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Cross-lagged panel model 1

Running head: CROSS-LAGGED PANEL MODEL

A critique of the cross-lagged panel model

E. L. Hamaker1,R.M.Kuiper

1andR.P.P.P.Grasman

2

1. Methodology and Statistics, Faculty of Social and Behavioural Sciences, Utrecht

University

2. Psychological Methodology, University of Amsterdam

Author Note:

This study was supported by the Netherlands Organization for Scientiﬁc Research

(NWO; VIDI Grant 452-10-007).

Correspondence concerning this paper should be addressed to E. L. Hamaker,

Methodology and Statistics, Faculty of Social and Behavioural Sciences, Utrecht

University, P.O. Box 80140, 3508 TC, Utrecht, The Netherlands. Email:

e.l.hamaker@uu.nl.

Published in 2015, in Psychological Methods, 20(1), 102-116.

doi: 10.1037/a0038889.

Cross-lagged panel model 2

Abstract

The cross-lagged panel model is believed by many to overcome the problems

associated with the use of cross-lagged correlations as a way to study causal

inﬂuences in longitudinal panel data. The current paper however shows that if

stability of constructs is to some extent of a trait-like, time-invariant nature, the

autoregressive relationships of the cross-lagged panel model fail to adequately

account for this. As a result, the lagged parameters that are obtained with the

cross-lagged panel model do not represent the actual within-person relationships

over time, and this may lead to erroneous conclusions regarding the presence,

predominance, and sign of causal inﬂuences.

We present an alternative model that separates the within-person process from

stable between-person diﬀerences, and discuss how this model is related to existing

structural equation models that include cross-lagged relationships. Furthermore, we

derive the analytical relationship between the cross-lagged parameters from this

alternative model and those from the cross-lagged panel model. Through

simulations we demonstrate the spurious results that may arise when using the

cross-lagged panel model to analyze data that include stable, trait-like individual

diﬀerences. This is followed by the presentation of a modeling strategy to avoid this

pitfall, which we illustrate using an empirical data set. The implications for existing

and future cross-lagged panel research are discussed.

Cross-lagged panel model 3

A critique of the cross-lagged panel model

In 1980, Rogosa’s seminal paper A critique of the cross-lagged correlation was

published, which successfully conveyed the message that comparing cross-lagged

correlations from longitudinal panel data is an inappropriate basis for making causal

inferences.1One of the key insights stemming from Rogosa’s paper is that, if two

constructs are characterized by diﬀerent degrees of stability, the comparison of

cross-lagged correlations may lead to spurious conclusions regarding the causal

mechanism. Since then, most researchers interested in causality in panel data have

abandoned cross-lagged correlations and endorsed what we will referred to in this

paper as the cross-lagged panel model (CLPM) instead. In the CLPM stability of

the constructs is controlled for through the inclusion of autoregressive relationships,

and it is therefore often believed that the cross-lagged regression parameters

obtained with this model are the most appropriate measures for studying causality

in longitudinal correlational data (e.g., Deary, Allerhand, & Der, 2009; Soenens,

Luyckx, Vansteekiste, Duriez, & Goossens, 2008; Wood, Maltby, Gillett, Linley, &

Joseph, 2008). Speciﬁcally, it is common practice to standardize the cross-lagged

regression coeﬃcients and compare their relative strength to determine which

variable has a stronger causal inﬂuence on the other (Bentler & Speckart, 1981).

The current paper forms a sequel to the warning given by Rogosa (1980), in

that it will be argued that not only should we account for stability, but we also need

to account for the right kind of stability. It will be shown that if stability of the

constructs is to some extent of a trait-like, time-invariant nature, the inclusion of

autoregressive parameters will fail to adequately control for this. As a result the

Cross-lagged panel model 4

estimates of the cross-lagged regression coeﬃcients will be biased, which may lead to

erroneous conclusions regarding the underlying causal pattern. This message is not

novel in itself: In fact, it has been recognized repeatedly that the “omitted variable

problem” may aﬀect the estimation of the cross-lagged coeﬃcients (e.g., Dwyer,

1983; Finkel, 1995; Heise, 1970), and diverse modeling strategies have been

proposed to account for unobserved variables that inﬂuence the observed variables.

However, given the popularity of the CLPM, it seems that either this warning has

been lost on a large group of substantive researchers, or many researchers are simply

not convinced that this could form a serious problem.

In the current paper, we therefore present a closely related alternative

structural equation modeling (SEM) approach that is inspired by considering

cross-lagged panel data from a multilevel perspective, implying we need to

distinguish between the within-person and the between person level. We show that

this alternative SEM approach can lead to very diﬀerent conclusions than the

traditional CLPM when considering the three major objectives of cross-lagged panel

research, that is: a) whether or not variables inﬂuence each other; b) which of the

variables is causally dominant; and c) what the sign of inﬂuence is. In doing so we

hope to raise awareness about the limitations of the traditional CLPM, and to

stimulate researchers to consider alternative SEM approaches.

This paper is organized as follows. In the ﬁrst section, two models for

investigating cross-lagged eﬀects are presented: the traditional CLPM and an

extension of this model based on taking a multilevel perspective. We discuss the

meaning of each model, the way they predict change, and the minimum number of

waves needed for identiﬁcation. In the second section, we discuss four other SEM

Cross-lagged panel model 5

approaches that include cross-lagged relationships and discuss how these are related

to the model we propose. In doing so, we sketch the broader context of the current

account and point the reader in the direction of other alternatives. The third

section consists of a more in-depth comparison of the traditional CLPM and the

proposed alternative. In the fourth section, a modeling strategy is proposed to

ensure that – if present – both forms of stability are accounted for and we illustrate

this using an empirical data set. The paper ends with summarizing the most

important ﬁndings of the present study, discussing the implications for longitudinal

research, and providing guidelines for future cross-lagged panel research.

Two models for studying reciprocal inﬂuences

Cross-lagged panel research is concerned with the eﬀect of two or more

variables on each other over time. To give an impression of the kinds of questions

researchers have tried to tackle using the CLPM, consider the following anthology:

Do maternal warmth and praise reduce internalizing and externalizing problems in

children with autism (Smith, Greenberg, Mailick Seltzer, & Hong, 2008)? Is the

relationship between parenting and adolescent delinquency bidirectional

(Gault-Sherman, 2012)? Does gratitude foster social support or vice versa (Wood et

al., 2008)? What is the direction of causality between intelligence and academic

achievement (Watkins, Lei, & Canivez, 2007)? Is processing speed a foundation for

successful cognitive aging (Deary et al., 2009)? What is the role of a pessimistic

explanatory style on developing and maintaining social support networks in

adolescents (Ciarrochi & Heaven, 2008)? What is the directional nature of the

relationship between the quality of the parent-child relationship and a child’s

Cross-lagged panel model 6

ADHD symptoms (Liﬀord, Harold, & Thapar, 2008)? And – at a macro

social-economic level – what is the direction of causality between intelligence and

economic welfare of nations (Rindermann, 2008)?

In this section the traditional CLPM is presented, which is the most typical

modeling approach for this kind of research. In addition, an alternative model is

presented, which we refer to as the random intercepts cross-lagged panel model

(RI-CLPM), that accounts for trait-like, time-invariant stability through the

inclusion of a random intercept (i.e., a factor with all loadings constrained to 1).

This random intercept partials out between-person variance such that the lagged

relationships in the RI-CLPM actually pertain to within-person (or within-dyad)

dynamics. We discuss how these models predict change, how many measurement

waves are needed for identiﬁcation, and how they are related to each other.

The CLPM

The CLPM can be used if two or more variables have been measured at two or

more occasions, and if the interest is in their inﬂuences on each other over time. Let

xand ydenote two distinct variables which were measured multiple times, and

which will be analyzed with the CLPM. While this approach typically consists of

modeling the covariance structure only, the means are included here as well; note

however that no constraints are imposed on them, which is equivalent to analyzing

the centered data.

A graphical representation of this model is given in the left panel of Figure 1

(see Appendix 1 for the corresponding SEM speciﬁcation). The measurement

Cross-lagged panel model 7

equations can then be expressed as

xit =μt+ξit (1a)

yit =πt+ηit (1b)

where ξit and ηit represent the individual’s temporal deviations from the temporal

group means μtand πtrespectively. These temporal deviations are modeled with

the structural equations

ξit =αtξi,t−1+βtηi,t−1+uit (1c)

ηit =δtηi,t−1+γtξi,t−1+vit.(1d)

The autoregressive parameter αtand δtare included to account for the stability of

the constructs: The closer these autoregressive parameters are to one, the more

stable the rank order of individuals is from one occasion to the next. However, even

when the stability coeﬃcients are very high, when enough time passes, the original

rank order will be lost. Hence, it is not stability of a trait-like nature, and it is

therefore often referred to as temporal stability instead (e.g., Heise, 1970).

Insert Figure 1 about here

The cross-lagged parameters βtand γtform the key to investigating reciprocal

causal eﬀects in this model (Rogosa, 1980): Through standardizing these

parameters, a comparison of the relative eﬀects of xand yon each other can be

made, which can then be used to determine causal predominance (Bentler &

Speckart, 1981). These parameters are often interpreted in terms of predicting

Cross-lagged panel model 8

change (e.g., Finkel, 1995; Ribeiro et al., 2011; Rindermann, 2008). To show the

reasoning behind this interpretation, we write

yit −yi,t−1=πt+ηit−πt−1+ηi,t−1

=πt−πt−1+δt−1ηi,t−1+γtξi,t−1+vit,(2)

which shows that the cross-lagged parameter γtindicates the extent to which the

change in ycan be predicted from the individual’s prior deviation from the group

mean on x(i.e., ξi,t−1=xi,t−1−μt−1), while controlling for the structural change in

y(i.e., πt−πt−1), and one’s prior deviation from the group mean on y(i.e.,

ηi,t−1=yi,t−1−πt−1).

The CLPM is just identiﬁed with only two waves of data, which makes it an

appealing modeling approach from a practical point of view: In fact, we found that

45% of the datasets published in 2012, which were used to estimate this model,

consisted of only two waves of data. In Figure 2 the distribution of all 117 datasets

from 2012 is given.2This is noteworthy, because it implies that in almost half of the

applications, the parameters of the CLPM and their standard errors can be

estimated, but it is not possible to evaluate whether the model provides a proper

description of the actual underlying mechanism (as the model is just identiﬁed and

will yield a perfect ﬁt, which is really not meaningful).

Insert Figure 2 about here

Cross-lagged panel model 9

The RI-CLPM

As described above, the CLPM only accounts for temporal stability through

the inclusion of autoregressive parameters. This implies that in this model it is

implicitly assumed that every person varies over time around the same means μt

and πt, and that there are no trait-like individual diﬀerences that endure. At closer

consideration, this is a rather problematic assumption, as it is diﬃcult to imagine a

psychological construct – whether behavioral, cognitive, emotional or

psychophysiological – that is not to some extent characterized by stable individual

diﬀerences (if not for the entire lifespan, then at least for the duration of the study).

Longitudinal data can actually be thought of as multilevel data, in which

occasions are nested within individuals (or other systems, like dyads). When

considering this perspective, it becomes clear that we need to separate the

within-person level from the between-person level. This idea motivated the

development of the alternative model we present here, which can be thought of as

an extension of the CLPM that accounts not only for temporal stability, but also for

time-invariant, trait-like stability through the inclusion of a random intercept. This

alternative model can be expressed as

xit =μt+κi+ξ∗

it (3a)

yit =πt+ωi+η∗

it (3b)

where μtand πtare again the temporal group means. The additional terms κiand

ωiare the individual’s trait-like deviations from these means: They can be thought

of as latent variables or factors whose factor loadings are all constrained to 1, as in

case of random intercepts in latent growth curve (LGC) modeling (with the

Cross-lagged panel model 10

diﬀerence that here the group means are allowed to vary freely over time). We have

added an asterisk to the temporal deviation terms ξ∗

it and η∗

it, to emphasize these

terms are diﬀerent from the individual deviation terms in the traditional CLPM: In

the current model they represent the individual’s temporal deviations from their

expected scores (i.e., μt+κiand πt+ωi), rather than from the group means (i.e., μt

and πt).

Subsequently these deviations are model as

ξ∗

it =α∗

tξ∗

i,t−1+β∗

tη∗

i,t−1+u∗

it (3c)

η∗

it =δ∗

tη∗

i,t−1+γ∗

tξ∗

i,t−1+v∗

it,(3d)

where the autoregressive and cross-lagged regression parameters diﬀer from the ones

in the CLPM, as indicated by the asterisks. That is, the autoregressive parameters

α∗

tand δ∗

tdo not represent the stability of the rank order of individuals from one

occasion to the next, but rather the amount of within-person carry-over eﬀect (cf.,

Hamaker, 2012; Kuppens, Allen, & Sheeber, 2010; Suls, Green, & Hillis, 1998): If it

is positive, it implies that occasions on which a person scored above his/her

expected score are likely to be followed by occasions on which he/she still scores

above the expected score again, and vice versa.3

The main interest here is however in the cross-lagged parameters β∗

tand γ∗

t,

which indicate the extent to which the two variables inﬂuence each other.

Speciﬁcally, γ∗

tindicates the degree by which deviations from an individual’s

expected score on y(i.e., η∗

it =yit −{πt+ωi}) can be predicted from preceding

deviations from one’s expected score on x(i.e., ξ∗

i,t−1=xi,t−1−{μt+κi}), while

controlling for the individual’s deviation of the preceding expected score on y(i.e.,

η∗

i,t−1=yi,t−1−{πt−1+ωi}). The cross-lagged relationships pertain to a process

Cross-lagged panel model 11

that takes place at the within-person level and they are therefore of key interest

when the interest is in reciprocal inﬂuences over time within individuals or dyads. A

graphical representation of this model is given in the right panel of Figure 1 (see

Appendix 1 for the corresponding SEM speciﬁcation).

Expressing change in the RI-CLPM, we can write

yit −yi,t−1=πt+ωi+η∗

it−πt−1+ωi+η∗

i,t−1

=πt−πt−1+δ∗

t−1η∗

i,t−1+γ∗

tξ∗

i,t−1+v∗

it,(4)

which shows that the cross-lagged parameter indicates the extent to which the

change in ycan be predicted from the individual’s prior deviation from his/her

expected score on the other variable (i.e., ξ∗

i,t−1=xi,t−1−{μt+κi}), while

controlling for the structural change in y(i.e., πt−πt−1)and the prior deviation

from one’s expected score on y(i.e., η∗

i,t−1=yi,t−1−{πt−1+ωi}).

The expressions in Equations 2 and 4 are similar, but unless κiand ωiare

zero, the CLPM predicts change from other aspects than the RI-CLPM. In fact, it is

easy to see that the traditional CLPM is nested under the current model, as it can

be obtained from the latter by ﬁxing the variances and covariance of κiand ωito

zero. To compare the two models statistically, a chi-square bar test should be used,

as it requires two parameters to be ﬁxed at the boundaries of the parameter space

(see for details: Stoel, Galindo Garre, Dolan, & Wittenboer, 2006).

While the CLPM requires only two waves of data, the RI-CLPM requires at

least three waves of data, in which case there is 1 degree of freedom (df).4If the

intervals are of the same size, and if we assume that the eﬀects the variables have on

each other remain stable over time, we could decide to constrain the lagged

parameters over time, giving us an additional 4 df (i.e., 5 df in total). Furthermore,

Cross-lagged panel model 12

we could investigate whether the means can be constrained over time, such that we

obtain another 4 df (resulting in 9 df in total). If on the other hand, we are not

willing to make these assumptions, and we are not sure whether the eﬀect of the

time-invariant stability components κiand ωiare equal over time, we may wish to

remove the constraint on the factor loadings. This relaxation may especially be of

interest when the observations are made further apart in time, and we expect that

we are also measuring some structural changes. However, this would imply that κi

and ωino longer represents random intercepts (as in multilevel modeling), but

rather represent latent variables or traits (as common in SEM). Even more so, it

would imply we need more waves of data to estimate this model.

Conclusion

The CLPM is nested under the RI-CLPM. The latter is an attempt to

disentangle the within-person process from stable between-person diﬀerences while

the former does not diﬀerentiate between these two levels that are likely to be

present in the data. The question thus rises what happens if the data were

generated by the RI-CLPM, but are analyzed using the CLPM: Most likely this will

lead to a contamination of the estimated within-person reciprocal eﬀects, but to

obtain more insight into this matter, we need to take a closer look at the

relationship between the cross-lagged parameters from both models.

However, before doing this, we consider how the RI-CLPM is connected to

other longitudinal SEM approaches that include cross-lagged relationships: In doing

so we aim to present a broader context for the current exposition and provide some

reference points for readers already familiar with (some of ) these SEM approaches.

Cross-lagged panel model 13

Relatedness to other existing SEM approaches

There are several other longitudinal SEM approaches that can be used for

bivariate data and which include cross-lagged relationships. Here we consider four of

these, that is: a) the Stable Trait Autoregressive Trait and State (STARTS) model

(Kenny & Zautra, 2001; Kenny & Zautra, 1995); b) the Autoregressive Latent

Trajectory (ALT) model (Bollen & Curran, 2006; Curran & Bollen, 2001); c) the

Latent Change Score (LCS) model (Hamagami & McArdle, 2001; McArdle &

Hamagami, 2001); and d) a modiﬁcation of the Latent State-Trait (LST) model

(Schmitt & Steyer, 1993; Steyer, Schwenkmezger, & Auer, 1990). In this section we

discuss the relatedness between the RI-CLPM and these four alternatives, focussing

on the substantive and methodological similarities and diﬀerences. Note that this

section is decidedly not meant as an in depth evaluation of these diverse alternatives:

The interested reader is referred to the included citations for further details.

STARTS model by Kenny and Zautra

The STARTS model by Kenny and Zautra (2001), is also known as the Trait

State Error (TSE) model (Kenny & Zautra, 1995). It allows the user to decompose

observed variance into three components: a) the stable trait, which does not change;

b) the autoregressive trait, which changes according to an autoregressive process;

and c) the state or error, which is unique to the occasion. Originally, Kenny and

Zautra (1995) included constraints over time in their model, such that the relative

contributions of these three components remains stable over time, but these

constraints may be relaxed if enough measurement waves are available (cf. Lucas &

Donnellan, 2007).

Cross-lagged panel model 14

Most applications of this model are based on univariate repeated

measurements, but Kenny and Zautra (1995) also present a bivariate extension of

their model. The RI-CLPM proposed in this paper diﬀers from the bivariate

STARTS model in that it does not include measurement error: The RI-CLPM can

thus be thought of as a special case of the STARTS model (without the constraints

on the lagged relationships over time), in which the observations are modeled

without measurement error.

Clearly, the inclusion of measurement error in itself is recommendable, as we

know that measurement error is likely to be present in psychological measurements.

However, Kenny and Zautra (2001) indicate that the model is often diﬃcult to

estimate, and that it may require 10 or more waves of data. Cole, Martin, and

Steiger (2005) performed a simulation study and concluded that the (univariate)

STARTS model frequently led to improper solutions that were diﬃcult to interpret

(i.e., negative variance estimates, or problems with convergence in the form of

singularity of the approximate Hessian matrix). They also discuss some of the

reasons for this: For instance, when the autoregressive parameter is very close to

zero, it becomes diﬃcult to distinguish between variance that is due to measurement

error, and variance that is the stochastic input of the autoregressive process. Thus,

while extending the model with measurement error may be preferable from a

theoretical point of view, the practical consequences (i.e., having to have many more

measurement waves), make it a less attractive alternative for the traditional CLPM.

ALT model by Curran and Bollen

The ALT model was developed by Curran and Bollen (2001; see also Bollen &

Curran, 2006), to “combine the best of two worlds”: It allows people to be

Cross-lagged panel model 15

characterized by their own trajectory over time (as in the LGC model), while their

observations may also exhibit some carry-over eﬀect from one occasion to the next

(as in the autoregressive or simplex model). In the bivariate extension of the ALT

model presented by Curran and Bollen (2001), the random eﬀects that describe the

individual trajectories may be correlated to each other across the variables (as is the

case in a bivariate LGC model), and there may also be cross-lagged inﬂuences

between the observations (as in the CLPM).

While this hybrid model seems to have a lot of potential, applying and

interpreting the ALT model is not as straight forward as one may be inclined to

think at ﬁrst: Because the lagged relationships are included in this model between

the observations, there is a recursiveness in the model, which has some adverse

eﬀects. First, it implies the process needs to be “started up”, for which Curran and

Bollen (2001) propose two solutions: Either the ﬁrst observation is treated as

exogenous, or nonlinear constraints are imposed on the loadings for the ﬁrst

occasion. While treating the ﬁrst occasion as exogenous is relatively easy, Jongerling

and Hamaker (2011) show that this may lead to rather unexpected growth curves:

For instance, in an ALT model with a random constant only (i.e., no linear trend

parameter), one may actually be modeling an increasing or decreasing trend over

time. Such undesirable eﬀects are not encountered when using the nonlinear

constraints to start up the process, but these require the assumption that the lagged

eﬀects are constant over time,5and are more diﬃcult to impose, especially in the

bivariate case.

Second, the recursiveness in the ALT model implies that the random constant

and the random change parameter no longer have the original role of individuals’

Cross-lagged panel model 16

intercepts and slopes (cf. Hamaker, 2005). For instance, the random constant not

only aﬀects an observation directly, but also indirectly through (all) previous

occasions. Hamaker (2005) has shown that under the assumption that the lagged

eﬀects are invariant over time, the ALT model can be rewritten as a LGC model

with autoregressive residuals, with the advantage that the random parameters in

this reparametrization serve as the random intercept and slope that describe the

underlying individuals’ deterministic trends. This result has also been extended to

multivariate processes, meaning that the bivariate ALT models can be rewritten as

a bivariate LGC model with residuals that are characterized by autoregressive and

cross-lagged regressive relationships (cf. Hamaker, 2005).

Considering this latter parametrization, the RI-CLPM is related to a bivariate

ALT model with only random intercepts and no random slopes. However, in the

RI-CLPM we do not constrain the mean structure, meaning that there may be

changes–possibly, but not necessarily linear–over time, which are identical for all

individuals. If the group means can be constrained to be equal over time, the

RI-CLPM is nested under the ALT model with only a random intercept and no

slope (using the parametrization proposed by Hamaker, 2005, to avoid the

recursiveness in the model).

LCS model by McArdle and Hamagami

The LCS model, also known as the Latent Diﬀerence Score (LDS) model, was

proposed by McArdle and Hamagami (2001; Hamagami & McArdle, 2001), and

forms a rather general modeling framework that includes many longitudinal SEM

approaches as special cases. What is characteristic of the LCS model is that latent

changes (i.e., the diﬀerences scores corrected for measurement error), from one

Cross-lagged panel model 17

occasion to the next are modeled as a function of a constant change parameter and a

proportional change parameter that depends on the preceding score: For this reason

the model is also referred to as the Dual Change Score model (McArdle, 2009).

In the bivariate extension of this model, change is not only a function of a

constant change parameter and the proportional change parameter, but also of the

preceding score on the other variable. The cross-lagged paths, going from one

variable to the change in the other, are referred to as coupling parameters,rather

than cross-lagged regression parameters. The interpretation is the same however, in

that signiﬁcant coupling parameters imply that one variable has the tendency to

impact changes in the other variable (McArdle & Grimm, 2010). But instead of

comparing standardized coeﬃcients in order to determine which variable is causally

dominant, the coupling parameters are used to set up a vector ﬁeld which depicts

the expected changes from one occasion to the next on both variables as a function

of the current state (see Boker & McArdle, 1995; McArdle, 2009; McArdle &

Grimm, 2010). This plot is then used to make statements like: “The resulting ﬂow

shows a dynamic process, where scores on Non-Verbal abilities have a tendency to

impact score changes on the Verbal scores, but there is no notable reverse eﬀect.”

(p. 348, McArdle, 2005).

The LCS model has been extended with what has been referred to as “dynamic

error”, to distinguish it from measurement error (see for instance McArdle, 2001):

While measurement error only aﬀects the observation at the current occasion,

dynamic error feeds forward through the lagged relationships, aﬀecting the

trajectory of the system and making it a stochastic rather than deterministic

process. The RI-CLPM can be thought of as closely related to the LCS model with

Cross-lagged panel model 18

dynamic error, but without measurement error or a constant change parameter.

However, the LCS model is characterized by a similar recursiveness as is present in

the ALT model, and therefore the random intercept term, which directly aﬀects the

ﬁrst latent score, also inﬂuences future occasions indirectly. Because the process is

not “started up” as is done in ALT modeling, the recursiveness is not dealt with in

such a way that we can ensure the process is stable in the absence of a constant

change parameter. As a result, the RI-CLPM is not a special case of the LCS

model, although they may be closely related in certain situations.

The LST model by Steyer and colleagues

The LST model was originally developed to distinguish between measurement

error and the true score (i.e., a latent variable), and to further decompose the true

score into a trait-like and a state-like part (Schmitt & Steyer, 1993; Steyer et al.,

1990). In practice this typically implies that it is assumed that there is an

underlying construct, which is measured by multiple indicators. This underlying

construct at a particular occasion is referred to as the state, which is then

decomposed into a trait-like part and an occasion-speciﬁc part: The trait-like part is

included as a second-order factor, relating the states, which are represented by the

ﬁrst-order factors, to each other. The occasion-speciﬁc part is the residual part of

the state factor, which was not accounted for by the trait.

The LST model has been extended with autoregressive relationships either

between the state factors (introducing a similar recursiveness as exists in the ALT

model and the LCS model), or between the occasion-speciﬁc components (to avoid

the detrimental recursiveness in the model): The latter has been coined the Trait

State Occasion (TSO) model (Cole et al., 2005). Recently, the TSO has been

Cross-lagged panel model 19

modiﬁed by Luhmann, Schimmack, and Eid (2011) to handle single indicator data.

In this modiﬁed model, the measurement error term is omitted, the trait factor is

modeled as a separate factor with free factor loadings over time (rather than a

second-order factor), and second-order autoregressive relationships are included.

Note that if the measurement error term had been kept (and the second-order

autoregressive relationships were omitted), the model would be identical to the

STARTS model.

Luhmann et al. (2011) also propose a bivariate version of the model, which

includes cross-lagged regression paths between the occasion-speciﬁc components

(and no second-order autoregressive relationships). The RI-CLPM can be seen as a

special case of this bivariate single indicator LST model, in which the factor

loadings for the traits are constrained to 1 over time. In applying this model to

empirical data, Luhmann et al. focus on decomposing the variance into separate

parts, as is also the goal in applying the STARTS model and the original LST

model. Furthermore, they decompose the covariance between the two variables into

a part accounted for by the traits, a part accounted for by the autoregressive and

cross-lagged regressive relationships, and a part due to the relationship between the

residuals of the occasion-speciﬁc factors.

Conclusion

Clearly, the models discussed above show some overlap with each other and

with the RI-CLPM presented in the current paper. When considering these diverse

modeling strategies, two observations seem of key importance. First, if researchers

are speciﬁcally interested in decomposing the variance into trait-like and state-like

components and the means are not of interest, the STARTS model and the models

Cross-lagged panel model 20

based on the LST model are most relevant. In contrast, if the interest is in

individual developmental trajectories, the ALT model and the LCS model are more

appropriate, as they are based on modeling both the mean structure and the

covariance structure and allow for individuals to have their own growth curves.

Second, the STARTS model, the ALT model and the LST model are most typically

applied to univariate data (even though the original LST model uses multiple

indicators); while bivariate (or multivariate) extensions are possible, they do not

form the core focus and the cross-lagged regression parameters are not the key

interest. In contrast, the LCS model is most typically used to investigate how two

variables inﬂuence each other (based on the expected change described with the

vector ﬁeld), although it can also be applied to univariate data.

The above observations are relevant, because they help pitting the RI-CLPM

against these alternatives. The main inspiration for proposing the RI-CLPM is that

we want to obtain estimates of cross-lagged regression parameters that truly reﬂect

the underlying reciprocal process that takes place at the within-person level. The

model thus requires bivariate (or multivariate) data, the mean structure is not

(necessarily) of interest, and the focus is on how (i.e., positive or negative

cross-lagged coeﬃcients), and how much (i.e., compare standardized absolute values

of cross-lagged coeﬃcients) the variables inﬂuence each other. Hence, because the

focus is on the covariance structure rather than the mean and covariance structures,

we could say that the RI-CLPM is more closely related to the STARTS model and

the LST and TSO models. However, the goal is not to decompose the variance and

covariance into trait-like and state-like parts, but to determine how the variables

inﬂuence each other through the cross-lagged relationships at the within-person,

Cross-lagged panel model 21

state-like level, while controlling for trait-like diﬀerences at the between-person

level. With this goal in mind, the RI-CLPM can be thought of as more closely

related to the bivariate ALT model or the LCS model, although there is no inherent

interest in individual developmental trajectories.

In sum, it can be stated that all models discussed in this section could serve as

alternatives to the CLPM: Each model forms an attempt to separate between-person

trait-like diﬀerences from the within-person reciprocal process. While some of these

models include desirable properties such as measurement error and/or diﬀerences in

developmental trajectories, the advantage of the RI-CLPM is that it is most closely

related to the CLPM and requires only three waves of data. Since two or three

waves of data are currently the norm in cross-lagged panel research, the RI-CLPM

is more likely to be considered by researchers as a feasible alternative than models

that require (many) more waves. In the following sections we focus on the CLPM

and the RI-CLPM, but we return to the issue of other alternatives in the discussion.

Comparing the cross-lagged parameters

Cross-lagged panel research is characterized by three major objectives: ﬁrst,

the aim is to determine whether the variables have a signiﬁcant eﬀect on each other;

second, the question is which variable is causally dominant; and third, researchers

want to know whether a variable has a positive or negative inﬂuence on the other

variable. If researchers use the CLPM when the data were actually generated by the

RI-CLPM, the question is whether this alters their conclusions with respect to these

three objectives. In this section we focus on these issues through considering the

cross-lagged regression parameters from both models analytically and in simulations.

Cross-lagged panel model 22

Analytical comparison

In Appendix 2 we show that the standardized cross-lagged regression

parameter in the CLPM from variable xto variable ycan be expressed as a function

of the parameters of the RI-CLPM, that is

γt

SD(xi,t−1)

SD(yit)=1−cov(ωi,κ

i)+cov(η∗

i,t−1,ξ

∗

i,t−1)2−1

×cov(ωi,κ

i)+δ∗

tcov(η∗

i,t−1,ξ

∗

i,t−1)+γ∗

tvar(ξ∗

i,t−1)

−cov(ωi,κ

i)+cov(η∗

i,t−1,ξ

∗

i,t−1)

×var(ωi)+δ∗

tvar(η∗

i,t−1)+γ∗

tcov(η∗

i,t−1,ξ

∗

i,t−1),(5)

which shows that it is a complex function of: a) the cross-lagged regression

coeﬃcient from variable xto variable y,thatisγ∗

t; b) the within-person

autoregressive parameter of variable y,thatisδ∗

t; c) the covariance between the

within-person deviations at the previous time point, that is cov(η∗

i,t−1,ξ

∗

i,t−1); d) the

variance of the within-person deviation at the preceding occasion, that is var(η∗

i,t−1);

e) the variance of the trait-like component, that is var(ωi); and f ) the covariance

between the trait-like components, that is cov(ωi,κ

i).

Considering the ﬁrst objective of cross-lagged panel research, that is, is there a

signiﬁcant eﬀect of one variable on the other, the relationship in Equation 5 is not

very informative, although it may be expected that the two models will not

necessarily lead to same conclusion regarding the presence of a cross-lagged

relationship.

With respect to the second objective, the question is whether the diﬀerence in

absolute values of the standard cross-lagged coeﬃcients is of the same sign across

Cross-lagged panel model 23

the two models. That is, the question is whether

|γSD(xi,t−1)

SD(yit)|−|βSD(yi,t−1)

SD(xit)|and |γ∗SD(ξ∗

i,t−1)

SD(η∗

it)|−|β∗SD(η∗

i,t−1)

SD(ξ∗

it)|.

are either both positive, leading to the conclusion that xis causally dominant, or

both negative, leading to the conclusion that yis causally dominant. If these

diﬀerences are not of the same sign, this implies that using one model leads to the

conclusion that xis causally dominant, while the other model leads to the

conclusion that yis causally dominant. Clearly, that is not a desirable situation.

For instance, when investigating the reciprocal inﬂuences of mothers’ harshness and

children’s behavioral problems, the RI-CLPM may indicated that the mothers are

causally dominant and form the driving force in this potentially negative spiral,

while the CLPM may point to the children as being the instigator of maladaptive

patterns. However, due to the rather complex relationships between the models’

diﬀerences of absolute standardized cross-lagged parameters, it is diﬃcult to

evaluate when these models will lead to conﬂicting conclusions, although in general

we may expect that larger trait-like diﬀerences are likely to have a stronger eﬀect

than in case of small between-person diﬀerences.

The third objective concerns the sign of the cross-lagged parameters. Thus the

question is: If γ∗>0, will γ>0, and when γ∗<0, will γ<0? Naturally, the same

question applies to β∗and β. Although this is not immediately apparent from the

expression in Equation 5, the many unrelated terms from the two levels strongly

suggest that γ∗and γnot necessarily have the same sign. This is again quite

disturbing, as it suggests that using the CLPM may lead to the conclusion that

mothers’ harshness has a damping eﬀect on children’s behavioral problems, while

the RI-CLPM may indicate that mothers’ harshness actually exacerbates the

Cross-lagged panel model 24

children’s behavioral problems.

Simulations

In order to further investigate the eﬀect of using the CLPM instead of the

RI-CLPM with respect to the three objectives of cross-lagged panel research

identiﬁed above, we performed a series of simulations based on four models. We

emphasize that the models used here were handpicked, to illustrate several speciﬁc

situations that can arise, and we do not claim that these are necessarily reﬂecting

realistic scenarios. Speciﬁcally, we used Mplus (Muth´en & Muth´en, 1998-2012), to

simulate two-wave bivariate data according to a RI-CLPM, which were subsequently

used to estimate the traditional CLPM. For each model, 1000 replications were

generated, of N= 200 each. Saving the parameter estimates in a separate ﬁle,

which we then imported into R (R Core Team, 2012), we computed the

standardized cross-lagged parameters (as Mplus does not allow for the computation

of standardized parameters in case of Monte Carlo simulations).

In the ﬁrst model, we had autoregressive parameters of .5 and no cross-lagged

regression coeﬃcients. The within-person variances of both variables was set to 1,

and the covariance between the two variables was .4. Since we made sure the

process was stationary (meaning the variances and covariances are stable over time;

cf. Hamilton, 1994), this implies that the residual variances at the second wave were

.75 and the residual covariance was .3. The between-person variances were set to 3

for each variable, and the covariance at this level was set to -2. Hence, this

represents a process which is characterized by a negative correlation at the

between-person level, while there is a positive correlation at the within-person

level,6which can be seen as an instance of Simpson’s paradox (cf., Kievit,

Cross-lagged panel model 25

Frankenhuis, Waldorp, & Borsboom, 2013). In the upper-left panel of Figure 3, the

standardized cross-lagged parameter estimates of this model are plotted. It clearly

shows that the point estimates are far from the generating values (indicated by the

diamond). The average βestimate was -.118 (SD=.036, average SE=.036), and the

average γestimate was -.120, (SD=.037, average SE=.036). Considering whether

the 95% conﬁdence intervals of these parameter estimates contained zero, we

obtained coverage rates of .105 for the βparameter, .103 for the γparameter, which

implies that in about 90% of the cases, the CLPM would lead to the conclusion that

there is at least one signiﬁcant negative cross-lagged parameter, although no

cross-lagged relationships were present in the model that generated the data.

Insert Figure 3 about here

The second model is based on autoregressive parameters of .5 and cross-lagged

regression parameters of .3. The within-person variances were set to 1, and the

within-person covariance to .5, which through the additional constraint of

stationarity implies that the innovations variances were .51 and the covariance

between the innovations was .03. The between-person variances were set to 2, and

the between-person covariance was set to -1. In the upper-right panel of Figure 3

the standardized cross-lagged parameter estimates are plotted. Based on 1000

replications, the average βestimate was -.003 (SD=.032, average SE=.034), and the

average γestimate was -.003 (SD=.034, average SE=.034). Coverage rates for the

95% conﬁdence interval containing zero were .951 and .945 respectively, indicating

that in about 95% of the cases it would be concluded that these parameters are

Cross-lagged panel model 26

non-signiﬁcantly diﬀerent from zero, although there were substantial cross-lagged

relationships in the model that generated the data.

The third model is based on autoregressive parameters of .5, and a

cross-lagged parameter βof -.3 (from variable yto variable x), and a cross-lagged

parameter γof .1 (from xto y). The within-person variances were both set to 1,

and the covariance to -.5, such that the innovation variances were .51 for the

x-variable and .79 for the y-variable, while their covariance was -.29 (to ensure

stationarity). The between-person variances were set to 2 and their covariance was

set to 1. The standardized cross-lagged parameter estimates are given in the

lower-left panel of Figure 3. It shows that while the original combination of

parameter values is in the area that is characterized by a standardized |γ|that is

smaller than the standardized |β|, indicating that variable yis causally dominant,

most point estimates fall in the area in which the standardized |γ|is actually larger

than the standardized |β|, leading to the opposite conclusion that variable xis

causally dominant. The average estimate for βis .002 (SD=.039, average SE=.040),

and for γit is .151 (SD=.034, average SE=.033). For β(which equalled -.3 in the

generating model), the coverage rate of the 95% conﬁdence interval containing zero

was .958, which implies that in about 95% of the cases the conclusion would be that

there is a nonsignﬁcant relationship from yto x. The coverage rate γ(where true γ

is .1) was .010, which implies that in 90% of the cases a signiﬁcant relationship from

variable xto ywould be detected. This further shows that the CLPM may result in

the wrong variable being identiﬁed as being causally dominant.

Finally, in the fourth model the autoregressive parameters were set to .5, the β

parameter (from variable ytox)to.3andaγparameter of .1. The within-person

Cross-lagged panel model 27

variances are both 1, the covariance is .5, which in combination with the restriction

of stationarity leads to the innovation variances of variables xand ybeing .72 and

.60 respectively, and a covariance between the innovations of -.056. The

between-person variances ware set to 3 and their covariance to -2. The standardized

point estimates of the cross-lagged parameters are presented in the lower-right panel

of Figure 3, showing that, while the generating cross-lagged parameters implied that

variable ywas causally dominant, the parameter estimates almost always lead to

the conclusion that variable xis causally dominant. The average point estimate for

βwas -.023 (SD=.037, average SE=.036), and for γit was -.093 (SD=.033, average

SE=.033). The 95% conﬁdence intervals included zero with a rate of .897 for β,and

.192 for γ, meaning that in almost 90% of the cases we would fail to detect the

relationship from variable yto x(which in reality was .3), while in more than 80%

of the cases we would detect a signiﬁcant negative relationship from variable xto y

(which in reality was .1). This illustrates another disturbing fact: The CLPM may

result in a signiﬁcant estimate of a cross-lagged parameter that actually has a

diﬀerent sign than the corresponding cross-lagged parameter in the generating

model.

Conclusion

While the algebraic relationship in Equation 5 shows that the cross-lagged

parameters from the two models are not necessarily identical, it is not easy to see

how they will diﬀer, especially in the light of the three objectives of cross-lagged

panel research. The simulations we presented here show however that the CLPM

can lead to spurious results regarding all three objectives in this line of research,

that is, it can be misleading with respect to: a) the presence of causal relationships

Cross-lagged panel model 28

(models 1 and 2); b) the causal priority of two variables (models 3 and 4); and c)

the sign of the causal relationship (model 4).

The simulations here were designed to illustrate these speciﬁc situations,

without the intention to represent typical psychological processes. The fact is that

we do not know what would be typical values for the parameter of the RI-CLPM,

because this is not a model that is currently used in practice. In the simulations here

the between-person variance was relatively large (i.e., two or three times as large as

the within-person variance), and in general it can be stated that the results from

the CLPM deviated more from the generating RI-CLPM when the between-person

variances increased. Furthermore, the correlation at the between-person level also

inﬂuences the results, especially if it is of the opposite sign of the correlation that

exists at the within-person level (i.e., in the presence of Simpson’s paradox).

Finally, sample size aﬀects the variability in estimates and their standard errors

(i.e., both are inversely related to sample size), but the bias resulting from

estimating a model that does not distinguish between within-person dynamics and

between-person trait-like diﬀerences does not vanish when sample size increases.

Modeling strategy

To avoid the pitfall exposed above, we propose a modeling strategy that allows

us to investigate whether there are trait-like, time-invariant individual diﬀerences

present in the constructs that are studied, which should be accounted for through

the inclusion of a random intercept. This strategy is based on the fact that the

CLPM is nested under the RI-CLPM, such that if three or more waves of data are

available, both models can be ﬁtted to the data and can be compared using a

Cross-lagged panel model 29

chi-bar-square test for the diﬀerence in chi-squares (Stoel et al., 2006). We illustrate

this strategy using data that are reported in Soenens et al. (2008), concerning the

eﬀect of diverse aspects of parenting style on depressive symptoms of adolescents

and vice versa. The data were obtained from 396 students and consist of three

waves, with intervals of one year, starting in the fall of the ﬁrst year in college.

We begin with considering the relationship between Parental Psychological

Control (based on items like “My parents are less friendly to me if I don’t see things

like they do”), and Adolescents’ Depressive Symptomatology.First,weﬁtamodelin

which the means of each variable are constrained over time (i.e., μt=μand πt=π),

while the covariance structure is unconstrained: Models in which the group means

do not change over time facilitate interpretation, although it time-invariant means

are no prerequisite for the models considered here. The ﬁt of this model is not

satisfactory (chi-square is 13.75, 3 df, p=.008; RMSEA = .078; CFI = .990; SRMR

=.024), and inspection of the means shows that especially the mean of Adolescents’

Depressive Symptomatology at the ﬁrst wave is higher than at the other two waves:

This measurement is from the ﬁrst semester that the participants are in college, and

the elevated average may thus reﬂect the diﬃculties associated with getting

adjusted to these new circumstances. Freeing this mean leads to appropriate model

ﬁt (χ2(3) = 3.33, p=.344; RMSEA= .017; CFI= 1.000; SRMR= .011). Although

constraining this ﬁrst mean does not aﬀect our results for the lagged parameters in

a substantive way, the results reported below are based on the model in which this

ﬁrst mean for Adolescents’ Depressive Symptomatology is not constrained to be

equal to the means at subsequent waves.

Second, we model the covariance structure using the RI-CLPM, while keeping

Cross-lagged panel model 30

the constraints on the means (except for the ﬁrst mean of Adolescents’ Depressive

Symptomatology ), and time-invariant lagged parameters. This model ﬁts well

(chi-square is 9.85, 8 df, p=.276; RMSEA = .024; CFI = .998; SRMR = .025).

Finally, we ﬁt the CLPM, with the same constraints on the means and lagged

parameters as used in the previous model. This model does not ﬁt well (chi-square

is 66.18, 11 df, p<.001; RMSEA = .113; CFI = .943; SRMR = .042), although

some SEM users may claim on the basis of the CFI and the SRMR that the model

ﬁts approximately. Note that since the null-model here consists of ﬁxing two

parameters on the boundary of the parameter space (i.e., two variances ﬁxed to

zero), the standard chi-square diﬀerence test will be too conservative (see Stoel et

al., 2006). The chi-square diﬀerence is 66.18 −9.85 = 56.33, with 3 df, which is

signiﬁcant at an αof .05 (that is, p<.01).

To show that the substantive interpretation of the underlying process depends

on the model one uses, we consider the standardized cross-lagged regression

parameter estimates from both models presented in Figure 4. It shows that both

models lead to signiﬁcant positive cross-lagged parameters. However, while the

RI-CLPM indicates that the eﬀect of Parental Psychological Control on Adolescents’

Depressive Symptomatology is only slightly larger than the reverse eﬀect (i.e., .240

versus .212 and .265 versus .205 between wave 1 and wave 2), the CLPM leads to

the conclusion that the eﬀect of parents on adolescents is much larger than that of

adolescents on their parents (i.e., .239 versus .139 and .248 versus .134 between

wave 1 and wave 2). Hence, using the CLPM would lead to the conclusion that

parents are causally dominant, while the RI-CLPM leads to the conclusion that the

reciprocal process is much more symmetric.

Cross-lagged panel model 31

Insert Figure 4 about here

We apply the same procedure for the variables Parental Responsiveness (based

on items like “My parents make me feel better after I discussed my worries with

them”), and Adolescents’ Depressive Symptomatology. Here, both the ﬁrst mean of

the adolescents’ variable, and the last mean of the parents’ variable were estimated

freely, in order to obtain a ﬁtting model (chi-square is .933, 2 df, p=.627; RMSEA

=.000; CFI = 1.000; SRMR = .006): The last mean of Parental Responsiveness was

signiﬁcantly lower than that at the other two measurement waves, which may reﬂect

the increasing independence of the adolescents in the third year of college. The

RI-CLPM ﬁtted well (chi-square is 11.86, 7 df, p=.105; RMSEA = .042; CFI

=.996; SRMR = .031), while the CLPM did not lead to a well ﬁtting model

(chi-square is 76.01, 10 df, p<.001; RMSEA = .129; CFI = .939; SRMR = .048),

although again, some SEM users may claim this model is not too far oﬀ. The

chi-square diﬀerence is 76.01 −11.86 = 64.15, with 3 df, which is signiﬁcant at an α

of .05 (that is, p<.01).

Comparing the standardized lagged parameter estimates from both models

given in Figure 4, the RI-CLPM leads to the conclusion that there are no reciprocal

inﬂuences between Parental Responsiveness and Adolescents’ Depressive

Symptomatology, whereas the CLPM leads to the conclusion that there is a

signiﬁcant negative eﬀect from Parental Responsiveness to subsequent Adolescents’

Depressive Symptomatology (and while there is no signiﬁcant eﬀect from adolescents

to parents, it would be concluded that parents are causally dominant here).

Cross-lagged panel model 32

In conclusion, the modeling strategy illustrated above shows that it is possible

to investigate whether the constructs are characterized by time-invariant, trait-like

individual diﬀerences, and that using the traditional CLPM can lead to erroneous

conclusions regarding the pattern of mutual inﬂuences. Hence, researchers should

make sure to use an alternative that decomposes the variance into between-person

diﬀerences and the within-person process. If the constructs are not characterized by

time-invariant, trait-like individual diﬀerences, running the RI-CLPM will not aﬀect

the results substantially, although in that case one can also use the simpler CLPM

instead.

Discussion

Rogosa summarized his critique on the cross-lagged correlation

methodology–which he referred to as CLC–saying: “CLC may indicate the absence

of direct causal inﬂuence when important causal inﬂuences, balanced or unbalanced,

are present. Also, CLC may indicate a causal predominance when no causal eﬀects

are present. Moreover, CLC may indicate a causal predominance opposite to that of

the actual structure of the data; that is, CLC may indicate that Xcauses Ywhen

the reverse is true.” (p. 246, Rogosa, 1980). In the current paper, similar problems

have been exposed in the context of the CLPM. That is, the CLPM may indicate

there are reciprocal eﬀects when these do not exist (model 1), and may fail to detect

them when they do exist (model 2). Furthermore, the CLPM may identify one

variable as being causally dominant, when in fact the other variable is (models 3

and 4). Finally, the CLPM may indicate a negative inﬂuence from one variable on

another, while in reality the eﬀect is positive (model 4).

Cross-lagged panel model 33

The source of these problems is the failure to adequately separate the

within-person and the between-person level in the presence of time-invariant,

trait-like individual diﬀerences. As a result, the estimates of lagged parameters are

confounded by the relationship that exists at the between-person level (see

Hamaker, 2012 for other situations in which this confounding may occur). As it is

reasonable to assume that most psychological constructs that are studied with

cross-lagged panel designs are to some extent characterized by time-invariant

stability reﬂecting a trait-like property (at least for the duration of the study), it

follows that many lagged parameters reported in the literature will not reﬂect the

actual within-person (causal) mechanism.

This is especially problematic if one wishes to use the results from cross-lagged

panel research as a basis for future interventions. For instance, the results obtained

with the traditional CLPM for adolescent depression and parental responsiveness in

this paper, would lead the researcher to conclude that increasing parental

responsiveness should result in a reduction in depressive symptoms on part of the

adolescent; however, the RI-CLPM shows that this result is an artefact, and that

there is actually no lagged eﬀect from parental responsiveness to adolescents’

depression. Note that this does not imply that the two variables are unrelated: In

fact, the trait-like individual diﬀerences are negatively correlated (estimated

correlation is -.443, SE= .067, p<.001), indicating that parents who tend to be

more responsive on average, tend to have adolescents who suﬀer less from depressive

symptomatology on average. However, we cannot derive a causal mechanism from

these results, which explains this relationship and that can be used as the

foundation for an intervention. This shows that “getting it right” with respect to

Cross-lagged panel model 34

the cross-lagged relationships is not just an academic concern.

We found that 45% of the studies that make use of the CLPM are based on

only two waves of data. In these cases, the CLPM is saturated, and hence no

statements regarding model ﬁt can be made: That is, the model will always ﬁt

perfectly, and the interest in estimating this model is simply in obtaining estimates

of the cross-lagged regression parameters which are corrected for the temporal

stability of the constructs. This implies that to date, it is impossible to tell what

portion of the results reported in the literature based on the CLPM provide truthful

reﬂections of the actual reciprocal mechanisms, and what portion is ﬂawed and if so,

how serious these errors are.

Researchers interested in studying lagged relationships are therefore well

advised to employ the following approach. First, a minimum of three measurement

waves are required: Only then can the within-person process be controlled for stable

between-person diﬀerences through the inclusion of a random intercept. Second,

start with a model in which only the means are constrained over time, while the

covariance structure is estimated freely: This allows one to determine whether there

are structural changes over time. If this model proves tenable, subsequent models

can be speciﬁed for the covariance structure, while leaving the means constrained

over time. If the ﬁrst model proves untenable however, the researcher should

identify the source of misﬁt, and consider freeing certain means (as we did in the

empirical applications included in this paper), or refrain to an alternative modeling

approach, such as LGC or ALT modeling (Hamaker, 2005). If there is no need to

refrain to an alternative model based on the mean structure, the researcher is

advised to continue with the extended models which were proposed in this paper,

Cross-lagged panel model 35

and compare the CLPM to the RI-CLPM in order to determine whether the

constructs are characterized by trait-like between-person diﬀerences, or that it can

be assumed that all individuals vary around the same mean or trend.

In addition, if researchers expect their measurements to contain measurement

error, they are advised to either obtain a large number of repeated measurements

(say >10) such that they can estimate a bivariate STARTS model, or to obtain

multiple indicators (e.g., test halves) such that they can ﬁt a bivariate TSO model.

In both cases they will be able to distinguish between the within-person process and

stable trait-like between-person diﬀerences, while controlling for measurement error.

Finally, although we restricted our focus here on an alternative model in which

the eﬀect of the between-person diﬀerences is constant over time (such that it can

be modeled as a random intercept and there is a clear connection with multilevel

modeling), we recognize that this will not always be a tenable assumption. That is,

when observations are taken further apart in time, the eﬀect of stable individual

diﬀerences may vary over time, such that the constraints on the factor loadings

should be relaxed. This implies that more waves of data are necessary, and that

other alternatives such as discussed in this paper should also be considered. Our

intention is not to try to convince researchers that the RI-CLPM is necessarily the

best alternative for the CLPM – there will be many instances where one of the other

alternatives is more suited – but rather, to raise awareness regarding the limitation

of the CLPM for uncovering within-person reciprocal processes.

Cross-lagged panel model 36

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Cross-lagged panel model 42

Appendix 1

Specifying a CLPM for three occasions can be done with the measurement

equation

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

xi1

yi1

xi2

yi2

xi3

yi3

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

μ1

π1

μ2

π2

μ3

π3

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

+

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

ξ1i

η1i

ξ2i

η2i

ξ3i

η31

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,(6a)

and structural equation

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

ξ1i

η1i

ξ2i

η2i

ξ3i

η31

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

000000

000000

α2β20000

γ2δ20000

00α3β300

00γ3δ300

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

ξ1i

η1i

ξ2i

η2i

ξ3i

η31

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

+

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

ξ1i

η1i

u2i

v2i

u3i

v31

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,(6b)

where the covariance matrix of the latter residual vector is

Ψ=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

σ2

x1

σx1y1σ2

y1

00σ2

u2

00σu2v2σ2

v2

0000σ2

u3

0000σu3v3σ2

v3

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

.(6c)

Cross-lagged panel model 43

Note that the variances and covariance between ξi1and ηi1are identical to those of

xi1and yi1in this model.

Specifying the RI-CLPM for three waves of data in SEM software is based on

the measurement equation

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

xi1

yi1

xi2

yi2

xi3

yi3

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

μ1

π1

μ2

π2

μ3

π3

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

+

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

10000010

01000001

00100010

00010001

00001010

00000101

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

ξ∗

1i

η∗

1i

ξ∗

2i

η∗

2i

ξ∗

3i

η∗

31

κi

ωi

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,(7a)

and structural equation

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

ξ∗

1i

η∗

1i

ξ∗

2i

η∗

2i

ξ∗

3i

η∗

3i

κi

ωi

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

00000000

00000000

α∗

2β∗

20 0 0000

γ∗

2δ∗

20 0 0000

00α∗

3β∗

30000

00γ∗

3δ∗

30000

00000000

00000000

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

ξ∗

1i

η∗

1i

ξ∗

2i

η∗

2i

ξ∗

3i

η∗

3i

κi

ωi

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

+

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

η∗

1i

ξ∗

1i

u∗

2i

v∗

2i

u∗

3i

v∗

3i

κi

ωi

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,(7b)

Cross-lagged panel model 44

where the covariance matrix of the latter residual vector is

Ψ=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

σ2

ξ∗

1

σξ∗

1η∗

1σ2

η∗

1

00σ2

u∗

2

00σu∗

2v∗

2σ2

v∗

2

0000σ2

u∗

3

0000σu∗

3v∗

3σ2

v∗

3

000000σ2

κ

000000σκ,ω σ2

ω

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

.(7c)

Note that in contrast to the previous model, here the variances and covariance of ξ∗

1i

and η∗

1iare not identical to those of xi1and yi1(unless κi=ωi=0foralli).

Cross-lagged panel model 45

Appendix 2

The standardized cross-lagged parameters in the traditional CLPM can be

expressed as partial correlations (e.g., Heise, 1970). Focussing on the cross-lagged

parameter γtfrom ξi,t−1to ηit, and making use of the fact that ξi,t and ηit are the

group mean centered variablesxit and yit , we can write

γt

σ(xi,t−1)

σ(yit)=ρ(xi,t−1yit)−ρ(yi,t−1xi,t−1)ρ(yi,t−1yit)

1−ρ(yi,t−1xi,t−1)2.(8)

In order to see how the cross-lagged parameter γfrom the traditional CLPM

is related to the cross-lagged parameters γ∗of the RI-CLPM, we need to express the

correlations used on the righthand side of Equation 8 in terms of the parameters of

the latter model. If we assume that all the observed variables are standardized, the

correlation between yi,t−1and yit can be expressed as

ρ(yi,t−1yit)=E

ωi+η∗

i,t−1ωi+η∗

it

=E

ω2

i+E

η∗

i,t−1η∗

it

=var(ωi)+E

η∗

i,t−1δ∗

tη∗

i,t−1+γ∗

tξ∗

i,t−1+v∗

it

=var(ωi)+E

δ∗

tη∗2

i,t−1+E

γ∗

tη∗

i,t−1ξ∗

i,t−1

=var(ωi)+δ∗

tvar(η∗

i,t−1)+γ∗

tcov(η∗

i,t−1,ξ

∗

i,t−1),(9)

while the correlation between yi,t−1and xi,t−1can be expressed as

ρy1x1=E

ωi+η∗

i,t−1κi+ξ∗

i,t−1

=E

ωiκi+E

η∗

i,t−1ξ∗

i,t−1

=cov(ωi,κ

i)+cov(η∗

i,t−1,ξ

∗

i,t−1),(10)

Cross-lagged panel model 46

and the correlation between yit and xi,t−1can be expressed as

ρ(xi,t−1yit)=E

κi+ξ∗

i,t−1ωi+η∗

it

=E

κiωi+E

ξ∗

i,t−1η∗

it

=cov(ωi,κ

i)+E

ξ∗

i,t−1δ∗

tη∗

i,t−1+γ∗

tξ∗

i,t−1+v∗

it

=cov(ωi,κ

i)+E

δ∗

tξ∗

i,t−1η∗

i,t−1+E

γ∗

tξ∗2

i,t−1

=cov(ωi,κ

i)+δ∗

tcov(η∗

i,t−1,ξ

∗

i,t−1)+γ∗

tvar(ξ∗

i,t−1) (11)

Using these expressions for the correlations in Equation 8, we can now write

γt

SD(xi,t−1)

SD(yit)=cov(ωi,κ

i)+δ∗

tcov(η∗

i,t−1,ξ

∗

i,t−1)+γ∗

tvar(ξ∗

i,t−1)

1−cov(ωi,κ

i)+cov(η∗

i,t−1,ξ

∗

i,t−1)2

−cov(ωi,κ

i)+cov(η∗

i,t−1,ξ

∗

i,t−1)var(ωi)+δ∗

tvar(η∗

i,t−1)+γ∗

tcov(η∗

i,t−1,ξ

∗

i,t−1)

1−cov(ωi,κ

i)+cov(η∗

i,t−1,ξ

∗

i,t−1)2.

Similarly, the relationship between βtand β∗

tcan be derived.

Cross-lagged panel model 47

Foot notes

1While the omitted variable problem implies that we cannot make strong

causal statements based on correlational data, it does not prohibit the use of the

concept of Granger causality (Granger, 1969). However, many researchers using

cross-lagged regression refrain from using the term causal, and use terms like

reciprocal relationship (Erickson, Wolfe, King, King, & Sharkansky, 2001; Lindwall,

Larsman, & Hagger, 2011), role (Ribeiro et al., 2011), cross-domain eﬀects (Burt,

Obradovi´c, Long, & Masten, 2008), exposure (Cole, Nolen-Hoeksma, Girgus, &

Paul, 2006), impact (Gault-Sherman, 2012), or inﬂuence (Green, Furrer, &

McAllister, 2011), instead. It may be argued however, that these alternative terms

also imply a causal mechanism, and even more so, that an interest in causality is

actually the driving force behind these studies. Therefore, we decided to use the

terms causal and causality in the current paper, although we acknowledging that

strong causal statements can only be based on experimental designs, and we should

conﬁne ourselves to the concept of Granger causality.

2We used PsychINFO and searched for peer reviewed papers that appeared in

2012 and which made reference to the term “cross-lagged” in either the title, the

abstract or the key words. We found 115 peer reviewed publications of which two

were on time series analysis, one on multilevel modeling, and one did not include an

application. The 111 remaining publications reported on 117 datasets.

3One could also say these autoregressive parameters indicate the stability of the

rank-order of individual deviations, but this is less appealing from a substantive

viewpoint.

4The number of observed statistics in the covariance matrix is equal to

Cross-lagged panel model 48

(6*7)/2=21, while the number of parameters for the covariance structure equals 20,

that is: 2 variance and 1 covariance for the between-person structure (i.e., the

random intercepts), 2 variances and 1 covariance for the ﬁrst occasion at the

within-person level, 4 lagged parameters for the ﬁrst interval, 4 lagged parameters

for the second interval, 2 residual variances and 1 residual covariance at the second

occasion at the within-person level, and 2 residual variances and 1 residual

covariance at the third occasion at the within-person level.

5Actually, one only has to assume the lagged relationships were invariant before

the observations started, which is rather abstract when considering the model as a

local description instead of an everlasting truth; hence, this is not a very restrictive

assumption in practice.

6A possible example could be the relationship between number of words typed

per minute and the number of typos: At the within-person level there is a positive

relationship, as a person tends to make more mistakes when (s)he types faster, while

at the between-person level there is a negative relationship as people who have more

experience tend to type faster while making fewer mistakes, and vice versa.

Cross-lagged panel model 49

Figure Captions

Figure 1. Two bivariate models for three waves of data: the standard CLPM, and

the alternative RI-CLPM. Squares denote observed variables; circles represent latent

variables; triangles represent means.

Figure 2. Histogram of number of waves per data set from 111 peer reviewed

publications referring to cross-lagged research in 2012.

Figure 3. Standardized cross-lagged parameter estimates obtained with the

traditional CLPM. Generating values from the RI-CLPM are denoted by the

diamond. Areas A indicate solutions in which |β|<|γ|such that variable xis

causally dominant; areas B indicate solutions in which β|>|γ|such that variable y

is causally dominant. Only 250 estimates (of the 1000 replications) per model are

plotted for reasons of clarity.

Figure 4. Standardized parameter estimates for Soenens data obtained with the

RI-CLPM (above the arrows) and the CLPM (below the arrows). Standard errors

are given between parentheses. ∗indicates signiﬁcant at α=.05; ∗∗ indicates

signiﬁcant at α=.01; ∗∗∗ indicates signiﬁcant at α=.001.

y1 y2 y3

η2 η3

δ2 δ3

v2

η1

1

1

π1 π2 π3

1 1

1 1

CLPM with means RI-CLPM with means

ξ2 ξ3

α2 α3

ξ1

v3

u2 u3

x1 x2 x3

1 1 1

1

μ2

μ1 μ3

β2

γ2

β3

γ3

y1 y2 y3

η2 η3

δ2 δ3

v2

η1

1 1 1

1 1

ξ2 ξ3

α2 α3

ξ1

v3

u2 u3

x1 x2 x3

1 1 1

β2

γ2

β3

γ3

κ

1

1 1

1

ω

1

1

1

1

π1

π2

π3

μ2

μ3

μ1

0

10

20

30

40

50

60

123456

Number of datasets

2 3 4 5 6 7

Number of waves

-0.4 -0.2 0.0 0.2 0.4

-0.4 -0.2 0.0 0.2 0.4

Model 1

Standardized beta

Standardized gamma

-0.4 -0.2 0.0 0.2 0.4

-0.4 -0.2 0.0 0.2 0.4

Model 4

Standardized beta

Standardized gamma

A

A

BB

-0.4 -0.2 0.0 0.2 0.4

-0.4 -0.2 0.0 0.2 0.4

Model 3

Standardized beta

Standardized gamma

A

A

BB

-0.4 -0.2 0.0 0.2 0.4

-0.4 -0.2 0.0 0.2 0.4

Model 2

Standardized beta

Standardized gamma

P1 .230 (.073)*

.657 (.024)*** P2

A1 A2

.122 (.077)

.368 (.033)***

P3

A3

.265 (.094)** *

.675 (.025)***

.113 (.076)

.357 (.035)***

Parental Control

P1 .132 (.130)*

.780 (.180)*** P2

A1 A2

.073 (.088)

.424 (.033)***

P3

A3

.101 (.119)*

.772 (.020)***

.063 (.079)

.414 (.035)***

Parental Responsiveness

Adolescent Depressive Symptomatology

Adolescent Depressive Symptomatology