STRUCTURAL EQUATION MODELING: A MULTIDISCIPLINARY JOURNAL
https://doi.org/10.1080/10705511.2020.1821690

On the Differences between General Cross-Lagged Panel Model andRandom-Intercept Cross-Lagged Panel Model: Interpretation of Cross-Lagged
Parameters and Model Choice
Satoshi Usami
University of Tokyo
ABSTRACT

Many methods have been developed to infer reciprocal relations between longitudinally observed
variables. Among them, the general cross-lagged panel model (GCLM) is the most recent development
as a variant of the cross-lagged panel model (CLPM), while the random-intercept CLPM (RI-CLPM) has
rapidly become a popular approach. In this article, we describe how common factors and cross-lagged
parameters included in these models can be interpreted, using a unified framework that was recently
developed. Because common factors are modeled with lagged effects in the GCLM, they have both direct
and indirect influences on observed scores, unlike stable trait factors included in the RI-CLPM. This
indicates that the GCLM does not control for stable traits as the RI-CLPM does, and that there areinterpretative differences in cross-lagged parameters between these models. We also explain that
including such common factors as well as moving-average terms in the GCLM makes this interpretation
very complicated.

Many researchers aim to uncover reciprocal (or, mutual orprospective) relations between longitudinally observed vari­ables, and we have seen an increased number of such studies
in the behavioral sciences. For example, during the past year
more than 1,000 published psychology papers have dealt withthis type of relation.1 For this analytic purpose, the use of thecross-lagged panel model (CLPM) and estimation via struc­tural equation modeling (SEM) have been a standard approachfor decades in the behavioral sciences. In this model, a crosslagged coefficient, which indicates a path from one variablemeasured at a time point t 1 to another measured at time
point t, is a key parameter. Many alternatives to the CLPM
have been proposed in various disciplines using SEM
approaches (see Orth et al., in press; Usami, Murayama et al.,2019). Notably, in the last few years the random-intercept
CLPM (RI-CLPM; Hamaker et al., 2015) has rapidly become
a popular approach among psychologists, reaching more than
600 citations on Google Scholar as of August 2020. A major
strength of this model is that it can account for stable trait
factors that control for stable individual differences, allowing
researchers to infer within-person relations between variables.
Usami, Murayama et al. (2019) proposed a unified frame­
work to clarify the mathematical and conceptual similarities
and differences among various longitudinal models. This fra­mework revealed that existing SEM-based longitudinal models
can be classified according to whether the model posits uniquefactors and/or (dynamic) residuals, and what types of common

factors are used to model changes. They argued that the latter isessential to understanding how cross-lagged parameters can beinterpreted in each model, and showed from the viewpoint ofa potential outcome (or counterfactual) approach (the Rubincausal model; Rubin, 1974) that including stable trait factors as
in the RI-CLPM is mathematically equivalent to controlling forlatent (unobserved) time-invariant confounders.
Although longitudinal designs have numerous advantages
over cross-sectional designs (e.g., McArdle & Nesselroade,2014), the issue of causal inference becomes complicated andchallenging in general for longitudinal studies, because research­ers must effectively account for time-varying and time-invariant
confounders. For this reason, efforts by researchers to devise
a better methodology are continuing (e.g., Asparouhov et al.,2018; Hamaker et al., 2015; Imai & Ratkovic, 2015; Robins, 1999;
Robins & Hernán, 2009; Zyphur et al., 2020a).
Among such methodologies, the general cross-lagged panel
model (GCLM; Zyphur et al., 2020a, 2020b) is a recent variant of
the CLPM. This model was not covered in the discussion of
Usami, Murayama et al. (2019). The GCLM as a SEM-based
approach assumes (time-varying) unit effects as well as moving
average (MA) and cross-lagged moving average (CLMA) terms,
aiming to increase the range of dynamic processes that can bemodeled. Zyphur et al. (2020b) discuss the relation between the
GCLM and other longitudinal models such as the latent growthmodel (LGM; Meredith & Tisak, 1984, 1990) and the autoregres­sive latent trajectory (ALT) model (Bollen & Curran, 2004, 2006;

CONTACT Satoshi Usami
usami_s@p.u-tokyo.ac.jp
University of Tokyo, Tokyo, Japan
This article has been republished with minor changes. These changes do not impact the academic content of the article.
1
Based on a literature search using the UTokyo REsource Explorer (TREE; http://tokyo.summon.serialssolutions.com/) search engine on Oct 28, 2019. TREE aggregates
information from many major databases, including Web of Science, PubMed, PsycINFO, Engineering Village, ERIC, and JSTOR, as well as electronic journals undercontract with the University of Tokyo. We used the keyword “cross-lagged,” searching papers published in English from Oct 2018 to Oct 2019. We limited our search to
only peer-reviewed papers, resulting in 1,065 papers found. Most were applied research, though some were research reviews or theoretical/methodological studies.
© 2020 The Author(s). Published with license by Taylor & Francis Group, LLC.
This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/),
which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way.

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Curran & Bollen, 2001). Many researchers have shown interest in
applying the GCLM, including more than 10 citations on Google
Scholar as of August 2020 (e.g., Bollmann, Rouzinov, Berchtold, &Rossier, 2019; Oswald, 2019; Zhang et al., 2019).
This article aims to elucidate how common factors and crosslagged parameters included in the GCLM can be interpretedusing the unified framework, and to highlight conceptual andmathematical differences among the GCLM, the RI-CLPM, andother longitudinal models. Despite interpretative differences
existing in cross-lagged parameters among models, it has long
been common practice for researchers to run a single model(typically, the CLPM) and evaluate its cross-lagged relations
without considering potential alternative models (Usami, Todo
et al., 2019). This article should better reveal the strengths andpotential weakness of the GCLM over other alternatives, helping
researchers to seek better methodologies while minimizing riskof wrong conclusions regarding reciprocal effects.
Importantly, we show that the mathematical relation
between the RI-CLPM and the GCLM as described in Zyphur
et al. (2020b, p. 13) requires correction, and that there areinterpretative differences in cross-lagged parameters and com­
mon factors in these models. Specifically, because common
factors are modeled with lagged effects in the GCLM, they
have both direct and indirect influences on observed scores,
meaning that the GCLM does not control for stable traits as
does the RI-CLPM. We will also show that including unit
effects as well as moving average terms in the GCLM greatly
complicates the interpretation of cross-lagged parameters.
In the next section, we provide a brief introduction for the
GCLM after a brief overview of some existing models: the RICLPM, the LGM, and the ALT model. In the third section, we
introduce the unified framework (Usami, Murayama et al.,2019). Readers who are already familiar with these models
and the framework may skip these two sections. In the fourth
section, we discuss how the GCLM can be characterized using
the unified framework, showing interpretative differences incommon factors and cross-lagged parameters from the RICLPM and other models. A conceptual diagram as well as
path diagrams of models are also provided to further clarify
relations between the models. The fifth section presents anempirical example to demonstrate how choice of the analysis
model might lead to different conclusions about reciprocal
relations. The sixth section briefly contrasts these SEM-based
approaches with potential alternatives to effectively account for
the influences of time-varying confounders. The final section
presents our conclusions and some areas for future research.

Existing models
We first give an overview of the RI-CLPM, the LGM, and the
ALT model, and then introduce the GCLM.
Random intercept cross-lagged panel model (RI-CLPM)
Throughout this article we assume that researchers are interested
in reciprocal relations between two variables X and Y. Let xit and2

yit be the measurements at time point t (1 . . . t . . . T) for person i(1 . . . i . . . N). In the RI-CLPM, xit and yit are first modeled asxit ¼ μxt þ Ixi þ xit�
yit ¼ μyt þ Iyi þ yit� :(1)Here, μxt and μyt are the temporal group means at time point t(i.e., Eðxit Þ ¼ μxt , Eðyit Þ ¼ μyt ). The terms Ixi and Iyi are (timeinvariant) stable trait factors (alternatively, random intercepts)
that represent a person’s trait-like deviations from the tem­poral group means. Trait factors Ixi and Iyi have means of 0 anda variance–covariance matrix. By accounting for stable trait
factors for each person, xit� and yit� represent temporal devia­
tions from the means of that person because they are sub­tracted from the expected scores of person i (i.e.,
μxit ¼ μxt þ Ixi and μyit ¼ μyt þ Iyi ). Accordingly, in the RICLPM, the time series xit� and y�it can be considered as withinperson fluctuation. Due to this statistical property in temporal�
deviations, at t ¼ 1 the initial deviation terms (xi1and y�i1 ) are
assumed to be uncorrelated with stable trait factors. Using
these within-person deviation terms, in the RI-CLPM the reci­
procal relations are modeled for t � 2 as��xit� ¼ βx xiðt1Þ þ γx yiðt 1Þ þ dxit��
�yit ¼ βy yiðt 1Þ þ γy xiðt 1Þ þ dyit ;(2)

where βx and βy are autoregressive parameters. γx and γy arecross-lagged parameters, which are key for inferring recipro­
cal relations between the variables.2 In t ¼ 1, the initial statesxi1 and yi1 are modeled as exogenous variables (i.e., theirvariances and covariance are assumed). The residuals dxit
and dyit are typically assumed to be normally distributedand correlated. If stable trait factors are omitted (i.e., if
Ixi ¼ Iyi ¼ 0), this version of the RI-CLPM is mathematicallyequivalent to the CLPM. The RI-CLPM is identified if two or
more variables have been measured at three or more time
points, whereas the CLPM requires only two time points (inwhich case it is saturated).
Because the RI-CLPM separates within-person fluctuations(temporal deviations) from stable between-person differences(stable trait factors) over time, cross-lagged relations inEquation 2 can be considered as those pertaining to a processthat takes place at the within-person level. Therefore, in the RICLPM, γx and γy can be interpreted as quantities that express
the extent to which the two variables influence each otherwithin persons. Hamaker et al. (2015) argued that parameter
estimates in the CLPM conflate between-person and withinperson processes, and that this model provides inaccurate
estimates for within-person reciprocal effects as a consequence.
Equations (1) and (2) are the formulation that were used in
Hamaker et al. (2015). However, there is another formulationof the (RI-)CLPM if intercepts (α) are included in the lagged
regressions (Equation 2) instead of excluding temporal groupmeans (μ) in Equation (1):xit ¼ Ixi þ xit� ;yit ¼ Iyi þ yit�

Note that the original introduction in Hamaker et al. (2015) assumed time-varying autoregressive and cross-lagged parameters, but we assume here, without loss ofgenerality, time-invariant parameters to keep the discussion concise.

STRUCTURAL EQUATION MODELING: A MULTIDISCIPLINARY JOURNAL
��xit� ¼ αxt þ βx xiðt1Þ þ γx yiðt 1Þ þ dxit ;���
yit ¼ αyt þ βy yiðt 1Þ þ γy xiðt 1Þ þ dyit :(3)

α and μ are not mathematically identical unless T ¼ 2, because theformer is modeled jointly with lagged effects and thus the influ­
ences of effects feed forward through the lagged relations. For
example, expected values for X and Y at t ¼ 3 can be calculated���
by substituting the relations xi2¼ αx2 þ βx xi1
þ γx yi1
þ dxi2 and��þ dyi2 into the equations xi3¼
y�i2 ¼ αy2 þ βy y�i1 þ γy xi1
�
�
�
�
�
αx3 þ βx xi2 þ γx yi2 þ dxi3 and yi3 ¼ αy3 þ βy yi2 þ γy xi2 þ dyi3 .
From this procedure, mathematical relations between α (inEquation (3)) and μ (in Equation (1)) can be expressed asEðxi3 Þ ¼ μx3 ¼ αx3 þ βx αx2 þ γx αy2
Eðyi3 Þ ¼ μy3 ¼ αy3 þ βy αy2 þ γy αx2 :(4)

By the same procedure, for t ¼ 4, we can derive the relations
between α and μ asEðxi4 Þ ¼ μx4 ¼ αx4 þ βx αx3 þ γx αy3 þ ½βx ðβx αx2 þ γx αy2 Þ
þγx ðβy αy2 þ γy αx2 Þ�þγy ðβx αx2 þ γx αy2 Þ�:(5)Note that numbers of parameters are independent of the choiceof α or μ, and this choice does not influence the estimation
results of other parameters, such as β and γ.
Latent growth model (LGM)
A bivariate version of the (linear) LGM can be expressed as1ÞSxi þ �xit
1ÞSyi þ �yit :(6)Here, Ixi and Iyi are intercept factors and Sxi and Syi are (linear)slope factors. The values �xit and �yit are unique factors (ormeasurement errors). In the LGM literature, I and S are often
called growth factors, and nonzero factor means (as well as
variances and covariances) are assumed.
The main difference between the LGM and the RI-CLPM isthat while the LGM explicitly models mean growth trajectories
via the intercept and slope factor means, this model instead
assumes there are no lagged effects. In other words, in the RICLPM temporal group means (μ) play a role in capturing mean
growth trajectories without making an explicit modelingassumption (e.g., that growth trajectories are linear); instead,(within-person) reciprocal relations are modeled using lagged
effects as well as stable trait factors.
Autoregressive latent trajectory (ALT) model
The ALT model was proposed by Curran and Bollen (2001),
aiming to synthesize the traditions of the CLPM and the LGM.
The bivariate (linear) ALT model can be expressed as
31ÞBxi þ βx xiðt 1Þ þ γx yiðt 1Þ þ dxit
1ÞByi þ βy yiðt 1Þ þ γy xiðt 1Þ þ dyit :(7)

As in the LGM, temporal group means (μ) are not included.
Instead, common factors A and B play a role in describing
growth trajectories (i.e., A and B have nonzero factor means).
As before, βx and βy are autoregressive parameters, γx and γyare cross-lagged parameters, and dxit and dyit are residuals. The
ALT is identified if two or more variables have been measured
at four or more time-points when stationarity of parameters is
assumed, while five or more time points are required undera non-stationarity assumption.
Obviously, the main difference between the ALT model andthe LGM is that the former assumes lagged effects, as in the RICLPM. However, as we will see later, the presence of lagged
effects causes the interpretations of common factors A and B in
the ALT model to differ from those of growth factors I and S in
the LGM.

General cross-lagged panel model (GCLM)

Eðyi4 Þ ¼ μy4 ¼ αy4 þ βy αy3 þ γy αx3 þ ½βy ðβy αy2 þ γy αx2 Þxit ¼ Ixi þ ðt
yit ¼ Iyi þ ðt

xit ¼ Axi þ ðt
yit ¼ Axi þ ðt

3

The GCLM was proposed as a generalization of the CLPM byincluding two aspects: (1) stable trait factors (i.e., unit effects),and (2) MA and CLMA terms (Zyphur et al., 2020a).3 The
latter idea was motivated by vector autoregressive moving
average (VARMA) models (Box et al., 2008; Browne &Nesselroade, 2005; as cited in Zyphur et al., 2020a), whichhelps to expand the range of dynamic processes that can bemodeled.
For the former idea, Zyphur et al. (2020a) cited the workof Hamaker et al. (2015) and noted the necessity of account­ing for stable trait factors from the view of causal inference:“[b]y failing to model stable factors, they will be confoundedwith the system dynamics that should be reflected by AR
and CL terms” (p. 8). Specifically, Hamaker et al. (2015)
argued that parameter estimates in the CLPM conflate
between-person and within-person processes, while the RICLPM is an alternative model that can separate withinperson processes from stable between-person differences.
Therefore, inference of within-person reciprocal (as well as
causal) effects by including stable trait factors, as in the RICLPM, was one of the central aims for Zyphur and collea­
gues in developing the GCLM.
Without loss of generality, here we can focus on a GCLM
that assumes first-order lags for autoregressive and crosslagged terms as well as for MA and CLMA terms. Zyphur
et al. (2020a) called this version of the GCLM the AR(1)MA(1)CL(1)CLMA(1) model (the numbers in parentheses indicate
lag orders), which can be expressed as4
xit ¼ αxt þ λxt Bxi þ βx xiðt 1Þ þ γx yiðt 1Þ þ δx dxiðt 1Þþζ x dyiðt 1Þ þ dxit
yit ¼ αyt þ λyt Byi þ βy yiðt 1Þ þ γy xiðt 1Þ þ δy dyiðt 1Þ

(8)þζ y dxiðt 1Þ þ dyit ;Zyphur et al. (2020a) used the term “stable factors” rather than “stable trait factors.”
For consistency between expressions of the introduced models, and to clarify the meanings of parameters and common factors, here we use symbols different fromðxÞ ðxÞ
ðxÞðxÞthose in Zyphur et al. (2020ab). Specifically, in this article we use symbols λxt , Bxi , γx , ζ x , and dxit , instead of using λt , ηi , βðxÞy , δy , and uit , respectively.

4

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for t � 3. The terms αxt and αyt are occasion-specific intercepts or
occasion effects. As before, βx and βy are (first-order) autoregres­sive parameters, and γx and γy are (first-order) cross-lagged
parameters. The terms Bxi and Byi are (unit-specific) common
factors, which Zyphur et al. (2020a, 2020b) called unit effects or
stable (trait) factors. The means of these factors are set to zero. In
addition, although not explicitly stated in Zyphur et al. (2020a,
2020b), these factors are assumed to be uncorrelated with theinitial states (xi1 and yi1 ), though this assumption can be relaxed.
λxt and λyt are weights or occasion-specific factors loadings,
expressing changes in the effects of common factors over time.
We will later discuss the meanings of the common factors (as wellas cross-lagged parameters) in the GCLM.
The terms δx and δy indicate MA effects, and ζ x and ζ y are
CLMA effects. They are included to make observations a directfunction of past impulses (i.e., residuals dðt 1Þ ). In this version of
GCLM, the short-run persistence for a variable becomes AR+MA,
and the short-run effect of one variable on another becomes CL+CLMA. Zyphur et al. (2020a, p. 12) argued that past impulsesdyiðt 1Þ and dxiðt 1Þ impact xit and yit via both CL and CLMA
paths, respectively, because dyiðt 1Þ and dxiðt 1Þ are the compo­
nents of yiðt 1Þ and xiðt 1Þ . Therefore, this is akin to estimating aneffect of dyiðt 1Þ and dxiðt 1Þ on xit and yit as the short-run effect byγx þ ζ x and γy þ ζ y , again, respectively (Zyphur et al.,2020a, p. 12).
This version of the GCLM (the AR(1)MA(1)CL(1)CLMA(1)model) can be extended straightforwardly by including second- or
higher-order terms. For example, we can include the additionalterms βx2 xiðt 2Þ and δx2 dxiðt 2Þ for explaining xit , and βy2 yiðt 2Þ
and δy2 dyiðt 2Þ for explaining yit . This version of the GCLM can benotated as AR(2)MA(2)CL(1)CLMA(1). The AR(1)MA(1)CL(1)
CLMA(1) model fixing occasion-specific factor loadings as
λxt ¼ λyt ¼ 1, which is applied in an empirical example below, isidentified if two or more variables have been measured at three ormore time-points when stationarity of parameters is assumed,
while four or more time points are required under a nonstationarity assumption.
We can find mathematical relations between the GCLM andsome of the longitudinal models we previously introduced. For
example, if we set the weights of B as λxt ¼ λyt ¼ t 1 andexclude the intercepts (i.e., αxt ¼ αyt ¼ 0) as well as MA and
CLMA terms (i.e., δx ¼ δy ¼ ζ x ¼ ζ y ¼ 0), and then include an
additional common factor (A) whose weight is fixed to one, thisversion of the GCLM is mathematically equivalent to the ALT
model. Likewise, by setting the weights of B as λxt ¼ λyt ¼ t 1
and excluding intercepts and all lagged effects (set the AR, CL,
MA, and CLMA terms to zero), and then instead including one
additional common factor whose weight is fixed to one, thisversion of the GCLM is mathematically equivalent to the LGM.
In the latter comparison, because no lagged effects are assumed to
be present, the common factor (B) in the GCLM plays a similarrole as the growth factor (S). In the former comparison, because
AR and CL effects are still present in the GCLM (or the ALT
model), the common factor (B) does not play the role as thegrowth factor (S) in the LGM. This is why we use the notation B(rather than S) to express common factors in the GCLM. This
point is closely related to how we should interpret the common

factors and cross-lagged parameters in each model. We revisit thisissue later in more detail.
The relation between the CLPM and the GCLM is more
obvious: excluding MA and CLMA terms as well as common
factors (i.e., unit effects) from the GCLM reduces it to the CLPM.
Therefore, the CLPM is a special case of the GCLM. This point
might make us wonder whether the relation between the RICLPM and the GCLM is also simple. In fact, Zyphur et al. (2020b,
p. 13) explain that a variant of the RI-CLPM (a special case of the
RI-CLPM that assumes specific weights (time-varying effects) for
stable trait factors I) is equivalent to the GCLM with MA and
CLMA terms eliminated. However, because lagged effects (i.e.,
the AR and CL terms) are present and they are jointly modeledwith common factors B in the GCLM, even if we exclude MA and
CLMA terms and fix the factor loadings to one, this version of the
GCLM is not mathematically equivalent to the RI-CLPM, inwhich common factors (stable trait factors I) are separately(rather than jointly) modeled with lagged effects (i.e., Equations(1) and (2)). Therefore, common factors (B) included in the
GCLM cannot be interpreted as the stable trait factors (I) in the
RI-CLPM. We discuss this point in detail using the unified
framework (Usami, Murayama et al., 2019) described in thenext section.

Unified framework and specifications of existing
longitudinal models
Unified framework
Usami, Murayama et al. (2019) provides a unified statistical frame­
work that clarifies mathematical and conceptual relations among
diverse SEM-based longitudinal models to examine reciprocal
effects, which can be specified through this framework as particu­
lar cases. Formulation of the unified framework consists of threesets of equations, which Usami, Murayama et al. (2019) calledmeasurement equations, decomposition equations, and dynamic
equations.
Measurement equations. The first set of equations can beused to separate the latent true scores from unique factors (ormeasurement errors) as
xit ¼ fxit þ �xit
yit ¼ fyit þ �yit :(9)These unique factors are typically assumed to be normally dis­tributed and possibly correlated. Among the models we haveintroduced, the LGM assumes unique factors. The GCLM as wellas the (RI-)CLPM and the ALT model assume residuals d in thelagged regressions, but they do not account for the presence of
unique factors (or measurement errors) in their formulations.
Some of the longitudinal models that include cross-lagged para­
meters (e.g., the stable trait autoregressive trait and state(STARTS) model; Kenny and Zautra (1995, 2001); and latentchange score (LCS) model; Hamagami and McArdle (2001);
McArdle and Hamagami (2001) assume unique factors.
Though the inclusion of unique factors is desirable on con­
ceptual grounds, it can easily lead to estimation problems due tothe strong dependency among the estimated parameters (Usami,
Murayama et al., 2019). Notably, the STARTS model, in which
both stable trait factors and unique factors are assumed, often

STRUCTURAL EQUATION MODELING: A MULTIDISCIPLINARY JOURNAL

suffers from improper solutions (e.g., Hamaker et al., 2015;
Usami, Todo et al., 2019), and a potential solution using
Bayesian estimation has been recently investigated (Lüdtke et al.,2018). Usami, Todo et al. (2019) provide a deeper discussion
about improper solutions when applying the STARTS modelthrough a simulation study that considers the presence of model
misspecifications. Orth et al. (in press) also compared the behaviorof several longitudinal models in ten datasets, empirically showingthat CLPM and the RI-CLPM converged in every sample, whereas
the other (ALT, LCS, and STARTS) models frequently failed toconverge or did not converge properly.
Decomposition equations. The second set of equations allow
for decomposition into an individual deterministic trend and�
a temporal deviation from this individual trend, denoted as fxit
�and fyit. The individual deterministic trend can depend on thetemporal group means μxt and μyt and/or on the random inter­
cepts and (linear) slopes (i.e., Ixi , Iyi , Sxi , and Syi ). Thus, we havefxit ¼ ½μxt þ fIxi þ ðt
fyit ¼ ½μyt þ fIyi þ ðt�1ÞSxi g� þ fxit�1ÞSyi g� þ fyit :�fxit¼ fAxi þ ðt�fyit ¼ fAyi þ ðt��1ÞBxi g þ βx fxiðt
1Þ þ γx fyiðt 1Þ þ dxit�
�1ÞByi g þ βy fyiðt
1Þ þ γy fxiðt 1Þ þ dyit :a general structure that helps to relate the many diverse mod­eling approaches (Usami, Murayama et al., 2019). Anotherimportant point is that in this framework MA and CLMA
terms were not originally included, though extending the fra­mework to include them is not difficult.

Specification of longitudinal models based on the unified
framework
Using the unified framework presented above we can easily seethat there are components that may or may not be included in themodel. Here we explain how the longitudinal models we haveintroduced so far can be expressed using the unified framework.
First, the RI-CLPM can be expressed within the unified
framework as
xit ¼ fxit ;�fxit ¼ μxt þ Ixi þ fxit;

(11)

Although these equations may look very similar to the ALT
model (Equation (7)), dynamic equations are defined to explain
temporal deviations ðf � Þ rather than observed scores. Because allterms in the dynamic equations contribute to the lagged pre­dictors, their influences feed forward through the lagged rela­tions and accumulate at later time points. This implies that theaccumulating factors A and B as well as (dynamic) residuals dhave direct and indirect effects on f � , and thus on the observed
scores. In contrast, the effects of stable trait or growth factors (Iand S) are temporal and they have only direct effects on scores:
their influences do not feed forward through the lagged relations
and do not accumulate at later time points (Usami, Murayama
et al., 2019). As we will discuss later, the choice to include
accumulating factors (A and/or B) or trait/growth factors (Iand/or S) in the model makes a difference in how we control(unobserved) confounders, resulting in different interpretationsof the cross-lagged parameters in each model.
Note that we cannot apply this unified model to longitudi­
nal data because it would be unidentified due to overparame­terization. The aim of introducing this framework is to provide

yit ¼ fyit
�fyit ¼ μyt þ Iyi þ fyit

(12)

�
�
�
fxit
¼ βx fxiðt
1Þ þ γx fyiðt 1Þ þ dxit ;���fyit ¼ βy fyiðt 1Þ þ γy fxiðt 1Þ þ dyit ;(10)

Importantly, as we will show later, the common factors
included in the decomposition equations (i.e., I and S) have
only direct effects on f , indicating these influences do not feed
forward to later time points. Thus, these common factors canbe characterized as stable trait (if S is omitted as in the RICLPM) or growth factors (i.e., random intercepts and (linear)slope factors as in the LGM).
Dynamic equations. Finally, the dynamics of the processes
are modeled with the dynamic equations, which include thelagged terms as autoregressive parameters βx and βy and crosslagged parameters γx and γy along with the (dynamic) residuals
dxit and dyit . In addition, they also include the common factors
A and B, which are called accumulating factors in Usami,
Murayama et al. (2019). This gives

5by excluding unique factors �, slope factors S, and accumulat­
ing factors A and B in the unified framework. As we have
explained, because in the RI-CLPM the common factors (i.e.,
stable trait factors) are modeled separately (rather than jointly)
from lagged effects (i.e., Equations (1) and (2)), they have only
direct effects on observed scores.
The (linear) LGM can be expressed asxit ¼ fxit þ �xit ; yit ¼ fyit þ �yit��
fxit ¼ Ixi þ ðt 1ÞSxi þ fxit
; fyit ¼ Iyi þ ðt 1ÞSyi þ fyit��
¼ 0; fyit
¼ 0;
fxit
(13)
by excluding temporal group means μ, accumulating factors A
and B, lagged effects (i.e., setting βx ¼ βy ¼ γx ¼ γy ¼ 0) and(dynamic) residuals d from the unified framework.
One feature of the LGM is that it does not include lagged
effects, implying that the distinction between the dynamic
equations and other equations becomes meaningless in thisspecial case. Therefore, if we temporarily disregard the original
definition that terms included in the dynamic equations are to
be modeled by lagged effects, we can find another expression ofthe (linear) LGM using different symbols:

�fxit¼ Axi þ ðt

xit ¼ fxit ;�fxit ¼ fxit;
1ÞBxi þ dxit ;yit ¼ fyit
�fyit ¼ fyit�fyit
¼ Ayi þ ðt

(14)1ÞByi þ dyit :Conceptually, the common factors (A and B) and d included inthis expression are to be interpreted as growth factors andunique factors, rather than as accumulating factors and(dynamic) residuals, respectively.
The ALT model can be expressed within the unified frame­
work asxit ¼ fxit ; yit ¼ fyit
��
fxit ¼ fxit
; fyit ¼ fyit
(15)
�
�
�
fxit ¼ fAxi þ ðt 1ÞBxi g þ βx fxiðt 1Þ þ γx fyiðt
þ
d
xit
1Þ

6

USAMI

Figure 1. Path diagrams of cross-lagged models. Notes. Residual covariances and covariances between common factors are omitted for clarity of presentation. In the
LGM and the unified framework, unique factors (or measurement errors) are indicated by arrows only. Note that the means of common factors in the RI-CLPM and the
GCLM are set to zero. In the GCLM, covariances between initial states and accumulating factors are not assumed.

STRUCTURAL EQUATION MODELING: A MULTIDISCIPLINARY JOURNAL�fyit¼ fAyi þ ðt��1ÞByi g þ βy fyiðt
1Þ þ γy fxiðt 1Þ þ dyit ;of accumulating factors B are fixed to t 1 in the GCLM. Then,the dynamic equations for the GCLM can be expressed asby excluding �, μ, and the growth factors I and S. We provide
path diagrams of the existing models (expressed using theunified framework) as well as the unified framework itself in
Figure 1. This should better clarify the properties of trait/
growth factors and accumulating factors: the former has only
direct effects on observed scores and its influences are tem­poral, while the latter has both direct and indirect effects andinfluences feed forward through the lagged relations and accu­
mulate at later time points.

Contrasting the GCLM and the RI-CLPM
Specification of the GCLM from the unified framework
As we have observed, one important key to understanding thedifferences among the models is whether common factors
included in the model are modeled with lagged effects, that is,whether common factors are included in either the decompo­
sition equations (e.g., the RI-CLPM) or the dynamic equations(e.g., the ALT model).
With this point in mind, in this section we first show how
the GCLM (i.e., Equation 8) can be expressed within the uni­
fied framework. Then we discuss the interpretative differencesof cross-lagged parameters between the GCLM and the RICLPM, which was overlooked in Zyphur et al. (2020a,
2020b). We also warn of the potential difficulty of its inter­
pretation in the GCLM.
Like the (RI-)CLPM and the ALT model, the GCLM does
not assume unique factors (or measurement errors), meaning�xit ¼ �yit ¼ 0. Therefore, the expression in the measurement
equations for the GCLM becomesxit ¼ fxit
yit ¼ fyit :(16)

In the GCLM, the common factors (B) are modeled jointly withlagged effects to explain observed scores (see Equation (8)),
indicating that common factors included in this model can beconsidered as accumulating factors rather than stable trait or
growth factors. Therefore, growth factors (I and S) in thedecomposition equation can be excluded for the specificationof the GCLM. Note that the original specification of the GCLM
(Equation (8)) includes occasion-specific intercepts α. Recall
that the mean structure in the model can be expressed bymodeling the temporal group means μ (rather than α) asa function of α and lagged effects (Equations (4) and (5)).
Therefore, we can express the decomposition equations for
the GCLM using μ as�fxit ¼ μxt þ fxit
�fyit ¼ μyt þ fyit :

7

(17)

As we have observed, the GCLM includes MA and CLMA terms
to explain observed scores. However, these terms were not origin­ally included in the unified framework. For concision, suppose weslightly extend the unified framework to include these terms in the
dynamic equations and assume the weights (i.e., factor loadings)

�
¼ ðt
fxit

�
�
1ÞBxi þ βx fxiðt
1Þ þ γx fyiðt 1Þ þ δx dxiðt 1Þþζ x dyiðt 1Þ þ dxit
�
fyit
¼ ðt

�
�
1ÞByi þ βy fyiðt
1Þ þ γy fxiðt 1Þ þ δy dyiðt 1Þ

(18)

þζ y dxiðt 1Þ þ dyit ;by excluding accumulating factors A. Note that the occasionspecific intercepts α, which are included in the original specifica­
tions of the GCLM (Equation (8)), are not modeled here because
temporal group means μ in the decomposition equations alreadyaccount for the mean structure. A path diagram of the GCLM
within the unified framework is provided in Figure 1.
To better clarify the relations among models including the
GCLM, a conceptual diagram is provided in Figure 2, which isan extension of the figure provided in Usami, Murayama et al.
(2019). Note that we assumed time-invariant autoregressive
and cross-lagged parameters in all the models here, though
this assumption can be relaxed.
As we have observed, the GCLM and the ALT model com­monly include accumulating factors, and a special case of the
GCLM (in which weights are set to λ ¼ t 1, intercepts as well
as MA and CLMA terms are excluded as zero, and one addi­
tional common factor A is included instead) is mathematicallyequivalent to the ALT model. Thus, we express the (condition­ally) nested relations between these two models in the diagram.
In sum, the GCLM can be viewed as a model in which all terms(excepts for temporal group means μ) are posited in the dynamic
equations, as in the ALT model. Thus, it is again obvious that thecommon factors included in the GCLM can be viewed as accu­
mulating factors rather than as stable trait or growth factors.
Therefore, one important conclusion of this article is that thecommon factors (unit effects) included in the GCLM cannot beinterpreted as the stable trait factors used in the RI-CLPM, andthat the GCLM does not control for the stable traits as in the RICLPM. In addition, the description that compares the RI-CLPM
and the GCLM in Zyphur et al. (2020b, p. 13) is incorrect, because
a variant of the RI-CLPM (i.e., a special case of the RI-CLPM thatassumes specific weights for stable trait factors I) is not mathema­tically equivalent to the GCLM with MA and CLMA terms
eliminated.
Mathematical definition of stable traits and illustration ofwhat the GCLM and the RI-CLPM control for
Psychometricians have used the terms “(stable) traits” and“within-person relations” in ambiguous ways when describing
SEM-based longitudinal models, despite mathematical and inter­
pretative differences existing between them. Common factors thatplay a role as accumulating factors have been called by different
names. The unified framework helps to resolve this problem andfacilitate comparison of different models. In this subsection we
provide a mathematical definition of stable traits, and mathema­tically sketch what the GCLM and the RI-CLPM control forthrough common factors.
Inspired by the formulation of the RI-CLPM and the unified
framework, Usami (2020) defined a stable trait factor (say, for

8

USAMI

Figure 2. Conceptual diagram clarifying relations among cross-lagged models as an extension of Usami, Murayama et al. (2019). Notes. Single-headed arrows indicate
nested relations, with dotted lines indicating relations that can be conditionally satisfied. Double-headed dotted lines indicate that models are statistically equivalent
under particular circumstances. Note that we suppose time-invariant autoregressive and cross-lagged parameters in all models. Hamaker (2005) compared the ALT
model and the LCM-SR, and McArdle (2009) explained that the LGM (or, latent curve model: LCM) is a special version of the LCS model. Usami, Hayes & McArdle (2015)
showed that the factor CLPM (i.e., CLPM that includes measurement errors) is a special version of the TCS model (i.e., LCS model that assumes time-varying factor
loadings for accumulating factors). GCLM: general cross-lagged panel model; CLPM: cross-lagged panel model; RI-CLPM: random-intercepts CLPM; STARTS: stable trait
autoregressive traitand state; LCM-SR: latent curve model with structured residuals; ALT: autoregressive latent trajectory; LCS: latent change score; TCS: triple change
score.

variable Y) for person i as the difference between expected value
for an observation (the true score) of this person at time t(expressed as μyit ) and the temporal group mean at time t (μyt )that are invariant over time as
Iyi ¼ μyit

μyt ;

(19)

where
1 < μyit < 1 and1 < μit < 1. Note thatEðIyi Þ ¼ Eðμyit μyt Þ ¼ μyt μyt ¼ 0. Then, the withinperson variability score yit� is also defined as the temporaldeviation of person i at time t (i.e., the difference between an
observation and its expected value) as
yit� ¼ yit

μyit ¼ yit

ðμyt þ Iyi Þ;(20)assuming that Eðy�it Þ=0 and Covðμyit ; yit� Þ ¼ 0 (i.e., expectedvalues of observations and within-person variability scores
are uncorrelated). Then, Usami (2020) proposed a generalprocedure for estimating causal effects of time-varying treat­ments or predictors on outcomes using within-person varia­
bility scores that are estimated by a factor analysis model.
As we have argued, the role of common factors criticallydepends on whether they are separated from lagged relations in
the model. Because accumulating factors included in dynamic
equations are not separated from the lagged relations in themodel, they do not satisfy the relation in Equation (19) (i.e., thedifference between the expected value of observation (true score)for person i and the temporal group mean is not invariant overtime). To illustrate this point, we dig into the observed score ofvariable Y at time t (yit ) in the RI-CLPM and the GCLM. In the
RI-CLPM, yit can be re-expressed using Equations (1) and (2) asyit ¼ μyt þ Iyi þ yit��
¼ μyt þ Iyi þ βyt y�iðt 1Þ þ γyt xiðt1Þ þ dyit��¼ μyt þ Iyi þ βyt ðβyðt 1Þ yiðt2Þ þ γyðt 1Þ xiðt 2Þ þ dyiðt 1Þ Þ��þ γyt ðβxðt 1Þ xiðt2Þ þ γxðt 1Þ yiðt 2Þ þ dxiðt 1Þ Þ þ dyit : (21)This expression shows that stable trait factor Iyi has a direct effecton yit , since it does not show up in the lagged terms in Equation(21). As a result, the expected score for person i at t is
μyit ¼ μyt þ Iyi . Thus, μyit μyt ¼ ðμyt þ Iyi Þ μyt ¼ Iyi , indicat­
ing that the stable trait factor in the RI-CLPM satisfies the defini­tion in Equation (19).
In contrast, in the GCLM, it can be shown that accumulat­
ing factor Byi has both direct and indirect effects on yit . Using
Equation (8), we can re-express yit in the GCLM asyit ¼ αyt þ λyt Byi þ βy yiðt 1Þ þ γy xiðt 1Þþδy dyiðt 1Þ þ ζ y dxiðt 1Þ þ dyit
¼ αyt þ λyt Byi þ βy ðαyðt 1Þ þ λyðt 1Þ Byi þ βy yiðt 2Þ
þγy xiðt 2Þ þ δy dyiðt 2Þ þ ζ y dxiðt 2Þ þ dyiðt 1Þ Þ
þγy ðαxðt 1Þ þ λxðt 1Þ Bxi þ βx xiðt 2Þ þ γx yiðt 2Þþδx dxiðt 2Þ þ ζ x dyiðt 2Þ þ dxiðt 1Þ Þ þ δy dyiðt 1Þþζ y dxiðt 1Þ þ dyit

STRUCTURAL EQUATION MODELING: A MULTIDISCIPLINARY JOURNAL¼ αyt þ ðλyt þ βy λyðt 1Þ ÞByi þ γy λxðt 1Þ Bxi þ βy ðαyðt 1Þ
þβy yiðt 2Þ þ γy xiðt 2Þ þ δy dyiðt 2Þ þ ζ y dxiðt 2Þþdyiðt 1Þ Þ þ δy dyiðt 1Þ þ ζ y dxiðt 1Þ þ dyit :(22)This shows that the effect of the common factor Byi is different
at each time point, and its function becomes increasingly more
complex at later time points. In the first line, where yit andyiðt 1Þ are contrasted, the effect of the accumulating factor
appears as λyt Byi . However, in the third line, where yit andyiðt 2Þ are contrasted, the effect becomes ðλyt þ βy λyðt 1Þ ÞByi ,which reflects both a direct effect and an indirect effect of Byi .
Bxi also impacts yit (γy λxðt 1Þ Bxi ). Furthermore, the third line
also suggests that the expected score for person i on variable Yat tð� 3Þ is a function of a) occasion-specific intercepts α, b)occasion-specific factor loadings λ, c) the common factors B, d)
the lagged parameters β and γ, and e) the initial states yi1 andxi1 . These suggest that the difference between the expectedvalue of observation (true score) for person i and the temporalgroup mean is not equal to Byi and not invariant over time.
Namely, the accumulating factor in the GCLM (or the ALT
model) does not satisfy the definition in Equation (19). Zyphur
et al. (2020a, p. 9) gave reasons for allowing the effect of
accumulating factor B to vary over time (λt �λ), and the defini­tion of stable trait factors Equation (19) might be expanded toallow such time-varying effects. Regardless of how we definestable traits, however, this observation reveals that mathema­tical roles differ between stable trait factors in the RI-CLPM
and the accumulating factors in the GCLM (or the ALT
model), and that the GCLM (or the ALT model) does notcontrol for stable traits as in the RI-CLPM.

Interpretation of cross-lagged parameters in the GCLM
Given the above, how can the cross-lagged parameters in the
GCLM be interpreted, and how is this different from other models,such as the RI-CLPM? Zyphur et al. (2020a) explained how therange of dynamic processes that can be modeled increases by
including MA and CLMA terms, and also discussed two threats
to causal inference (trends and regime changes). However, they
did not provide a clear explanation of how to interpret crosslagged parameters in the GCLM. Below we discuss the potential
difficulty of interpreting cross-lagged parameters in the GCLM
from two aspects: including the accumulating factors, and the MA
and CLMA terms. Here we partly refer to Usami, Murayama et al.
(2019) for the former point.
Accumulating factors. As we have argued, the accumulating
factors have both a direct effect and an indirect effect on out­
comes Equation (22), and the same is true even if occasionspecific factor loadings are fixed to 1 (λ ¼ 1). Therefore, the
GCLM (or the ALT model) does not control for stable traits asin the RI-CLPM. Because accumulating factor B correlates withobservations and its effect differs at each time point, one could
5

9

say that the GCLM (or the ALT model) implicitly controls forunobserved time-varying confounders, and that those influ­
ences feed forward through lagged relations. However, one
potential risk of the GCLM is that estimates of cross-lagged
parameters are biased and cannot be interpreted as causal
estimates unless influences of time-varying confounders are
precisely expressed by the complex function of B and λ, aswell as β and γ Equation (22). Correct specification of a highly
structured model such as the GCLM might be a strong assump­tion in general, and even minor model misspecifications cancause severely biased estimates of cross-lagged parameters.
Even if no model misspecification occurs, the assumptionthat the GCLM (or the ALT model) controls for unobserved
time-varying confounders may be inappropriate for someinstances, potentially leading to erroneous conclusions.
Specifically, if individual differences in growth captured byaccumulating factors in one of the observed variables is actually(in part) the result of growth in another observed variable, thenusing this model, which might cause unnecessary adjustment(i.e., overadjustment5) of growth, is likely to result in biased
estimates of reciprocal (as well as causal) effects (Usami,
Murayama et al., 2019). Namely, there is a great risk that
accumulating factors wrongly account for individual differ­ences in growth, resulting in biased estimates of reciprocal
effects if such individual differences are considered to be con­stituent components of these effects.6 More importantly, it isvery difficult in general for researchers to precisely know what
the accumulating factor B as a time-varying latent variable
actually represents, causing interpretative difficulty of thecross-lagged parameters if this is included in the model.
In contrast, the RI-CLPM allows for a group-level trajectory(expressed as μt ) that can take on any shape, and each person
deviates from this trajectory by a constant distance (i.e., I). The
reciprocal effects are then modeled between the residuals, thatis, the deviations from the expected scores, without controllingfor persons’ growths by slope (S) or accumulating factors (Aand B). Hence, in this approach, stable trait factors can beclearly interpreted, and (time-varying) individual differencesin growth remain in the reciprocal parts of the model, unlike in
the GCLM (or the ALT model). However, if individual differ­ences in growth are actually caused by unobserved timevarying confounders, failing to include them in the model
may also cause biased estimates of cross-lagged parameters(Usami, Murayama et al., 2019).
In other words, the critical point of model selection andinterpretation of cross-lagged parameters lies in what aspects
are considered critical components of reciprocal or causal effects.
If (time-varying) individual differences in growth trajectories are
considered a critical component of these effects, then the RICLPM might be an appropriate choice (Usami, Murayama et al.,2019). On the other hand, if researchers assume that the influ­
ences of unobserved time-varying confounders can be perfectly
captured by accumulating factors, then the GCLM (or the ALT

We use the terms unnecessary adjustment or overadjustment to describe a variable that increases net bias and/or decreases precision, while some researchers use
these to indicate only the latter meaning. See Schisterman et al. (2009) regarding this point.
6
As a variant of the ALT model, we have seen the recent development of a latent variable-autoregressive latent trajectory (LV-ALT) model (Bianconcini & Bollen, 2018),
which aims to provide a framework for comparing different longitudinal models and allows researchers to explore alternative structures to best model theirlongitudinal data. One may wish to apply the bivariate version of this model to evaluate reciprocal effects between variables. However, the risk that accumulating
factors wrongly account for (time-varying) individual differences in growth remains in this model.

10

USAMI

model) might be a better choice. However, in many cases one
can see that growth aspects are considered to be a critical com­ponent of reciprocal or causal effects rather than mere unob­
served time-varying confounders, so choosing the GCLM (or the
ALT model) might not be appropriate. This point implies thattwo primary analytic purposes of applying longitudinal models
that include reciprocal relations-namely, inferring reciprocal
effects between variables and modeling individual differencesin growth trajectories by common factors-are intertwined(Usami, Murayama et al., 2019).
MA and CLMA terms. We have discussed the potential
limitation of interpreting cross-lagged coefficients if accumulat­
ing factors are modeled as in the GCLM. However, this problem
becomes more complicated in the GCLM because it assumes
MA and CLMA terms in addition to accumulating factors.
(CL)MA terms are composed by (dynamic) residuals. The
residuals mean that the components of the observed scores thatcannot be explained by lagged effects (i.e., deviations ofobserved scores from temporal group means in the previous
time point) as well as accumulating factors (i.e., unit effects).
Considering also that the role of the residuals in general is toaccount for all sources of variation unexplained by the fitted
model (e.g., possible model misspecification, unobserved con­founders and (dynamic) errors), it is not surprepsilong that theexact meanings of (CL)MA terms (or residuals) are obscureand their interpretations are very difficult in general. This point
makes interpretation of cross-lagged parameters much moredifficult in the GCLM, and can also cause biased estimates
because of overadjustment.
A critical point is that (dynamic) residuals in the CLMA
terms are already accounted for as one component of temporaldeviations in the CL term. Specifically, say dyðt 1Þ , which is
a component of yt 1 in the CL term to explain xt , is also
accounted for by the CLMA term. This implies that in the
GCLM, the same residual d is accounted for twice. Likewise, inthe unified framework expression for the GCLM (Figure 1d), fxt�is expressed by (i) the direct effect of dyðt 1Þ (a path dyðt 1Þ ! fxt� )and (ii) an indirect effect of dyðt 1Þ (a path trace��dyðt 1Þ ! fyðt1Þ ! fxt ). This redundancy can cause high corre­
lation between the CL term (yt 1 ) and the CLMA term (dyðt 1Þ ).
As a result, not only biases in cross-lagged parameter estimates
but also multicollinearity (inflated standard errors) might arise.
In sum, accumulating factors and (CL)MA terms included in
the GCLM increase the risks of bias, multicollinearity (inflated
standard errors), and interpretative difficulty in cross-lagged para­
meters to estimate reciprocal or causal effects, even if no model
misspecification occurs. Although the RI-CLPM is not a perfectprocedure for every situation, if the model can be correctly speci­
fied and if time-varying confounders can be appropriately con­trolled for, this choice better infers reciprocal or causal effects
occurring at the within-person level. We revisit the issue of timevarying confounders from the view of causal inference later in thisarticle.
Our goal in this article is not to suggest that researchers
completely avoid using the model. The GCLM might be a good
choice for researchers interested in building a useful linear
7

https://www.icpsr.umich.edu/icpsrweb/ICPSR/studies/36282

model to predict observed scores, because this model canincrease the range of dynamic processes that can be modeled.
However, if the interpretation of cross-lagged parameters(inferring reciprocal effects) or controlling for stable traits(like in the RI-CLPM) is key in the application, applying the
GCLM cannot be recommended.

An example using empirical data
This section presents an example using empirical data to show
how estimates of reciprocal effects differ depending on model
choice. Specifically, we focus on the GCLM and the RI-CLPM toillustrate (i) how differences in common factors included in themodels, and (ii) how inclusion of (CL)MA terms in the GCLM
influence estimates of reciprocal effects. In this example we inves­tigate the reciprocal relation between adolescents’ exposure tosmoking in movies (X) and their smoking intensity (Y), using
data from the Minnesota Adolescent Community Cohort(MACC) Study 2000–2013. The MACC Study is a prospective
cohort study designed to expand understanding of the transitional
process from nonsmoking to smoking during adolescence and to
examine the effect of state- and local-level tobacco prevention andcontrol programs for youth in Minnesota (Choi, Forster,
Erickson, Lazovich, & Southwell, 2012). For illustrative purposes
we used a sample of 4,671 adolescents aged 15 to 20 years who
were surveyed from every six months in most years of the survey.
When participants responded to two surveys in a year, only
response data from the first survey were used to construct thedataset of T ¼ 6. More detailed information about the study
design and population in the MACC study is available in Choiet al. (2012) and from the website of the Inter-university
Consortium for Political and Social Research.7 Usami,
Murayama et al. (2019) used the same dataset to compare esti­mates of reciprocal effects among various models, though they did
not include the GCLM in this comparison. In this example we
newly fit the GCLM to the dataset for illustrative purposes.
Exposure to smoking in movies and smoking intensity wereassessed during each round of data collection. Participants wereasked to report how often they saw actors and actresses smoking
when they watched movies, with four response options: most of
the time (4), some of the time (3), hardly ever (2) and never (1).
The data also included an index for six levels of smoking inten­
sity, which was created by five measures (see Choi et al., 2012, fordetails). There were two sources of missing data, those missing
by design and those by attrition. We used all available data whenestimating parameters of the models, but we removed data forone participant with missing data for all variables at all time
points. All analyses were conducted using the lavaan package(Rosseel, 2012) in R with the full information maximum like­lihood estimation method. The lavaan source code used in thisexample is available in the Online Supplemental Materials.
To clarify the comparison between the models, in thisexample occasion-specific factor loadings are all fixed to 1
(λxt ¼ 1 and λyt ¼ 1), and covariances between the first states(xi1 and yi1 ) and accumulating factors B were not assumed in
the GCLM. We then fit two kinds of first-order GCLMs to the

STRUCTURAL EQUATION MODELING: A MULTIDISCIPLINARY JOURNAL

11

Table 1. Parameter estimates and model fit indices from different longitudinal models (N = 4,670).
RI-CLPM

GCLM
AR(1)MA(1)CL(1)CLMA(1)Parameters and model fit indices
βy
γy
βx
γx
CFI
AIC
BIC
RMSEA [95% CI]
SRMR
Degrees of freedom
Number of parameters

Est.
0.700
−0.0050.166−0.008

SE
0.012
0.022
0.011
0.007
0.959
72224.166
72385.389
0.045 [0.042, 0.048]
0.066
65
25

p
0.000
0.824
0.000
0.250

Est.
0.708
−0.181
0.273−0.083

SE
0.028
0.044
0.025
0.013
0.975
72000.100
72238.710
0.039 [0.036, 0.043]
0.054
53
37

AR(1)CL(1)
p
0.000
0.000
0.000
0.000

Est.
0.525
−0.0310.142−0.021

SE
0.016
0.020
0.010
0.007
0.962
72181.582
72342.805
0.043 [0.040, 0.047]
0.068
65
25

p
0.000
0.122
0.000
0.001

Note. Analysis result of the RI-CLPM is the same from Usami, Murayama et al. (2019). In the GCLM occasion-specific factors loadings were all fixed to 1, and covariances
between the initial states and accumulating factor were not assumed. Because the AR(1)MA(1)CL(1)CLMA(1) model showed the improper solutions, the stationarity
assumption was relaxed for residual variance of X and residual covariance. RI-CLPM random-intercept cross-lagged panel model; GCLM; general cross-lagged panel
model; CFI comparative fit index; AIC Akaike information criterion; BIC Bayesian information criterion; RMSEA root mean square error of approximation; CI confidence
interval; SRMR standardized root mean square residual.

dataset: AR(1)MA(1)CL(1)CLMA(1) and AR(1)CL(1).
Comparison between these two GCLMs illustrates how inclu­
sion of the (CL)MA terms affects estimates of cross-lagged
parameters in the GCLM. Because the number of parametersis the same and the kind of common factor included is the only
difference between AR(1)CL(1) and the RI-CLPM, comparing
estimates from these two models illustrates how choice ofcommon factors included in the model affects estimates ofcross-lagged parameters. In both the GCLMs and the RICLPM, we assume stationarity of parameters (e.g., equality of
AR, CL, and CL(MA) effects and residual (co)variances overtime). However, because the AR(1)MA(1)CL(1)CLMA(1)model resulted in an improper solution, we assumed nonstationarity of parameters for residual variances of X andresidual covariances in this model.
Table 1 shows parameter estimates and fit indices from the
RI-CLPM and the two GCLMs. The RI-CLPM showed non­significant estimates for both cross-lagged parameters (expo­
sure to smoking in movies X did not predict later smoking
intensity Y, and vice versa), while the two GCLMs showedsignificant estimates. The AR(1)MA(1)CL(1)CLMA(1) fit bet­
ter than did the other models, and resulted in significant
estimates for both cross-lagged parameters. Note that thismodel also showed relatively larger standard errors, implying
the influence of multicollinearity caused by correlations among
accumulating factors (B), (lagged) observations (xiðt 1Þ andyiðt 1Þ ), and (lagged) residuals (dxiðt 1Þ and dyiðt 1Þ ).
This example clearly demonstrates the risk of drawing differ­ent conclusions based on the cross-lagged parameters from dif­ferent models. Focusing on model fit, the AR(1)MA(1)CL(1)
CLMA(1) model would be the most appropriate for these data.
Moreover, since model fits of the two GCLMs were much better
than that of the RI-CLPM, it seems safe to conclude that there areomitted time-varying confounders affecting both observed
variables.
As we have argued, however, cross-lagged parameters in theAR(1)MA(1)CL(1)CLMA(1) model are very difficult to interpretand also pose greater risks of bias in its estimates due to inclusionof (CL)MA terms and accumulating factors. If researchers areinterested in estimating reciprocal or causal effects and canreasonably expect that influences of unobserved time-varying

confounders can be perfectly expressed as accumulating factor
B, choosing the estimation results from the AR(1)CL(1) model
should be more reasonable than use of the AR(1)MA(1)CL(1)
CLMA(1) model. In most cases, however, researchers are uncer­
tain regarding how the impacts of unobserved time-varying
confounders shift over time. If longitudinal changes in smoking
intensity and exposure to smoking in movies are considered
critical components of reciprocal effects, AR(1)CL(1) is an inap­
propriate option, and the RI-CLPM could be an option. In thiscase, it would be reasonable to conclude that there are no sig­
nificant reciprocal effects between these variables. However,from the view of causal inference, we need to consider that
their estimates of reciprocal effects in the RI-CLPM might bebiased due to omitted time-varying confounders. To mitigate thisrisk, including observed time-varying confounders in this model
should be a useful strategy, and this is discussed further in thenext section.

Estimating causal effects by the RI-CLPM and recent
potential outcome approaches
As we have argued, including accumulating factors and (CL)MA
terms as in the GCLM should increase risk of bias in reciprocal or
causal effect estimates. The RI-CLPM, which does not include
these components, separates within-person fluctuations (temporaldeviations) from stable between-person differences (stable trait
factors) over time, and cross-lagged relations in Equation (2) can
be considered as those pertaining to a process that takes place at
the within-person level. From the view of the potential outcome
approach, which is currently the standard framework for defining
causal effects, within-person reciprocal effects estimated in the RICLPM can represent causal effects under assumption of no model
errors and no unobserved confounders (see Usami, Murayama
et al., 2019; Usami, 2020 for more details). If measurement errors(violation of the consistency assumption) are expected to be pre­sent, using the STARTS model, which is a simple extension of the
RI-CLPM to allow measurement errors, might be an option.
However, see Usami, Todo et al. (2019) and Orth et al. (in press)
for further discussion about improper solutions frequently appear­ing in the STARTS model.

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USAMI

Regardless of which model we take, estimates of cross-lagged
parameters are biased if there are omitted unobserved confoun­ders. When time-varying observed confounders are available,
a typical approach is to directly include them into the model likean analysis of covariance (ANCOVA). This approach requires
assumptions of linearity and additivity of relations between out­
comes and confounders. Namely, even if observed confounders
are included when applying the RI-CLPM (or the STARTS
model), it requires correctly specified linear regressions to connect
variables at the within-person level. However, the linearity that is
typically assumed in path modeling and SEM has often been
criticized in the causal inference literature (e.g., Hong, 2015).
To mitigate this problem, non-SEM approaches such as mar­
ginal structural models (MSMs; Robins, 1998, 1999; Robins et al.,2000) or structural nested mean models (SNMMs; e.g., Robins,
1994, 1999) with G-estimators are useful. They have been
applied in epidemiology to estimate the causal effects ofsequences of time-varying treatments or predictors
Aðt 1Þ ¼ ðAðt 1Þ ; Aðt 2Þ ; . . . ; A1 Þt on outcomes Yt . Althoughthese methods originally considered the situation where one isinterested in evaluating a unidirectional relation (the effect ofa treatment on an outcome) rather than reciprocal relations, they
can be extended in a straightforward manner.
In MSMs, researchers specify a treatment assignment modelf ðAðt 1Þ jLðt 1Þ Þ at time point t 1 (and previous time points)
to express the probability that one receives a certain level oftreatment or predictors Aðt 1Þ ¼ aðt 1Þ using the history ofobserved confounders Lðt 1Þ ¼ ðLðt 1Þ ; Lðt 2Þ ; . . . ; L1 Þt . Next,
the inverse probability weights (IPW) required for estimating
an outcome model f ðYt j Aðt 1Þ Þ are calculated using informationof inverse probability 1=^f ðAðt 1Þ j Lðt 1Þ Þ at time point t 1 (andprevious time points) under the assumption of no unobserved
confounders or sequential ignorability. Causal effects are thenestimated by fitting a weighted outcome model with an IPW
estimator. Unlike ANCOVA, MSMs do not demand that
researchers model the relation between outcomes and observed
confounders. In general, MSMs can be easily understood and fit
with standard, off-the-shelf software that allows for weights(Vansteelandt & Joffe, 2014). However, it is also well-known
that MSMs can be highly sensitive to misspecification of thetreatment assignment model, even when there is a moderate
number of time points (e.g., Hong, 2015; Lefebvre et al., 2008).
Imai and Ratkovic (2015) proposed a covariate balancing pro­pensity score methodology for robust IPW estimation.
Although actual applications have been relatively infrequent,mainly due to a lack of the off-the-shelf software (however, see
Wallace et al., 2017 as an exception), SNMMs with G-estimators
are a better approach for handling violation of assumption of nounobserved
confoundersorsequential
ignorability(Vansteelandt & Joffe, 2014). Specifically, by solving estimating
equations constructed based on this assumption, consistent esti­mates of causal parameters can be obtained when eithera treatment assignment model or a model for outcome thatwould be observed if the treatment were stopped froma specific time can be correctly specified (the doubly robust
property; see Vansteelandt & Joffe, 2014; Usami, 2020 for
details). In addition, SNMMs can allow direct modeling of theinteractions and moderation effects of treatments or predictors
with observed confounders. Another advantage of SNMMs isthat the variance of locally efficient IPW estimators in MSMs
exceeds that of G-estimators in SNMMs, unless treatments or
predictors and observed confounders are independent.
Note that observed confounders are typically included inapplications of MSMs and SNMMs. That is, they do not often
explicitly include latent variables or common factors like the stable
trait factors. This implies that stable individual differences might
not be adequately controlled for in usual applications of MSMs
and SNMMs. Usami (2020) proposes a two-step analysis method
for within-person variability scores-based causal inference to esti­
mate joint effects of time-varying treatments or predictors bycontrolling for stable traits as time-invariant unobserved confoun­ders. In this method, within-person variability scores for each
person Equation (20), which are disaggregated from stable traitsof that person, are first calculated through a factor analysis model.
Causal parameters are then estimated via a potential outcome
approach, either MSMs or SNMMs, using calculated withinperson variability scores. Through simulation and empirical appli­
cation, it was shown that the proposed method can recover causal
parameters well and that causal estimates might be severely biased
if one does not properly account for stable traits. It should be
beneficial for researchers to take such non-SEM-based approaches
as possible alternatives in future research, especially if they aim toeffectively and flexibly control for time-varying confounders.

Conclusion
We discussed how common factors and cross-lagged parameters
included in the GCLM can be interpreted using a unified frame­
work, highlighting the conceptual and mathematical differencesamong the GCLM, the RI-CLPM, and other longitudinal models.
Our conclusions can be summarized as follows: (1) Common
factors included in the GCLM are not stable trait factors (asincluded in the RI-CLPM), but are accumulating ones (asincluded in the ALT model), which have both direct and indirect
influences on observed scores, meaning that the GCLM does notcontrol for stable traits of persons as does the RI-CLPM. (2)Including accumulating factors as well as (CL)MA terms, which
are main features of the GCLM, makes the interpretation of crosslagged parameters very complicated in general. (3) Even if nomodel misspecification occurs, seriously biased estimates of crosslagged parameters (and inflated standard errors) might beobtained in the GCLM when (time-varying) individual differencesin the growth trajectories accounted for by accumulating factors
are critical components of reciprocal or causal effects. (4)Although the GCLM might be an option if one is especiallyinterested in predicting outcomes, applying this approach entails
great risk if uncovering reciprocal or causal effects is the primary
focus. (5) Although the RI-CLPM (and the STARTS model) is nota perfect procedure for every situation, if the model can becorrectly specified and if time-varying confounders can be appro­
priately controlled for, this choice is better to infer reciprocal or
causal effects that occur at a within-person level. Finally, (6) If one
aims to effectively and flexibly control for time-varying confoun­ders, the RI-CLPM (or the STARTS model) with ANCOVA
approach is not the only option, and it is beneficial to take nonNote. The assumption of stationarity indicates that autoregressive parameters, cross-lagged parameters, moving-average parameters and residual (co)variances are time invariant, whereas non-stationarity indicates time-varying.
*In the GCLM, time-varying loadings from accumulating factors B were assumed to be prespecified as λt = 1, and covariances between the initial states and accumulating factor were not assumed. CLPM cross-lagged panel
model; RI-CLPM random-intercept CLPM; ALT autoregressive latent trajectory model; GCLM general cross-lagged panel model; MSMs marginal structural models; SNMMs structural nested mean models

NoNo
Within-person variability scoresbased method with MSMs andSNMMs (Usami, 2020)

3

Stable trait factor

Yes

No

Yes
Yes
Yes
No
Yes
Yes
No
No
Accumulating factor
Accumulating factor
3* (4*)
3* (3*)

Method
CLPM
RI-CLPM
ALT
GCLM
AR(1)MA(1)CL(1)CLMA(1)AR(1)CL(1)Table 2. Summary of methods.

Minimum T required for
Can control for stable Can control for unobserved time-varying confounders Has interpretative Is SEM-based approach (also requires
identification in stationarity
traits (Equation 19) as by common factors (has also a risk that factors wrongly difficulty of cross- correctly specified [linear] regression(non-stationarity)
Commonunobserved timeaccount for critical components [individual differences lagged parameters models between [observed] timeassumption
factor controlled for invariant confoundersof growth] in reciprocal or causal effects)
due to MA terms
varying confounders and outcomes)
2 (2)No factor
No
No
No
Yes
3 (3)
Stable trait factor
Yes
No
No
Yes
4 (5)
Accumulating factor
No
Yes
No
Yes

STRUCTURAL EQUATION MODELING: A MULTIDISCIPLINARY JOURNAL

13

SEM-based approaches such as MSMs and SNMMs as possible
alternatives. Table 2 summarizes our discussion of each model.
Notably, regarding the last point, SNMMs with G-estimators
are a better approach for handling violation of assumption of nounobserved confounders or sequential ignorability, because of
their doubly robust property. However, MSMs and SNMMs ori­
ginate from epidemiology and thus have not been broadly used in
the behavioral sciences. In future studies, we plan to contrast thesemethods with various longitudinal models from both conceptualand mathematical viewpoints, and to introduce the within-person
variability scores-based causal inference approach (Usami, 2020).
Because estimation performance of these methods has not been
exhaustively compared with the RI-CLPM (with an ANCOVA
approach), this point should be investigated in future studies. We
hope this corner and related future studies will help researchers
choose better methodologies when aiming to uncover reciprocal
or causal effects with minimal risk of obtaining wrong
conclusions.

Funding
This work was supported by the Japan Society for the Promotion of
Science [19K14378].

ORCID
Satoshi Usami

http://orcid.org/0000-0002-5670-2242

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