Endogenous vs Exogenous Variables in Simultaneous Equations Models

Ever get stuck in a “chicken and egg” debate? Does a high price for coffee cause farmers to plant more, or does a bad harvest that limits supply cause the high price? In economics, the answer is often “yes.” Both are true, and they are happening at the same time. This tangled web of cause-and-effect is at the heart of many economic questions, and to unravel it, we use a powerful tool called Simultaneous Equations Models (SEMs). But before you can build one of these models, you have to become a detective and sort your variables into two critical groups: the ‘insiders’ and the ‘outsiders’.

In the language of econometrics, these are called endogenous and exogenous variables. Understanding this difference isn’t just academic jargon; it’s the fundamental step that determines whether your model’s conclusions are brilliant insights or complete nonsense. Getting this wrong is like trying to solve a mystery when you’ve mixed up the suspects with the bystanders. So, let’s dive in and clear up what these terms mean and why they matter so much.

What exactly is an endogenous variable? 🔗

Endogenous variables are the ‘insiders’ of your model. They are the stars of the show, the variables whose values are determined within the system of equations you’ve built. Think of them as the main characters in a story whose fates are intertwined. You can’t understand one’s story without understanding the other’s.

The ‘insiders’ determined by the model 🔗

The word “endogenous” comes from Greek: *endo-* meaning “within” and *-genous* meaning “producing.” These variables are “produced within” the model itself. In a model, they are the key dependent variables that not only are influenced by other variables but also influence them back. This mutual, simultaneous relationship is what we call a feedback loop.

Let’s take the most classic example: the market for coffee.

• A demand equation tells us that the quantity of coffee people want to buy depends on its price. (High price = less demand).
• A supply equation tells us that the quantity of coffee farmers are willing to sell also depends on its price. (High price = more supply).

In this system, Price (P) and Quantity (Q) are both endogenous. Price affects quantity, and quantity (specifically, the interplay of supply and demand) determines price. They are “jointly determined.” You can’t find the equilibrium price without also finding the equilibrium quantity, and vice versa. They are locked together in a simultaneous dance.

The classic Keynesian example 🔗

Another foundational example in macroeconomics is the simple Keynesian model of national income. This model often has two core equations:

1. The Consumption Function: $Consumption = f(Income)$
2. The Income Identity: $Income = Consumption + Investment$

In this system, Consumption and Income are the endogenous variables. Your model is trying to explain both of them. Why? Because they have a feedback loop. When people earn more income, they spend more (consumption goes up). But when people spend more, that spending becomes income for someone else (income goes up). This mutual causation is the defining feature of endogeneity. These variables are the core mystery the model is trying to solve.

Understanding exogenous and predetermined variables 🔗

If endogenous variables are the ‘insiders’ caught in a feedback loop, exogenous variables are the ‘outsiders’ that come in and shake things up. They are the external forces that influence the system but are not, in turn, influenced by it.

The ‘outsiders’ that shape the system 🔗

The word “exogenous” means “produced outside.” These variables are determined by factors outside the scope of your model. Their values are taken as a given. They are the “unmoved movers” that can push and pull the endogenous variables around, but the endogenous variables can’t push back.

Let’s go back to our coffee market. What would be an exogenous variable?

• Weather: A sudden, severe frost in Brazil (a major coffee producer) would wipe out a huge portion of the coffee supply. This frost is an external “shock” to the system. It will drastically affect the price and quantity of coffee, but-and this is the key-the price of coffee in Chicago has absolutely no effect on the weather in Brazil. The causal arrow points in only one direction.
• Government Policy: The government decides to impose a new $1 per-pound tax on all coffee imports. This policy decision is made “outside” the market’s supply-demand dance. It will influence the final price and quantity, but the market’s price doesn’t (in this model) change the tax law.
• Consumer Tastes: A new viral health study is published claiming coffee extends your life. This is an external event that would shift the entire demand curve, affecting both $P$ and $Q$, but the price of coffee itself didn’t cause the study to be published.

Predetermined variables: The all-important ‘given’ inputs 🔗

This brings us to a slightly broader and very important term: predetermined variables. This category includes all variables in the model whose values are “given” and not determined within the system in the current time period. They are the inputs you use to solve for the endogenous ‘insiders’.

The “predetermined” bucket contains three distinct types of variables:

1. Current Exogenous Variables: This is what we just discussed. The frost that happens *this month*, or the tax law passed *this week*. Their values are set externally.
2. Lagged Exogenous Variables: These are exogenous variables from a previous time period. For example, the amount of government spending from *last year*. It’s an external factor, and since it already happened, its value is a fixed historical fact.
3. Lagged Endogenous Variables: This one is a bit tricky but crucial. It’s an endogenous variable… but from the past. Think about *last month’s* price of coffee. While *today’s* price is endogenous (determined right now), *last month’s* price is a “done deal.” It’s a historical number you can look up. It’s no longer being determined. It’s “predetermined.”

Why do we care? Because these lagged endogenous variables can be powerful predictors. Last month’s price might heavily influence what suppliers expect this month’s price to be, thereby affecting *today’s* supply. In this way, a variable that was an “insider” last period becomes a “given input” for this period.

Notation in a general SEM framework 🔗

In econometrics, we like to formalize this. Don’t let the letters scare you; it’s just a way of organizing the concepts we’ve already covered. A general simultaneous equations model is often described as a system containing:

• M endogenous variables (the ‘Y’s): These are the variables we are trying to explain (e.g., $Y_1$ = Price, $Y_2$ = Quantity).
• K predetermined variables (the ‘X’s): These are all the ‘given’ inputs-all the exogenous and lagged variables (e.g., $X_1$ = Rainfall, $X_2$ = Consumer Income, $X_3$ = Last Month’s Price).

The structural equations 🔗

The model itself is written as a set of structural equations. “Structural” just means it’s based on an underlying economic theory of how things work. There is one equation for each endogenous variable ($Y$).

Sticking with our coffee market, the structural equations might look something like this:

Equation 1 (Demand): $Y_1 \text{ (Quantity)} = \beta_{10} + \beta_{12}Y_2 \text{ (Price)} + \gamma_{11}X_1 \text{ (Consumer Income)} + u_1$

Equation 2 (Supply): $Y_1 \text{ (Quantity)} = \beta_{20} + \beta_{22}Y_2 \text{ (Price)} + \gamma_{22}X_2 \text{ (Rainfall)} + u_2$

Let’s break that down:

• The $Y$ variables (Price and Quantity) appear in *both* equations, showing they are intertwined. This is the feedback loop.
• The $X$ variables (Income and Rainfall) are the predetermined inputs. Notice that “Income” ($X_1$) is in the demand equation (theory says it affects demand) but not the supply one. “Rainfall” ($X_2$) is in the supply equation (theory says it affects supply) but not the demand one.
• The $u$ variables ($u_1$ and $u_2$) are the stochastic error terms. They represent all the random, unobserved factors that we couldn’t measure (like a random shipping delay or a sudden fad).

The “problem” of simultaneous equations is that you can’t just use standard regression (like Ordinary Least Squares) to estimate these equations. Because $Y_1$ and $Y_2$ are linked, the error term in one equation (like $u_1$) will inevitably be correlated with the $Y$ variable in that same equation. This violates a key assumption of OLS and leads to biased results. This is the whole reason we have to identify the ‘insiders’ and ‘outsiders’ properly in the first place.

The role of theory in model specification 🔗

This leads to the final, and perhaps most important, question: How do we *know* which variables are which? How did we decide that consumer income is exogenous in our coffee model? What if higher coffee prices make people work more, raising their income? What if…?

The answer is: Economic theory.

A variable is not “born” endogenous or exogenous. Its classification is a choice made by the researcher based on the theory they are testing. This act of building the equations and classifying the variables is called model specification. When a researcher builds a model, they must create a compelling, logical argument for their choices. This is the “art” of econometrics.

Logically defending your ‘world’ 🔗

When you classify a variable as exogenous, you are making a strong claim. You are saying that it is “excluded” from being influenced by the system. These are called “exclusion restrictions” and they must be justified.

• Why is ‘Rainfall’ exogenous? Justification: This is easy. Meteorology and economics are separate fields. The price of corn in Iowa does not change the weather patterns in the Pacific. The causal link is one-way. This is a very strong and believable exclusion.
• Why is ‘Consumer Income’ exogenous? Justification: This is an assumption, but a reasonable one for a simple market model. We assume that the total income of all coffee drinkers in a country is so large that a small change in the price of coffee won’t meaningfully impact their overall income. We are *assuming* the feedback loop from coffee price to national income is negligible.
• Why is ‘Government Investment’ exogenous in a macro model? Justification: A researcher might argue that government spending is a political decision set by policymakers, based on election cycles or long-term plans, not by the day-to-day changes in national consumption.

These theoretical choices are everything. For example, researchers at the Reserve Bank of India building a model of inflation must make hard choices. Is the exchange rate endogenous (determined by inflation) or exogenous (set by global capital flows)? Is the oil price exogenous (set by OPEC) or endogenous? The answers they choose, based on their theoretical understanding of the Indian economy, will completely change the model’s results and policy recommendations.

Ultimately, sorting variables into these two camps is the first and most critical step. It’s how you define the “world” of your model, separating the interconnected parts you want to explain from the external forces you’ll use to explain them.

What do you think? Can you think of another real-world example of a “chicken and egg” problem where two or more variables seem to cause each other simultaneously? If a researcher mistakenly treats an ‘insider’ (like price) as an ‘outsider’ (an exogenous variable), what kind of flawed conclusions might they reach?

References

1. https://blogs.worldbank.org/impactevaluations/what-are-endogenous-and-exogenous-variables
2. https://www.econometrics-with-r.org/12-1-simultaneous-equations-models.html
3. https://statisticsbyjim.com/regression/endogeneity-exogeneity/
4. https://www.rbi.org.in/Scripts/PublicationsView.aspx?id=19904