Fixed-effects (FE) regression isolates within-person changes over time to estimate how these changes relate to variations in a given outcome. By using individual-specific data transformations — such as demeaning (subtracting each individual’s own average), first-differencing (subtracting previous observations), or individual dummy variables — FE regression effectively removes any stable personal characteristics (e.g., family background or genetic traits) that might confound the estimation of causal effects through omitted variable bias (Angrist & Pischke, 2009). Importantly, these stable, potentially confounding factors may be observed or they may be unobserved — that is, not measured, not included in the dataset being analyzed, or indeed, possibly not known to the researcher. Due to its capacity to control for these time-invariant confounders, FE regression has become a foundational analytical tool in longitudinal research across the social sciences, public health, and other fields.

The core idea of FE regression can also be expressed within a Structural Equation Modeling (SEM) framework. Rindskopf (1984) initially showed that stable, unobserved factors (often referred to as “unobserved heterogeneity”) could be represented as a latent (or phantom) variable. Building on this idea, Allison (2009) and Bollen and Brand (2010) further extended the approach specifically for longitudinal data analysis, creating the Fixed-effects Structural Equation Model (FE-SEM), as illustrated in Figure 1. The figure is a path diagram with four time points, where Ui represents the unobserved time-invariant factors for each unit i (i.e., “unobserved heterogeneity” or individual-specific unit effects); Xit is an independent variable that changes over time for each unit.

The causal processes in the figure can be expressed in the following equation:

Specifically, Y_it is a function of X_it, with the latent U_i representing the set of stable unobservables that are constant for each unit i. ε_it refers to idiosyncratic error term that varies across units and time. The latent (or phantom) variable U_i serves as a proxy for all time-invariant omitted confounders (Allison, 2009; Finkel, 1995) and thus is allowed to covary with X_it which distinguishes the fixed-effects model from the traditional random-effects approach (Bollen & Brand, 2010). In the traditional FE model, it is usually assumed that the effect of X on Y (β1) remains consistent over time, yielding averaged effects across time points. Additionally, it posits that residuals are independent, there is no autoregressive parameter of Y_it, no reverse effects of Y_it-1 on X_it, and no measurement errors in the variables.

FE-SEM offers key advantages over traditional regression, including model fit statistics (e.g., chi-square, RMSEA) and efficient handling of missing data using maximum likelihood methods. It also allows tests for residual autocorrelation and allows β1 to vary over time. Further, if X or Y is measured by multiple items, we can directly model measurement errors of each latent construct (Allison, 2009; Allison et al., 2017; Bollen & Brand, 2010). Finally, with multiwave data it is possible to relax the initial assumption that the effects of the time-invariant U_i in Equation (1) are equal (and set arbitrarily to 1).

Conceptually, a time-invariant characteristic such as U_i (e.g., sex or race/ethnicity) may still exert time-varying influences on outcomes. By this we mean that the slope linking U_i to X_it or Y_it can change over time even though U_i itself is constant. For example, if women’s political knowledge or academic performance increases more sharply over time than men’s, the effect of sex on the level of knowledge or achievement necessarily differs between waves — even though sex itself does not change. In this sense, the slope associated with U_i can vary across time, reflecting contextual or developmental shifts in how a time-invariant factor relates to a time-varying outcome (see also Ren & Allison, 2025). The FE-SEM framework is flexible enough to accommodate such scenarios (Allison et al., 2017; Zyphur et al., 2020).

As noted, the critical assumption in FE-SEM is that the potential unobserved confounders (U_i) are stable over time. For example, estimating the effects of voluntary group memberships on social trust may be confounded by unobserved stable personality attributes such as extroversion that are related both to an individual’s propensity to join groups and to trust other people (Finkel, 2002). But the assumption that all potential unobserved confounders are time-invariant is almost certainly unrealistic.

For example, in investigating the effects of obesity status on academic achievement among school-age children, unobserved aspects of the child’s home environment may be essentially stable. But as children move from kindergarten to elementary school, their peer and teacher relationships, and their participation in physical or cultural activities will also change, and to the extent that these factors are related to both health and education outcomes (Carbonaro & Maloney, 2019; Umberson & Karas Montez, 2010), these factors may represent unobserved time-varying confounders in the causal process. In that case, the conventional fixed-effects models will yield biased estimates.

We display a model with unobserved time-invariant and time-varying confounders in Figure 2, and in Equation (2) below. To simplify exposition, we consider only two time points and assume contemporaneous effects as in the traditional fixed-effects panel model.

Figure 2 depicts a new variable representing time-varying confounding (Zit) along with the effects of the time-invariant U_i. Since the time-varying Zit confounds the relationship between X and Y at each time point, above and beyond the influences of Ui, adjustment for Z_it is of necessity in order to estimate the causal direct effect of X on Y (β1). However, in observational longitudinal studies, it is challenging to identify and control for a set of all potential confounders at each time point.¹1

An alternative method for dealing with time-varying confounding is an instrumental variable analysis where an outside “instrument” serves as a proxy for an explanatory variable whose effects may be confounded by omitted variable bias. In observational studies, however, it can be challenging to identify suitable IVs that meet the assumptions of the model (Sajons, 2020). Methods like inverse-probability-of-treatment weighted (IPTW) and G-computation require consistent measurement of confounders and are thus limited in their applicability in control for unobserved time-varying confounding (Daniel et al., 2013).

A straightforward alternative way to include the effects of a set of time-varying unobservable confounders is suggested by Dormann (2001), who builds on the early common factor models for assessing “spuriousness” in cross-lagged panel models (Finkel, 1995; Kenny, 1975). In these measurement-like models, X and Y are treated as indicators of a latent construct which accounts for their observed association at given points in time. Our TVC-SEM can be viewed as a dynamic common-factor specification in which the latent factor plays the same conceptual role that Dormann’s (2001) time-varying common factor did — namely, accounting for any “third-variable” association between X and Y — but is now allowed to evolve autoregressively across waves to summarily capture both time-varying and time-invariant unobserved confounding. In this way, we are bringing back the common-factor idea but repurposing it as a robustness test that allows for unmeasured time-varying confounding in modern causal inference parlance. We show the time-varying confounding SEM (TVC-SEM) for the four-wave case in Figure 3 and Equation (3) below.

The model contains a time-varying latent Zit that has varying direct effects on Xit and Yit at each wave. However, as opposed to Figure 2 and its accompanying Equation (2), the γt effect of time-varying Z_it subsumes the effects of both unobserved time-varying and time-invariant confounding; that is, the effects of Z_it in Figure 3 represent the effects of both U_i and Z_it in Figure 2. This is the key idea of the approach that allows us to identify the model, since models with two latent variables that represent time-invariant and time-varying latent variables respectively are not identifiable. Most importantly, the autoregressive parameters of Z_it (Z_it-1 → Z_it) capture the stability of unobserved confounding and thus provide an estimate of the relative prevalence of stable versus time-varying confounders in the causal process. The model thus provides estimates of the effect of X_it on Y_it controlling for possible unobserved time-invariant and time-varying confounding that may exist in the process.²2

Our model could be conceptualized as having both “between” effects from the unobservables (what U_i would be from Figure 2) and “within” effects from Z_it, with these effects being permitted to differ. To the extent that the autoregressive effect of Z_it is 1, this would imply that only the “between” effect of the stable unobservables is relevant, whereas an autoregressive effect of Z_it of 0 would imply a model with only “within” effects. To the extent that the Z_it are relevant but not perfectly stable, the autoregressive effect of Z_it would fall between 0 and 1, and the estimated effect of Z_it would be a mixture of the unobserved confounders’ within and between effects.

At first glance, the TVC-SEM bears resemblance to a traditional FE-SEM model with autocorrelated (AR (1)) disturbances, where the Z_it proxies for both time-invariant confounding (U_i) and the AR (1) process of the disturbances in Y_it. However, in contrast to the fixed-effects AR (1) model, we directly model the effect of the latent variable Z_it on the X_it as well, thus accounting for the confounding of the causal effects from X_it to Y_it that is due to their joint association with Z_it (and U_i as well). In this sense, the FE-SEM model with AR (1) disturbances, as the other FE-SEM variants, falls short in controlling for unobserved time-varying confounding.³3

In the following parts of the paper, we compare the performance of the TVC-SEM with the AR (1) model, as well as the other variants of the FE-SEM model we have presented thus far.

In the model proposed by Dormann, which incorporates autoregressive effects (often referred to as “dynamic” effects) of both the independent variable (X_it) and the dependent variable (Y_it), identifying the model requires the inclusion of three endogenous variables. Additionally, to ensure model identification, there must be equality constraints on the autoregressive parameters as well as stability in the variance of the latent variable (Z_it) over time.

In our case, the traditional fixed-effects model does not contain dynamic components for either X or Y. This allows for a simpler version of the Dormann model, where only two endogenous variables are needed, provided there are at least three waves of data collection. Consequently, we are not required to impose homogeneity restrictions on the variances of the time-varying latent variable Zit. This adaptation follows the practices of traditional fixed-effects SEMs (Bollen & Brand, 2010).

As can be seen from the coefficient from Z_it to Y_it in Figure 3, however, we restrict the paths at 1 for model identification (as in Bollen & Brand, 2010), which also serves as a reference to gauge time-varying effects of Z_it at each wave. We also build correlations between X_it to account for remaining associations among predictors as shown in Figure 1 (for the model specification see SM 1).

All of the model parameters can be freely estimated without additional constraint subject to the following assumptions regarding Z_t and its relationship with X_t and Y_t (for notational brevity, in what follows we suppress the individual index i):

As in the traditional FE model, we also assume that residuals are independent, there is no autoregressive parameter of Yt, no reverse causal effects of Yt−1 on Xt, and no measurement errors in the variables.

Using a Monte Carlo simulation and a real data example, we test whether the TVC-SEM model with time-varying Zit provides less biased estimates of causal direct effects than the traditional FE-SEM models as well as a naïve model without any of the FE corrections for stable unobserved confounding variables. We show that TVC-SEM can replicate FE-SEM estimates under many conditions; more importantly, it provides substantially less biased estimates than FE-SEM whenever time-varying confounding is present, thus making the model an appropriate robustness test for the presence of time-varying confounding and the possible biases that would result in causal estimates from traditional fixed-effects models.

Figure 5a illustrates the results from a simulation where there is no time-varying confounding from Z_it at all, and where there are varied magnitudes of confounding from time-invariant U_i. The effects of U_i in this simulation are also constrained to be equal over time. This simulation corresponds exactly to the assumptions underlying the traditional fixed-effects model, and thus should be the most favorable condition for FE-SEM1 (the corresponding tables to Figure 5 are illustrated in Supplementary Material (SM) 4–5).

The zero value of Ui effects (in Figure 5a) reflects the condition of no unobserved heterogeneity that relates to both X and Y. Under such conditions, naïve SEM and the variants of the FE-SEM model, i.e., with time-invariant and time-varying effects of Ui, provide unbiased estimates, while TVC-SEM model with Zit provides a biased estimate (.13) along with a very high rate of convergence failure (85%) as well as a poor SE/SD ratio (.54) in SM 4. This occurs because TVC-SEM attempts to capture both (non-existent) Ui and Zit effects in the simulated data. Likewise, while FE-SEM1 and FE-SEM3 do not show any model convergence issue, FE-SEM2 introducing time-varying latent Ui also has some degree of convergence failure when the true U_i effects are null (U_i = 0, 37%) to small (U_i = .2, 11%).

As the influence of Ui increases, FE-SEM1-3 recover the exact causal effect, while naïve SEM produces upwardly biased estimates of up to 60%. Importantly, TVC-SEM also recovers true values without a serious convergence failure whenever the effects of U_i exceed .3 (to capture the degree of bias in detail, we illustrate findings with an increase of one decimal unit on Ui in Figure 5a). This is an important result: TVC-SEM with Z_it can recover causal effects as well as all FE-SEMs even when all confounders are time-invariant, so long as the U_i effects are at least of relatively modest magnitude (≥ .3). Only when the true population model has effects of Ui which are null to small in magnitude and the model has no time-varying confounding at all are traditional FE-SEMs better able to recover the causal direct effect of interest.

In the previous section, we assume that Ui effects are constant over time. As noted above, however, even fixed unobserved characteristics may have differential effects on X and Y over time (as, for example, in Ren & Allison, 2025). Such a model would indicate that the stable unobservables influence both the level and the change in the outcome over time. We thus vary the influences of Ui on X and Y at each wave with incremental/decremental changes from the lowest/highest values of unobserved heterogeneity in Table 1.

From the results, as seen in Figure 5b with full results in SM 5, in the presence of time-varying effects of Ui, even FE-SEM1 and FE-SEM3 produce biased estimates of up to 23%. As expected, the FE-SEM2 allowing for time-varying effects of Ui successfully recovers the true values. The TVC-SEM with time-varying effects of Zit, however, also performs very well compared to both the Naïve and FE-SEM1 and FE-SEM3 with a minor convergence failure (6%). In other words, when stable unobservables have varying effects on outcomes over time, TVC-SEM performs better in terms of bias than both the traditional FE-SEM1 that assumes time-invariant effects of Ui, better than the FE-SEM3 with an AR (1) disturbance, and it performs nearly as well as the model that conforms to the true data generating process in the population.

We model this condition by varying the effect of Zit on Xit and Yit at different waves, fixing all other parameters at their baseline values. The results are shown in Figure 6 (also in SM 6). In this scenario, all of the FE-SEM models perform extremely poorly, with biases up to 55%.⁶6

We also observe that FE-SEM2 with time-varying effects of Ui sometimes produces more upwardly biased estimates than FE-SEM1, especially in more extreme conditions (see also bolded values in SM 2).

We also test further the five models by varying the autoregressive parameters of Zit (i.e., its stability). The results, illustrated in SM 3, show broadly similar patterns that TVC-SEM with time-varying Zit tends to provide less biased estimates under most conditions. Yet, when the stability of Zit is too low (.2) or the fluctuation in the stability of Zit is present (∆ .2), SEM with time-varying Zit yields more biased estimates. But in every case the biases are less than any of the alternative models.

It is also worth noting that TVC-SEM generally produced larger (more conservative) standard errors than FE-based alternatives, reflecting its richer parameterization — e.g., wave-specific Z→X and Z→Y paths and serial covariance terms. Accordingly, we position TVC-SEM primarily as a bias-robustness tool, while FE models may be preferable when efficiency is paramount and when their assumptions — i.e., when only time-invariant confounders are relevant to the causal processes — are credible (see the conclusion for further discussion).

We now apply the five models, including the naïve SEM and both versions of FE-SEM and TVC-SEM, in the ECLS-K and RSVP data across four waves.⁸8

We provide the Mplus syntax used for each model in SM 8. In dealing with a potential violation of normal distribution assumption in the proposed models, bootstrap methods are preferred.

In Table 2, naïve SEM (M1) shows a significant negative association between BMI and math scores for boys (-.06) and girls (-.05) at p < .001. Notably, in the traditional FE-SEM1 (M2) the observed direct effects reduce to about half of the coefficients in the naïve model. For girls, the effects are reduced further and are no longer statistically significant in FE-SEM2 (M3) that allows for time-varying effects of the stable unobservables. FE-SEM3 with an AR (1) disturbance (M4) also yields the same estimates as that of FE-SEM2, though model fit is slightly improved.

The drop in coefficient magnitudes is predictable because, while omitted stable confounders such as family/school SES (within a relatively short interval) are likely to be positively (+) correlated with math achievement, they tend to be negatively (-) correlated with BMI or overweight status. Since there is an open negative (-) backdoor path between BMI and math scores, M1 is likely to produce a downward bias relative to FE-SEMs.

Under the assumption that some time-varying unobserved confounding exists, M5 shows BMI-math relationships in M1 and M2 drop and lose significance for boys and girls. Moreover, a cross-model Wald test (Clogg et al., 1995) comparing FE-SEM1 (M2) and TVC-SEM (M5) for boys shows that the .04 difference in effects is statistically significant; given the larger standard errors in the TVC-SEM model, this kind of comparison is perhaps a more important indication of the possible impact of time-varying unobserved confounding than the examination of the TVC-SEM model on its own. Yet the corresponding cross-model Wald test for girls shows a statistically insignificant change (Δ = .04) in the causal impact of BMI on math achievement.

In Table 3, we present the year-to-year relationship between crime victimization and offending from the RSVP data. The positive association between victimization and offending decreases from .36 in naïve M1 to .26 in FE-SEMs and further to .18 in TVC-SEM — a 50% reduction in magnitude. In contrast to the ECLS-K results, the causal effect of interest is statistically significant in both the FE-SEM and TVC-SEM model estimations, but the cross-model test comparing the TVC-SEM with FE-SEM estimates nevertheless shows a significant reduction in the effect, once the possibility of time-varying unobserved confounding is allowed.