Reciprocal relationships, reverse causality, and temporal ordering

Testing theories with cross-lagged panel models

Charles C. Lanfear

University of Cambridge

Thiago R. Oliveira

University of Manchester

What is this?

We were invited to write a paper on proper use of Cross-Lagged Panel Models (CLPM) for the Journal of Developmental and Life Course Criminology1

 

Our current plan:

  • Review common problems in applied work
  • Focus on aligning theory with estimation
  • Suggest robust default practices

The Cross-Lagged Panel Model

 

Cross-lags enforce temporal order to address reciprocality

Reciprocality

Two forms:

  • Theoretical reciprocality
  • Empirical reciprocality (i.e., simultaneity)

Our position:

  • The world is recursive
    • Reciprocal theory is usually a symptom of ignoring time and/or mechanisms
  • Reciprocality is an empirical problem

We suggest acyclic graphs for theory and SEM diagrams for estimation

Time DAG

 

Reality and theory operate at one recursive pace:

 

Time Estimator

But you observe data—and must estimate models—at another that results in simultaneity:

Nonrecursive models

True simultaneous effect models are just a class of this:

But they come with strong assumptions

Theory comes first

Clear theory is a prerequisite before specifying an estimator

You should:

  • Derive recursive model from theory
    • Use a DAG or recursive equation
    • Include unobserved mechanisms when appropriate
  • Use theoretical model to specify estimator
    • Contemporaneous (fast) vs. lagged (slow) effects in structure
    • Covariances to address ambiguity and minimize undesirable assumptions
    • Non-recursive simultaneous equations as last resort

Three common applied problems

Temporal Order

 

As illustrated by Vaisey & Miles (2019), if…

  • True model: \(y = \beta x_t + \alpha_i + e_{it}\)
  • Estimated model: \(y = \beta^* x_{t-1} + \alpha_i + e_{it}\)
  • Resulting bias: \(E(\beta^*) = -0.5\beta\)

Incorrect temporal order can reverse signs

Example paper

 

 

Be very suspicious of unexpected reversed signs

Solutions

  • Use strong theory to get timing right
  • Use robust estimators, e.g., Allison’s approach
  • If ambiguous contemporaneous path matters, you’ll need strong instrument(s)

Unobserved stable heterogeneity

 

  • Most common concern is stable traits1
  • Autoregressive term alone does not solve problem
  • Can’t just toss in fixed effects due to Nickell bias from endogenous lag
    • Bias shrinks in proportion to T
  • Generally well-recognized

Example: Lanfear & Kirk (2024)

Collective efficacy (CE)
Crime (C)
Opportunity (O)

Airbnb properties (A)
Time-invariant unobservables (U)

Solutions

 

  • More robust estimators1
    • Psychological approaches: RI-CLPM
      • Still learning these!
    • Econometric approaches: Arellano-Bond, ML-SEM
      • Also relax strict exogeneity

These solutions commonly lead to our third problem…

Low Inter-temporal variation

 

  • \(Var(Y_2|Y_1) \rightarrow 0\) as \(\rho(Y_1,Y_2) \rightarrow 1\)
  • Error and bias become proportionally larger components
  • Common with short observation times and stable constructs

Example paper

 

Sometimes a near-perfect multicollinearity problem:

Solutions

  • Address during data collection
    • Oversample for change
    • Look for shocks
  • Hybrid Mundlak model
  • Consider different time lags or units of analysis
  • Error can be dealt with using measurement models
    • Should do this anyway as outcomes are regressors, so measurement error attenuates estimates
  • Give up, go get a pint

Takeaways

Our advice

  • Separate theory problems from estimation problems
    • Theory comes first
    • Then clear model
    • Then estimator
  • Default to robust estimators

 

Feedback and Questions

What else should we cover?

  • Sequential ignorability
  • Counterfactual causality
  • Predetermined vs. strictly exogenous regressors

Contact:

Charles C. Lanfear
Institute of Criminology
University of Cambridge
cl948@cam.ac.uk