Do We Need a Mundlak Specification Test in AMMM3?
Background
Classical panel econometrics uses a Mundlak specification test to assess the mean-independence restriction behind a random-effects (RE) model. In its usual form, the test evaluates whether the coefficients on declared unit-level regressor summaries are jointly zero.
A rejection is evidence against that particular RE restriction under the test assumptions. A failure to reject is not proof that RE is adequate: weak variation, collinearity, finite samples, or an incomplete summary basis can leave the test uninformative.
Why AMMM3 Does Not Reproduce the Frequentist Test
AMMM3 fits Bayesian models. It does not attach an asymptotic Wald test or a chi-squared reference distribution to the Mundlak coefficients. Posterior inference on those coefficients answers a different question and depends on the declared priors and summary basis.
Two interpretations must be avoided:
- A posterior interval containing zero does not establish that the RE mean-independence assumption is adequate.
- A posterior interval excluding zero indicates a conditional association with the declared summaries. It does not identify the amount of confounding or establish causal identification.
This distinction is especially important in marketing mix modelling, where media transformations are estimated and the available between-unit variation may be weak.
Legacy Mundlak Surface and Named CRE Preset
use_mundlak_cre=True is the retained low-level panel surface. It adds legacy
Mundlak terms to an unlabelled dimensioned PanelMMM. It is not an alias for
the named cre estimator preset.
The named CRE preset has a separate, explicit contract. Its media summaries use the declared transformed exposure basis and it records estimability evidence. The released v1 surface retains explicit limits on prediction and unsupported downstream operations.
Do not transfer an interpretation or diagnostic result from one surface to the other without checking the actual fitted summary basis.
What to Inspect
Posterior summaries
For a legacy Mundlak fit, inspect the coefficients and their joint posterior geometry:
import arviz as az
az.summary(
mmm.idata,
var_names=["gamma_channel_mundlak", "gamma_control_mundlak"],
)
Treat these summaries as evidence about associations conditional on the fitted model. Check effective sample size, R-hat, posterior correlations, prior sensitivity, and the amount of within- and between-unit variation before interpreting their magnitude.
Estimability and sensitivity
Before relying on the adjustment:
- Confirm that the declared summaries have non-zero between-unit variation.
- Inspect rank, collinearity and condition-number diagnostics.
- Compare posterior results under defensible prior alternatives.
- Check that substantive conclusions are not driven by one summary-basis choice.
- Use prior and posterior predictive checks to detect implausible model behaviour.
Predictive comparison
A predeclared held-out comparison can test whether adding the adjustment improves prediction for the intended forecasting task. Use a split that respects panel and temporal dependence. Predictive improvement does not by itself validate the identifying assumption or convert observational associations into causal effects.
Summary
| Evidence | Supported conclusion | Unsupported conclusion |
|---|---|---|
| Adjustment interval includes zero | The data and prior do not clearly separate the coefficient from zero | RE is adequate |
| Adjustment interval excludes zero | Association with the declared unit-summary basis | Identified confounding or causal correction |
| Predictive comparison improves | Better prediction for the declared holdout task | Correct causal structure |
| Estimability diagnostics pass | No detected defect under the implemented screens | Global or posterior-wide identification |
AMMM3 should therefore retain explicit posterior, estimability, sensitivity and predictive evidence. It should not turn the Bayesian adjustment into a binary RE-versus-CRE adequacy test.
References
- Mundlak, Y. (1978). “On the Pooling of Time Series and Cross Section Data.” Econometrica, 46(1), 69–85.
- Vehtari, A., Gelman, A., & Gabry, J. (2017). “Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC.” Statistics and Computing, 27(5), 1413–1432.