CD is likely to be the rule rather than the exception, and it sometimes goes beyond WCD due to omitted variables as documented in GX2021. It is then reasonable to call for a complete toolkit set that is robust to the presence of dependence.

To better present our motivations, we start with four datasets, of which $N$ and $T$ stand for the number of individuals and the number of time periods respectively.

We shall provide numerical evidences demonstrating that the four datasets exemplify both WCD and SCD, highlighting the need for a unified framework to model these varying dependencies.

For each dataset, we observe $\{x_{it}\}$ as defined in (ref), which yields the pairwise correlation $r_{ij}$ for $\forall i,j\in [N]$. For $\forall\tau \in[0,1]$, we can calculate:

where $I(\cdot)$ stands for the indicator function. This allows us to plot $p(\tau)$ against $\tau$, as shown in Figure (ref) below. Here, $\tau$ represents a specific correlation threshold, while $p(\tau)$ measures the percentage of absolute correlation values that exceed the threshold. We also calculate the following measure:

which is adopted from Assumption C of BN2002, and represents the magnitude of cross-sectional dependence of a panel dataset. Having presented these measures, we proceed.

Dataset 1 --- We examine a time period spanning from October 2003 to September 2023, resulting in a total of $T = 240$ observations along the time dimension. After removing variables with missing values, we are left with $N = 127$ macro variables. In the first sub-figure of Figure (ref), we observe that around 20% of the absolute correlations $\{|r_{ij}|\mid i>j\}$ are greater than 0.8, and roughly more than 50% of these correlations have absolute values exceeding 0.5. Additionally, $\overline{\rho}=66$ is approximately $N/2$. Thus, we find a high degree of correlation among these macro variables. An intuitive thought is that many of these macro variables are generated by the same set of unobservable shocks.

Dataset 2 --- We analyze temperature and sunshine data from the U.K. Meteorological office. There are 37 stations in total, widely distributed across the U.K. Each station reports both temperature and sunshine monthly, resulting in $N = 74$ individual time series. After handling missing values, we focus on the period from January 1950 to February 2023, resulting in $T = 878$ observations. In the second sub-figure of Figure (ref), over 80% of the absolute correlations $\{|r_{ij}|\mid i>j\}$ are greater than 0.5, and approximately 30% of these correlations exceed 0.8. We have $\overline{\rho}=52$, which is almost the same as the sample size of the individual dimension. Thus, it is evident that these climate data exhibit a high degree of correlation.

Dataset 3 --- We study a dataset of real bank equity returns for 46 advanced and emerging economies ($N=46$) in the period from 1870 to 2016 ($T=147$). This dataset, constructed by Baron2021, illustrates the influence of banking crises on subsequent output gaps and credit contractions. The estimated values for $p(\tau)$ are presented in the third sub-figure of Figure (ref). Obviously, this dataset has the weakest cross-sectional dependence among the four datasets, and most of $r_{ij}$'s are less than 0.6, which might be a signal of WCD. The value $\overline{\rho}$ of $\eqref{def.rho}$ is 13 which is relatively small.

Dataset 4 --- A final example is a realized volatility dataset, which exhibits strong connections across different equity markets. This dataset consists of realized volatility data for 16 international stock market indices ($N=16$) and is computed using tick-by-tick stock index data from Refinitiv DataScope Select. The dataset covers 3809 common trading days ($T=3809$) over a 16-year period from January 4, 2005, to February 26, 2021. For this dataset, we compute the pairwise correlations between realized volatilities and present the estimated values of $p(\tau)$ in the fourth sub-figure of Figure (ref). More than 50% of the correlations are above 0.5, indicating a strong connectedness in realized volatility across markets. The value $\overline{\rho}$ of $\eqref{def.rho}$ is 8.

Four datasets offer examples of CD with different magnitude. The issue at hand is certainly a cause for concern, as inferring $\mu_i$ of (ref) is a cornerstone of data analysis (e.g., Hamilton1994; Lazarus; CP2024). Without a grasp of the dependence magnitude, the existing literature offers scant guidance on how to infer the homogenous/heterogeneous means, let alone more intricate scenarios including homogenous/heterogeneous trends (ROBINSON20124,WSX2023). See Example (ref) and Example (ref) of Section (ref) for the purpose of demonstration. To our knowledge, although CP2015 formalize the definitions of WCD and SCD, only Assumption 3 of goncalves_2011 addresses this issue using a set of high-level conditions. Yet, the underlying data generating mechanism remains underexplored. In related research, ROBINSON2011, ROBINSON2012 and LEE2016 impose a linear system structure to represent cross-sectional dependence. This approach, while convenient in theory, is infeasible in practice due to the fact that there is no natural ordering in the cross-sectional dimension. When solely WCD is of interest, Assumption C of BN2002, which regulates dependence across cross-sections and time using various moments, is often cited. Nonetheless, the underlying data generating process remains somewhat obscure.

Considering the aforementioned points, our contributions in this paper are as follows:

The remainder of this paper is structured as follows. In Section (ref), we present the underlying data generating process in detail, and show its practical relevance. The corresponding asymptotic properties under homogenous (i.e., $\mathbb{H}_0$ of (ref)) and heterogeneous (i.e., $\mathbb{H}_1$ of (ref)) settings are given in Sections (ref) and (ref) respectively. Building on Sections (ref) and (ref), we provide the test statistic to exam (ref) in Section (ref). In Section (ref), we revisit the CCE estimators of Pesaran2006, and some results presented in PM1999 to showcase the practical relevance of the results of Section (ref). Section (ref) conducts extensive simulation studies to exam our theoretical results. An empirical study is given in Section (ref) to exam whether the rational expectations of financial markets are homogeneous or heterogeneous. Section (ref) concludes with a few remarks. The preliminary lemmas and proofs are regulated to the online appendices.

Notations --- Before proceeding, we introduce some notations and present a few useful facts to facilitate development. Throughout, vectors and matrices are always in bold font. We let $\mathsf{i}$ be the imaginary unit; for a matrix $\mathbf{A}=\{a_{ij}\}_{m\times n}$, let $\mathbf{A}^+$ define Moore-Penrose inverse, and let

define respectively its Spectral norm, column norm, row norm, and entry wise norm; moreover, we always use $^\dag$ and $^\sharp$ to represent the column and row of a matrix, e.g.,

\[ \mathbf{A} =(\mathbf{a}_1^\dag,\ldots,\mathbf{a}_n^\dag)=(\mathbf{a}_1^\sharp,\ldots,\mathbf{a}_m^\sharp)^\top . \] Given two conformable matrices $\mathbf{A}$ and $\mathbf{B}$, we let $\mathbf{A}\circ \mathbf{B}$ denote its Hadamard product. For a vector $\mathbf{v} =(v_1,\ldots, v_p)^\top$, we let $|\mathbf{v}|_\infty\coloneqq \max_{i}v_i$. $\mathbf{I}_p$ stands for a $p\times p$ identify matrix, and when no misunderstanding arises, we write $\mathbf{I}$. $\mathbf{1}_p$ stands for a $p\times 1$ vector of ones. $\mathbf{e}_j$ always stands for a selection column vector with the $j^{th}$ individual being 1 and others being 0. For two positive constants $a$ and $b$, $a\asymp b$ stands for $a=O(b)$ and $b=O(a)$; for $a,b\in \mathbb{R}$, $a\wedge b=\min \{a,b\}$ and $a\vee b=\max \{a,b\}$. We always let $\Phi(x)$ and $\phi(x)$ be the CDF and PDF of the standard normal distribution, and let $\widetilde{\phi}(x)\coloneqq \exp(-x^2/2)$ for notational simplicity. Thus, $\sqrt{2\pi}\phi(x)=\widetilde{\phi}(x)$. The cumulant generating function of a random variable $x$ is defined by $C(u)=\log E[\exp(u x)],$ and we have

where $\psi(u)$ defines the characteristic function of $x$. Finally, $E^*(\cdot)$ and $\text{Pr}^*(\cdot)$ always refer to the operations induced by the sample space.

The Setup and Asymptotic Properties

We firstly present the main results achieved in this paper, and then outline the establishment of some key results before presenting any asymptotic results.

To proceed, we recall the notations defined in the end of Section (ref), and present a result which is independent of the assumptions to be adopted, and facilitates the Edgeworth expansion for both homogeneous and heterogeneous cases.

The result is built on the generating function of the Probabilist's Hermite polynomials, and, together with (ref), naturally connects cumulants of a random variable with the density function of a standard normal distribution.

We are now ready to formulate our ideas. To exam (ref), we need to have a good understanding about both homogenous (i.e., $\mathbb{H}_0$) and heterogeneous (i.e., $\mathbb{H}_1$) cases. The investigation does not only offer post-testing inference under either $\mathbb{H}_0$ or $\mathbb{H}_1$, but also helps to establish the test statistic. Having said that, we respectively investigate (ref) under the null $\mathbb{H}_0$ in Section (ref), and under the alternative $\mathbb{H}_1$ in Section (ref). In Section (ref), we assemble the results of both sections to finalize the test statistic.

Firstly, we explain the necessity of accounting for WCD and SCD in a unified framework. For simplicity, suppose that $\mathbb{H}_0$ holds, and define the following HDMA($\infty$) process for $\{x_{it}\}$ of (ref):

where $\mathbf{x}_t\coloneqq (x_{1t},\ldots, x_{Nt})^\top$, $\mathbf{B}(L)\coloneqq \sum_{\ell=0}^{\infty} \mathbf{B}_\ell L^\ell$ with $L$ being the lag operator, $\{\mathbf{B}_\ell\mid \ell\ge 0\}$ is a set of $N\times N$ matrices, $\pmb{\varepsilon}_t =(\varepsilon_{1t},\ldots, \varepsilon_{Nt})^\top$, and $\{\varepsilon_{it}\}$ are independent and identically distributed (i.i.d.) over both $(i,t)$ with mean 0. Each $\mathbf{B}_\ell$ admits the following representation:

The HDMA($\infty$) of (ref) offers the flexibility to account for different types of dependence. For example, simple algebra (Appendix (ref)) shows three types of dependence as follows:

Loosely speaking, we require the following conditions to hold: (i) CD does not necessarily shrink as $|i-j|$ increases, which is evident in view of Example (ref) below; (ii) TSA shrinks as $t-s$ increases, and may vary with respect to $i$; (iii) $\text{CD}+\text{TSA}$ inherits the properties of (i) and (ii).

To see the difference between WCD and SCD, we provide the following examples.

A few findings emerge in view of both examples. First, (ref) extends ROBINSON2011 and offers detailed data generating process for goncalves_2011. Second, inferring $\mu$ under WCD and SCD respectively requires different normalizers to construct the standard deviations. A challenge arises naturally, as one needs to decide the magnitude of CD prior to analysis. Third, the magnitude of cross-sectional dependence will mathematically influence the Spectral norm of $\mathbf{B}_\ell$'s from the modelling perspective. In what follows, we shall account for these findings in our investigation.

Inference under $\mathbb{H}_0$

Firstly, we infer the homogenous mean via the following statistic:

where $L_N$ is generic notation, and varies with respect to the magnitude of CD. Obviously, $L_N=N$ for Example (ref), and $L_N=N^2$ for Example (ref). Here, $\sigma_x^2 \coloneqq \lim_{N,T}\frac{1}{L_N T}E[S_{NT}^2]$.

To facilitate development, we present the first assumption.

Assumption (ref).1 is standard. See saulis for detailed definition of cumulants. In general, $\mathbf{B}_\ell\ne \mathbf{0}$ for $\ell \ge 1$, so we need to regulate the elements of $\mathbf{B}_\ell$ using Assumption (ref).2, which ensures that after suitable normalization, the Spectral norms of $\mathbf{B}_\ell$'s are summable. Obviously, Assumption (ref).2 is satisfied for both Examples (ref) and (ref). While $L_N\in (N, N^2)$, the magnitude of cross-sectional dependence is in between both cases. By Lemma (ref) of the appendix, Assumption (ref) entails that

which does not only incorporate long run covariance along the time dimension, but also accounts for WCD and SCD automatically.

Having these conditions in hand, we present the first main result of this paper.

Theorem (ref) gives an Edgeworth expansion for $\widetilde{S}_{NT}$, and a considerably simplified form will be presented in Corollary (ref) under a slightly more restrictive condition. The detailed definition of $\beta_3$ is omitted here due to its cumbersome notation. It is worth mentioning that if the skewness of $\varepsilon_{it}$ is 0 (i.e., $\kappa_3= 0$), the term $\frac{\beta_3}{6} (1-u^2)\phi(u)$ automatically vanishes. Then $F_{NT}(u)$ converges to $\Phi(u)$ in a much faster rate. In general, we do not have $\kappa_3= 0$, so we keep the statement of Theorem (ref) as it is.

It is noteworthy that for either Example (ref) or Example (ref), the result reduces to

where $|\beta_3|=O(\frac{1}{\sqrt{NT}} \vee \frac{1}{T^{3/2}})$. Sequentially, for both examples, Theorem (ref) gives the following Berry-Esseen bound:

which says that, for the panel data with TSA and WCD/SCD, the Berry-Esseen bound is usually at the order of $\frac{1}{\sqrt{NT}}$ unless $N\gg T$ such that $\frac{T}{\sqrt{N}}\to 0$. The presence of the term $\frac{1}{T^{3/2}}$ is due to the BN decomposition which introduces a truncation residual along the time dimension. In a typical panel data setting $N\asymp T$, the truncation residual is negligible. Our result complements the work of fan2015power, who specifically study cases where $\frac{T}{\sqrt{N}}\to 0$, by considering a broader range of scenarios and relaxing the independence assumptions employed in PY2008 and YU2024105458.

With a minor additional restriction, the following corollary holds.

The condition $\frac{N}{T^2}\to 0$ is rather common in the literature of panel data analysis, and is apparently fulfilled given $N\asymp T$. Using Assumption (ref).2, it is straightforward to see that $\frac{\sqrt{N}\|\mathbf{B}\|_1^2}{L_NT} =\frac{N\|\mathbf{B}\|_1^2}{L_N}\cdot \frac{1}{\sqrt{N}T}\to 0.$

Bootstrap Inference --- Below, we provide a bootstrap procedure to conduct inference. Given the nature of panel data, an intuitive thought is to draw a set of random variables, say, $\{\zeta_{it}\mid i\in [N], t\in [T] \}$ under certain restrictions, and construct the bootstrap counterpart of $S_{NT}$ as follows:

It is then straightforward to obtain that

where $\pmb{\zeta}_t=(\zeta_{1t},\ldots, \zeta_{Nt})^\top$, $\mathbf{W}_{ts} =E[\pmb{\zeta}_t\pmb{\zeta}_s^\top]$ for $t\ne s$, and $\mathbf{W}_{0}\coloneqq \mathbf{W}_{tt}$ for simplicity. Correspondingly, we can write $\widetilde{S}_{NT}^2 $ as follows:

Comparing the right hand sides of $E^*(\widetilde{S}_{NT}^{*2})$ and $\widetilde{S}_{NT}^2$, it is obvious that we need (ref) and (ref) to mimic (ref) and (ref) respectively. As $\mathbf{1}_N \mathbf{1}_N^\top $ has rank one, by Lemma (ref) the most obvious form of $\mathbf{W}_{ts}$ should be

where $\alpha_{ts}$ is a scalar varying with respect to the distance between $t$ and $s$. Therefore, we conclude that, to have a valid bootstrap procedure, one should replace $\zeta_{it}$ with $\zeta_t$ to ensure there is no cross-sectional variation. The finding nicely fits our study, as we assume no prior information about the magnitude of CD. By doing so, we preserve the dependence along the individual dimension in the bootstrap draws, and also avoid imposing certain order on individuals implicitly.

Formally, the bootstrap procedure is as follows.

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Accordingly, we impose the following assumption.

Assumption (ref) is a typical assumption in the literature of dependent wild bootstrap. We refer interested readers to Shao2015 for a comprehensive review of this line of research. Several conventional kernel functions satisfy such conditions. For example, for the Bartlett kernel, $q_a= 1$ and $C_1 = 1$; for the Parzen, Tukey-Hanning, QS kernels, and the trapezoidal functions, $q_a= 2$ and the values of $C_2$ vary and all satisfy $C_2<\infty$. See Andrews1991 for comments on different kernel functions. Notably, when the Bartlett kernel is adopted, the condition $\limsup_{N}\sum_{\ell =0}^{\infty}\ell^{q_a} \cdot C_{N\ell}<\infty$ reduces to Assumption (ref).2.a.

Using Assumptions (ref)-(ref), the following lemma and theorem hold for the bootstrap procedure.

Lemma (ref) summarizes some basic statistical properties to facilitate the investigation on the bootstrap statistic for the homogeneous cases. With them in hand, we present the following theorem.

Theorem (ref).1 establishes the first-order Edgeworth expansion for the bootstrap statistic $\widetilde{S}_{NT}^*$, extending the results of tik1981 to the panel data framework. Conditional on the sample, the rates of the first two results of Theorem (ref) are optimal, as the bootstrap draws $\{\zeta_t\}$ are $m$-dependent time series data only. Theorem (ref).3 demonstrates that the bootstrap covariance estimator can consistently estimate the true covariance, and infers that MSE is minimized at $$m_{\text{opt}}= \big(\frac{C_{q_a}\Delta_{q_a}}{2\sigma_x^4\int_{-1}^1 a^2(u)du}\big)^{2/(2q_a+1)} T^{1/(2q_a+1)}.$$

Inference under $\mathbb{H}_1$

Under $\mathbb{H}_1$, the model (ref) admits the following vector form:

where $\pmb{\mu}=(\mu_1,\ldots, \mu_N)^\top$, and the rest settings are identical to those in (ref).

Inferring heterogeneity is slightly more complicated, as there are two options:

It is noteworthy that, when studing each $\mu_i$, the conditional expectations and probabilities involved in Lemma (ref) and Theorem (ref) are still random variables, we therefore do not take $\max$ over $i$ in these results. In order to study the uniform inference over all individuals (i.e., inferring $\pmb{\mu}$ as a whole), we provide Theorems (ref) and (ref), which will help us establish the corresponding test statistic in Section (ref).

We start with the first choice, and consider the following quantify:

where $\widetilde{p}_i\coloneqq \frac{1}{\sigma_{p,i}\sqrt{T}}p_i\coloneqq \frac{1}{\sigma_{p,i}\sqrt{T}} \sum_{t=1}^T(\mathbf{e}_i^\top \mathbf{x}_t-\mu_i)$, $\sigma_{p,i}^2 \coloneqq \lim_N\mathbf{e}_i^\top \mathbf{B}\mathbf{B}^\top \mathbf{e}_i>0$, and $\mathbf{e}_i$ is a selection vector as defined in Section (ref).

To proceed, we need more structures to investigate heterogeneity.

Assumption (ref) regulates the rows of $\mathbf{B}_\ell$'s and $\mathbf{B}$, and is rather minor in view of Examples (ref) and (ref). Under this condition, we are able to present the following theorem.

Again, if the skewness of $\varepsilon_{it}$ is 0, the term $\frac{\beta_{i3}}{6} (1-u^2)\phi(u) $ vanishes. Then $F_i(u)$ converges to $\Phi(u)$ at a faster rate. In general, we do not have $\kappa_3= 0$, so we keep the statement of Theorem (ref) as it is.

To infer the distribution of $\widetilde{p}_i$ in practice, we provide a heterogeneous version of the dependent wild bootstrap procedure.

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For the heterogeneous case, we present the following lemma to facilitate the development of the asymptotic distribution of $\widetilde{p}^\ast_{i}$ conditional on the sample.

Using Lemma (ref), we are able to establish the asymptotic distribution of $\widetilde{p}^\ast_{i}$ conditional on the sample, which is an extension of Theorem (ref) for the heterogeneous bootstrap statistics.

Theorems (ref) and (ref) jointly imply that we are able to infer every single $\mu_i$. The discussion under Theorem (ref) still applies here.

We then explore the second option under (ref) when inferring heterogeneity. Mathematically, it means that one is concerned with $\max_i \widetilde{p}_i$ rather than any individual $\widetilde{p}_i$. This is useful, as there is an increasing literature concerning about the uniform inference in the panel data setting (e.g., LLS2024, SJW_2024). The following result contributes to this line of research.

Theorem (ref) establishes a Gaussian approximation within the panel data framework. Notably, this approximation remains valid irrespective of whether the cross-sectional dependence is WCD or SCD. Also, it is worth mentioning that under the sub-Gaussian condition of $\varepsilon_{it}$, the term $N^{2/J}$ in the above theorem can be replaced by $\log N$. Thus if we impose an exponential tail assumption, $N$ can even diverge at an exponential rate of $T$.

To have a practically feasible version of Theorem (ref), we need to know $\mathrm{Var}(\mathbf{z}_t) = \mathbf{B}\mathbf{B}^\top \eqqcolon \pmb{\Omega}$. Thus, define the high-dimensional long-run covariance matrix estimator by

and accordingly, define the Gaussian multiplier bootstrap approximate by

where $\{\mathbf{z}_t^* \mid t\in [T]\}$ is a vector of i.i.d. $N$-dimensional Gaussian random variables with $\mathbf{z}_t^*\sim N(\mathbf{0},\mathbf{I}_N)$.

Theorem (ref) builds upon the robust inference of high-dimensional covariance matrix estimation presented in gao2024robust and the continuity of the maximum of a Gaussian distribution. Similar to Theorem (ref), Theorem (ref) accommodates various types of CD.

The condition $\max_{i}\sum_{\ell=1}^{\infty}\ell^{q_\alpha}\|\mathbf{b}_{\ell i}^{\sharp}\|_2<\infty$ is slightly more restrictive than those required in Assumption (ref). This is not surprising, as we now need to investigate the uniform inference rather than derive CLT for any individual. Nevertheless, we only require an algebraic decay rate of the temporal dependence. $\widetilde{m}\asymp T^{1/(2q_{\alpha}+1)}$ corresponds to the optimal level of bandwidth in terms of minimizing the asymptotic mean squared error of each element in $\widehat{\bm{\Omega}}$. The condition $\frac{N^2T\log T}{(T\widetilde{m}\log N)^{J/4}}\to 0$ imposes restrictions on $(\widetilde{m}, N, T)$ jointly, which are easy to realize. If $\varepsilon_{it}$ is sub-Gaussian, $J$ can be arbitrarily large. Then the restrictions can be much simplified.

In the following subsection, we show that the above results offer the theoretical framework for the purpose of inference.

Test Statistic

According to the results established in Section (ref) and (ref), we are now ready to consider an $L_\infty$-based test statistic:

$$ Q_{NT} = \max_i \sqrt{T}(\overline{x}_i - \overline{x}), $$ where $\overline{x}_i = \frac{1}{T}\sum_{t=1}^{T}x_{it}$ and $\overline{x} = \frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}x_{it}$. Under $\mathbb{H}_0$, assume $N \leq L_N < N^2$ which implies that $\overline{x}-\mu = o_P(1/\sqrt{T})$, then simple algebra yields that $$ Q_{NT} = \left|\frac{1}{\sqrt{T}}\sum_{t=1}^{T}(\mathbf{x}_t-\mu\cdot \mathbf{1}_N) \right|_{\infty} + o_P(1). $$ Hence, we can calculate the critical value of $Q_{NT}$ using Theorem (ref). In addition, our test statistic can detect a class of sparse local alternatives at the rate $T^{-1/2}$. Specifically, under the sparse local alternatives such that $\mathbb{H}_1:\ \mu_i = \mu + a_{T}\delta_i$ with $a_T = 1/\sqrt{T}$ and $|\delta_i| > 0$ for some $i$ (our $L_\infty$-based test is still powerful even only one individual violates the null hypothesis), we have $$ Q_{NT} = \max_i\left(\frac{1}{\sqrt{T}}\sum_{t=1}^{T}(x_{it} - \mu) + \delta_i\right) + o_P(1), $$ and thus $Q_{NT}$ diverges to infinity if $ \sqrt{T}a_T \to \infty$. We now state the following proposition.

The distributional approximation established in Proposition 1 enables us to explore the known features of the partial sum of the bootstrapped versions of Gaussian samplers for inferential purposes.

Extensions

In this section, we consider three extensions. We firstly connect the above investigation with the literature on grouping structure analysis such as those surveyed in BM2015 and SSP2016. In the second extension, we relax some restrictions imposed on the panel data unit root processes of PM1999. Finally, we revisit the model of Pesaran2006 and the corresponding CCE estimators using the results established above.

Connection with Grouping Analysis

In this subsection, we explain how the above study can be connected with the current literature on grouping structure analysis. The numerical implementation of grouping can be easily done via $K$-mean or the agglomerative hierarchical clustering algorithm (hastie2009elements). In what follows, we explain how the above results are connected with the grouping analysis.

Note that $\mu_i$ can be estimated by $\overline{x}_i$ as in Section (ref), so $\mu_i$ is already known approximately. To proceed, we introduce the grouping structure, and impose the necessary conditions. Denote $J_0$ sets of indices (i.e., $\mathscr{G}_1,\ldots, \mathscr{G}_{J_0}$) such that

where $J_0 \ge 1$ is a fixed positive integer, and $\sharp \mathscr{G}_{j}$ stands for the cardinality of $\mathscr{G}_{j}$.

Typically, one assumes that $\mu_{i}\equiv \overline{\mu}_j$ for all $i\in \mathscr{G}_{j}$, so $c_{N}\equiv 0$. Assumption (ref).1 relaxes this restriction slightly. Assumption (ref).2 requires all groups to be well partitioned. If $J_0$ is known, we can estimate the group structure as follows:

where $S(\mathcal{G}, \pmb{\nu}) \coloneqq \frac{1}{N}\sum_{j=1}^{J_0}\sum_{i\in \mathcal{G}_j}| \overline{x}_i- \nu_j|^2$, $\mathcal{G} \coloneqq (\mathcal{G}_1,\ldots, \mathcal{G}_{J_0})$, $\pmb{\nu} \coloneqq(\nu_1,\ldots, \nu_{J_0})$, $\widehat{\mathcal{G}}\coloneqq (\widehat{\mathcal{G}}_1,\ldots, \widehat{\mathcal{G}}_{J_0})$, and $\widehat{\pmb{\nu}}\coloneqq (\widehat{\nu}_1,\ldots, \widehat{\nu}_{J_0})$. Here $\mathcal{G}_1,\ldots, \mathcal{G}_{J_0}$ are $J_0$ sets of indices such that

In (ref), $\widehat{\mathcal{G}}$ and $\widehat{\pmb{\nu}}$ estimate $\mathscr{G}\coloneqq (\mathscr{G}_1,\ldots, \mathscr{G}_N)$ and $\overline{\pmb{\mu}} \coloneqq (\overline{\mu}_1,\ldots, \overline{\mu}_{J_0})$, respectively.

Since $J$ is not known in practice, we estimate $J_0$ using the following information criterion:

where $\text{IC}(J)\coloneqq S(\widehat{\mathcal{G}}_{\mid J}, \widehat{\pmb{\nu}}_{\mid J}) + \rho_{NT}J$, $J^*$ is a user-specified large fixed constant, and $\rho_{NT}$ is a tuning parameter satisfying that

In (ref), $\widehat{\pmb{\nu}}_{\mid J}\coloneqq (\widehat{\nu}_{1\mid J},\ldots,\widehat{\nu}_{J\mid J})$ and $\widehat{\mathcal{G}}_{\mid J}\coloneqq (\widehat{\mathcal{G}}_{1\mid J},\ldots, \widehat{\mathcal{G}}_{J\mid J})$ are obtained via (ref) by assuming the number of groups being $J$. A natural choice of $\rho_{NT}$ is $[\log(N+T)]^{-1}$.

The first result of this proposition is evident building on the investigation in Section (ref) and Assumption (ref).1. After successfully partitioning the individuals, one can further investigate the homogeneity/heterogeneity for the $j^{th}$ group, e.g.,

using the results developed in Sections (ref)-(ref).

Nonstationary Panel Data

We now revisit the nonstationary panel data model studied in PM1999. Consider the following data generating process:

where $\mathbf{y}_t=(y_{1t},\ldots, y_{Nt})^\top$, $\mathbf{y}_0=\mathbf{0}$ without loss of generality, and $\mathbf{B}(L)\pmb{\varepsilon}_t$ is the same as that in (ref). As presented in Section 2 of PM1999, when studying nonstationary panel data, a key quantity is

which is also the foundation of some basic results of BAI200982 and DGP2021. Using the results of Section (ref), we are now able to account for the cross-sectional dependence of these unit root processes, and present the following proposition.

Previously, one has to impose cross-sectional independence such as PM1999, DGP2021 and HUANG2021198 due to technical constraints.

CCE Estimators

We now revisit the model of Pesaran2006 and the corresponding CCE estimators using the results established above. Accordingly, we let $L_N\equiv N$ in what follows. The model is as follows:

where only $\{(y_{it}, \mathbf{w}_{it})\mid i\in [N], t\in [T] \}$ are observable, $\mathbf{w}_{it}$ is a $k\times 1$ vector, $\mathbf{f}_t$ is an $m\times 1$ vector, and the dimensions of the other variables are defined accordingly. Both $m$ and $k$ are finite. The model admits a vector form:

where we have $\mathbf{Y}_i =(y_{i1},\ldots, y_{iT})^\top$, $\mathbf{W}_i =(\mathbf{w}_{i1},\ldots, \mathbf{w}_{iT})^\top$, $\mathbf{F} =(\mathbf{f}_1,\ldots, \mathbf{f}_T)^\top$, and $\pmb{\epsilon}_i =(\epsilon_{i1},\ldots, \epsilon_{iT})^\top$. In the homogenous setting (e.g., Westerlund2018 and references therein), one further assumes that

It is then natural to question whether (ref) holds practically.

To exam (ref), we briefly review the CCE approach. When eliminating the unobservable factor structure, the CCE approach utilizes the following form:

and for notational simplicity, we rewrite (ref) as

where the definitions of $\mathbf{z}_{it}$, $\mathbf{C}_i$, and $\mathbf{u}_{it}$ are self-evident. Simple algebra yields that

where $\overline{\mathbf{C}}\coloneqq \frac{1}{N}\sum_{i=1}^N \mathbf{C}_i $, $\overline{\mathbf{Z}}\coloneqq \frac{1}{N}\sum_{i=1}^N \mathbf{Z}_i $ with $\mathbf{Z}_i =(\mathbf{z}_{i1},\ldots, \mathbf{z}_{iT})^\top$, and $\overline{\mathbf{U}}\coloneqq \frac{1}{N}\sum_{i=1}^N \mathbf{U}_i $ with $\mathbf{U}_i =(\mathbf{u}_{i1},\ldots, \mathbf{u}_{iT})^\top$. Consequently, the CCE estimators of $\pmb{\theta}_i$ and $\pmb{\theta}$ are respectively defined by

where $\mathbf{M}_{\overline{\mathbf{Z}}} =\mathbf{I}_T-\overline{\mathbf{Z}}(\overline{\mathbf{Z}}^\top \overline{\mathbf{Z}})^+ \overline{\mathbf{Z}}^\top$.

The key part of the CCE approach is that $\mathbf{M}_{\overline{\mathbf{Z}}} $ offers a good approximation of $\mathbf{M}_{\mathbf{F}}$, because

\[ \overline{\mathbf{C}}^+ =\overline{\mathbf{C}}^\top(\overline{\mathbf{C}}\, \overline{\mathbf{C}}^\top)^{-1} \] under the conditions: (1) $\overline{\mathbf{C}}$ has full row rank, and (2) $\overline{\mathbf{U}}$ is asymptotically negligible.

Based on (ref), we construct the following quantities for all $j\in [k]$

where $\mathbf{e}_j$ is a selection vector as defined in Section (ref). We further let

where $(\widehat{\pmb{\xi}}_{j1},\ldots,\widehat{\pmb{\xi}}_{jT}) = (\widehat{\pmb{\mathbf{U}}}_{1,1}\circ \widehat{\pmb{\mathbf{U}}}_{1,1+j},\ldots, \widehat{\pmb{\mathbf{U}}}_{N,1}\circ \widehat{\pmb{\mathbf{U}}}_{N,1+j})^\top$, $\widehat{\pmb{\mathbf{U}}}_i= \mathbf{M}_{\overline{\mathbf{Z}}} \mathbf{Z}_i,$ and $\widehat{\pmb{\mathbf{U}}}_{i,j}$ stands for the $j^{th}$ column of $\widehat{\pmb{\mathbf{U}}}_{i}$.

To facilitate the development, we impose the following conditions.

The first two conditions of Assumption (ref).1 are standard. In the third condition, $\frac{1}{T}\overline{\mathbf{U}}^\top \overline{\mathbf{U}}=O_P(\frac{1}{N})$ and $\frac{1}{T} \mathbf{F}^\top \overline{\mathbf{U}}=O_P(\frac{1}{\sqrt{NT}})$ are rather standard in view of the development of Section (ref), and the following expansions:

where $\mathbf{U}_t=(\mathbf{u}_{1t},\ldots, \mathbf{u}_{Nt})^\top$. The requirement $\frac{1}{T}\mathbf{V}_i^\top \mathbf{M}_{\mathbf{F}} \pmb{\epsilon}_i=\frac{1}{T}\mathbf{V}_i^\top \pmb{\epsilon}_i+O_P(\frac{1}{T})$ can be verified by the development of Section (ref). Assumption (ref).2 allows $\pmb{\epsilon}_t$ and $\mathbf{v}_t$ to be weakly dependent.

Under these condition, the following proposition holds.

According to Proposition (ref), all $Q_j$'s should not fall in the rejection region yielded by the bootstrap draws under (ref). We further examine this result in the simulation studies.

On top of these extensions, we may also revisit the time trend analyses of gh2006, cgl2012, ROBINSON20124 and WSX2023, which are of course mathematically involved due to the nonparametric nature. We leave them for future study.

Simulation

In this section, we conduct simulation studies to examine the theoretical results about testing and inference of Section (ref), and also verify our argument about the CCE estimators in Section (ref).

Simulation 1 (Testing) --- The data generating process (DGP) is simplified as follows:

where $t=-200, \ldots,0, 1,\ldots, T$, $\rho_x=0.3$, $\pmb{\mu}\coloneqq (\mu_1,\ldots, \mu_N)^\top$, $\mathbf{x}_t\coloneqq (x_{1t},\ldots, x_{Nt})^\top$, and $\pmb{\nu}_t\coloneqq (\nu_{1t},\ldots, \nu_{Nt})^\top$. As well understood, the AR(1) process admits an MA($\infty$) representation thus suiting the definition of (ref). To introduce cross-sectional dependence, we let $\pmb{\Sigma}_\nu=\{\rho_\nu^{|i-j|}\}_{N\times N}$. The observations from $t=-200,\ldots, 0$ are burn-in sample in order to eliminate the impact of the initial value. We let $\{\nu_{it} \}$ be i.i.d. over both $i$ and $t$, and be generated in three cases:

Case 1 is a symmetric distribution with thin tails; Case 2 is a symmetric distribution with heavy tails; and Case 3 is an asymmetric distribution. To exam the size and power of the proposed test in Section (ref), for each case we consider three scenarios for $\pmb{\mu}$:

Scenarios (a)-(c) are designed to examine size, local power, and global power respectively.

Notably, the above DGP is a special case of PY2008 and YU2024105458. Thus, for the purpose of comparison, we also consider the approaches of these two papers (referred to as PY and YYX respectively). To put everything on equal footing, we consider the null of YU2024105458 (i.e., $\mathbb{H}_0:\mu_i\equiv 0$ for all $i$), and modify the test statistic of PY2008 accordingly in an obvious manner. For the sake of space, we refer interested readers to their papers for detailed implementation. As PY and YYX methods calculate the asymptotic variances neglecting the dependence of the residuals (e.g., PY2008 and YU2024105458), we anticipate some distorted size or power. Additionally, YU2024105458 rely on the Gaussian assumption, so we anticipate the DGPs of Cases 2 and 3 will further distort size or power. For our method (referred to as GLPY), we calculate $Q_{NT}$ under the null for each dataset, and obtain the 95% confidence interval (denoted by $\text{CI}_\infty$) via $ |\widehat{\bm{\Omega}}^{1/2}\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\mathbf{z}_t^* |_{\infty}$ based on 399 bootstrap replications. After $R$ replications, we calculate the rejection rate as follows:

where the subindex $j$ stands for the corresponding values obtained in the $j^{th}$ simulation replication. Similarly, we will report the rejection rates for PY and YYX approaches. For our method, we expect that $\Delta_{test}$ is sufficiently close to 0.05 for the scenario (a), is reasonably close to 1 for the scenario (c), and is in between 0 and 1 for the scenario (b). For simplicity, we let $a(\cdot)$ be Bartlett kernel, and take suggestions from gao2024robust to set $\widetilde{m}=\lfloor 1.75 T^{1/3}\rfloor$ for simplicity. Additionally, we let $R=1000$, $N \in \{100,150,200 \}$, $T\in \{ 200, 400 \}$, and $\rho_\nu \in \{0.5, 0.95 \}$.

The results are summarized in Table (ref). Overall, our approach has reasonable size, local power, and global power irrespective to the magnitude of CD (i.e., the value of $\rho_\nu$) as expected. Due to omitting dependence, PY and YYX methods tend to over reject, which is evident in view of the rejection rates of scenario (a) of Cases 1-3.

Simulation 2 (Inference) --- In this simulation, we examine the bootstrap inferences documented in Sections (ref) and (ref). For simplicity, we consider the three cases identical to Simulation 1 with $\pmb{\mu}=\mathbf{0}_{N\times 1}$, and we set $m=\lfloor 1.75 T^{1/3}\rfloor$. For each dataset, we infer a few quantities. For the homogenous case, we calculate $\widetilde{S}_{NT}$ of Section (ref), and simulate its distribution via $\widetilde{S}_{NT}^*$ based on 399 bootstrap replications, where $\widetilde{S}_{NT}^*$ is self-normalized by construction and does not require any prior knowledge about $L_N$. Using the bootstrap draws, we construct the 95% confidence interval of $\widetilde{S}_{NT}$, denoted by $\text{CI}_{S}$. Second, for each $i$, we calculate $\widetilde{p}_i$ of Section (ref), and construct its distribution via $\widetilde{p}_i^*$ based on 399 bootstrap replications. Accordingly, we construct the 95% confidence interval of each $\widetilde{p}_i$, denoted by $\text{CI}_{p_i}$.

After $R$ simulation replications, we calculate the following measures:

where again $j$ indexes the $j^{th}$ simulation replication. We anticipate that $\Delta_{\text{HM}}$ and $\Delta_{\text{HE}} $ are close to 0.05, and $\text{Sd}_{\text{HE}}$ is close to 0 indicating the proposed method in Section (ref) is stable for all $i$'s.

Table (ref) shows that most values are as expected. While the DGPs cover a symmetric distribution with thin tails, a symmetric distribution with heavy tails, and an asymmetric distribution, the above results are reasonably good irrespective of the magnitude of CD.

Simulation 3 (CCE) --- We then consider the following panel data model:

where all elements are scalar for simplicity. In this case, $m=1$ and $k=1$, so $m\le k+1$ (a typical requirement for CCE estimators) is fulfilled. We examine the size and power of Proposition (ref).

The DGP is as follows. $\{\epsilon_{it}\}$ follows the identical DGP of $\{x_{it}\}$ as in Cases 1-3 of Simulation 1 with $\pmb{\mu}=\mathbf{0}_{N\times 1}$ and $\rho_{\nu}=0.5$. We let $f_t\sim N(0,1.5)$, $\gamma_i \sim N(0.8, 1)$, $\Gamma_i\sim N(-0.2, 2)$, and $\{v_{it}\coloneqq \epsilon_{i,t-1} \}$. For each case of $\{\epsilon_{it}\}$, we further consider three scenarios for $\{\theta_i\}$:

Similar to Simulation 1, scenarios (a)-(c) are designed to evaluate size, local power, and global power. For each dataset, we calculate $Q_1$ and generate the corresponding confidence interval (say, $\text{CI}_{Q_1}$) as in Proposition (ref). After $R$ simulation replications, we report the following measure:

where $j$ still indexes the $j^{th}$ simulation replication. $\Delta_{\text{Q}}$ should be close to 0.05 and 1 for the scenarios (a) and (c) respectively, and should be in between 0 and 1 for the scenario (b). As shown in Table (ref), the results are as expected. For scenario (a) of Case 3, our approach is slightly under-size with large $N$, and it might be due to the fact that the error component of Case 3 is skewed.

A Case Study

Heterogeneous expectations, which may arise in financial markets because investors interpret and react to available information differently, are crucial for understanding variations in asset prices, portfolio allocations, and market dynamics. As a result, the study of heterogeneous expectations has become an important focus in recent years, with a number of works highlighting its implications for financial theory and practice ame2020, Brun2021, Giglio2021. Notably, Giglio2021 document significant and persistent cross-sectional variations in individual investors' return expectations, which can only be partially explained by demographic factors. Similarly, DI2024 reveal substantial heterogeneity in equity return expectations among institutional investors and investment consultants and it can be linked to asset managers' portfolios. These findings challenge traditional finance models based on homogeneous rational expectations and motivate the development of alternative frameworks.

In this study, we revisit some prior work in this area and formally test the heterogeneity in equity return expectations using the newly proposed inference method. Furthermore, we investigate the relationships between equity premium expectations and equity valuations, as explored by DI2024.

Variables and Data

Following DI2024, we examine three types of subjective return expectations held by asset managers for U.S. equities. The equity return expectation ($ere_{it}$) is defined as the (geometric) nominal equity return forecast for large-cap U.S. equities over a 10-year horizon, as reported in public disclosures\footnote{As noted by DI2024, the forecast horizons for return expectation data range from 1 to 50 years. However, most asset managers provide forecasts close to a 10-year horizon. Consequently, this study focuses on 10-year horizon forecasts.}. The equity premium expectation over yield ($epey_{it}$) is calculated by subtracting the horizon-matched log nominal Treasury yield from the nominal equity return expectation. Similarly, the equity premium expectation over cash ($epec_{it}$) is derived by subtracting the expected annualized return on cash (over the next 10 years) from the equity return expectation.

For the equity valuation, we utilize the cyclically adjusted price-to-earnings ratio ($cape_t$), defined as the log ratio of real equity prices to real earnings averaged over the past decade. Introduced by campbell1988, this measure of equity valuation is widely recognized in the finance literature JL2019. Additionally, we include the past 12-month return of the S&P 500 index ($pr_t$) to capture momentum effects and the horizon-matched Treasury yield ($rf_t$) to account for the risk-free rate.

The data used to construct these variables are available on the website maintained by the authors of DI2024. By focusing on a fixed 10-year forecast horizon, we obtain a panel dataset comprising observations from 45 asset managers across 109 months, covering the period from November 1997 to April 2021. Descriptive statistics for these variables are presented in Table (ref).

Model Specifications and Inference

To analyze the heterogeneity in equity return expectations, we estimate three econometric models, each progressively incorporating more explanatory variables to account for the potential determinants of the expectations:

where $y_{it}\in\{epey_{it}, epec_{it}, ere_{it}\}$. Model 1 is the baseline model that captures the individual-specific mean levels of equity return expectations, where $\mu_i$ represents the time-invariant mean for each asset manager $i$. In Model 2, the equity return expectations are modeled as a function of the $cape_t$, with $\beta_i$ representing the sensitivity of each manager's expectations to equity valuations. DI2024 reveal countercyclical expectations for U.S. equity returns, by showing that the expectations are negatively associated with $cape_t$. In this study, we further investigate whether the relationship between return expectations and valuations differs across managers. The inclusion of $pr_t$ and$rf_t$ in Model 3 further enables us to examine whether these additional factors, such as historical market performance, can help explain cross-sectional differences in expectations.

For each specification, we conduct hypothesis tests to evaluate the presence of heterogeneity in key parameters. Specifically:

These tests provide evidence regarding both the variation in average return expectations and the diversity in how managers incorporate fundamental and market-driven information into their forecasts. Rejecting these null hypotheses would indicate the heterogeneity of return expectations among the institutional investors and investment consultants.

Estimation and Testing Results

We first estimate three specifications of equity return expectations using both heterogeneous and homogeneous regression models. The estimated coefficients for each model, along with their 95% confidence intervals, are presented in Table (ref). Across all models, the cyclically adjusted price-to-earnings ratio is found to be significantly negatively associated with asset managers' forecasts of equity returns and premiums, indicating countercyclical patterns in equity return expectations: when equity valuations are higher, asset managers expect lower future returns. This result aligns with the findings of DI2024 and contrasts with the procyclical expectations among retail investors revealed by GS2014.

Comparing the outcomes from heterogeneous and homogeneous estimations, we can identify noticeable differences. In particular, most homogeneous coefficient estimates for the intercept and $cape$ are larger (in absolute values) than the corresponding average values from heterogeneous models. This discrepancy highlights the importance of allowing for heterogeneity in the data. Homogeneous models, by assuming no diversity across asset managers, may overestimate the influence of equity valuation on return expectations.

Furthermore, both estimation approaches consistently suggest that past returns exert no significant influence on asset managers' expectations regarding future equity premiums. This finding suggests that asset managers may not rely on recent return trends when forming long-term return expectations, potentially due to the forward-looking nature of their strategies. It also indicates that their expectations are primarily driven by fundamental factors, such as valuations and macroeconomic conditions, rather than past market performance. Importantly, these results hold consistently across all three measures of equity return expectations.

To further investigate the heterogeneity in asset managers' expectations for U.S. equity returns, we perform a sequence of heterogeneity tests as outlined in Section (ref). The results of these tests are summarized in Panel A of Table (ref). For Model 1, the null hypothesis of homogeneous means is rejected at the 0.01 significance level for all three types of expectations: equity returns, equity premium over yield, and equity premium over cash. This indicates substantial variation in the average levels of return expectations across asset managers.

For Models 2 and 3, the heterogeneity in expectations remains significant even after accounting for equity valuations, past returns, and risk-free rates. The results for $\mathbb{H}_0^3$ further reveal that the relationship between equity return expectations and equity valuations varies significantly across institutional investors and investment consultants. This finding indicates that differences among asset managers extend beyond simple average return expectations; they also exhibit diverse sensitivities to fundamental factors such as equity evaluations. Collectively, these findings confirm the presence of heterogeneity in U.S. equity premium expectations, which aligns with recent literature emphasizing the role of diverse beliefs and preferences in shaping market outcomes Brun2021.

In order to illustrate the heterogeneity test results for the intercept in each model, we provide the distributions of bootstrap test statistics in Figure (ref). For comparison, the test statistics and critical values for the homogeneity test of slope coefficients proposed by PY2008 (PY) and the homogeneity test of intercepts developed by YU2024105458 (YYX) are also computed for $\mathbb{H}_0^3$ and $\mathbb{H}_0^2$, respectively, under Model 2. The results are reported in Panel A of Table (ref). Notably, the YU2024105458's test originally examines heterogeneity in intercepts under the null hypothesis of known homogeneous intercepts. To adapt this framework to our setting, we modify their statistics to test heterogeneity against an unknown constant by specifying the intercept in the null as its homogeneous estimator. This adjustment ensures comparability with our proposed methodology while maintaining the robustness of their test.

As a robustness check, we follow DI2024 and expand the dataset to include observations with forecast horizons close to ten years, rather than limiting the sample to exactly 10-year-horizon. The heterogeneity test results for this expanded sample are reported in Panel B of Table (ref). Most tests continue to reject the null hypothesis of homogeneity at the 0.01 significance level. The only exception is the one for the mean value of equity premiums over cash, which indicates a slightly weaker rejection (at a 0.05 significance level) of the homogeneity. Nevertheless, the overall robustness of the test results confirms that our findings are not sensitive to the inclusion of additional observations with wider forecast horizons.

Conclusion

In this paper, we introduce an underlying data generating process that allows for different magnitude of CD, along with TSA. This is achieved via high-dimensional moving average processes of infinite order (HDMA($\infty$)), which automatically generalizes the spatial structure introduced by Robinson and his co-authors in recent years. The framework is important in the sense that as noted by brockwell1991time and FanYao, the Wold decomposition theorem ensures a formal linear representation exists for any stationary time series with no deterministic components, and HDMA($\infty$) naturally incorporates this result into a panel data framework.

Our setup and investigation significantly integrates and enhances both homogenous and heterogeneous panel data modelling and testing (such as Pesaran2006, PY2008, fan2015power, YU2024105458). To study HDMA($\infty$), we extend the BN decomposition (e.g., BN1981,PS1992) to a high-dimensional time series setting, and derive a complete set of toolkit. It is worth mentioning our investigation complements the work of fan2015power, who specifically study cases where $\frac{T}{\sqrt{N}}\to 0$, by considering a broader range of scenarios and relaxing the independence assumptions employed in PY2008 and YU2024105458.

We exam homogeneity against heterogeneity using Gaussian approximation, a prevalent technique for establishing uniform inference (e.g., chernozhuokov2022improved, and references therein). For post-testing inference, we derive Central Limit theorems through Edgeworth expansions for both homogenous and heterogeneous settings. Notably, the demand for Gaussian approximation in panel data analysis has been increasing recently, as exemplified in Section 4 of SJW_2024 and Section 4 of LLS2024. Our study also contributes to this research direction by providing a set of foundational conditions and deriving a set of useful basic results.

We showcase the practical relevance of the established asymptotic theory by (1). connecting our results with the literature on grouping structure analysis such as those surveyed in BM2015 and SSP2016, (2). examining a nonstationary panel data generating process presented in PM1999, and (3). revisiting the common correlated effects (CCE) estimators of Pesaran2006. Typically, when investigating nonstationary panel data, one has to impose cross-sectional independence such as PM1999, DGP2021 and HUANG2021198 due to technical constraints. Our study offers a set of toolkit to account for the dependence of unit root precesses.

Finally, we verify our theoretical findings via extensive numerical studies using both simulated and real datasets.

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Throughout the proofs, we suppose that without loss of generality, $\mu \equiv 0$ for the homogenous case, and $\mu_i\equiv 0$ for all $ i\in [N]$ for the heterogeneous case. We shall not mention them again unless misunderstanding may arise.

This document is structured as follows: Appendix (ref) points out that the proposed inference methods can accommodate unbalanced panels after some necessary modifications; Appendix (ref) provides some foundational facts used throughout the proofs; Appendix (ref) contains all preliminary lemmas; Appendix (ref) presents the theoretical proofs of the main results; and Appendix (ref) details the proofs of the preliminary lemmas.

Unbalanced Panel Data

More often than not, one encounters unbalanced panel data practically. Given the missing proportion is asymptotically negligible, the proposed inference methods can accommodate unbalanced panels after some necessary modifications.

For instance, we let $\mathbb{T}_i$ denote the sample set for individual $i$ and $T_i\coloneqq \sharp \mathbb{T}_i$. The $L_\infty$-based test statistic is then redefined as

$$ Q_{NT} = \max_i\left|\sqrt{T_i}(\overline{x}_i - \overline{x})\right|, $$ where $\overline{x}_i = \frac{1}{T_i}\sum_{t=1}^{T_i}x_{it}$ and $\overline{x} = \frac{1}{N}\sum_{i=1}^{N}\frac{1}{T_i}\sum_{t=1}^{T_i}x_{it}$.

Additionally, the high-dimensional long-run covariance matrix estimator for unbalanced panels is defined as $ \widehat{\bm{\Omega}}\coloneqq (\widehat{\Omega}_{in})$, where

and the Gaussian multiplier bootstrap approximation is accordingly defined by

where $\{z_{it}^* \mid t\in\mathbb{T}_i \}$ is a sequence of i.i.d. $ N(0,1)$.

For the CCE estimation, the estimators and test statistics for unbalanced panels can also be updated. Let $\mathbb{N}_t$ contain the indices for individuals that have valid observations at time $t$ and let $N_t$ be the number of indices in this set. Additionally, for each $i$, define $\overline{\mathbf{Z}}_{\cdot i}\coloneqq (\overline{\mathbf{z}}_{t_1},\cdots, \overline{\mathbf{z}}_{t_{T_i}})^\top$, where $t_1,\cdots,t_{T_i}\in \mathbb{T}_i$ and $\overline{\mathbf{z}}_t\coloneqq \frac{1}{N_t}\sum_{i\in\mathbb{N}_t}\mathbf{z}_{it}$.

The CCE estimators of $\pmb{\theta}_i$ and $\pmb{\theta}$ can be then defined as

where $\mathbf{M}_{\overline{\mathbf{Z}}_{\cdot i}} =\mathbf{I}_T-\overline{\mathbf{Z}}_{\cdot i}(\overline{\mathbf{Z}}_{\cdot i}^\top \overline{\mathbf{Z}}_{\cdot i})^{-1} \overline{\mathbf{Z}}_{\cdot i}^\top $.

The heterogeneity test statistic is then given by

where $\mathbf{e}_j$ is a selection vector. For constructing the bootstrap statistics, we use $\widehat{\pmb{\Omega}}_{j}=(\widehat{\pmb{\Omega}}_{j,in})$, where

where $\widehat{\xi}_{j,it}$ denotes the $t$-th element of $\widehat{\pmb{\mathbf{U}}}_{i,1}\circ \widehat{\pmb{\mathbf{U}}}_{i,1+j}$, with $\widehat{\pmb{\mathbf{U}}}_i= \mathbf{M}_{\overline{\mathbf{Z}}_{\cdot i}} \mathbf{Z}_i,$ and $\widehat{\pmb{\mathbf{U}}}_{i,j}$ standing for the $j^{th}$ column of $\widehat{\pmb{\mathbf{U}}}_{i}$. The distribution of $Q_j$ is then approximated by

where $\{z_{it}^* \mid t\in\mathbb{T}_i \}$ is a sequence of i.i.d. $ N(0,1)$.

Some Facts

We present a few facts in this section, which will be repeatedly used in the proofs.

On Cumulant --- Note that for a generic cumulant, we have for a constant $c$

We refer the interested reader to saulis for more details about cumulants.

On BN Decomposition --- Simple algebra shows the Beveridge and Nelson (BN) decomposition under the high-dimensional (HD) setting is as follows:

where $\widetilde{\mathbf{B}}(L)\coloneqq\sum_{\ell=0}^{\infty}\widetilde{\mathbf{B}}_\ell L^\ell$ with $\widetilde{\mathbf{B}}_\ell\coloneqq \sum_{k=\ell+1}^{\infty}\mathbf{B}_k$.

On Dependence --- To calculate CD, for $\forall i,j\in [N]$ it is easy to obtain that

To calculate TSA, for $t> s$ we write

To calculate CD + TSA, for $t> s$ we write

On Cumulants --- Recall the notation of Section (ref). Let $\widetilde{\phi}(x)$ be the characteristic function of a standard normal distribution. We further define its cumulants by $\gamma_r$ for $r\ge 1$. We now consider a generic distribution function $G(x)$ with a characteristic functions $\chi(u)$ and cumulants $\beta_r$. Then by Taylor expansion, we have

\[ \log \frac{\chi(u)}{\widetilde{\phi}(x)} =\log \chi(u) - \log \widetilde{\phi}(x)=\sum_{r=1}^\infty (\beta_r-\gamma_r)\cdot \frac{(\mathsf{i} u)^r}{r!} \] and

On Exponential function --- By Taylor expansion,

Taylor theorem yields that

where $\widetilde{u}$ is between 0 and $u$.

Preliminary Lemmas

In this appendix, we provide some useful preliminary lemmas. The first three lemmas are either obvious or have been carefully proved in the literature, so we do not provide the proofs herewith. We will provide the proofs for Lemmas (ref) to (ref).

See Chapter XVI, section 3 of feller1971introduction for details about Esseen's smoothing Lemma.

See tik1981 for example.

Write

where the definition of $\mathbf{B}_v^*(L)$ for $v\ge 0$ should be obvious. In connection with the BN decomposition of (ref), we have

where

Proofs of the Main Results

Proofs of the Preliminary Lemmas