What Agent-Based Models See That Regressions Miss

The Trap of Spurious Correlation

In 1974, Granger and Newbold demonstrated that two independent random walks, when regressed against each other, produce high $R^2$ values and significant $t$-statistics with probability approaching one as the sample size grows. The regression looks convincing. The relationship is entirely fabricated.

This is the canonical spurious regression problem. But there is a subtler and more insidious version: two phenomena that are genuinely causally linked at the micro level, yet appear unrelated — or even negatively correlated — when measured at the aggregate level and subjected to standard regression. The primitives of statistical inference are not equipped to see the mechanism. They can only see the covariance structure, and covariance is a marginal statistic that integrates out everything that matters.

Agent-based models operate differently. They do not estimate relationships between aggregate variables. They generate those relationships as emergent properties of explicitly modelled micro-level rules. What looks like noise at the macro level resolves into structure at the micro level — structure that is invisible to regression but transparent to simulation.

Why Standard Regression Fails Here

The classical linear model assumes that the conditional expectation $\mathbb{E}[Y \mid X]$ is a useful summary of the relationship between $X$ and $Y$. For many physical and economic systems, it is not — because the relationship is non-linear, path-dependent, or mediated by a distribution of micro states that the aggregate variables $X$ and $Y$ do not capture.

Consider a system of $N$ interacting agents with individual states $\omega_i \in \Omega$. The aggregate observable is a function $X = \Phi(\boldsymbol{\omega})$ where $\boldsymbol{\omega} = (\omega_1, \ldots, \omega_N)$. The macro-level distribution satisfies:

$$p(X) = \sum_{\boldsymbol{\omega}\, :\, \Phi(\boldsymbol{\omega}) = X} \prod_{i=1}^{N} p_i(\omega_i \mid \boldsymbol{\omega}_{-i}).$$

$(1)$

Equation (1) is a marginalisation over an astronomically large micro state space. Two different micro configurations can produce identical macro observables; two similar macro observables can be generated by micro dynamics that are causally opposite. Any regression of $X$ on $Y$ conflates all of this. The ABM preserves it.

Four Cases Where the Structure Was Hidden

Wealth Inequality from Random Exchange

The yard-sale model places $N$ agents on a network, each with initial wealth $w_i > 0$. At each step, two agents $i$ and $j$ engage in a transaction: one wins and one loses a fixed fraction of the poorer agent’s current wealth:

$$w_i \leftarrow w_i + \epsilon\,\min(w_i, w_j), \qquad w_j \leftarrow w_j – \epsilon\,\min(w_i, w_j),$$

$(2)$

with the winner chosen at random and $\epsilon \in (0, 1)$ fixed. Every transaction is symmetric and fair in expectation. The agents are identical.

The macro outcome: all wealth concentrates in a single agent with probability one, regardless of $\epsilon$ or $N$, as time goes to infinity. The Gini coefficient converges to 1.

A naive regression of wealth on “productivity”, “effort”, or any individual characteristic would find nothing — there is nothing there to find. The inequality is not a consequence of individual differences. It is a consequence of the multiplicative structure of the exchange rule and the resulting absorbing barrier at $w = 0$. No regression sees this. The ABM makes it transparent on the first simulation run.

Segregation from Mild Preferences — Schelling (1971)

Thomas Schelling placed agents of two types on a grid. Each agent has a mild preference: it is satisfied if at least a fraction $\theta$ of its neighbours share its type. If unsatisfied, it moves to a random vacant cell. The threshold is intentionally set low — $\theta = 1/3$, meaning the agent tolerates having two-thirds of its neighbours be different.

The formal satisfaction condition for agent $i$ is:

$$f_i(\mathbf{s}) = \frac{\bigl|\{j \in \mathcal{N}(i) : s_j = s_i\}\bigr|}{|\mathcal{N}(i)|} \geq \theta.$$

$(3)$

An analyst who regressed “neighbourhood segregation index” on “individual preference intensity” $\theta$ would find a strongly nonlinear, threshold-sensitive relationship that looks like noise across most of the parameter space and then jumps discontinuously near a critical $\theta^*$. The aggregate statistic bears no simple relationship to the individual parameter.

The ABM reveals the mechanism: cascading relocations. When one agent moves, it changes the satisfaction of its neighbours, who may then move, triggering further relocations. The macro outcome — near-total segregation — emerges from a preference that is, by construction, integrationist. This cannot be captured by any regression because the causal chain runs through a cascade of interactions, not a direct link between $\theta$ and the segregation index.

Phantom Traffic Jams

Traffic engineers have long observed that congestion waves can propagate backwards through a highway — against the direction of travel — at roughly 15 km/h, even on roads with no bottleneck and no external cause. The phenomenon is called a jamiton (Sugiyama et al., 2008).

In a regression framework: regress “vehicle speed at point $x$” on “vehicle density at point $x$”. You find a negative correlation — higher density, lower speed. Obvious. What you do not find is the dynamic: a small perturbation (one driver brakes slightly) propagates backward through the traffic as a density wave, growing in amplitude until it becomes a standing jam. The regression captures the cross-sectional correlation but is blind to the propagation dynamics.

The ABM — specifically a cellular automaton or car-following model — generates the jamiton spontaneously from nothing more than a rule that says “maintain a safe following distance”. The macro phenomenon (backward-travelling jam) is a genuine emergent property of the micro rule (local reaction to proximity).

Bank Runs and Solvent Failure

A bank with fundamentally sound assets can fail — not because its assets deteriorate, but because enough depositors believe that enough other depositors will withdraw, making the failure self-fulfilling. This is the Diamond-Dybvig equilibrium (1983).

Standard regression of “bank failure” on “asset quality” misses this entirely, and not merely because the sample is too small. The failure event is not a consequence of asset quality. It is a consequence of a coordination game that has two Nash equilibria — run and no-run — and the regression conflates both equilibria in a single sample. The resulting coefficient is an average of two causally distinct regimes, and it is meaningless.

An ABM models the network of depositors, their beliefs about each other’s beliefs, and the withdrawal decision as a threshold rule. It generates both equilibria, simulates the transition dynamics between them, and allows the analyst to identify the conditions under which the run equilibrium is selected. The regression cannot do this, not in principle.

The Formal Advantage

The ABM’s formal advantage over regression is not computational — it is epistemological. A regression asks: given the data, what function best describes the relationship between $X$ and $Y$? An ABM asks: given a model of micro behaviour, what aggregate relationship does the system generate?

The first question is answered by minimising a loss function over observed data. The second is answered by forward simulation. When the true data-generating process involves interaction effects, non-linearities, path dependence, or multiple equilibria, the first question has a misleading answer. The second question does not.

The practical upshot: whenever you observe a time series or cross-section and a standard regression produces insignificant, unstable, or sign-switching coefficients — do not conclude that there is no relationship. Conclude that the relationship, if it exists, may only be visible at the micro level.

Takeaway

Spurious regression is usually discussed as a statistical nuisance to be corrected with better tests. The deeper point is different: even well-specified regressions between non-integrated series can fail to detect genuine causal structure if that structure lives at the micro level and is invisible at the aggregate. Agent-based models are not a replacement for statistical inference — they are a different tool, operating on a different level of description, and asking a different question. The combination of both is what produces reliable insight.

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