Paper 2 Argument Specification
Paper 2 Argument Specification
Working specification — unfinished and unapproved.
Scientific Problem
Researcher-Facing Problem
Panel researchers need to relate causal assumptions about stable and changing context to the random-effect structures used in psychological models. Existing representations tend to show only one side of that relation:
- a nonparametric causal graph may compress unobserved heterogeneity into a generic latent cause pointing to many repeated variables;
- a random-effects path diagram and covariance matrix expose the modeled heterogeneity, but do not by themselves identify which dependencies confound a chosen causal effect.
The candidate problem is therefore graphical and inferential: how can a panel framework preserve the relevant person-level and time-level structure while distinguishing the full random-effect dependence structure from the covariance channels relevant to a specified causal query?
The principal conceptual task is to construct and defend that graph. Dynamic estimands and comparisons among fitted models should show what follows from the graphical framework; neither should replace it as the paper’s organizing idea.
Technical Tension
A full random-effect covariance matrix contains more dependencies than may be relevant to one causal effect. A generic latent common cause contains too little structure to distinguish random intercepts, random slopes, random variances, person-level context, and time-level context.
The proposed distinction lies between these extremes. The structural causal model is the data-generating process. A causal query states which intervention and outcome relation the analysis asks the model to answer. Together, the model and query determine the query-specific causal graph used in Paper 2.
A random-effect covariance belongs to the full statistical model, but does not by itself justify an edge in this query-specific graph. It receives causal backdoor meaning only when the structural roles of the two random effects create a backdoor channel for the specified query. A covariance between components that belong to the same outcome equation can therefore remain in the covariance matrix and path diagram without becoming an edge in the query-specific causal graph.
An estimator enters only after the model, query, and graph have been defined. Its population target and bias decomposition show which of the graph’s causal channels the estimator removes, models, transforms, or leaves active. The estimator does not create an estimator-specific causal graph.
The paper will use bias, protection, robustness, and susceptibility as formal terms only after fixing the target estimand and data-generating model.
Panel-Specific Pressure
Panel data may contain two shared contexts: persons and measurement occasions or historical time. Occasion effects may be central when many persons share the same waves or periods. This differs from many intensive longitudinal settings, where an occasion index need not denote shared calendar time.
For now, a time-level random effect is a placeholder for context shared across persons at a measurement occasion. Before formalizing it, the paper must decide whether that context represents sampled occasions, period shocks, historical events, cohort-time conditions, or another object.
Provisional Scientific Direction
Candidate Umbrella Claim
Paper 2 connects nonparametric and parametric panel SCMs to the graphical and statistical representations used in psychology. For a specified causal query, it distinguishes the causal covariance channels from the full random-effect dependence structure. It then uses those channels to assess the vulnerability of panel estimators.
This claim is deliberately broad. It remains a container until the scientific problem, graph ontology, and contribution hierarchy survive collaborator review.
Contribution Hierarchy
The four candidate contributions are:
- Nonparametric-to-parametric translation. Relate a nonparametric panel SCM to parametric panel SCMs with person and time context, random intercepts, random slopes, random variances, and their dependence.
- Query-specific graph construction. Define how a causal query selects the covariance relations represented as causal channels, while the random-effects path diagram and covariance matrix retain the full parameterized dependence structure.
- Person-by-time representation. Show person-level and shared-time context without forcing all structure into one unreadable graph.
- Estimator vulnerability profiles. Use population targets and bias decompositions to show which fixed channels different estimators remove, model, transform, or leave active.
Comparisons among centering, FE, TWFE, CRE, Mundlak/Chamberlain, and random-slope models are derived applications. Dynamic estimands and the boundary between one-time impulses and treatment histories remain optional extensions.
The provisional order is SCM and query definition, query-specific graph construction, separation from the path diagram, estimator vulnerability, person-by-time representation, and derived implications. Reopen this order if the graph rule cannot be defined independently of the estimator.
Core Graphical Ontology
Formal Organizing Frame
A structural causal model (SCM) is the data-generating object: it states the endogenous variables, their structural functions, the exogenous variables, and the joint distribution of those exogenous variables. A causal query \(q\) specifies the intervention, outcome, contrast, horizon, target population, and averaging operation whose answer is sought under that model. The query is not a fixed-effect coefficient and it is not an estimator.
For example, let \(q_k(x,x')\) ask for the average change in \(Y\) at horizon \(k\) when \(X\) at occasion \(t\) is set to \(x\) rather than \(x'\), averaged over persons and exogenous variation in target population \(\mathcal P\). The model answers that query with the causal estimand
\[ \begin{aligned} &\tau_{q_k(x,x')}(\mathcal M_\theta) \\ &\quad = \mathbb E_{\mathcal M_\theta,\mathcal P}\!\left[Y_{i,t+k}\mid do(X_{i,t}=x)\right] \\ &\qquad - \mathbb E_{\mathcal M_\theta,\mathcal P}\!\left[Y_{i,t+k}\mid do(X_{i,t}=x')\right]. \end{aligned} \]
In a one-step linear model with a homogeneous cross-lagged coefficient, this contrast may reduce to \((x-x')\beta_{YX}\), and to the fixed effect \(\beta_{YX}\) for a unit contrast. With random slopes, nonlinear propagation, or longer horizons, the same query generally need not reduce to one fixed-effect coefficient. An estimator \(\widehat\tau_q\) is introduced only afterward to estimate the model-defined target \(\tau_q(\mathcal M_\theta)\).
A nonparametric SCM leaves the relevant functions and distributions unrestricted. The working symbol for that object is \(\mathcal M\). Its proposed query-focused object is \(G_{\mathcal M}^{[q]}\):
\[ (\mathcal M,q) \longmapsto G_{\mathcal M}^{[q]}. \]
A parametric SCM restricts the functions and distributions to a family indexed by population-level parameters \(\theta\). The working symbol is \(\mathcal M_\theta\). Its proposed query-focused object is \(G_{\mathcal M_\theta}^{[q]}\):
\[ (\mathcal M_\theta,q) \longmapsto G_{\mathcal M_\theta}^{[q]}. \]
The arrows denote Paper 2’s proposed association among an SCM, a causal query, and a query-focused analytical display. Paper 2 has not yet defined this association as a mathematical mapping or graph operation. The output may be a DAG, an ADMG, another mixed graph, or a display derived from a model-level graph, depending on the formal rule that is ultimately established.
Pearl and Mackenzie distinguish a diagrammatic language of knowledge from a symbolic language of queries. Their causal inference engine treats the model, query, and data as separate inputs. Paper 2 adopts that separation. Their framework motivates the use of \(q\), but it does not establish Paper 2’s query-specific edge-selection rule. The term query-specific causal graph is therefore a working Paper 2 term.
Checked source: Pearl and Mackenzie (2018), Introduction, pp. 13–19, especially the languages of knowledge and queries and the causal inference engine in Figure I.1.
Forré and Mooij define an SCM as a model-level tuple of variables, domains, an exogenous distribution, and causal mechanisms, and define a parameterized SCM family separately. They then derive graphical representations from the functional parent relations in the SCM. This supports treating structural equations and graphs as different representations of one model. It does not support deleting model-level graph edges because they are irrelevant to one query.
Checked source: Forré and Mooij (2025), Definitions 6.2.1–6.2.2, pp. 172–173; Section 6.4, pp. 186–190; and Definitions 6.9.1–6.9.2 and Remark 6.9.3, pp. 210–212.
The framework separates three dimensions:
| Dimension | Question | Distinction |
|---|---|---|
| Model restriction | How strongly does the SCM restrict its structural functions and exogenous distribution? | Nonparametric SCM \(\mathcal M\) versus parametric SCM \(\mathcal M_\theta\). |
| Model representation | How is the structure of the SCM expressed? | Structural equations, a model-level causal graph, a random-effects path diagram, or a covariance matrix retain different kinds and amounts of information about the same model. |
| Causal query | Which intervention effect should the SCM answer? | The intervention, outcome, contrast, horizon, target population, and averaging operation collected in \(q\). |
Parametricity is a restriction on the SCM, not a representation type. The same parametric SCM can be written as structural equations and represented in several graphical or matrix forms. Paper 2’s proposed \(G_{\mathcal M_\theta}^{[q]}\) is therefore not automatically the model-level causal graph of \(\mathcal M_\theta\). It is a candidate query-focused analytical display whose relation to the standard model-level graph must be defined. The random-effects path diagram and \(\mathbf\Psi\) continue to represent the full parameterized random-effect structure.
Paper 1 prepares this notation without using it in exactly this form. It writes the causal functions as \(f_\theta\) and the exogenous distribution as \(P_\theta\), but does not explicitly name the whole parametric SCM \(\mathcal M_\theta\). Paper 2 can introduce \(\mathcal M_\theta\) as the natural model-level continuation, but should not claim that the symbol itself was already established in Paper 1.
From Paper 1’s Formal Construction to Paper 2’s Refinement
Paper 1 establishes a formal sequence for translating a multilevel statistical model into a parametric structural causal model and an associated causal graph. Paper 2 inherits this sequence but separates three operations that Paper 1 does not distinguish consistently: parametric model specification, graph construction for a causal query, and estimator assessment.
| Step | Formal construction in Paper 1 | Inheritance and refinement in Paper 2 |
|---|---|---|
| 1. Define the SCM | Endogenous variables are generated by structural functions \(f\); randomness enters through a joint distribution \(P\) over exogenous variables. Parametric restrictions are expressed through \(f_\theta\) and \(P_\theta\). | Retain this definition and name the complete nonparametric model \(\mathcal M\) and the complete parametric model \(\mathcal M_\theta\). |
| 2. State the varying-parameter model | Level-1 equations describe \(X\) and \(Y\) through cluster-specific intercepts or locations and a cluster-specific causal slope. | Extend the same construction to longitudinal panel equations with person and shared-time context, autoregressive and cross-lagged coefficients, and innovation scales or variances. |
| 3. Introduce auxiliary equations | Auxiliary equations decompose each varying parameter into a population-level parameter and an exogenous random effect. | Retain this decomposition while distinguishing the roles of location, intercept, slope, scale, person, and time components more explicitly. |
| 4. Derive the mixed structural equations | Substitution produces structural equations that separate endogenous variables, exogenous random variables, and population-level causal parameters. | Retain the substitution step. The resulting observation equations are linear Gaussian conditional on the random effects, while coefficient-by-predictor and scale-by-error products make the marginal system generally nonlinear or nonadditive. [needs derivation] |
| 5. Specify the exogenous distribution | Paper 1 specifies the complete joint Gaussian distribution of the random effects through the covariance matrix \(\mathbf\Psi\). | Retain the complete random-effect distribution. Every modeled covariance remains in \(\mathbf\Psi\) and may appear in the random-effects path diagram. |
| 6. Define and analyze the causal query | After constructing the model-wide graph, Paper 1 defines an intervention on \(X\), derives the corresponding causal effect, and examines the backdoor paths for \(X\rightarrow Y\). | Move this step before construction of the analytical graph. The query \(q\) fixes the intervention and outcome relation that the graph must answer. |
| 7. Construct the analytical graph | Directed edges in Paper 1 follow from the structural functions. Every nonzero covariance between exogenous variables is then represented by a bidirected edge before the focal effect is analyzed. | Directed parent relations continue to follow from the structural functions. Given \((\mathcal M_\theta,q)\), Paper 2 applies a query-specific rule to determine which random-effect dependencies form causal covariance or backdoor channels in \(G_{\mathcal M_\theta}^{[q]}\). The exact graph operation remains to be formalized. |
| 8. Assess estimators | Paper 1 notes that a regression coefficient may differ from the population-level causal effect when heterogeneous slopes receive nonrepresentative weights. | Derive each estimator’s population target and vulnerability profile after fixing \(\mathcal M_\theta\), \(q\), and \(G_{\mathcal M_\theta}^{[q]}\). Estimators may remove, model, transform, or leave active different channels without changing the graph. |
The resulting ownership structure is straightforward. The model \(\mathcal M_\theta\) fixes the structural equations, the random-effect distribution summarized by \(\mathbf\Psi\), and the corresponding model-level graphical and statistical representations. The query \(q\) fixes the causal target. The pair \((\mathcal M_\theta,q)\) supplies the inputs to Paper 2’s proposed query-focused display \(G_{\mathcal M_\theta}^{[q]}\). Only after these objects have been fixed is an estimator \(\widehat\tau_q\) assigned a vulnerability profile.
This sequence gives algebra two distinct tasks:
| Algebraic task | Input | Question answered | Output |
|---|---|---|---|
| Model- and query-level decomposition | \((\mathcal M_\theta,q)\) | Which random-effect dependencies create causal backdoor channels for the specified query? | The working edge set of \(G_{\mathcal M_\theta}^{[q]}\). [needs derivation] [needs source] |
| Estimator-specific bias decomposition | \((\mathcal M_\theta,q,\widehat\tau_q)\) | Which of the previously defined channels remain active in the estimator’s population target? | The estimator vulnerability profile. [needs derivation] |
Paper 2’s refinement occurs at the graph-construction step. Paper 1’s model section already distinguishes the two covariance relations that confound \(X\rightarrow Y\) from \(\operatorname{Cov}(U^{Y\mu},U^{YX})\), which links two random components of the outcome equation. Its general graph rule and final figure nevertheless represent all three covariances as bidirected edges. Its subsequent backdoor analysis returns to the narrower two-channel interpretation.
For \(q_{X\rightarrow Y}\), Paper 2 therefore retains
\[ \operatorname{Cov}(U^{Y\mu},U^{YX}) \]
in \(\mathbf\Psi\) and the random-effects path diagram but omits
\[ U^{Y\mu}\leftrightarrow U^{YX} \]
from \(G_{\mathcal M_\theta}^{[q_{X\rightarrow Y}]}\). This is the selected working interpretation. Its compatibility with standard SCM and ADMG semantics, the formal edge-selection operation, and the term query-specific causal graph still require a bounded literature check.
Three Analytical Levels
The argument needs three levels that must remain separate.
- Data-generating process and model representations. The nonparametric SCM \(\mathcal M\) or parametric SCM \(\mathcal M_\theta\) states what generates the data. Structural equations and the model-level causal graph \(G(\mathcal M)\) or \(G(\mathcal M_\theta)\) represent its functional and qualitative structure. For \(\mathcal M_\theta\), the random-effects path diagram and \(\mathbf\Psi\) additionally show the full parameterized random-effect structure.
- Causal query and proposed analytical display. The query \(q\) states which intervention effect the model must answer. Paper 2’s \(G_{\mathcal M}^{[q]}\) or \(G_{\mathcal M_\theta}^{[q]}\) is a proposed query-focused display derived from the model-query pair. Whether it is formally a causal graph, a subgraph, or a different display remains to be established.
- Estimator assessment. An estimator \(\widehat\tau_q\) is assessed against the causal effect defined in the data-generating model. Its population target and bias decomposition yield an estimator vulnerability profile: the set of causal covariance channels to which that estimator remains exposed. This profile may be shown by highlighting paths on the fixed query-specific graph or in a separate method-by-channel display. It is not a second causal graph.
In compact form, the model and query first determine the proposed query-focused display. The estimator \(\widehat\tau_q\) is then assessed against the target \(\tau_q(\mathcal M_\theta)\) without changing that display.
Two estimators applied to the same parametric SCM and causal query may therefore have different vulnerability profiles while sharing the same query-specific causal graph. Changing \(q\) may change the graph.
Working Query-Specific Graph Rule
The framework separates graph construction from estimator evaluation. Graph construction asks which relations in the data-generating SCM carry causal meaning for \(q\). Estimator evaluation asks which of those relations affect the population target of a particular estimator. The estimator cannot change the answer to the first question.
Within \(\mathcal M_\theta\), the structural functions determine the directed parent relations. The query fixes which causal relation the graph must answer. The exogenous distribution supplies the complete random-effect covariance matrix \(\mathbf\Psi\). The working Paper 2 rule evaluates each off-diagonal entry in \(\mathbf\Psi\) by the structural roles of the two random effects. For \(q_{X\rightarrow Y}\), a covariance forms a causal covariance channel when it connects a cause of \(X\) to a cause or effect modifier of \(Y\), thereby opening a backdoor path. The query-specific graph includes that relation. The random-effects path diagram continues to show every modeled covariance.
Paper 1’s three random effects illustrate the proposed rule:
| Covariance relation | Structural roles | Proposed causal interpretation for \(X\rightarrow Y\) |
|---|---|---|
| \(U^{X\mu}\leftrightarrow U^{Y\mu}\) | \(U^{X\mu}\) causes \(X\); \(U^{Y\mu}\) contributes to the level or intercept of \(Y\). | Creates the backdoor path \(X\leftarrow U^{X\mu}\leftrightarrow U^{Y\mu}\rightarrow Y\). |
| \(U^{X\mu}\leftrightarrow U^{YX}\) | \(U^{X\mu}\) causes \(X\); \(U^{YX}\) modifies the effect of \(X\) on \(Y\). | Creates the backdoor path \(X\leftarrow U^{X\mu}\leftrightarrow U^{YX}\rightarrow Y\). |
| \(U^{Y\mu}\leftrightarrow U^{YX}\) | Both random effects enter the structural equation for \(Y\); neither creates a route into \(X\). | Remains in \(\mathbf\Psi\) and the path diagram but is absent from \(G_{\mathcal M_\theta}^{[q_{X\rightarrow Y}]}\). |
Only after fixing the model, query, and graph does Paper 2 introduce an estimator \(\widehat\tau_q\). Its population target and bias formula show whether that estimator remains vulnerable to either backdoor channel. If a channel is absent from one estimator’s bias formula, the estimator is insensitive to that channel under the stated assumptions. The query-specific causal graph remains unchanged.
The special working class makes this distinction concrete. Conditional on the random effects, the observation equations are linear Gaussian. The random effects themselves have a finite-dimensional covariance structure. For a fixed \(q\), the query-specific causal graph uses the subset of covariance relations that carry causal backdoor meaning. The path diagram and covariance matrix retain the full set.
Query-Specific Graph Decision and Remaining Terminology Work
The working design uses the query-specific reading. For \(q_{X\rightarrow Y}\), \(U^{Y\mu}\leftrightarrow U^{YX}\) is absent from \(G_{\mathcal M_\theta}^{[q_{X\rightarrow Y}]}\) because the relation does not create a route into \(X\). It remains in \(\mathbf\Psi\) and the random-effects path diagram. The estimator does not determine this edge set.
| Formal reading | Treatment of \(U^{Y\mu}\leftrightarrow U^{YX}\) | Disposition |
|---|---|---|
| Full model-level SCM graph | Retains the edge because the exogenous variables are dependent, while treating it as irrelevant to the backdoor paths for \(X\rightarrow Y\). | Not selected for Paper 2’s analytical graph. Retained as the serious standard-semantics alternative. |
| Query-specific causal graph | Retains the covariance in \(\mathbf\Psi\) and the path diagram but omits the edge from \(G_{\mathcal M_\theta}^{[q_{X\rightarrow Y}]}\). | Selected as the Paper 2 working interpretation. |
Pearl and Mackenzie’s distinction between a language of knowledge and a language of queries supports separating \(\mathcal M_\theta\) from \(q\). It does not establish the edge-selection operation above. A bounded SCM and ADMG literature check must determine whether query-specific causal graph is an accepted term, a new object that Paper 2 must define, or should be replaced by query-specific causal display. The interpretation is selected; the terminology and formal graph operation remain open.
Terminology Under Review
The following terms express the intended distinctions. They are working terms for the specification, not yet settled manuscript terminology.
| Formal object or role | Recommended working term | Meaning and boundary |
|---|---|---|
| \(\mathcal M\) | nonparametric structural causal model | Data-generating SCM without the parametric restrictions introduced for Paper 2. |
| \(q\) | causal query | Specification of the intervention, outcome, contrast, horizon, target population, and averaging operation. It is neither a fixed-effect coefficient nor an estimator. |
| \(G(\mathcal M)\) | model-level causal graph | Graphical representation derived from the functional parent and latent-common-cause structure of \(\mathcal M\). |
| \(G_{\mathcal M}^{[q]}\) | nonparametric query-specific causal graph or display | Working object derived from \(\mathcal M\) for \(q\). Its exact relation to \(G(\mathcal M)\) remains to be established. |
| \(\mathcal M_\theta\) | parametric structural causal model | Data-generating SCM whose functions and exogenous distribution are indexed by \(\theta\). |
| \(G(\mathcal M_\theta)\) | model-level parametric causal graph | Graphical representation of the qualitative causal structure of \(\mathcal M_\theta\); it does not by itself encode every parameter value. |
| \(G_{\mathcal M_\theta}^{[q]}\) | parametric query-specific causal graph or display | Working object derived from \((\mathcal M_\theta,q)\); the formal operation and graph status remain to be established. |
| Full graphical display of \(\mathcal M_\theta\)’s random components | random-effects path diagram | Statistical model visualization; may show the entire covariance structure. |
| Full second-moment structure of the random effects | random-effect covariance structure or covariance matrix | Parameterized statistical dependence structure; it retains entries omitted from \(G_{\mathcal M_\theta}^{[q]}\). |
| A covariance relation that creates a causal backdoor route | causal covariance channel or backdoor channel | Covariance relation that opens a backdoor path for the specified causal relation. |
| Estimator-specific exposure to existing channels | estimator vulnerability profile | Records which graph channels an estimator removes, models, transforms, or leaves active. |
Do not use bias-channel graph as the name of the causal object. That label makes the graph sound estimator-specific. Use query-specific causal graph as the working term and describe highlighted estimator-relevant paths literally. Do not use projection or subgraph unless Paper 2 defines that operation.
Notation Problem and Options
The notation must identify the process, parameter role, and indexing level. It must also distinguish means, intercepts, location components, slopes, and scales. Paper 1 exposes the problem: it uses \(\eta_j^{X\mu}\) for the varying mean of \(X\), but \(\eta_j^{Y\mu}\) for the varying intercept of \(Y\), which equals a group mean only under the relevant centering condition. The superscript \(\mu\) therefore does not have one invariant meaning. Paper 2 adds autoregressive and cross-lagged slopes, innovation scales or variances, and person and time indices, so the ambiguity would grow.
Three coherent directions remain open:
| Option | Illustrative notation | What it solves | Cost |
|---|---|---|---|
| Paper 1 continuity | \(U^{X\mu}, U^{Y\mu}, U^{YX}, U^{X\sigma}\) | Keeps direct continuity with the published equations and figures. | \(\mu\) switches between mean and intercept; \(\sigma\) may blur scale, standard deviation, and variance. |
| Semantic role labels | \(U^{X,\mathrm{loc}}, U^{Y,\mathrm{loc}}, U^{YX,\mathrm{slope}}, U^{X,\mathrm{scale}}\) | Names the role of each random component directly and scales to richer models. | Longer notation; loc remains unsettled and may feel foreign in the SEM lineage. |
| Coefficient-slot notation | \(U^{X0}, U^{Y0}, U^{YX}, U^{X\sigma}\) | Uses \(0\) for the intercept slot and preserves the compact directional slope label. | Less explicit about location and scale roles; still needs a convention for dynamic, person-level, and time-level indices. |
No option is currently recommended. In particular, loc is a candidate, not a decision. The notation will be fixed only after comparing its use across the SCM, multilevel SEM, DSEM, GCLM, and dynamic-panel lineages.
Working Observed-Variable Convention
The current graph convention draws lagged causal arrows directly between the observed panel variables. It shows the latent random components required by the structural equations and includes their causal relations when they are relevant to \(q\). It does not add a latent within-person deviation beneath every observed score merely to reproduce the usual RI-CLPM decomposition. Unique disturbances may remain implicit.
Christian Gische’s dissertation provides a checked precedent. Its random-intercept panel graph draws autoregressive and cross-lagged arrows among the observed variables, with random intercepts as latent causal parents (Figure 5, p. 45). Its formal linear-SCM treatment draws an ADMG without explicit disturbances (Figure 10) and a corresponding SEM path diagram with explicit errors and error variances (Figure 11). The ADMG represents assumptions about the data-generating mechanism; the path diagram represents the analytic SEM.
Checked source: Christian Gische’s 2021 dissertation, especially Figures 5, 10, and 11 and the surrounding discussion of random-intercept panel graphs, ADMGs, and SEM path diagrams.
This precedent does not prove that the observed-variable model is identical to the conventional RI-CLPM. The equations must establish whether it is an exact reparameterization of the within-between RI-CLPM or a related random-intercept cross-lagged model. Until then, RI-CLPM names the literature reference point, not a proven model identity.
RI-CLPM readers may expect lagged arrows between latent within-person components. The proposed response is that a statistical decomposition of an observed score does not automatically define separate causal variables. A small companion display can map the causal graph to the within-between parameterization used for estimation, but only after the equations establish the mapping and its restrictions.
Working Mathematical Class
The baseline observation equations are linear Gaussian conditional on the random effects. Random coefficients and scales enter through coefficient-by-predictor and scale-by-error products, so the marginal model is generally nonlinear and nonadditive. Integrating over the random effects may also produce a non-Gaussian mixture. [needs derivation] [needs source]
The tractability question is specific: which causal effects, moments, and estimator targets remain available in closed form under a finite-dimensional Gaussian random-effect distribution? Component-wise linearity alone does not answer that question. [needs derivation]
Scientific Boundaries to Prior Work
Gische, Völkle, and West: Person-Specific Effects Manuscript
Checked source: Gische, Völkle, and West’s unpublished person-specific effects manuscript, version 6, dated 2023-12-05.
The manuscript already:
- formulates a general nonparametric panel SCM;
- defines average and person-specific intervention effects;
- introduces parametric random-intercept and random-coefficient panel models;
- links random coefficients to unobserved confounding and effect heterogeneity;
- discusses identification and maximum-likelihood estimation;
- distinguishes causal models from statistical models.
Paper 2 may inherit the movement from nonparametric to parametric models and the random-coefficient material needed for its own argument. Its candidate addition combines random variances with random slopes and their dependence. It also adds query-specific causal graphs, estimator vulnerability profiles, crossed person and historical-time context, and the distinction between a random-effects path diagram and a parametric query-specific causal graph.
Gische, West, and Völkle: Forecasting Intervention Effects
Checked source: Gische, West, and Völkle’s 2021 paper on forecasting causal effects of interventions rather than predicting future outcomes.
This paper distinguishes intervention forecasts from predictions of future observations and develops interventional distributions in a panel setting. It can support the causal interpretation of dynamic intervention quantities and the distinction between observing a value and intervening on it.
The main paper defines the marginal interventional distribution \(P(Y_{t+k}\mid do(x_t))\), and thus an outcome at a later horizon after a single intervention. It does not define a treatment-history or cumulative causal estimand. The supplement refers to the long-run cumulative effect of a unit impulse only when parameterizing the data-generating process. Paper 2 must therefore define any treatment-history or cumulative estimand it uses rather than inherit one from this source.
Paper 1: From Multilevel Structural Equations to Causal Graphs
Checked source: Paper 1’s treatment of graph-based causal models, multilevel SCMs, multilevel causal graphs, its empirical example, final causal graph, proof, and supplements.
Paper 1 supplies the formal construction summarized above. It also distinguishes multilevel SEM path diagrams from causal graphs, adds SCM and mixed-graph semantics, and establishes the cross-sectional multilevel grammar that Paper 2 inherits. Its model already includes varying predictor and outcome levels, a varying predictor-to-outcome slope, a varying predictor variance, and dependence between predictor variance and the heterogeneous slope.
Paper 2 preserves that construction but extends it to a longitudinal panel system with person and time context, dynamic heterogeneity, random autoregressive and cross-lagged slopes, and random innovation variances. Its formal refinement is the explicit query-specific rule for selecting causal covariance channels. The non-lagged location-scale model itself is not a new Paper 2 contribution.
Paper 1 contains a checked internal inconsistency that Paper 2 must not smooth away:
| Paper 1 component | Checked statement | Consequence |
|---|---|---|
| Multilevel-SCM section | \(\psi_{U^{X\mu}U^{Y\mu}}\) and \(\psi_{U^{YX}U^{X\mu}}\) represent level-2 unobserved confounding, whereas \(\psi_{U^{YX}U^{Y\mu}}\) does not; the latter links random components belonging to the same outcome equation. | The model text already distinguishes causal confounding from other random-effect covariance. |
| Multilevel-causal-graph section | Any nonzero covariance between two exogenous variables is said to imply a bidirected edge reflecting unobserved confounding. | This general rule is broader than the preceding model interpretation. |
| Final causal graph | The full covariance triangle among the three shared random components is drawn. | The figure implements the broader rule rather than the narrower causal distinction. |
| Backdoor analysis | Only the paths through \(U^{X\mu}\leftrightarrow U^{Y\mu}\) and \(U^{X\mu}\leftrightarrow U^{YX}\) are listed for \(X\rightarrow Y\). | The causal analysis follows the narrower two-channel interpretation and does not use the third covariance as a backdoor path. |
The contradiction concerns the relation between the covariance matrix and the causal graph. It does not, on the evidence checked so far, overturn Paper 1’s listed backdoor paths or adjustment argument. That narrower claim still needs a formal recheck if Paper 2 relies on it.
The scientific and rhetorical decisions are separate. Under a full model-level SCM reading, Paper 1’s problem is the interpretation of every bidirected edge as confounding of \(X\rightarrow Y\), not necessarily the presence of the third edge. Under Paper 2’s selected query-specific reading, the final figure contains one edge too many for \(q_{X\rightarrow Y}\). Paper 2 must state which reading it adopts before deciding how explicitly to describe the difference.
Three response strategies were considered:
| Strategy | What Paper 2 would do | Advantage | Risk |
|---|---|---|---|
| Explicit correction | State the inconsistency that follows from the selected graph semantics: Paper 1’s final figure draws one covariance edge too broadly for \(q_{X\rightarrow Y}\). | Maximally transparent and theoretically clean. | May turn a local refinement into an unnecessary public correction narrative. |
| Silent refinement | Use the new distinction without naming the Paper 1 inconsistency. | Keeps Paper 2 focused on its own contribution. | An attentive reader can see the mismatch, and the continuity claim becomes harder to defend. |
| Continuity with a transparent boundary note | Retain Paper 1’s SCM and notation lineage, state that Paper 2 distinguishes the full random-effect covariance display from the causal graph more strictly, and briefly explain which covariance does not represent a backdoor channel. | Preserves continuity without hiding the substantive difference. | Requires careful wording so that the clarification is neither evasive nor disproportionately self-critical. |
Working position: preserve continuity through a transparent boundary note. Paper 2 retains Paper 1’s SCM and notation lineage while stating the refined distinction directly. It also states accurately that Paper 1 did not apply this distinction consistently. The boundary belongs wherever the formal graph rule is easiest to verify, whether in the main text, a footnote, or the figure comparison.
Paper 3: Intensive Longitudinal Causal Graphs
Checked source: the current Paper 3 handbook chapter, especially its treatment of time-indexed DAGs and longitudinal confounding.
Paper 3 already provides the researcher-problem opening, basic time-indexed DAG reading, and the distinction among stable person-level, shared time-level, and within-person time-varying confounding. It intentionally omits the full panel model taxonomy, fixed-versus-random-effects debate, path coefficients, residual covariances, and impulse-response derivations.
Paper 2 should use only the minimum graphical background required for its technical extension. Its distinct material is the parametric model structure, query-specific causal covariance channels, and estimator vulnerability profiles.
Imai and Kim
Imai and Kim (2019), When Should We Use Unit Fixed Effects Regression Models for Causal Inference with Longitudinal Data?, is the stronger nonparametric graph precedent. It pairs an unrestricted outcome function containing treatment, a unit-specific latent cause, and a time-specific disturbance with a three-wave DAG that exposes excluded dynamic arrows.
Imai and Kim (2021), On the Use of Two-Way Fixed Effects Regression Models for Causal Inference with Panel Data, supplies a complementary person-by-time mapping through unit-specific and time-specific latent confounders and their additively separable effects.
Checked sources: Imai and Kim (2019), especially Assumption 1 and Figure 1; Imai and Kim (2021), especially their treatment of unit- and time-specific latent confounders and additive separability.
Dynamic Panel and GCLM Comparison Set
The relevant comparison set may include cross-lagged panel models, the role of time in psychological mechanisms, continuous-time modeling, and GCLM. The exact sources have not been selected.
The GCLM literature distinguishes short-run and long-run dynamic effects, uses impulse-response functions, and models occasion effects, time-varying unit effects, trends, and richer dynamic covariance structures. Paper 2 may use this literature as a target for causal interpretation, but its precise relation to GCLM remains unresolved until the relevant sources establish the novelty and model-structure boundaries.
Model and Figure Sequence
Cross-Lagged Opening, Univariate Explanation, and Bivariate Return
The preferred sequence opens with the familiar cross-lagged problem, explains the heterogeneity mechanism in a univariate autoregressive model, and then returns to the bivariate panel:
- A generic latent cause points to repeated variables in a bivariate one-lag panel with autoregressive and cross-lagged arrows.
- A random-intercept panel graph gives that stable context a more specific parametric role.
- A univariate autoregressive model retains the random intercept, then adds a random autoregressive slope, a random innovation variance or scale, and their dependence.
- An optional bridge adds a contemporaneous predictor to the autoregressive outcome if the resulting joint weighting problem improves the explanation.
- The bivariate return adds the second outcome equation and reciprocal lagged effects without reteaching every heterogeneity component.
The opening graphs should form one compact move, not two complete model stages. The univariate model has one job: make the random-slope–random-variance channel and the distinction between statistical and causal graphs readable. The paper should not repeat every random-effect extension across every temporal model.
Alternative Model Orders
| Sequence | Advantage | Cost | Disposition and reopening condition |
|---|---|---|---|
| Begin with a univariate model containing the full random-effect set, then move to contemporaneous and cross-lagged bivariate models. | The smallest model may already contain the main intercept, slope, variance, covariance, and weighting channels. | The full heterogeneity structure arrives immediately, and later models may repeat derivations. | Not preferred. Reopen if the cross-lagged opening creates a misleading scientific problem. |
| Stay inside the cross-lagged family and add random intercepts, slopes, and variances cumulatively. | Readers remain in a familiar model family. | Each addition enters an already dense dynamic graph. | Not preferred. Reopen if the univariate detour disrupts the argument more than the dense graph does. |
| Open cross-lagged, zoom into the univariate mechanism, then return to the bivariate model. | Combines a familiar opening with a readable central mechanism. | Requires a precise transition into and out of the univariate model. | Provisionally selected. Reopen if those transitions cannot be stated as scientific questions. |
Opening Display
The first paired display uses matched three-wave unrolled panels:
- both panels show the same observed variables and time layout;
- autoregressive and cross-lagged arrows appear from the first graph;
- the first panel uses a generic latent cause;
- the second gives stable context a random-intercept representation;
- plate notation is deferred until the repeated lag structure is understood.
Each panel should add a more specific representation or answer the next scientific question. The comparison should not be framed as one graph merely correcting a deficient predecessor.
Companion Mapping to the RI-CLPM
Three explanations were considered:
| Approach | Advantage | Cost | Disposition |
|---|---|---|---|
| Prose and equations only | Uses the least space. | RI-CLPM readers may miss the correspondence. | Rejected as the sole explanation. |
| A small companion mapping diagram | Makes the reparameterization visible without another full model stage. | Requires a formally correct mapping before it can be drawn. | Provisionally selected. |
| A full conventional RI-CLPM path diagram beside the causal graph | Makes both conventions explicit. | Repeats prior material and shifts attention toward the analytic decomposition. | Rejected for the main sequence; reopen only if the small mapping remains ambiguous. |
The selected companion uses one generic bivariate transition from \(t-1\) to \(t\). It should show the decomposition of both observed processes and the autoregressive and cross-lagged relations among their within-person components. It should not add a third wave, plate notation, the complete disturbance structure, the complete query-specific causal graph, or an estimator-relevant path display.
Before this companion is drawn, the equations must establish the exact mapping and its restrictions. The visual must keep the analytic within-between decomposition distinct from the causal graph.
Univariate Heterogeneity Core
The univariate random-effects path diagram contains the random intercept, random autoregressive slope, random innovation variance or scale, and their allowed dependence. The covariance matrix records all modeled heterogeneity. For a fixed \(q\), the associated query-specific causal graph retains only the covariance relations that meet the graph rule. A later estimator analysis highlights the subset of those existing channels to which the chosen estimator is vulnerable.
Intended distinction: in the specific conditionally-on-random-effects linear-Gaussian class, random-intercept–random-variance dependence that does not create a backdoor route remains in the covariance matrix and path diagram but is absent from \(G_{\mathcal M_\theta}^{[q]}\). Random-slope–random-variance dependence may instead form a causal channel because the innovation scale shapes the regressor distribution while the random slope shapes the causal response. The exact claim requires the variance or scale component, causal query, temporal role of the lagged outcome, and distributional assumptions to be fixed. The subsequent estimator derivation must then ask whether that estimator is exposed to the channel; it must not be used to create the channel. [needs derivation]
Random variances matter because dependence between within-person predictor variance and a random slope can change how observations or persons weight heterogeneous effects. The exact channel and estimator target remain to be derived.
Optional Autoregressive Bridge
The optional bridge adds a contemporaneous time-varying predictor to an autoregressive outcome. Its only candidate main-text job is to move from a scalar weighting result for one heterogeneous coefficient to the joint weighting of an autoregressive and a contemporaneous coefficient. With two predictors, the pooled coefficient vector is weighted by the full within-person regressor covariance matrix. Separable scalar weights require additional covariance or orthogonalization restrictions. [needs derivation]
Three neighboring bias problems must remain distinct:
- omitted-lag dynamic bias in an otherwise static fixed-effects equation;
- Nickell bias after the lagged outcome enters a short-panel fixed-effects equation;
- the pooled target induced by person-specific autoregressive and contemporaneous slopes.
Only the third directly extends the proposed graphical treatment of random slopes, random variances, and covariance channels. Klosin, Nickell, and Pesaran and Smith use related equation skeletons but isolate different mechanisms and asymptotic regimes. They must not be cited as interchangeable results.
Checked source boundary: Klosin (2024) addresses omitted-lag dynamic bias; Nickell (1981) addresses short-panel fixed-effects bias; Pesaran and Smith (1995) address heterogeneous dynamic coefficients. Asparouhov, Hamaker, and Muthén (2018) connect Nickell bias to latent centering in psychological DSEM. Detailed source locators remain in the evidence record.
An exploratory version of this bridge contains random intercepts, random autoregressive and contemporaneous slopes, and random innovation scales. Its claim that the two slope-scale biases are additive and independent remains unverified: the analytical formulas are missing, and the general matrix result shows that independence cannot be assumed when the regressors covary. [needs derivation] [needs simulation]
Retain the bridge only if it clarifies the scalar-to-joint weighting transition. Otherwise move directly from the univariate core to the bivariate return.
Person-by-Time Representation
The model must represent person-level and time-level context together. A single graph may become unreadable when it combines both contexts, dynamic lags, random slopes, random variances, and all relevant dependencies.
Serious figure alternatives are:
- one staged sequence of unrolled graphs;
- separate person and shared-time views;
- plate notation after one unrolled case is understood;
- paired random-effects path diagram and covariance matrix displays;
- the associated query-specific causal graph, with estimator-relevant paths highlighted when useful.
The formal model may use fully crossed random effects, a nested approximation, or a staged treatment of the two contexts. Reopen the figure sequence if the chosen formal structure cannot be represented without conflating person and time roles.
Candidate Figure Sequence
- Matched three-wave generic-latent and random-intercept panel graphs.
- A one-transition companion mapping from the RI-CLPM within-between parameterization to the observed-variable causal graph.
- A univariate random-effects path diagram and covariance matrix containing intercept, autoregressive slope, innovation scale, and their dependence.
- The corresponding parametric query-specific causal graph after the SCM, causal query, and graph rule establish its edges.
- If needed, a scalar-to-joint weighting display for the optional bridge.
- A bivariate return showing the additional channels introduced by reciprocal lagged effects and person-by-time context.
Two conditional figures may follow:
- A method-by-channel comparison, if it adds information beyond the estimator derivations.
- An impulse-response extension, if it adds a distinct causal query.
Each figure needs one analytical job. Decorative duplication of Paper 1 or Paper 3 is outside the scientific case.
Estimands and Model Consequences
Method-by-Channel Comparison
Centering is one member of the comparison, not the umbrella contribution. Candidate method families include:
- group-mean and grand-mean centering;
- unit fixed effects and two-way fixed effects;
- Mundlak and Chamberlain auxiliary models;
- correlated-random-effects models;
- random-intercept and random-slope models;
- restrictions on the random-effect covariance matrix;
- separate or mean-group estimation where relevant.
The comparison should state which channels each method removes, models, transforms, or leaves active. The intercept-slope channel remains potentially active until a specified transformation removes it; group-mean and grand-mean centering need not remove the same terms. [needs derivation]
Provisional comparison: fixed-effects and two-way fixed-effects models that omit random slopes may remain exposed to a random-slope–random-variance channel. Parametric random-slope models may address that channel under explicitly named covariance and distributional assumptions. [needs derivation] [needs simulation]
Dynamic and Impulse-Response Estimands
A causal impulse response tracks the consequences of a one-time intervention across future horizons. It is relevant to GCLM, VAR, and related panel models, but should enter only after a one-step effect establishes the basic graphical logic. Each additional lag may introduce new random dynamic coefficients and covariances.
The smallest useful horizon is the first horizon at which a new random dynamic coefficient, covariance channel, or estimator ranking appears. A one-time impulse and the joint effect of a treatment history answer different causal queries; neither is a special case of the other without a formal argument.
Evidence Still Needed
Source Checks
- The dynamic panel and GCLM sources that define the prior-work boundary remain to be selected.
- Novelty and model-structure claims relative to GCLM remain unresolved.
- Pearl and Mackenzie’s language-of-queries distinction supports separating \(\mathcal M_\theta\) from \(q\), but it does not establish Paper 2’s query-specific edge-selection rule.
- The standard SCM and ADMG literature must establish whether query-specific causal graph is an accepted term, a new defined graph operation, or should be replaced by query-specific causal display.
- The SCM, multilevel SEM, DSEM, GCLM, and dynamic-panel lineages must be compared before choosing between \(\mu\),
loc, and coefficient-slot notation. - Compatibility among the graph-equation conventions in Imai and Kim, Paper 1, the checked panel precedents, and the proposed ontology remains unresolved.
Derivations
- Define the early causal effect and estimator.
- Formalize how \((\mathcal M_\theta,q)\) selects causal covariance channels from the full random-effect covariance structure without referring to an estimator.
- Derive the estimator’s population target under the stated random-effect structure.
- Separate query-specific channels involving random intercepts, random slopes, random variances, and person-by-time context from estimator-specific exposure to those channels.
- Test whether intercept–variance dependence drops out while slope–variance dependence remains in the first univariate case.
- Derive the joint matrix weighting in the optional bridge and the restrictions that reduce it to separable scalar weights.
- Establish how centering, FE/TWFE, CRE, Mundlak/Chamberlain, and random-slope modeling change the relevant terms.
- Establish the exact observed-variable/RI-CLPM reparameterization.
- Identify the tractability boundary of the conditionally linear-Gaussian, bilinear, location-scale class.
Simulations
Any simulation should test a named formal or interpretive claim. Candidate contrasts are:
- fixed effects without random slopes versus correctly specified random-slope models;
- explicitly named random-effects distribution misspecifications versus omitted random-slope structure;
- estimated versus restricted random-effect covariance entries;
- person effects only versus person and time effects;
- one-step effects versus a bounded response horizon.
No contrast is required merely because it appears here. Robustness claims must name the data-generating process, estimator, causal effect, and performance criterion.
Work Required Before the Design Is Complete
Before the design is complete, it must:
- fix the intervention, outcome, target population, and averaging operation for the first causal query \(q\);
- select the estimator and population target for the first vulnerability profile;
- define the formal operation that constructs \(G_{\mathcal M_\theta}^{[q]}\) and settle whether query-specific causal graph is the correct term;
- establish the exact relation between the observed-variable model and the intended RI-CLPM;
- determine whether person and shared-time context require complementary views; and
- choose notation that preserves useful continuity with Paper 1 without retaining the ambiguity of \(\mu\).