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Exogenous and endogenous variables
Exogenous and endogenous variables
Exogenous and endogenous variables
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In an economic model, an exogenous variable is one whose measure is determined outside the model and is imposed on the model, and an exogenous change is a change in an exogenous variable.[1]: p. 8 [2]: p. 202 [3]: p. 8  In contrast, an endogenous variable is a variable whose measure is determined by the model. An endogenous change is a change in an endogenous variable in response to an exogenous change that is imposed upon the model.[1]: p. 8 [3]: p. 8 

The term 'endogeneity' in econometrics has a related but distinct meaning. An endogenous random variable is correlated with the error term in the econometric model, while an exogenous variable is not.[4]

Examples

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In the LM model of interest rate determination,[1]: pp. 261–7  the supply of and demand for money determine the interest rate contingent on the level of the money supply, so the money supply is an exogenous variable and the interest rate is an endogenous variable.

Sub-models and models

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An economic variable can be exogenous in some models and endogenous in others. In particular this can happen when one model also serves as a component of a broader model. For example, the IS model of only the goods market[1]: pp. 250–260  derives the market-clearing (and thus endogenous) level of output depending on the exogenously imposed level of interest rates, since interest rates affect the physical investment component of the demand for goods. In contrast, the LM model of only the money market takes income (which identically equals output) as exogenously given and affecting money demand; here equilibrium of money supply and money demand endogenously determines the interest rate. But when the IS model and the LM model are combined to give the IS-LM model,[1]: pp. 268–9  both the interest rate and output are endogenously determined.

See also

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References

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Exogenous and endogenous variables

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In mathematical and econometric modeling, exogenous variables are those whose values are determined externally to the model and treated as fixed inputs that influence other variables without being affected by them in return, while endogenous variables are those whose values are determined internally through the model's equations and relationships, often involving interdependence or feedback effects. This distinction is essential for structuring models that capture causal relationships and equilibrium outcomes, particularly in fields like economics where variables interact dynamically. The classification originates from the need to differentiate between factors controlled or imposed from outside a —such as variables or external shocks—and those resolved endogenously within it, like market prices or outputs in supply-demand frameworks. In (SEMs), exogenous variables appear solely on the right-hand side of equations as independent predictors, whereas endogenous variables feature on , requiring joint solution to avoid inconsistencies in . For instance, in a , interest rates set by a may serve as exogenous, directly impacting endogenous variables like and GDP through behavioral equations. This framework extends beyond to in statistics and social sciences, where endogeneity can introduce biases if not addressed, such as through assumptions of no between exogenous variables and terms. Proper identification of variable types ensures model validity, influencing techniques like ordinary least squares versus more advanced methods for handling simultaneity. Overall, the exogenous-endogenous dichotomy underpins rigorous analysis of complex systems by clarifying and external influences.

Definitions and Fundamentals

Exogenous Variables

In mathematical modeling, particularly in and , an exogenous variable is defined as one whose value is determined outside the model and treated as given or fixed, independent of the model's internal mechanisms. This contrasts with endogenous variables, which are outputs determined internally by the model's equations. Key characteristics of exogenous variables include their lack of influence from other variables within the model, positioning them as parameters or external shocks that do not participate in feedback loops. They serve to provide initial conditions or driving forces that propel the system's dynamics, allowing the model to simulate responses to external changes without altering the inputs themselves. The concept of exogenous variables originated in through the work of in the 1930s, where he introduced the distinction to differentiate external factors in early econometric models, particularly in his analysis of dynamic economic systems. In linear models, such as those in simultaneous equations frameworks, exogenous variables typically appear on the right-hand side of the equations, contributing to the determination of endogenous outcomes without being simultaneously determined by them.

Endogenous Variables

In economic and econometric models, endogenous variables are those whose values are determined internally by the equations and relationships within the model itself, rather than being externally imposed. These variables serve as the primary outcomes or dependent factors that the model seeks to explain through its structural specifications. Key characteristics of endogenous variables include their dependence on other elements of the system, such as fellow endogenous variables and exogenous inputs, often leading to interdependence or feedback effects among them. Unlike exogenous variables, which act as fixed influences from outside the model, endogenous variables arise from the model's internal dynamics and are not predetermined independently. Endogenous variables fulfill the role of capturing the model's predicted outcomes or equilibrium conditions, reflecting the resolved state of the under given assumptions. In particular, within simultaneous systems, their values must be derived by jointly solving the interconnected , as opposed to being directly observed or assumed constant. This contrasts with parameters, which remain fixed constants unaffected by model interactions, whereas endogenous variables fluctuate based on the evolving relationships they embody.

Mathematical Formulation

Model Representation

In mathematical modeling, particularly within , variables are partitioned into endogenous and exogenous sets to represent dependencies within a . Endogenous variables, denoted as y\mathbf{y}, are those whose values are jointly determined by the equations of the model, while exogenous variables, denoted as x\mathbf{x}, are predetermined and independent of the model's disturbances. This partitioning allows the model to capture internal interactions among dependent variables and external influences from independent ones. The structural form of such a model is expressed as a defining the relationships among variables. In matrix notation, it takes the general form y=Ay+Bx+ε,\mathbf{y} = A \mathbf{y} + B \mathbf{x} + \boldsymbol{\varepsilon}, where y\mathbf{y} is a vector of endogenous variables, x\mathbf{x} is a vector of exogenous variables, AA is the capturing interactions among endogenous variables, BB is the for exogenous variables, and ε\boldsymbol{\varepsilon} is a vector of error terms representing unobserved factors. This form assumes linearity in parameters for analytical tractability, enabling the representation of simultaneous relationships without specifying nonlinear complexities. To solve for the endogenous variables explicitly, the model is rearranged into its by isolating y\mathbf{y}: (I−A)y=Bx+ε,(I - A) \mathbf{y} = B \mathbf{x} + \boldsymbol{\varepsilon}, yielding y=(I−A)−1(Bx+ε),\mathbf{y} = (I - A)^{-1} (B \mathbf{x} + \boldsymbol{\varepsilon}), which expresses the endogenous variables solely as functions of the exogenous variables and errors, assuming I−AI - A is invertible. The highlights how exogenous factors drive the system's outcomes through the composite matrix (I−A)−1B(I - A)^{-1} B. A key assumption underlying this representation is the exogeneity condition, which requires that the error terms are uncorrelated with the , formally E(ε∣x)=0E(\boldsymbol{\varepsilon} \mid \mathbf{x}) = 0. This ensures that exogenous variables serve as valid predictors without feedback from the model's disturbances, facilitating consistent and .

Identification and Simultaneity

In simultaneous equation models, simultaneity arises when two or more endogenous variables mutually influence each other, creating circular causation that violates the assumptions of (OLS) estimation. This mutual dependence means that an explanatory variable is correlated with the error term in its , leading to inconsistent and biased parameter estimates if treated as exogenous. For instance, in a supply-demand system, and are simultaneously determined, so regressing quantity on price yields simultaneity , as price responds to quantity shocks as well. The identification problem addresses whether structural parameters in a can be uniquely recovered from the reduced-form parameters observable in . A necessary but not sufficient condition for identification is the order condition, which requires that the number of exogenous variables excluded from an equals or exceeds the number of endogenous variables included (minus one). The rank condition, which is necessary and sufficient, ensures that the coefficients on the excluded exogenous variables form a matrix of full rank equal to the number of included endogenous regressors. These conditions, formalized in the of linear simultaneous systems, prevent under-identification where multiple structural forms map to the same . Treating an endogenous variable as exogenous introduces bias akin to , where the correlation between the regressor and the error term distorts estimates, often inflating or deflating coefficients unpredictably. In over-identified systems, where the number of valid instruments exceeds the number of endogenous regressors, this setup enables tests for instrument validity and exogeneity, such as the Sargan or Hansen J-test, which assess whether over-identifying restrictions hold under the null of instrument exogeneity. A primary solution to simultaneity and related identification issues is the use of instrumental variables (IV), where instruments are exogenous variables correlated with the endogenous regressors but uncorrelated with the model errors. Methods like two-stage least squares (2SLS) implement IV by first regressing endogenous variables on instruments to obtain predicted values, then using these in the structural equation, yielding consistent estimates under valid instruments. This approach, widely applied since the Cowles Commission era, directly counters biases from mutual causation or omitted factors.

Applications in Economics

Microeconomic Models

In microeconomic models, the framework illustrates the roles of exogenous and endogenous variables in determining market outcomes at the individual or firm level. and are endogenous variables, jointly determined by the interaction between buyers' and sellers' supply. Exogenous variables, such as consumer income, tastes, and production technology, shift the or supply curves but are assumed fixed within the model. The equilibrium concept arises at the point where supply equals , setting the values of the endogenous variables with no excess or supply. This intersection predicts prices and quantities based on the exogenous parameters. Shifts in exogenous variables, such as a on sellers, cause the supply curve to shift leftward, leading to a new endogenous equilibrium with higher prices and lower quantities. analyzes these effects, quantifying how changes in exogenous factors—like increases boosting —alter the endogenous equilibrium outcomes. These models draw on the general mathematical , where endogenous variables solve a parameterized by exogenous inputs. In game-theoretic extensions of microeconomic , payoffs function as exogenous parameters defining the strategic payoffs for each action profile, while strategies are endogenous, emerging from players' rational responses to achieve .

Macroeconomic Models

In macroeconomic models, exogenous variables represent external forces or policy instruments that drive fluctuations in aggregate output, employment, and prices, while endogenous variables capture the internal responses of the to these influences. These distinctions are central to frameworks that analyze economy-wide interactions, such as the balance between goods and money markets or long-term growth dynamics. For instance, fiscal and monetary policies often serve as exogenous shocks that shift equilibrium outcomes determined by endogenous conditions. The IS-LM model, developed by John Hicks to represent Keynesian theory graphically, illustrates this framework by treating output (Y) and the interest rate (r) as endogenous variables determined simultaneously through the intersection of the investment-savings (IS) curve and the liquidity preference-money supply (LM) curve. Exogenous factors, such as changes in government spending or money supply from fiscal and monetary policy, shift these curves and thereby influence the endogenous equilibrium of output and interest rates. For example, an increase in exogenous government expenditure shifts the IS curve rightward, raising both output and interest rates endogenously. In growth models like the Solow-Swan model, (K) and output per worker (y) are endogenous, evolving dynamically based on production functions and , while parameters such as the savings rate (s), rate (n), and technological progress (A) are treated as exogenous. These exogenous elements determine the steady-state levels of capital and output, with deviations arising from initial conditions or shocks that the adjusts to endogenously over time. The model's exogenous technological progress underscores long-run growth as externally driven, contrasting with endogenous growth theories that internalize innovation. Dynamic stochastic general equilibrium (DSGE) models extend this by incorporating microeconomic foundations—such as optimizing households and firms—into stochastic environments, where technology shocks are typically modeled as exogenous processes that propagate through endogenous variables like consumption, , and labor supply. In these frameworks, random exogenous disturbances to productivity generate business cycles, with endogenous responses amplified by frictions like sticky prices. Exogenous policy variables, such as or rates, play a key role in influencing endogenous business cycles by altering and supply responses in these models. For instance, countercyclical fiscal expansions can dampen endogenous fluctuations in output during recessions, stabilizing the through shifts in exogenous instruments. Under , macroeconomic models assume agents form unbiased forecasts of future endogenous variables using all available information, incorporating forward-looking behavior that makes current endogenous outcomes dependent on anticipated future states. This , integral to New Keynesian and real business cycle models, ensures that endogenous variables like and output reflect optimal responses to both current exogenous shocks and expected paths, avoiding systematic forecast errors.

Applications in Other Disciplines

Econometrics and Statistics

In and statistics, the distinction between exogenous and endogenous variables is central to , where exogenous regressors are assumed to be uncorrelated with the error term in the model, ensuring the consistency and unbiasedness of ordinary least squares (OLS) estimates under standard assumptions. This exogeneity condition implies that the explanatory variables are determined independently of the model's disturbances, allowing for valid inference on causal relationships. In contrast, endogenous regressors—those correlated with the error term—arise from issues such as , measurement error, or simultaneity, leading to inconsistent OLS estimates that fail to recover the true population parameters even in large samples. A key diagnostic for detecting endogeneity is the Hausman test, which compares the OLS estimator (efficient under exogeneity but inconsistent otherwise) with an instrumental variables (IV) estimator (consistent but less efficient under exogeneity). If the two estimates differ significantly, the of exogeneity is rejected, indicating the presence of endogenous regressors and the need for alternative estimation strategies like IV or (). This test, introduced by Jerry A. Hausman, has become a cornerstone for specification testing in econometric models, particularly in cross-sectional and settings where unobserved confounders may correlate with regressors. In causal inference, exogenous variables play a pivotal role by enabling the identification of treatment effects, as they provide the necessary variation that is independent of potential confounders, allowing researchers to isolate causal impacts without bias from selection or reverse . For instance, in potential outcomes frameworks, strict exogeneity ensures that treatment assignment is orthogonal to unobserved errors, facilitating the estimation of average treatment effects through methods like regression discontinuity or difference-in-differences. This property is essential in observational data studies, where is absent, and exogenous shocks or instruments serve as quasi-experimental levers for credible inference. In time-series analysis, exogeneity is often assessed in the Granger sense, where a variable XX is exogenous with respect to YY if past values of the error term in YY's equation do not help predict XX, meaning XX has no predictive power derived from YY's disturbances. This weak form of exogeneity, distinct from strict exogeneity in simultaneous systems, supports and by confirming that XX can be treated as given without feedback from YY's innovations. Granger's framework, originally developed for multivariate , underpins tests like the Granger causality test, which evaluates whether lagged values of one variable improve predictions of another beyond its own lags. An important extension in panel data econometrics involves fixed effects models, which treat unobserved unit-specific heterogeneity as exogenous by differencing out or demeaning the data to eliminate time-invariant individual effects that could otherwise correlate with regressors. This approach assumes strict exogeneity of the remaining time-varying covariates conditional on the fixed effects, allowing consistent estimation of within-unit causal relationships while controlling for omitted variables that are constant over time. Widely used in applied research, fixed effects estimation mitigates bias from unobserved heterogeneity, such as ability in labor economics panels, provided the exogeneity assumption holds for deviations from individual means.

Systems and Control Theory

In systems and , exogenous and endogenous variables are fundamental to modeling dynamic systems, particularly in representations where the structure illustrates interactions between external influences and internal states. Exogenous variables typically represent inputs or disturbances that originate outside the , such as reference signals or environmental perturbations, depicted as arrows entering the block from external sources. In contrast, endogenous variables correspond to the 's internal states or outputs, which evolve based on these inputs and the 's dynamics, shown as signals propagating within the feedback loops or output paths of the diagram. A formulation in control systems is the state-space model, which explicitly distinguishes exogenous inputs from endogenous states. In discrete-time linear systems, the dynamics are captured by the equation xt+1=Axt+But,\mathbf{x}_{t+1} = A \mathbf{x}_t + B \mathbf{u}_t, where xt\mathbf{x}_t denotes the endogenous state vector at time tt, representing internal variables like position or in a mechanical system; ut\mathbf{u}_t is the exogenous input vector, such as control forces or disturbances; and AA and BB are system matrices defining the endogenous evolution and input coupling, respectively. This representation allows for the analysis of how external inputs drive the system's internal behavior over time. Feedback mechanisms in control systems enable regulation of endogenous variables in response to exogenous noise or disturbances. Controllers, often designed using state feedback, adjust the input ut\mathbf{u}_t based on measured states xt\mathbf{x}_t to counteract unpredictable exogenous effects, such as sensor noise or load variations, thereby stabilizing the system's output. This closed-loop configuration ensures that endogenous states track desired trajectories despite external perturbations. Stability analysis in these systems focuses on ensuring that endogenous dynamics converge or remain bounded irrespective of bounded exogenous inputs. For linear time-invariant systems, asymptotic stability of the matrix AA guarantees that states xt\mathbf{x}_t approach zero for zero input, while (ISS) extends this to non-zero exogenous inputs, preventing unbounded growth from disturbances like wind gusts in flight control. Such properties are verified using Lyapunov functions or eigenvalue analysis. In applications like biological or ecological modeling, exogenous variables often include environmental factors such as periodic rainfall or fluctuations, while endogenous variables represent sizes or levels that respond through density-dependent feedbacks. For instance, in dryland models, rainfall acts as an exogenous driver exciting damped oscillatory modes in water and plant (endogenous states), with revealing how external forcing amplifies or synchronizes .

Examples and Illustrations

Simple Economic Example

Consider the market for apples as a straightforward illustration of exogenous and endogenous variables. In this model, the equilibrium price PP and quantity QQ are endogenous, determined by the intersection of within the system. Exogenous variables, such as consumer income and weather conditions, are determined externally and shift the demand or supply curves. The demand function is specified as Qd=a−bP+c⋅Income,Q_d = a - bP + c \cdot \text{Income}, where a>0a > 0, b>0b > 0, and c>0c > 0 are parameters, capturing how higher prices reduce quantity demanded and higher income boosts it. The supply function is Qs=d+eP+f⋅Weather,Q_s = d + eP + f \cdot \text{Weather}, with dd, e>0e > 0, and f>0f > 0 as parameters, where favorable weather (e.g., rainfall) increases supply. Market equilibrium requires Qd=Qs=QQ_d = Q_s = Q, allowing solution for the endogenous variables as functions of the exogenous ones. Equating demand and supply yields the equilibrium price P∗=a−d+c⋅Income−f⋅Weatherb+eP^* = \frac{a - d + c \cdot \text{Income} - f \cdot \text{Weather}}{b + e} and quantity Q∗=d+eP∗+f⋅Weather.Q^* = d + eP^* + f \cdot \text{Weather}. These solutions demonstrate that changes in exogenous income or weather directly influence the endogenous P∗P^* and Q∗Q^*; for example, higher income shifts demand rightward, increasing both price and quantity. A exemplifies an exogenous shock, reducing the weather variable and shifting supply leftward, which raises P∗P^* and lowers Q∗Q^*. This highlights how external factors propagate through the model to affect market outcomes. The example also embodies the assumption, holding exogenous variables constant to isolate the price-quantity relationship along a single curve.

Non-Economic Example

In the physics of simple motion, the θ\theta serves as an endogenous variable, as its value evolves dynamically within the system based on the governing . The initial displacement and the gg are treated as exogenous variables, determined externally to the pendulum's oscillatory behavior. The fundamental equation describing this motion is d2θdt2+glsin⁡θ=0,\frac{d^2 \theta}{dt^2} + \frac{g}{l} \sin \theta = 0, where ll is the pendulum length, and the parameter g/lg/l is exogenous, fixed by environmental and structural factors independent of θ\theta itself. The endogenous variable θ\theta thus evolves over time from the given exogenous initial conditions, producing periodic oscillations that reflect the internal dynamics driven by these external inputs. In such physical systems, exogenous parameters like dictate the frequency and of the motion without being altered by it. A parallel illustration appears in biological through the Lotka-Volterra predator-prey model, where the numbers of predators and prey populations are endogenous variables, fluctuating interdependently based on their interactions. In contrast, parameters such as the per capita birth rates of prey and death rates of predators are exogenous, set by external ecological or environmental factors outside the model's core feedback loop. This interpretation underscores how, in non-economic domains like physics and , exogenous forces—such as conditions or vital rates—drive the of internal, endogenous oscillations or cycles within the .

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