All Models, but What's the Difference Between Econometric Models, Mathematical Models, and Economic Models?

1. Opening: A Confusing Scenario

You've probably heard phrases like: "Your model isn't accurate," "This model's assumptions are too idealized," or "Let's run the model on the data."

The same word—"model"—can refer to completely different things in different contexts. Today, we'll use one article (or one video episode) to thoroughly distinguish among three types of models: mathematical models, economic models, and econometric models.

Let's start with the conclusion: Mathematical models pursue "logical necessity"; economic models pursue "given assumptions, behavior should be thus"; econometric models pursue "what the data tells us about whether it actually is thus."

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2. Mathematical Models: A Purely Logical World

A mathematical model is a self-consistent logical system. It does not depend on real-world data; it depends only on axioms and rules of derivation.

For example, Euler's formula $e^{i\pi} + 1 = 0$. This formula does not require you to collect any data to verify it—it follows rigorously from the definitions of complex numbers, exponential functions, and π. It simply "is true," existing in the logical world.

Typical features of mathematical models:

• No empirical content: They do not discuss the real world; they only discuss logical relationships between definitions and propositions.
• No randomness: Conclusions are deterministic; there is no such thing as "significant at the 95% confidence level."
• The criterion is truth or falsehood: Whether the proof is correct, whether the derivation has gaps. There is no "goodness of fit."

In one sentence: The question a mathematical model answers is—"If the premises hold, must the conclusion be true?"

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3. Economic Models: Translating Economic Intuition into Mathematical Language

Economic models are "cousins" of mathematical models—they borrow mathematical tools, but their goal is entirely different: they aim to explain human economic behavior.

Take a classic example: Gary Becker's economic model of crime. Becker's reasoning was simple—criminals are also rational agents who compare the expected benefits of crime against the expected costs (probability of being caught × severity of punishment) and then make the "optimal" choice.

Becker's model used mathematical tools such as first-order conditions (marginal benefit = marginal cost), but he was not doing pure mathematics—he was constructing a theoretical framework about human behavior.

Typical features of economic models:

• Assumptions with economic meaning: Not arbitrary axioms, but characterizations of human nature, institutions, and markets (e.g., "rational agents," "perfect competition," "symmetric information").
• Produce testable hypotheses: Becker's model predicts that "the more severe the punishment, the less crime"—a proposition that can be tested with data.
• Do not touch data themselves: Whether a model is good depends first on whether the logic is self-consistent and the intuition is compelling, not on R².

But here's the problem: once you have testable hypotheses, what do you use to test them?

That's when the third type of model enters the stage.

In one sentence: The question an economic model answers is—"Given assumptions about human nature, how will people act?"

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4. Econometric Models: Letting Theory and Data Talk to Each Other

An econometric model is a bridge between theory and reality. It transforms the abstract variables in economic models into measurable indicators, and then uses statistical methods to let the data "speak."

A classic empirical study will illustrate this far better than any abstract definition.

Empirical Case: Returns to Education—The Mincer Equation

Economists have long wanted to know: how much does an extra year of schooling raise wages?

Jacob Mincer proposed a concise econometric model in 1974:

$\ln(\text{wage}) = \beta_0 + \beta_1 \cdot \text{education} + \beta_2 \cdot \text{experience} + \beta_3 \cdot \text{experience}^2 + \varepsilon$

Every symbol here has meaning:

Element	Meaning
$\ln(\text{wage})$	Log wage—economic theory tells us to use the logarithmic form
education	Years of schooling—a prediction from the economic model
experience	Work experience—a control variable
$\beta_1$	Return to education—the parameter we most want to know
$\varepsilon$	Random disturbance term—this is unique to econometric models!

Pay attention to the symbol $\varepsilon$—it is the key distinction between econometric models and economic models.

In economic models, there is no random disturbance term. All behavior is a deterministic optimal solution. But in econometric models, you must acknowledge that you do not know all the influencing factors—some are measurement error, some are omitted variables, and some are purely individual differences. $\varepsilon$ is the formal acknowledgment of these "unknowns."

Using U.S. data from the 1970s, the estimated $\beta_1$ is approximately 0.07–0.10. This means: each additional year of schooling raises wages by roughly 7%–10%.

This result, in turn, spawned new economic models: Why is the return to education so high? Is it human capital accumulation (you learned real skills), or is it signaling (you are merely proving you are smart)?—This sparked the long-standing debate between human capital theory and signaling theory.

You see, this is a perfect cycle: economic theory → econometric testing → empirical findings → new theory.

In one sentence: The question an econometric model answers is—"Acknowledging that our understanding of the world is incomplete, what conclusions do the data support?"

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5. Core Differences: A Table That Says It All

Dimension	Mathematical Model	Economic Model	Econometric Model
Core purpose	Prove logical truth	Explain economic behavior	Test theory with data, estimate parameters
Involves empirical content	No	Hypotheses are testable, but does not touch data	Yes, relies on real data
Includes randomness	No, deterministic	Usually not	Yes, $\varepsilon$ is a core element
Evaluation criterion	Whether the proof is correct	Whether assumptions are reasonable and logic is self-consistent	Whether estimates are consistent, significant, and robust
Typical output	Theorems	Comparative static analysis, testable hypotheses	Regression coefficients, confidence intervals, causal effects
Typical question	"Does a unique equilibrium solution exist?"	"Will a tax increase reduce labor supply?"	"Is the reduction in labor supply from a tax increase statistically significant? How large is the effect?"

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6. Another Example to Help You Fully Understand: Minimum Wage and Employment

This is one of the most famous debates in economics, and it perfectly demonstrates the division of labor among the three types of models.

Step 1: Economic Model (Theoretical Prediction)

The standard competitive model predicts: setting a minimum wage above the equilibrium wage → labor demand decreases → employment falls. This is mathematical comparative static analysis—given supply and demand curves, the result follows logically with necessity.

But the problem is: in reality, there may also be monopsonistic labor markets, in which case firms are wage-setters rather than wage-takers—in this scenario, the minimum wage might actually increase employment.

Both theories are logically self-consistent. Who is right? Theory cannot answer this question—we must look at the data.

Step 2: Econometric Model (Empirical Testing)

This brings us to one of the most famous studies in economics from 1994—Card and Krueger's minimum wage study.

In 1992, New Jersey raised its minimum wage from $4.25 to$5.05, while neighboring Pennsylvania did not. This created a perfect natural experiment.

The econometric model they used was difference-in-differences (DID):

$\text{Employment}_{it} = \beta_0 + \beta_1 \cdot \text{NJ}_i + \beta_2 \cdot \text{After}_t + \beta_3 \cdot (\text{NJ}_i \times \text{After}_t) + \varepsilon_{it}$

The key coefficient is $\beta_3$—it measures "the change in employment at New Jersey fast-food restaurants after the policy change, minus the change in employment in Pennsylvania over the same period." This is the causal effect of the minimum wage on employment.

Step 3: Results and Aftermath

Card and Krueger found that $\beta_3$ was not significantly negative—the minimum wage increase did not significantly reduce employment.

This result shook the economics profession. It did not overturn any mathematical theorem—the theorems remain true. But it powerfully challenged the applicability of the competitive labor market economic model and spurred the development of new theories such as search and matching models. In 2021, David Card was awarded the Nobel Prize in Economics for this work.

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7. Summary: Three Models, One Knowledge Production Chain

These three models are not in a "which is more advanced" relationship; rather, they constitute a complete research cycle:

Economic model (produces theoretical hypotheses)
      ↓
Econometric model (tests hypotheses with data)
      ↓
Empirical findings (support, overturn, or refine theory)
      ↓
New economic model (explains new findings) → New econometric tests → ……

• Without economic models, econometric analysis is like a headless fly—you wouldn't even know which variables to study.
• Without econometric models, economic theory would forever remain just a "story"—it sounds plausible but cannot be verified.
• And mathematics is the common grammar of both—without it, economics could never become a rigorous science.

So the next time someone says, "Economics is all models, and they're inaccurate," you can tell them: The value of a model lies not in whether it is "accurate," but in whether it offers clear predictions that can be overturned by facts—and the overturning itself is the progress of science.

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8. Presentation Suggestions for Bilibili/WeChat Official Account

• Bilibili video: Use animations to show the relationships among the three models, especially the "subtract, then subtract again" logic of DID. Comparing employment changes before and after the policy in New Jersey and Pennsylvania with two bar charts is very intuitive.
• WeChat official account: Add handwritten-style formula images at the Mincer equation and the DID formula to reduce reading pressure; turn the knowledge cycle diagram at the end into a flowchart.
• Key quote (suitable for cover/title cards):

  "Mathematical models ask 'must it be so?', economic models ask 'should it be so?', and econometric models ask 'is it really so?'"