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Abstract

Fixed effects estimation of a static model with robust or panel corrected standard errors is commonly used to model large N, large T panel data. However, this approach is biased and inconsistent in the presence of dynamic misspecification, slope heterogeneity, and cross-sectional dependence. Common correlated effects estimation of a dynamic model has been advanced to address these issues but is rarely used in sociology. Here, I provide an overview of the large N, large T panel data literature, and I conduct an array of Monte Carlo experiments to compare the fixed effects estimator to the common correlated effects estimator regarding the aforementioned issues. I show that fixed effects estimation with robust or panel corrected standard errors do not address these problems, which is most evident with high levels of slope heterogeneity and lag misspecification, and its performance worsens as the time dimension expands. In contrast, the common correlated effects estimator produces superior estimates as T increases and is robust to slope heterogeneity and cross-sectional dependence. Following the experiments, I present an example by examining the drivers of fossil fuel consumption at the U.S. state level from 1960 to 2018, and I conclude by presenting a decision-making framework for researchers to use to make informed decisions when modeling large N, large T panel data.

Keywords

panel data quantitative methods

Introduction

Large N (cross-sections), large T (time periods) data is an increasingly common data structure analyzed by sociologists. In statistical theory, large N, large T panel data are a data structure where N and T grow at the same rate to infinity (Ditzen 2018; Pesaran 2006; Pesaran and Smith 1995). In practice, this means that T is large enough so that a time-series regression can be estimated for each unit in the analysis and N is large enough so that averaging across units produces consistent results and central limit theorems hold (Pesaran and Smith 1995). In this article, an N and T of at least 30 is considered large.¹

This type of data structure is common in macro-level research that analyzes cross-national or meso-level (states and provinces) data. For instance, the World Bank’s (2021) world development indicators database extends back to 1960 in many cases, as does state-level energy and income data in the United States (Bureau of Economic Analysis 2021; U.S. Energy Information Administration 2021). Although these data are typically annual, more granular temporal data are available on a monthly, weekly, or daily basis in some cases (e.g., financial, employment, and industry-level data).

As interest in big data increases and the temporal depth of widely used data sets expands, large N, large T data are becoming more ubiquitous, making it paramount that researchers have the appropriate tools to analyze these data. To date, most statistical work in this area comes from econometrics (Chudik and Pesaran 2015a; Pesaran 2006; Pesaran and Smith 1995) and has yet to be widely used in sociology.² Part of this is due to these advances not being readily available in popular software programs, but recently, particularly in Stata, a plethora of user-written commands emerged that eases this burden (Bersvendsen and Ditzen 2021; Ditzen 2018, 2021). But that is not the only issue because there are limited resources available to bring these methodological techniques to applied researchers in an approachable way.

To fill this gap, this article coalesces the literature on large N and large T data to guide researchers through analyzing these data. Most importantly, I build on the prior literature and show using Monte Carlo experiments that researchers must correctly model dynamics (including the correct number of lags), slope heterogeneity (whether effects vary across units), and cross-sectional dependence (units are correlated with each other because they share a common factor like technology or are commonly impacted by an event like a pandemic or recession) and that failing to address these issues can lead to biased and inconsistent estimates. The experiments indicate that using robust or panel corrected standard errors with fixed effects estimation do not mitigate these issues, and the performance of these approaches depend on the degree of lag misspecification, the presence of slope heterogeneity and cross-sectional dependence, and the number of time periods in the analysis. In contrast, the common correlated effects estimator produces superior estimates and is robust to slope heterogeneity and cross-sectional dependence, and it performs just as well as the fixed effects estimator without these issues present. When T = 30, the two-way fixed effects and common correlated effects estimators perform similarly, but the performance of the common correlated effects estimator is superior as T expands. Based on these findings, I conclude with a decision-making framework and a set of recommendations for researchers to consider when modeling large N, large T data.

The article is separated into four sections. The first section outlines the modeling issues to consider with large N, large T data, along with a discussion of the fixed effects and common correlated effects estimators. The second section reports a series of Monte Carlo experiments to compare several versions of the fixed effects estimator to the common correlated effects estimator. The third section provides an example by examining the drivers of fossil fuel consumption in U.S. states from 1960 to 2018. I then conclude with the decision-making framework and a set of recommendations for researchers.

Modeling Large N, Large T Panel Data

In the following section, I first outline fixed effects estimation of a static model, which serves as the workhorse modeling approach in sociology. Following this discussion, I review dynamic models and the implications of dynamic misspecification. I then outline the issues associated with the fixed effects estimator in the cases of slope heterogeneity and strong cross-sectional dependence, followed by introducing the common correlated effects estimator which can address these concerns.

The Workhorse: Fixed Effects Estimation of a Static Model

Fixed Effects estimation of a static model (Equation 1) is the most commonly used approach to analyzing large N, large T data in sociology:

$y_{i, t} = α_{i} + β_{1} x_{i, t} + u_{t} + ε_{i, t},$ (1)

where $y_{i, t}$ is the dependent variable, $x_{i, t}$ is a time-varying independent variable, $α_{i}$ are the unit-specific fixed effects, $u_{t}$ are the time-specific fixed effects, and $ε_{i, t}$ is random variation for each unit at each point in time. This model is referred to as static because it models the contemporaneous effect of the independent variable on the dependent variable and assumes that the effect does not persist over time (De Boef and Keele 2008).

Fixed effects estimation can occur in two ways (Allison 2009). The first way is through the mean deviation approach, which works by constructing case-specific mean deviations for the dependent and independent variables and subtracting the individual-specific mean from each observed value:

$\begin{matrix} y_{i, t}^{*} = y_{i, t} - {\bar{y}}_{i} \\ x_{i, t}^{*} = x_{i, t} - {\bar{x}}_{i}, \end{matrix}$ (2)

and regressing $y_{i, t}^{*}$ on $x_{i, t}^{*}$ (Allison 2009).

The second way fixed effects can be estimated is by including a set of dummy variables for each case, which is also known as the least square dummy variable (LSDV) estimator (Allison 2009). This method allows a fixed effect coefficient to be estimated for each case (the mean-deviation approach does not allow for this), but it does constrain one of the units to zero, which drops it from the analysis. The LSDV estimator and the mean deviation approach produce equivalent estimates.

Fixed effects estimation of a static model has several limitations that can make it perform poorly. First, researchers tend to give little consideration to whether the static model is appropriate—often using it by default. Thus, many models are potentially misspecified due to incorrectly modeling the lag structure (i.e., not including the appropriate number of lags of the dependent and independent variables; De Boef and Keele 2008; Plümper and Troeger 2019). Second, even if the dynamics are properly specified, fixed effects estimation of a dynamic model in the presence of slope heterogeneity produces inconsistent estimates because the slope heterogeneity gets captured in the error term (Pesaran and Smith 1995). Third, failing to appropriately model cross-sectional dependence can lead to inconsistent and biased estimates because the cross-sectional dependence will become part of the error term and is likely correlated with the regressors (Pesaran 2006, 2015).

Dynamic Models and Dynamic Misspecification

Although the static model is commonly used, it is often statistically and theoretically inappropriate (Thombs, Huang, and Fitzgerald 2022). This model assumes that any potential autocorrelation is handled as a nuisance (e.g., measurement error)—rather than as part of the data generating process (i.e., that the estimated model is incorrect). When autocorrelation (and other issues like heteroskedasticity and cross-sectional dependence) arise, a robust variance estimator like clustered robust standard errors, panel corrected standard errors, and/or a generalized least squares transformation like Prais-Winsten are often used (Beck and Katz 1995). However, if the aforementioned issues are actually part of the data generating process, such corrections do little to mitigate their impact on estimation because the estimated model is not changed.³ Panel data are very likely to be autoregressive given that we expect social processes to be a function of the past. It would be unusual for any social theory to argue that history does not matter, which is what is assumed in a static model.

To see why applying a standard error correction to a static model likely fails to appropriately model the true data generating process, we can start by discussing how standard errors are estimated with ordinary least squares (OLS):

${(\hat{V})}_{O L S} = {\hat{σ}}^{2} {(X^{'} X)}^{- 1} .$ (3)

OLS assumes that the variance ( ${\hat{σ}}^{2}$ ) is constant across units and uncorrelated (no autocorrelation, homoskedastic, and cross-sectionally independent). However, in the panel data context, it is likely these assumptions are violated because observations tend to be correlated over time and across units. To correct for autocorrelation and heteroskedasticity, researchers often turn to robust standard errors. The clustered robust variance–covariance matrix is as follows:

${(\hat{V})}_{C l u s t e r e d} = (X' X)^{- 1} (\sum_{n = 1}^{N} X'_{n} {\hat{u}}_{n} \hat{u}'_{n} X_{n}) (X' X)^{- 1},$ (4)

where n represents a specific cluster, $u_{n}$ is a vector of residuals in each cluster, and $X_{n}$ is a matrix of the independent variables for the observations in each cluster. The idea here is that the OLS estimates of the coefficients are unbiased and consistent but are inefficient, so a robust variance estimator is used instead of the OLS variance estimator (Cameron and Trivedi 2010). This results in the standard errors being corrected, but the coefficients remain unchanged.

Using a Prais-Winsten transformation is another common way to account for autocorrelation. This approach works by first estimating the autocorrelation in the residuals of a static model:

$\begin{matrix} y_{i, t} = β_{1} x_{i, t} + ε_{i, t} \\ ε_{i, t} = ρ ε_{i, t - 1} + ξ_{i, t}, \end{matrix}$ (5)

then applying the Cochrane-Orcutt transformation to observations for T $> 1$ with $\hat{ρ}$ :

$y_{i, t} - \hat{ρ} y_{i, t - 1} = β (x_{i, t} - \hat{ρ} x_{i, t}) + υ_{i, t},$ (6)

Then, to preserve the first time period, the following transformation is done to T = 1:

$\sqrt{1 - {\hat{ρ}}^{2}} y_{1} = β (\sqrt{1 - {\hat{ρ}}^{2}} x_{1}) + \sqrt{1 - {\hat{ρ}}^{2}} υ_{i, t} .$ (7)

The Prais-Winsten transformation is commonly used in tandem with panel corrected standard errors because they do not correct for autocorrelation (e.g., Jorgenson and Clark 2012; Kelly 2020; Kollmeyer and Peters 2019).

The underlying assumption of clustered robust standard errors (or panel corrected standard errors) and the Prais-Winsten transformation is that the estimated model is specified correctly and any heteroskedasticity or correlation in the errors are nuisances to correct for. However, as Plümper and Troeger (2019:19) lucidly argue, “This error structure exists not because nature invented a complex error process that ought to be controlled away, but because of a dynamic misspecification in the underlying data-generating process.” In other words, rather than treating autocorrelation (or other correlations in the errors) as a nuisance to get rid of, the presence of these issues is likely telling us something about our model. Thus, in the case of autocorrelation, researchers can explicitly model it by using a dynamic model. Dynamic models model the memory of a time series, where effects are distributed over time. In contrast to static models, dynamic models estimate both short-run (immediate) effects and long-run (cumulative) effects.

The defining feature of a dynamic model is that it includes the lag of the dependent variable. The most basic dynamic model is the lagged dependent variable (LDV) model, which is also referred to as the partial adjustment model or the Koyck model (De Boef and Keele 2008):

$y_{i, t} = α_{i} + ϕ_{1} y_{i, t - 1} + β_{1} x_{i, t} + e_{i, t} .$ (8)

In addition to the LDV, lags of the independent variables can be included, which is referred to as an autoregressive distributed lag (ARDL) model (see Table 1 for a summary of the different dynamic models). The number of lagged dependent variables in the model is denoted as p and the number of lags of the independent variable as q: ARDL(p,q). A common ARDL model is the ARDL(1,1):

$y_{i, t} = α_{i} + ϕ_{1} y_{i, t - 1} + β_{1} x_{i, t} + β_{2} x_{i, t - 1} + e_{i, t} .$ (9)

The ARDL model can also be written as the algebraically equivalent error-correction model (ECM), where the contemporaneous values are modeled in first differences and the lags are in levels:

$Δ y_{i, t} = α_{i} - ϕ_{1}^{*} (y_{i, t - 1} - λ x_{i, t - 1}) + κ_{1}^{*} Δ x_{i, t} + e_{i, t} .$ (10)

$y_{i, t - 1} - λ x_{i, t - 1}$ is the cointegrating relationship between x and y. Cointegration is when a linear combination of two nonstationary series make a stationary process (see Engle and Granger 1987). $ϕ_{1}^{*}$ is the error-correction rate, indicating how quickly the model returns to equilibrium following a shock to an independent variable, and $λ$ is the long-run effect of $x_{i, t}$ (Webb, Linn, and Lebo 2020). The ECM can be rewritten as a generalized ECM (GECM) by multiplying through $- ϕ_{1}$ in Equation 10 to get Equation 11:

$Δ y_{i, t} = α_{i} + θ_{1}^{*} y_{i, t - 1} + β_{1}^{*} Δ x_{i, t} + β_{2}^{*} x_{i, t - 1} + e_{i, t} .$ (11)

Equations 9 through 11 are algebraically equivalent to one another. $ϕ_{1} - 1 = ϕ_{1}^{*} = θ_{1}^{*}$ , and $β_{1} + β_{2}$ in Equation 9 is equivalent to $ϕ_{1}^{*} λ$ in Equation 10 and $β_{2}^{*}$ in Equation 11. The short-run effect of $x_{i, t}$ is $β_{1} = κ_{1}^{*} = β_{1}^{*}$ in Equations 9 through 11, respectively. The long-run effect of $x_{i, t}$ is $\frac{{\hat{β}}_{1} + {\hat{β}}_{2}}{1 - {\hat{ϕ}}_{1}}$ in Equation 9, which is equivalent to $λ$ in Equation 10 and $- \frac{{\hat{β}}_{2}^{*}}{{\hat{θ}}_{1}^{*}}$ in Equation 11.

Table 1.

Dynamic Models.

Model Name	Model	Long-Run Effect
LDV	$y_{i, t} = α_{i} + ϕ_{1} y_{i, t - 1} + β_{1} x_{i, t} + e_{i, t}$	$\frac{{\hat{β}}_{1}}{1 - {\hat{ϕ}}_{1}}$
ARDL	$y_{i, t} = α_{i} + ϕ_{1} y_{i, t - 1} + β_{1} x_{i, t} + β_{2} x_{i, t - 1} + e_{i, t}$	$\frac{{\hat{β}}_{1} + {\hat{β}}_{2}}{1 - {\hat{ϕ}}_{1}}$
ECM	$Δ y_{i, t} = α_{i} - ϕ_{1}^{} (y_{i, t - 1} - λ x_{i, t - 1}) + κ_{1}^{} Δ x_{i, t} + e_{i, t}$	$\hat{λ}$
GECM	$Δ y_{i, t} = α_{i} + θ_{1}^{} y_{i, t - 1} + β_{1}^{} Δ x_{i, t} + β_{2}^{*} x_{i, t - 1} + e_{i, t}$	$- \frac{{\hat{β}}_{2}^{}}{{\hat{θ}}_{1}^{}}$

Note: LDV = lagged dependent variable; ARDL = autoregressive distributed lag; ECM = error-correction model; GECM = generalized error-correction model.

Estimating the Distribution of the Long-Run Effect

Researchers can calculate the distribution of the long-run effect over time by using the model estimates. In the calculation of the long-run effects, the denominator tells us how much the outcome variable changes over time given a shock to an independent variable (known as the speed of adjustment). In the LDV and ARDL models, an autoregressive coefficient near unity indicates that the dependent variable is highly persistent (and near zero for the ECM),⁴ and therefore, the long-run effect is distributed over a longer period of time. In contrast, an autoregressive coefficient near zero indicates that most of the effect occurs immediately.

We can manipulate the autoregressive coefficient in Equation 9 to illustrate this point. In a case where the dependent variable is highly persistent, $ϕ_{1}$ = 0.9 and $β_{1}$ = 0.4 and $β_{2}$ = 0.2, then by construction, the long-run effect is $\frac{0.4 + 0.2}{1 - 0.9} = 6$ .⁵ We can then calculate the proportion of the long-run effect occurring in each time period. Assuming a one-unit shock to $x_{i, t}$ , the proportion of the effect in the contemporaneous period is calculated by dividing the short-run effect by the long-run effect (0.4 / 6 = .067). In other words, 6.7 percent of the long-run effect occurs in the first period. Then in the second time period, we calculate the proportion of the long-run as $y_{i, t + 1}$ = 0.9(0.4) + 0.2 = 0.56. This suggests that 0.56 / 6 = .093, or 9.3 percent of the long-run effect occurs in the second period. Thus, after two periods, 6.7 percent + 9.3 percent = 16 percent of the long-run effect occurred. The proportion of the effect in the third period (and for the remaining time periods) is given by $y_{i, t + 2}$ = 0.9(0.56) = .50, indicating that .50 / 6 = .083, or 8.3 percent occurs in the third period. Thus, the long-run effect transpires relatively slowly over time.

If $ϕ_{1}$ = 0.1, we can see that the long-run effect is now $\frac{0.4 + 0.2}{1 - 0.1} = 0.67$ , and 59.7 percent (0.4 / 0.67 = .597) of the long-run effect occurs in the first period. For the second period, the effect is $y_{i, t + 1}$ = 0.1(0.4) + 0.2 = 0.24. This suggests that 0.24 / 0.67 = .358, or 35.8 percent of the long-run effect occurs in the second period, and $y_{i, t + 2}$ = 0.1(0.24) = .024 or .036 / .67 = .036, or 3.6 percent in the third period. In other words, the long-run effect transpires quickly relative to when $ϕ_{1}$ = 0.9. Figure 1 illustrates this by plotting the distribution of the long-run effect over 10 periods of time for each of these examples. As the figure illustrates, nearly all of the long-run effect occurs within the first three time periods when $ϕ_{1}$ = 0.1 but only 63.8 percent of the effect by T = 10 when $ϕ_{1}$ = 0.9.

Figure 1.

Distribution of the long-run effect over time by autoregressive coefficient.

Slope Heterogeneity

Slope heterogeneity, which is where the effect of a variable differs across units (e.g., the effect of a variable in a cross-national study is different in the United States than in Nigeria), is theoretically and statistically relevant for sociologists. In terms of theory testing, many macro-level sociological theories make arguments regarding heterogeneity—making it important to model to appropriately test different hypotheses. Statistically, slope heterogeneity makes the fixed effects estimator (and other pooled estimators) problematic to use.

We can see this by considering the following dynamic model with heterogeneous slopes that is estimated by pooled, fixed effects regression:

$\begin{matrix} y_{i, t} = λ_{1, i} y_{i, t - 1} + β_{1, i} x_{i, t} + α_{i} + ε_{i, t} \\ λ_{1, i} = λ + η_{1, i} \\ β_{1, i} = β + η_{2, i} \\ ε_{i, t} = η_{1, i} y_{i, t - 1} + η_{2, i} x_{i, t} + ξ_{i, t} . \end{matrix}$ (12)

Regardless of whether the lags are properly specified, fixed effects estimation of a dynamic model in the presence of slope heterogeneity produces inconsistent estimates because the heterogeneity gets captured in the error term (Pesaran and Smith 1995).⁶ The fixed effects estimator will produce unbiased and consistent (but inefficient) estimates in the presence of slope heterogeneity (Pesaran and Smith 1995). Even if $λ_{1}$ is truly homogeneous, slope heterogeneity in $x_{i, t}$ will lead $λ_{1}$ to tend toward unity and $β_{1}$ to 0 (Pesaran and Smith 1995). When lags of the regressors are included in the model, $β_{1}$ and the coefficient of $x_{i, t - 1}$ tend to take on the equal and opposite sign, and $λ_{1}$ will still tend toward 1 (Pesaran and Smith 1995). Thus, the long-run effect tends toward 0 because, based on the equation $(\frac{{\hat{β}}_{1} + {\hat{β}}_{2}}{1 - {\hat{λ}}_{1}}$ ), the numerator and the denominator tend toward 0 (Pesaran and Smith 1995:100).

To model slope heterogeneity, Pesaran and Smith (1995) propose the mean group (MG) estimator.⁷ The MG estimator runs a time-series regression on each unit and then takes the mean of all the estimates. MG estimation of the common ARDL(1,1) model is as follows:

$y_{i, t} = ϕ_{1, i} y_{i, t - 1} + β_{1, i} x_{i, t} + β_{2, i} x_{i, t - 1} + α_{i} + ε_{i, t},$ (13)

where the mean group coefficients are the average of the unit-specific coefficients (Ditzen 2018; Thombs et al. 2022):

$δ_{M G} = (ϕ_{M G}, β_{1, M G}, β_{2, M G}) = \frac{1}{N} \sum_{i = 1}^{N} δ_{i} .$ (14)

Cross-Sectional Dependence

Addressing cross-sectional dependence (cross-sectional units are correlated with one another) with large N, large T data is of growing interest in panel data econometrics (Bailey, Kapetanios, and Pesaran 2016; Chudik and Pesaran 2015a; Everaert and De Groote 2016; Pesaran 2015; Thombs et al. 2022). This expanding body of literature shows that failing to correct for it leads to inconsistent and biased estimates (Ditzen 2018; Pesaran 2006). Cross-sectional dependence can generally be characterized as weak or strong (see Chudik and Pesaran 2015b).Weak cross-sectional dependence is when the correlation between cross-sectional units converges to zero as N increases to infinity, whereas the correlation converges to a nonzero constant as N increases to infinity with strong cross-sectional dependence (Chudik, Pesaran, and Tosetti 2011; Ditzen 2018, 2021). Examples of weak cross-sectional dependence are when the units are correlated due to spatial distance, and spatial regression can be used to model it. On the other hand, strong cross-sectional dependence stems from unobserved common factors that tie units together. Common examples are pandemics, recessions, or technological change. In the large N, large T context, only weak cross-sectional dependence is needed to obtain consistent estimates, rather than the more restrictive assumption of cross-sectional independence, because correlation between a small number of units in the analysis does not impact estimates (Pesaran 2015).

Traditionally, cross-sectional dependence is dealt with using time fixed effects. Time fixed effects model the cross-sectional dependence by including T – 1 dummy variables of the time dimension. However, this approach makes strong assumptions that are often unrealistic, such as assuming common factors affect each unit in the analysis homogeneously. Such an assumption contrasts with an array of macro-comparative work that shows that structural factors have heterogeneous effects across nations and regions (Beckfield 2003; Chase-Dunn 1998; Givens, Huang, and Jorgenson 2019; Jorgenson and Clark 2012; VanHeuvelen and Summers 2019). Thus, time fixed effects may not adequately model the cross-sectional dependence in many cases.

Another approach is to use panel corrected standard errors, which adjust the standard errors but not the estimated coefficients (like robust standard errors).⁸ A fundamental difference between the two is that clustered robust standard errors assume that errors are not correlated across units. Beck and Katz (1995) first developed panel corrected standard errors and showed that they performed favorably to feasible generalized least squares, which were commonly used at the time. Assuming there are balanced data, panel corrected standard errors are estimated as follows:

${(\hat{V})}_{P C S E} = (X' X)^{- 1} (X' Ω X) (X' X)^{- 1},$ (15)

where Ω $Ω$ is a NT $\times NT$ block matrix with the contemporaneous covariances on the diagonal denoted by $Σ$ , which is defined as:

${\hat{Σ}}_{i j} = \frac{ϵ_{i}' ϵ_{j}}{T_{i j}},$ (16)

where $ϵ_{i}$ and $ϵ_{j}$ are the residuals for unit i and unit j and T is the number of time periods.

However, treating the cross-sectional dependence as a nuisance in the error term suffers from the same problems discussed in the previous section regarding autocorrelation. These approaches are sufficient if the cross-sectional dependence is truly part of the error term, but they will not adequately model the dependence if it is truly part of the data generating process. Given these considerations, we can turn to the common correlated effects estimator that offers a way to directly model the cross-sectional dependence.

Common Correlated Effects

Pesaran (2006) advanced the common correlated effects estimator to eliminate the effects of the unobserved common factors by using cross-sectional averages with OLS. This method contrasts with other common approaches that rely on correcting standard errors rather than explicitly modeling the cross-sectional dependence. The common correlated effects estimator allows these factors to affect each unit differently (Pesaran 2006). Assuming a model with heterogeneous slopes with one unobserved common factor, the estimator is based on Equation 17:

$\begin{matrix} y_{i, t} = α_{i} + β_{1, i} x_{i, t} + u_{i, t} \\ u_{i, t} = γ_{i} f_{t} + ε_{i, t}, \end{matrix}$ (17)

where $y_{i, t}$ is a function of time-varying variable $x_{i, t}$ , $α_{i}$ is the unit-specific fixed effect, and $u_{i, t}$ is the error term that consists of the unobserved common factor $f_{t}$ , with heterogeneous factor loadings $γ_{i}$ , and an independent and identically distributed error term, $ε_{i, t}$ , that is unit-specific (Ditzen 2018; Pesaran 2006). The actual equation the common correlated effects estimator estimates is Equation 18:

$y_{i, t} = α_{i} + β_{1, i} x_{i, t} + γ_{i}^{'} {\bar{υ}}_{t} + ε_{i, t},$ (18)

where ${\bar{υ}}_{t} = ({\bar{x}}_{t}, {\bar{y}}_{t})$ and $γ_{i}^{'}$ are the unit-specific factor loadings. Pesaran (2006) demonstrates that the cross-sectional averages will consistently estimate the common factors as long as $x_{i, t}$ is strictly exogenous. Chudik and Pesaran (2015a) show that the common correlated effects estimator is consistent with a dynamic model by adding a minimum of $\sqrt[3]{T}$ lags of the cross-sectional averages. Thus, the equation for an ARDL(1,1) model estimated by the common correlated effects estimator with p lags of the cross-sectional averages is as follows:

$y_{i, t} = λ_{1, i} y_{i, t - 1} + β_{1, i} x_{i, t} + β_{2, i} x_{i, t - 1} + \sum_{l = 0}^{p} γ_{i} {\bar{V}}_{t - l} + α_{i} + ε_{i, t} .$ (19)

Monte Carlo Experiments and Evidence

Monte Carlo Experiments

Five Monte Carlo experiments were conducted to compare the robustness of the fixed effects estimator and the common correlated effects estimator to autoregression (or persistence), slope heterogeneity, and cross-sectional dependence.⁹ The data generating process of the experiments are based on the following ARDL (p,q) model:

$y_{i, t} = λ_{1, i} y_{i, t - p} + β_{1, i} x_{i, t} + β_{2, i} x_{i, t - q} + γ_{i} f_{t} + c_{y i} + u_{i, t},$

where $y_{i, t}$ is the dependent variable, $x_{i, t}$ is the independent variable, $c_{y i}$ is the unit-specific fixed effect, and $γ_{i}$ are the heterogeneous factor loadings for the common factor $f_{t}$ . $λ_{1 i}$ , $β_{1 i}$ , and $β_{2 i}$ are the coefficients for the lagged dependent variable, the contemporaneous value of $x_{i, t}$ , and the lagged value of $x_{i, t}$ , respectively. $x_{i, t}$ is autoregressive ( $α_{x i}$ = 0.5) and a function of a unit-specific fixed effect ( $c_{x i}$ ) and a common factor ( $f_{t}$ ):

$x_{i, t} = α_{x i} x_{i, t - 1} + c_{x i} + γ_{x i} f_{t} + e_{i, t} .$

The unit-specific fixed effects are generated as:

$\begin{array}{l} c_{y i} \sim I I D N (1, 1) \\ c_{x i} = c_{y i} + ϱ_{i}, ϱ_{i} \sim I I D N (0, 1) . \end{array}$

The common factor is autoregressive:

$\begin{array}{l} f_{t} = ρ_{f} f_{t - 1} + ς_{f t}, ς_{f t} \sim I I D N (0, 1 - ρ^{2}) \\ ρ_{f} = 0.7, \end{array}$

and the error terms are independent and identically distributed:

$\begin{array}{l} u_{i, t} ~ I I D N (0, 1) \\ e_{i, t} ~ I I D N (0, 1 .) \end{array}$

I performed five experiments based on Equation 20, which I summarize in Table 2, Table 3, and Table 4. Experiments 1 through 3 follow an ARDL (1,1) and include a common factor and differ according to the degree of slope heterogeneity. Experiment 1 allows for the most variation in the slope coefficients, whereas Experiment 3 allows for the least, and Experiment 2 falls in the middle. Experiment 4 also follows an ARDL (1,1), but the slope coefficients are homogeneous. Experiment 5 restricts Equation 20 to an ARDL (0,0), that is, a static model, where the slope coefficients are homogeneous and there is no cross-sectional dependence. Experiment 5 is a “best case” scenario for the one-way fixed effects estimator. One thousand simulations are performed for each combination of N and T of 30, 60, and 90. Robust standard errors and panel corrected standard errors (potentially combined with a Prais-Winsten transformation) are used with the fixed effects estimator to test the robustness of these approaches (see Table 4). One hundred initial observations are generated and then dropped for each panel to prevent issues with initial conditions.

Table 2.

Summary of Monte Carlo Experiments.

Experiment	Model	Slope Heterogeneity	Common Factors
1	Dynamic	Yes	1
2	Dynamic	Yes	1
3	Dynamic	Yes	1
4	Dynamic	No	0
5	Static	No	0

Table 3.

Coefficients in Monte Carlo Experiments.

Coefficients	1	2	3	4	5
$λ_{1, i}$ (autoregressive coefficient)	IIDU (0, 0.9)	IIDU (0.2, 0.7)	IIDU (0.4, 0.6)	0.5	—
$β_{1, i}$ (short-run effect)	IIDU (0, 1)	IIDU (0.25, 0.75)	IIDU (0.4, 0.6)	0.5	0.5
$β_{2, i}$ (coefficient of $x_{i, t - 1}$ )	IIDU (0.2, 0.9)	IIDU (0.3, 0.7)	IIDU (0.4, 0.6)	0.5	—
$θ$ (long-run effect)	$\frac{{\hat{β}}_{1, i} + {\hat{β}}_{2, i}}{1 - {\hat{λ}}_{1, i}}$	$\frac{{\hat{β}}_{1, i} + {\hat{β}}_{2, i}}{1 - {\hat{λ}}_{1, i}}$	$\frac{{\hat{β}}_{1, i} + {\hat{β}}_{2, i}}{1 - {\hat{λ}}_{1, i}}$	$\frac{{\hat{β}}_{1} + {\hat{β}}_{2}}{1 - {\hat{λ}}_{1}}$	—
$γ_{i}, γ_{x i}$ (factor loadings)	IIDU (0.5, 1.5)	IIDU (0.5, 1.5)	IIDU (0.5, 1.5)	—	—

Table 4.

Estimators and Models used in the Monte Carlo Experiments.

Estimator	Model
1FE^R	$y_{i, t} = β_{1} x_{i, t} + α_{i} + ε_{i, t}$
2DFE^R	$y_{i, t} = λ_{1} y_{i, t - 1} + β_{1} x_{i, t} + β_{2} x_{i, t - 1} + α_{i} + ε_{i, t}$
2FE^R	$y_{i, t} = β_{1} x_{i, t} + α_{i} + u_{t} + ε_{i, t}$
2FE^PCSE/PSAR	$y_{i, t} = β_{1} x_{i, t} + α_{i} + u_{t} + ε_{i, t}$ ; $ε_{i, t} = ρ_{i} ε_{i, t - 1} + ξ_{i, t}$
2DFE^PCSE	$y_{i, t} = λ_{1} y_{i, t - 1} + β_{1} x_{i, t} + β_{2} x_{i, t - 1} + α_{i} + u_{t} + ε_{i, t}$
DFE^R	$y_{i, t} = λ_{1} y_{i, t - 1} + β_{1} x_{i, t} + β_{2} x_{i, t - 1} + α_{i} + u_{t} + ε_{i, t}$
CCE	$y_{i, t} = β_{1, i} x_{i, t} + γ_{i} {\bar{V}}_{t} + α_{i} + ε_{i, t}$
DCCE	$y_{i, t} = λ_{1, i} y_{i, t - 1} + β_{1, i} x_{i, t} + β_{2, i} x_{i, t - 1} + \sum_{l = 0}^{p} γ_{i} {\bar{V}}_{t - l} + α_{i} + ε_{i, t}$

Note: 1FE = one-way fixed effects; 2FE = two-way fixed effects; R = clustered robust standard errors; PCSE = panel corrected standard errors; PSAR = panel-specific autoregressive(1) correction; CCE = common correlated effects; D = dynamic.

Monte Carlo Results

The results of the Monte Carlo experiments are summarized for each experiment in the following. The results are reported in tabular format in the supplemental material. The bias, root mean square error (RMSE), and coverage are reported for each estimator. The bias reports the average difference between the estimator and the true parameter in percentage terms. The RMSE provides information on how much each estimator deviates from the true parameter, which is a function of how biased and inefficient an estimator is, and the coverage is the proportion of the confidence intervals that include the true parameter.

Experiment 1: dynamic, high slope heterogeneity, and cross-sectional dependence

As expected, the DCCE¹⁰ performs the best of the estimators in Experiment 1. The dynamic fixed effects estimators and the DCCE perform similarly when T = 30, but the DCCE is far superior as T increases. For the long-run effect ( $θ_{i}$ ), the bias and the RMSE of the dynamic fixed effects estimators increase as T increases, whereas the coverage decreases as T increases. For all DFE models when T = 90, the bias is greater than 38 percent, the RMSE is greater than 99, and the coverage does not exceed 86 percent. Furthermore, although the 2DFE^R and the 2DFE^PCSE produce the same bias and RMSE, the 2DFE^R performs better in terms of coverage. This suggests that PCSEs tend to be smaller than robust standard errors, which exacerbates the consequences of misspecification. In contrast, the bias and RMSE of the DCCE decrease as T increases, and the coverage increases as T increases. For the DCCE when T = 90, the bias does not excced ‒5 percent, the RMSE is less than 15, and the coverage is greater than 99 percent.

The results for $β_{1, i}$ , $λ_{1, i}$ , and $β_{2, i}$ shed light on how the short-run effects are driving the long-run estimates of each estimator. As expected for $β_{1, i}$ , the 2DFE and the DCCE have minimal bias (near zero), low RMSEs, and high coverage. However, the same is not true for $λ_{1, i}$ and $β_{2, i}$ . The bias of the DCCE is negative and tends toward 0 as T increases (as does the RMSE), while the dynamic fixed effects estimator is positively biased (tends toward 1), and the bias increases as T increases. When T = 90, the bias is greater than 69 percent for all DFE models. The inverse occurs for $β_{2, i}$ , where the dynamic fixed effects estimator produces a negative bias and the DCCE a positive bias.

It is also notable that the dynamic models perform much better than the static models in estimating $β_{1, i}$ . The 1FE^R performs by far the worst of any of the estimators (bias is greater than 194 percent for all combinations of N and T), which is expected because it does not account for the appropriate lag structure, slope heterogeneity, or cross-sectional dependence. The 2FE^R tends to produce positively biased coefficients (>71 percent for all combinations of N and T), while the 2FE^PCSE/PSAR produces negatively biased estimates. In absolute terms, the 2FE^PCSE/PSAR performs the best of the static models because it does not produce a bias greater than ‒29 percent for any combination of N and T.

Experiment 2: dynamic, medium slope heterogeneity, and cross-sectional dependence

The results for the long-run effect ( $θ_{i}$ ) in Experiment 2 are similar to those of Experiment 1, but the biases of the dynamic fixed effects estimators are not as great, which is expected given that there is less slope heterogeneity. The 2DFE produces slightly less bias (in absolute terms) than the DCCE when T = 30 (6.5 percent compared to ‒9.8 percent), but the bias and RMSE of the 2DFE increase as T increases. When T = 90, the bias is between 10.6 percent and 11.1 percent, and the RMSE is between 20.9 and 21.9. In contrast, the bias and RMSE for the DCCE tend toward zero as T increases. When T = 90, the bias is between ‒2.7 percent and ‒2.4 percent, and the RMSE is between 5 and 6.5. Like in Experiment 1, the coverage is greater for the 2DFE^R than for the 2DFE^PCSE, with the coverage for the 2DFE^R being between 45 and 61 percentage points higher.

The results for $β_{1, i}$ , $λ_{1, i}$ , and $β_{2, i}$ are also similar to Experiment 1. For $β_{1, i}$ , the 2DFE and the DCCE have minimal bias (near zero), low RMSEs, and high coverage. For $λ_{1, i}$ , the bias of the dynamic fixed effects estimators is again positive, which increases as T increases, but is smaller in magnitude than in Experiment 1. When T = 90, the bias of $λ_{1, i}$ is roughly 32 percent. The bias for $β_{2, i}$ in the 2DFE models is between ‒25.7 percent and ‒26.2 percent. In comparison, the bias for $λ_{1, i}$ is approximately ‒7 percent for the DCCE when T = 90, and the bias for $β_{2, i}$ is approximately 5 percent.

Like Experiment 1, the dynamic models are superior to the static models with regards to estimating $β_{1, i}$ . The 1FE^R performs by far the worst of any of the estimators (the bias is greater than 196 percent for all combinations of N and T), while the 2FE^PCSE/PSAR performs the best of the static models. The bias for the 2FE^PCSE/PSAR does not exceed ‒7.67 percent.

Experiment 3: dynamic, low slope heterogeneity, and cross-sectional dependence

The results for the long-run effect ( $θ_{i}$ ) in Experiment 3 follow a similar trend as Experiments 1 and 2, but the 2DFE exhibits minimal bias. The 2DFE performs well in estimating the long-run effect, with a bias less than 2 percent for all combinations of N and T, and the coverage is greater than 91 percent for all combinations of N and T. In comparison, the DCCE also performs well, and its bias and RMSE decrease as T increases, while its coverage increases as T increases. The 1DFE^R performs poorly compared to the other dynamic estimators because it exhibits bias (>23 percent for all combinations of N and T) and has very poor coverage (0 percent for all combinations of N and T >60). Overall, these results indicate that the 2DFE performs well with low levels of slope heterogeneity but is inconsistent as T increases.

The results for $β_{1, i}$ , $λ_{1, i}$ , and $β_{2, i}$ are also similar to Experiments 1 and 2, but the impact of slope heterogeneity is relatively small. For $β_{1, i}$ , the dynamic two-way fixed effects estimator and the DCCE have minimal bias (near zero), low RMSEs, and high coverage. For $λ_{1, i}$ , the bias of the dynamic fixed effects estimator remains positive (tends toward 1) but is significantly smaller in magnitude compared to the high slope heterogeneity case. For example, when N and T = 90, the bias of the 1DFE is 9.34 percent, and the bias for the 2DFE models is 5.35 percent. In comparison, the bias for $λ_{1, i}$ in the DCCE model is negative and tends toward 0 as T increases.

For $β_{2, i}$ , the bias of the dynamic fixed effects estimator increases as T increases but less so than in the first two experiments. The bias remains negative, indicating the coefficient tends toward the equal and opposite sign of $β_{1, i}$ . For example, when N and T = 90, the bias for the 1DFE is ‒14.69 percent, and the bias for the 2DFE models is ‒4.93 percent. In contrast, the DCCE produces a positive bias, and it converges to the true parameter as T increases, as evidenced by the bias decreasing from approximately 20 percent when T = 30 to approximately 5.9 percent when T = 90 for all combinations of N.

Like Experiments 1 and 2, the dynamic models outperform the static models in estimating $β_{1, i}$ . The 1FE^R performs the worst of any of the estimators, with a bias greater than 209 percent for all combinations of N and T. This suggests that the 1FE^R tends to perform poorly regardless of the level of slope heterogeneity. The 2FE^PCSE/PSAR performs the best of the static models, with a bias between ‒2.16 percent and ‒3.68 percent for all combinations of N and T. Its RMSE and coverage also far outperform the 1FE^R and 2FE^R.

Experiment 4: dynamic, no slope heterogeneity, and cross-sectional dependence

Experiment 4 differs from the first three experiments because the slope coefficients are homogeneous. The 2DFE performs the best in terms of estimating the long-run effect ( $θ_{i}$ ), and the bias (<–3.15 percent for all combinations of N and T) and the RMSE tend toward 0 as T increases. The DCCE also performs well, with the bias (<–8.76 percent for all combinations of N and T) and RMSE tending toward 0 as T increases. The 1DFE^R performs poorly, with the bias and RMSE increasing as T increases, and it has very poor coverage. The bias (between 23.24 percent and 28.07 percent) is similar to its bias in the other experiments with slope heterogeneity.

For $β_{1, i}$ , the 2DFE and the DCCE have minimal bias, low RMSEs, and high coverage. For $λ_{1, i}$ and $β_{2, i}$ the biases of the 2DFE and the DCCE tend toward 0 as T increases. However, the DCCE has poor coverage for $λ_{1, i}$ (<34.8 percent for all combinations of N and T). For $β_{2, i}$ , the 1DFE^R produces negatively biased coefficients that are more biased as N increases, and in contrast to the experiments with slope heterogeneity, the 2DFE produces positively biased coefficients (between 1.62 percent and 5.01 percent for all combinations of N and T) where the bias decreases as T increases. The DCCE produces positively biased coefficients, which decrease as T increases but remain larger than the 2DFE for all combinations of N and T.

Moving to the static models, the 2FE^PCSE/PSAR performs the best of the static models, with biases near 0 for all combinations of N and T, and it has a much lower RMSE and higher coverage compared to the other static models. The 1FE^R performs the worst of any of the estimators, with a bias greater than 209 percent for all combinations of N and T.

Experiment 5: static, no slope heterogeneity, and no cross-sectional dependence

Experiment 5 is a “best case” scenario for the one-way fixed effects estimator because there is no slope heterogeneity or cross-sectional dependence. The estimator does perform well, with biases near 0, a low RMSE, and high coverage for all combinations of N and T for the static and dynamic models. However, two-way fixed effects and CCE estimation of dynamic and static models perform just as well. This indicates that the one-way fixed effects estimator does not offer superior performance compared to the other estimators even in a best case scenario. A summary table of the estimators/models across the experiments is found in Table 5.

Table 5.

Summary of the Monte Carlo Experiments by Estimator and Model.

Estimator/Model	Summary
1DFE^R	Performs poorly in the presence of slope heterogeneity and cross-sectional dependence. It produces highly biased estimates, and the bias increases as T increases.
2DFE^R	Performs poorly with a high degree of slope heterogeneity but does pretty well with medium to low levels of heterogeneity. Tends to outperform the DCCE when T = 30, but its performance worsens as T increases.
2DFE^PCSE	Performs the same as the 2DFE^R in terms of bias and RMSE because it produces the same estimated coefficient. Its coverage is worse than the 2DFE^R because it tends to underestimate the standard errors.
DCCE	Performs the best in terms of bias, RMSE, and coverage when there is slope heterogeneity and strong cross-sectional dependence, particularly as T increases.
1FE^R	Strong assumptions are needed for it to perform well (no autoregression, no slope heterogeneity, and no strong cross-sectional dependence). Should be used with extreme caution.
2FE^R	Performs poorly (but better than the 1FE^R) because it assumes no autoregression, no slope heterogeneity, and homogeneous factor loadings.
2FE^PCSE/PSAR	Performs the best of the static models and produces negatively biased coefficients, whereas the other static models produce positive bias. However, like the other static models, its performance worsens as T increases.
CCE	Although it models slope heterogeneity and cross-sectional dependence, it produces highly biased estimates (because it assumes no autoregression). Its performance worsens as T increases.

An Example: Drivers of U.S. State-Level Fossil Fuel Consumption

Large N, large T data are commonly used in environmental sociology to study the drivers of environmental change (Dietz 2017; Jorgenson and Clark 2012; Jorgenson et al. 2019). I extend this literature by examining the drivers of fossil fuel consumption in the United States from 1960 to 2018. Fossil fuel consumption data are obtained from the U.S. Energy Information Agency (2021). Drawing from the Stochastic Impacts by Regression on Population, Affluence and Technology (STIRPAT) framework (York et al. 2003), I include real disposable personal income deflated by the CPI-U-US (Bureau of Economic Analysis 2021), total population (U.S. Energy Information Administration 2021), the industrial sector’s share of energy consumption (U.S. Energy Information Administration 2021), the share of income going to the top 10 percent of earners (Frank 2021), and the percentage energy consumption from renewables (U.S. Energy Information Administration 2021) as independent variables.¹¹ In the following, I report the dynamic and static models for the one-way fixed effects estimator with robust standard errors (1FE^R), the two-way fixed effects estimator with robust standard errors (2FE^R), the two-way fixed effects estimator with panel-corrected standard errors (2FE^PCSE),¹² and the common correlated effects (CCE) estimator. Models that are dynamic start with the letter “D.” The descriptive statistics are reported in Table 6, and the slope heterogeneity tests are reported in Table 7. The variables exhibit strong cross-sectional dependence (see Table 6), and the slope coefficients are heterogeneous according to all three slope heterogeneity tests.

Table 6.

Descriptive Statistics.

Variable	Measurement Unit	Mean (SD)	Between SD	Within SD	CD
Fossil fuel consumption	British thermal units (millions)	1.39 (1.62)	1.57	.43	174.17*
Personal income	Per capita	30,882.33 (10,283.30)	4,329.06	9,347.03	269.84*
Population	Total	4,945.89 (5,577.46)	5,380.77	1,647.32	231.75*
Industrial share	% of energy consumption	35.08 (12.91)	11.39	6.27	117.55*
Top 10% share	% of income	28.42 (6.39)	3.04	5.64	248.58*
Renewable energy (%)	% of energy consumption	9.73 (11.36)	10.89	3.59	88.03*

Note: Data are perfectly balanced with 3,009 total observations (N = 51, T = 59). Generated with the xtsum and xtcd2 (Ditzen 2018) commands in Stata 17. CD = Statistic for Pesaran test for weak cross-sectional dependence (H₀ = weak cross-sectional dependence).

p <.05.

Table 7.

Slope Heterogeneity Tests.

Test	Adjusted Delta Statistic
Blomquist and Westerlund HAC	11.11*
Pesaran-Yamagata test with CSAs	9.75*
Pesaran-Yamagata test AR adjusted	15.02*

Note: $H_{0}$ = slope homogeneity. The Bartlett-Westerlund HAC test is estimated with bandwidth of 3. The Pesaran-Yamagata test with cross-sectional averages incorporates three lags of each contemporaneous variable. Generated with the xthst (Bersvendsen and Ditzen 2021) command in Stata 17. AR = autoregressive; CSA = cross-sectional average; HAC = heteroscedasticity and autocorrelation consistent.

p < .05.

The results of the dynamic models are reported in Table 8, and the static models are reported in Table 9 for reference. For the dynamic models, the coefficients behave as expected. The autoregressive coefficient is near unity in the fixed effects models, while it is .504 for the DCCE model. The short-run coefficients and the coefficients on the lags of the independent variables tend to take on equal and opposite signs in the fixed effects models, which is also expected. Pesaran’s test for weak cross-sectional dependence (reported as the CD-statistic in Table 8) indicates strong cross-sectional dependence in the 1DFE. The CD statistic is less in magnitude with the inclusion of time fixed effects, but the statistic is significant at the .05 level. The use of PCSEs does little to mitigate the strong cross-sectional dependence because the CD statistic remains the same value.

Table 8.

Dynamic Models of Fossil Fuel Consumption, 1960–2018.

	1DFE^R	2DFE^R	2DFE^PCSE	DCCE
${Fossil fuel consumption}_{t - 1}$	.915* (.010)	.925* (.012)	.925* (.009)	.508* (.023)
Short-run effects
Personal income	.301* (.078)	.064 (.047)	.064 (.048)	–.024 (.054)
Population	.558* (.119)	.697* (.139)	.697* (.140)	.343# (.207)
Industrial share	.133* (.051)	.128* (.046)	.128* (.016)	.214* (.042)
Top 10% share	.032 (.028)	.061* (.029)	.061* (.029)	.005 (.026)
Renewable energy (%)	–.085* (.010)	–.067* (.009)	–.067* (.007)	–.089* (.014)
$x_{i, t - 1}$
Personal income	–.274* (.074)	–.029 (.046)	–.029 (.047)	.040 (.047)
Population	–.519* (.116)	–.653* (.135)	–.653* (.139)	–.155 (.217)
Industrial share	–.102* (.048)	–.104* (.044)	–.104* (.016)	–.082* (.032)
Top 10% share	–.033 (.026)	–.065* (.032)	–.065* (.029)	.025 (.027)
Renewable energy (%)	.074* (.010)	.062* (.010)	.062* (.007)	.042* (.010)
Long-run effects
Personal income	.311* (.071)	.472* (.181)	.472* (.249)	.062 (.112)
Population	.457* (.118)	.579* (.137)	.579* (.107)	.486* (.209)
Industrial share	.370* (.057)	.326* (.045)	.326* (.075)	.310* (.074)
Top 10% share	–.017 (.177)	–.051 (.272)	–.051 (.264)	.063 (.091)
Renewable energy (%)	–.126* (.026)	–.065# (.036)	–.065# (.034)	–.112* (.030)
Observations (N/T)	2,958 (51/58)	2,958 (51/58)	2,958 (51/58)	2,805 (51/55)
CD statistic	70.650*	–3.875*	–3.875*	–.460
CIPS	–6.164*	–6.060*	–6.075*	–6.162*
CSA variables (lags)	Fossil fuel consumption, renewable energy (%) (3)

Note: Standard errors are in parentheses. Generated with the xtreg, xtpcse, and xtdcce2 (Ditzen 2018) commands in Stata 17. 1FE = one-way fixed effects; 2FE = two-way fixed effects; PCSE = panel corrected standard errors; CCE= common correlated effects; CCE = common correlated effects; D = dynamic; CD-Statistic = Test Statistic for Weak Cross-Sectional Dependence Test; CIPS = Pesaran Panel Unit Root Test in the Presence of Cross-Sectional Dependence; CSA = crosssectional average.

$p <$ .05; ^#p < .10.

Table 9.

Static Models of Fossil Fuel Consumption, 1960–2018.

	1FE^R	2FE^R	2FE^PCSE/PSAR	CCE
Δ Personal income	.384* (.092)	.060 (.046)	.069 (.045)	.024 (.042)
Δ Population	.780* (.139)	.786* (.161)	.803* (.121)	.520* (.107)
Δ Industrial share	.127* (.052)	.121* (.047)	.118* (.016)	.210* (.041)
Δ Top 10% share	.035 (.028)	.070* (.030)	.071* (.028)	–.014 (.023)
Δ Renewable energy (%)	–.086* (.011)	–.066* (.011)	–.064* (.007)	–.088* (.013)
Observations (N/T)	2,958 (51/58)	2,958 (51/58)	2,958 (51/58)	2,958 (51/58)
CD statistic	76.859*	–3.792*	177.513*	–.232
CIPS	–6.151*	–6.161*	–2.565*	–6.132*
CSA variables	Δ Fossil fuel consumption,Δ renewable energy (%)

Note: ∆ = first difference. Standard errors are in parentheses. Generated with the xtreg, xtpcse, and xtdcce2 (Ditzen 2018) commands in Stata 17. 1FE = one-way fixed effects; 2FE = two-way fixed effects; PCSE = panel corrected standard errors; PSAR(1) = panel-specific autoregressive(1) correction; CCE= common correlated effects; CD-Statistic = Test Statistic for Weak Cross-Sectional Dependence Test; CIPS = Pesaran Panel Unit Root Test in the Presence of Cross-Sectional Dependence; CSA = crosssectional average.

p < .05.

Turning to the short-run effects, industrial share and population are positively associated with fossil fuel consumption, while renewable energy is negatively associated with fossil fuel consumption across all four models. There is disagreement between the models regarding the short-run impacts of personal income and the top 10 percent share. The 1DFE^R model finds that personal income is positive and statistically significant, whereas the 2DFE models and the DCCE find no statistically significant relationship. The top 10 percent share coefficient is positive and statistically significant for both 2DFE models, but it is not statistically significant in the 1DFE or DCCE model.

The long-run effects are similar across the four models, but there are some notable differences between the estimators. The DCCE finds that personal income is not associated with fossil fuel consumption, but the 1DFE and 2DFE estimate a sizeable long-run association between income and fossil fuel consumption. The discrepancy between the 1/2DFE and DCCE arises primarily from the autoregressive coefficient tending toward unity, which creates highly misleading results, as theory predicts (Pesaran and Smith 1995) and the Monte Carlo experiments showed. Likewise, although population is statistically significant across all of the models, the 1DFE and 2DFE estimates are counterintuitive because they indicate that the short-run effect is greater than the long-run effect. For the 2DFE models, a 1 percent increase in population is associated with a .697 percent increase in fossil fuel consumption at T = 1, but the effect is .579 percent in the long run. Given that prior research consistently finds that population has at least a proportionate impact on environmental outcomes, it is unlikely that the effect would become smaller over time (Rosa and Dietz 2012).

In the static models (Table 9), the differences between the models are similar to the dynamic case. First differences are used because the 2FE models in levels are nonstationary according to the Pesaran panel unit-root test in the presence of cross-section dependence (available on request). Personal income is positive and statistically significant for the 1FE but not for the 2FE models or the CCE. The top 10 percent share coefficient is statistically significant in both 2FE models but not the other models. The other coefficients are similar across the models. Both 2FE models suffer from strong cross-sectional dependence, and the CD statistic for 2FEPCSE/PSAR is even larger in absolute magnitude than the 1FE model—suggesting the use of PCSEs does not mitigate strong cross-sectional dependence. A summary of the results across Table 8 and Table 9 is found in Table 10.

Table 10.

Summary of Results: Drivers of U.S. State-Level Fossil Fuel Consumption.

					Static Models
Dynamic Models	1DFE^R	2DFE^R	2DFE^PCSE	DCCE	1FE^R	2FE^R	2FE^PCSE/PSAR	CCE
Short-run effects
Personal income	+				+
Population	+	+	+	+	+	+	+	+
Industrial share	+	+	+	+	+	+	+	+
Top 10% share		+	+			+	+
Renewable energy (%)	–	–	–	–	–	–	–	–
Long-run effects
Personal income	+	+	+
Population	+	+	+	+
Industrial share	+	+	+	+
Top 10% share
Renewable energy (%)	–	–	–	–
Strong CSD?	Yes	Yes	Yes	No	Yes	Yes	Yes	No

Note: Space is left blank if coefficient is not statistically significant at .10 level. 1FE = one-way fixed effects; 2FE = two-way fixed effects; PCSE = panel corrected standard errors; PSAR(1) = panel-specific autoregressive(1) correction; CCE= common correlated effects; CCE= common correlated effects; D = dynamic; strong CSD = strong cross-sectional dependence.

Conclusion and Recommendations for Modeling Large N, Large T Data

There are three key takeaways from the Monte Carlo experiments and the example presented. First, failing to correctly model the lag structure, slope heterogeneity, and cross-sectional dependence produces biased and inconsistent results. Second, the performance of the dynamic two-way fixed effects estimator relative to the common correlated effects estimator depends on T. Third and finally, use of the one-way fixed effects estimator requires the researcher to make strong assumptions that are often implausible. Even when it is theoretically appropriate to use, it performs similarly to other estimators. Its use is only appropriate when the researcher has strong evidence that there is no slope heterogeneity or cross-sectional dependence.

Based on the theoretical discussion earlier in the article and the takeaways from the experiments, I present a decision-making framework in flowchart form in Figure 2. As the framework illustrates, there are theoretical and statistical decisions the researcher must make. Researchers must first decide the appropriate ARDL(p, q) or ECM(p, q) model to estimate. In Step 2, they should test for slope heterogeneity and then, based on these results, estimate a model using the MG estimator (3a) or 1FE estimator (3b). Following estimation, researchers should test for strong cross-sectional dependence in the residuals. If there is no strong cross-sectional dependence, then researchers can report their estimates from 3a or 3b. If there is strong cross-sectional dependence, then they should use the CCE (5a) or the pooled CCE (assumes homogeneous slopes) or the 2FE estimator (5b) depending on whether it eliminates the strong cross-sectional dependence.

Figure 2.

Decision-making framework for modeling large N, large T data.

To conclude, I make several remarks and suggestions regarding the framework:

1. Use theory to guide the modeling approach. Rather than assuming fixed effects estimation of a static model is appropriate, researchers should explicitly connect their modeling strategy to the theories they are testing. Researchers should consider whether theories are static or dynamic in nature and whether effects are heterogeneous.

If there is uncertainty regarding lag specification, the best way to tell whether a dynamic model is appropriate is to include lags of the dependent (and independent) variables in the model. Including them will at once model the dynamic process and correct the issue of autocorrelation (Pickup 2015). If a standard error correction is used instead, it will be misspecified if the data generating process is truly dynamic. Thus, researchers should always start with a dynamic model unless they have a strong theoretical reason to believe otherwise.

Moreover, many sociological theories expect that social processes function differently across spatial and social contexts. Thus, the homogeneity assumptions of the fixed effects estimator often are too restrictive, making the flexibility of heterogeneous panel estimators like the CCE more apt to use in many cases.

2. Estimate several models to consider the impact of lag misspecification. Although using theory can give researchers a general idea whether processes are static or dynamic, they should always estimate several different models to derive robust estimates. Estimating multiple models can help researchers better understand their data and results, and it provides transparency to readers. Following a general-to-specific approach to modeling (Hendry 1995), researchers should start by specifying a general model that serves as a probable representation of the underlying data generating process and then restrict the model as necessary if lags are not statistically significant. Starting with an ARDL(1,1) model is common. However, more lags can be added, and researchers should examine how changing the lag length influences their results. Of course, this is not a flawless approach. Using too few lags can lead to biased estimates, and too many lags can reduce efficiency and affect power (Chudik et al. 2016).

3. Testing for slope heterogeneity and strong cross-sectional dependence. Testing for strong cross-sectional dependence and slope heterogeneity is easy to do in common software programs like Stata. Pesaran’s test for weak cross-sectional dependence (the CD test) can be implemented with the xtcd2 package (Ditzen 2018), and the exponent test for cross-sectional dependence developed by Bailey et al. (2016) can be implemented with the xtcse2 command (Ditzen 2021). The exponent test estimates the strength of the common factors responsible for cross-sectional dependence. For slope heterogeneity, Bersvendsen and Ditzen’s (2021) xthst command can be used. Finding that the fixed effects estimator produces a coefficient on $y_{i, t - 1}$ close to 1 and equal and opposite coefficients on $x_{i, t}$ and $x_{i, t - 1}$ can also alert researchers that slope heterogeneity is likely an issue.

4. The number of time periods matters. Taken together, these recommendations suggest that the common correlated effects estimator is more appropriate to use in many cases. However, in addition to the aforementioned considerations, the estimator of choice will depend on the number of regressors in the model and the length of the temporal dimension. The available degrees of freedom can be small with the DCCE when there are a large number of regressors in the model. This is particularly an issue when T ~ 30. For example, if an ARDL(1,1) is estimated with three regressors, 30 waves of data, and $\sqrt[3]{30}$ = 3 lags of the cross-sectional averages for all of the variables, then the degrees of freedom are 30 – 4 (time points dropped because of lags) – 1 (constant) – 7 (regressors) – 16 (contemporaneous and lagged cross-sectional averages) = 2. The Monte Carlo experiments also suggest that the bias of the two estimators are similar when T ~ 30, even with high slope heterogeneity, so the fixed effects estimator may be more appropriate to use in this case.

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Footnotes

I thank Andrew Jorgenson for helpful comments and feedback.

ORCID iD

Ryan P. Thombs

Supplemental Material

Supplemental material for this article is available online.

Author Biography

Ryan P. Thombs is a PhD candidate in the sociology department at Boston College. His research examines the drivers of global environmental change,the drivers of health disparities and population health outcomes,and quantitative methodology,particularly related to panel data and time-series modeling. His work appears in journals such as Sociological Methodology,the Journal of Health and Social Behavior,Sociological Forum,The Sociological Quarterly,Sociology of Development,Climatic Change,and Energy Research & Social Science.

References

Allison

Paul D.

2009. Fixed Effects Regression Models. Thousand Oaks, CA: SAGE.

Allison

Paul D.

Williams

Richard

Moral-Benito

Enrique

. 2017. “Maximum Likelihood for Cross-Lagged Panel Models with Fixed Effects.” Socius 3. doi:10.1177/2378023117710578.

Bailey

Natalia

Kapetanios

George

Pesaran

M. Hashem

. 2016. “Exponent of Cross-Sectional Dependence: Estimation and Inference.” Journal of Applied Econometrics 31(6):929–60.

Beck

Nathaniel

Katz

Jonathan N.

1995. “What to Do (and Not to Do) with Time-Series Cross-Section Data.” The American Political Science Review 89(3):634–47.

Beckfield

Jason

. 2003. “Inequality in the World Polity: The Structure of International Organization.” American Sociological Review 68(3):401–24.

Bersvendsen

Tore

Ditzen

Jan

. 2021. “Testing for Slope Heterogeneity in Stata.” The Stata Journal 21(1):51–80.

Bureau of Economic Analysis. 2021. “Personal Income by State.” https://www.bea.gov/data/income-saving/personal-income-by-state.

Cameron

A. Colin

Trivedi

Pravin K.

2010. Microeconometrics Using Stata. College Station, TX: Stata Press.

Chase-Dunn

Christopher

. 1998. Global Formation: Structures of the World-Economy. Lanham, MD: Rowman & Littlefield.

10.

Chudik

Alexander

Mohaddes

Kamiar

Pesaran

M. Hashem

Raissi

Mehdi

. 2016. “Long-Run Effects in Large Heterogeneous Panel Data Models with Cross-Sectionally Correlated Errors.” Pp. 85–135 in Essays in Honor of Man Ullah. Advances in Econometrics, vol. 36, edited by GonzÁlez-Rivera

Hill

R. Carter

Lee

T.-H.

Bingley, UK: Emerald Group Publishing Limited.

11.

Chudik

Alexander

Pesaran

M. Hashem

. 2015a. “Common Correlated Effects Estimation of Heterogeneous Dynamic Panel Data Models with Weakly Exogenous Regressors.” Journal of Econometrics 188(2):393–420.

12.

Chudik

Alexander

Pesaran

M. Hashem

. 2015b. “Large Panel Data Models with Cross-Sectional Dependence: A Survey.” Pp. 2–45 in The Oxford Handbook of Panel Data, edited by B. H. Baltagi. Oxford, UK: Oxford University Press.

13.

Chudik

Alexander

Pesaran

M. Hashem

Tosetti

Elisa

. 2011. “Weak and Strong Cross-Section Dependence and Estimation of Large Panels.” The Econometrics Journal 14(1):C45–90.

14.

De Boef

Suzanna

Keele

Luke

. 2008. “Taking Time Seriously.” American Journal of Political Science 52(1):184–200.

15.

Dietz

Thomas

. 2017. “Drivers of Human Stress on the Environment in the Twenty-First Century.” Annual Review of Environment and Resources 42(1):189–213.

16.

Ditzen

Jan

. 2018. “Estimating Dynamic Common-Correlated Effects in Stata.” The Stata Journal 18(3):585–617.

17.

Ditzen

Jan

. 2021. “Estimating Long-Run Effects and the Exponent of Cross-Sectional Dependence: An Update to xtdcce2.” The Stata Journal 21(2):687–707.

18.

Driscoll

John C.

Kraay

Aart C.

1998. “Consistent Covariance Matrix Estimation with Spatially Dependent Panel Data.” The Review of Economics and Statistics 80(4):549–60.

19.

Engle

Robert F.

Granger

C. W. J.

1987. “Co-Integration and Error Correction: Representation, Estimation, and Testing.” Econometrica 55:251–76.

20.

Everaert

Gerdie

De Groote

Tom

. 2016. “Common Correlated Effects Estimation of Dynamic Panels with Cross-Sectional Dependence.” Econometric Reviews 35(3):428–63.

21.

Frank

Mark W.

2021. “U.S. State Level Income Inequality Data.” https://www.shsu.edu/ecomwf/inequality.html.

22.

Givens

Jennifer E.

Huang

Xiaorui

Jorgenson

Andrew K.

2019. “Ecologically Unequal Exchange: A Theory of Global Environmental Injustice.” Sociology Compass 13(5):e12693. doi:10.1111/soc4.12693.

23.

Hendry

David F.

1995. Dynamic Econometrics. Oxford, UK: Oxford University Press.

24.

Jorgenson

Andrew K.

Clark

Brett

. 2012. “Are the Economy and the Environment Decoupling? A Comparative International Study, 1960–2005.” American Journal of Sociology 118(1):1–44.

25.

Jorgenson

Andrew K.

Fiske

Shirley

Hubacek

Klaus

Jia

McGovern

Tom

Rick

Torben

Schor

Juliet B.

Solecki

William

York

Richard

Zycherman

Ariela

. 2019. “Social Science Perspectives on Drivers of and Responses to Global Climate Change.” Wiley Interdisciplinary Reviews: Climate Change 10(1):e554. doi:10.1002/wcc.554.

26.

Kelly

Orla

. 2020. “The Silver Bullet? Assessing the Role of Education for Sustainability.” Social Forces 99(1):178–204.

27.

Kollmeyer

Christopher

Peters

John

. 2019. “Financialization and the Decline of Organized Labor: A Study of 18 Advanced Capitalist Countries, 1970–2012.” Social Forces 98:1–30.

28.

Leszczensky

Lars

Wolbring

Tobias

. 2022. “How to Deal with Reverse Causality Using Panel Data? Recommendations for Researchers Based on a Simulation Study” Sociological Methods & Research 51(2):837–65.

29.

Pesaran

M. Hashem

. 2006. “Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure.” Econometrica 74(4):967–1012.

30.

Pesaran

M. Hashem

. 2015. “Testing Weak Cross-Sectional Dependence in Large Panels.” Econometric Reviews 34:1089–117.

31.

Pesaran

M. Hashem

Shin

Yongcheol

Smith

Ron P.

1999. “Pooled Mean Group Estimation of Dynamic Heterogeneous Panels.” Journal of the American Statistical Association 94(446):621–34.

32.

Pesaran

M. Hashem

Smith

Ron

. 1995. “Estimating Long-Run Relationships from Dynamic Heterogeneous Panels.” Journal of Econometrics 68(1):79–113.

33.

Pickup

Mark

. 2015. Introduction to Time Series Analysis. Thousand Oaks, CA: SAGE.

34.

Plümper

Thomas

Troeger

Vera E.

2019. “Not So Harmless After All: The Fixed-Effects Model.” Political Analysis 27(1):21–45.

35.

Rosa

Eugene A.

Dietz

Thomas

. 2012. “Human Drivers of National Greenhouse-Gas Emissions.” Nature Climate Change 2:581–86.

36.

Thombs

Ryan P.

Huang

Xiaorui

Fitzgerald

Jared B.

2022. “What Goes Up Might Not Come Down: Modeling Directional Asymmetry with Large N, Large T Data.” Sociological Methodology 52(1):1–29.

37.

U.S. Energy Information Administration. 2021. “State Energy Data System (SEDS).” https://www.eia.gov/state/seds/seds-data-complete.php?sid=US.

38.

Vaisey

Stephen

Miles

Andrew

. 2017. “What You Can—and Can’t—Do With Three-Wave Panel Data.” Sociological Methods & Research 46(1):44–67.

39.

VanHeuvelen

Tom

Summers

Nik

. 2019. “Divergent Roads: A Cross-National Intercohort Analysis of Affluence and Environmental Concern.” Social Science Research 82:72–91.

40.

Webb

Clayton

Linn

Suzanna

Lebo

Matthew J.

2020. “Beyond the Unit Root Question: Uncertainty and Inference.” American Journal of Political Science 64(2):275–92.

41.

World Bank. 2021. “World Development Indicators.” https://databank.worldbank.org/source/world-development-indicators.

42.

York

Richard

Rosa

Eugene A.

Dietz

Thomas

. 2003. “Footprints on the Earth: The Environmental Consequences of Modernity.” American Sociological Review 68(2):279–300.

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