Simultaneous Equation Relationships

1. Core Concept Overview & Intuition

1.1 Introduction

In single-equation regression analysis, least-squares theory rests on the assumption that causation runs one way: the endogenous variable $Y$ depends on the exogenous variables $X$, but the $X$ do not in turn depend on $Y$. Formally, this is captured by the orthogonality condition

$$
E(XU)=0.
$$

When this assumption fails — that is, when there is double causation, so that $X$ also depends on $Y$ — the condition $E(XU)=0$ is violated and is no longer equal to zero. In such a scenario the function cannot be treated in isolation as a single-equation model. Instead it belongs to a wider system of equations describing the relationship among the relevant variables, $Y=f(X)$ and $X=f(Y)$. This wider system is a multi-equation model in which $Y$ and $X$ both appear as endogenous variables. A system describing the joint dependence of variables is called a system of simultaneous equations.

Simultaneous equations are studied on their own, despite the exhaustive treatment of single-equation models, because parameter estimation here has special features not available in single-equation models — for example, some explanatory variables are related to stochastic variables. Under such circumstances the ordinary least squares (OLS) method results in biased and inconsistent estimators. Methods must therefore be devised that give unbiased and consistent estimators. The major topics of concern are the dependence of the error term on the regressor(s), simultaneous-equation bias, and the structural and reduced forms of simultaneous-equation models.

1.2 Learning outcomes

By the end of this lesson the learner will be able to:

1. describe the structure of a system of simultaneous equations;
2. write the structural form of simultaneous equations into reduced forms.

1.3 Simultaneous equation relationships

Suppose we want to estimate the demand for food. This demand depends on its price and on the consumer's income, so we write the demand function

$$
Q_t=\beta_1+\beta_2P_t+\beta_3Y_t+u_t, \tag{1.1}
$$

where $Q_t$ is the quantity demanded, $P_t$ the price, $Y_t$ the consumer's income and $u_t$ the error term, with the usual assumptions

$$
E(u_t)=0,\quad E(u_t^2)=\sigma^2,\quad E(u_tu_{t'})=0\ (t\neq t'),\quad E(u_tY_t)=0.
$$

If OLS is applied to this equation the estimates will be biased, because even if the price $P$ is treated as an independent variable in this model, there exists another equation in which the price $P$ is the dependent variable; hence the price may be correlated with the error term, that is $E(P_tu_t)\neq 0$.

The price equation in the system is the supply (or price-determination) relation

$$
P_t=\gamma_1+\gamma_2Q_t+v_t, \tag{1.2}
$$

with

$$
E(v_t)=0,\quad E(v_t^2)=\sigma_v^2,\quad E(v_tv_{t'})=0\ (t\neq t'),\quad E(v_tu_t)=0.
$$

Thus the quantity depends on the price and the price on the quantity. This scenario is called double causation, and the two equations are solved simultaneously.

The intuition is that in a market, price and quantity are determined jointly by the interaction of demand and supply. Neither variable can be treated as if it were fixed by forces outside the system. Any attempt to estimate one equation in isolation ignores the feedback running through the other equation, and that feedback is precisely what contaminates the regressor with the error term.

2. Mathematical Formalization & Core Definitions

2.1 Endogenous and exogenous variables

> Definition (Endogenous variable). An endogenous variable is a variable whose value is determined within the system of simultaneous equations. In the income determination model, both consumption $C_t$ and income $Y_t$ are endogenous.

> Definition (Exogenous variable). An exogenous variable is a variable whose value is determined outside the system and which therefore enters the equations as a predetermined quantity. In the income determination model, investment $I_t$ is exogenous.

> Definition (Structural form). The structural form of a simultaneous-equation model is the set of equations written in terms of the original behavioural parameters, with the endogenous variables appearing on the left-hand side and the endogenous and exogenous variables on the right-hand side. Equations (1.1) and (1.2), and equations (1.3) and (1.4), are structural equations.

> Definition (Reduced form). The reduced form of a simultaneous-equation model expresses each endogenous variable solely as a function of the exogenous variables and the stochastic disturbances. It is obtained by solving the structural system for the endogenous variables.

> Definition (Double causation). Double causation occurs when two variables each depend on the other, so that $Y=f(X)$ and $X=f(Y)$ simultaneously. It is the defining feature of a simultaneous-equation relationship.

2.2 The income determination model

The income determination model has the form

$$
C_t=\beta_1+\beta_2Y_t+u_t \tag{1.3}
$$

$$
Y_t=C_t+I_t \tag{1.4}
$$

where $C_t$ and $Y_t$ are endogenous variables; $I_t$ is an exogenous variable; $u_t$ represents the error term; $\beta_1$ and $\beta_2$ are the parameters of the system to be estimated; and $t$ represents time. The two equations represent the consumption function and the national income identity respectively.

The error term satisfies

$$
E(u_t)=0,\quad E(u_t^2)=\sigma^2,\quad E(u_tu_{t'})=0,\quad E(u_tI_t)=0.
$$

If OLS were applied to estimate the parameters, the appropriate assumptions must be checked. The assumptions of non-autocorrelation and homoscedasticity are satisfied from the above. The critical assumption is that of independence between the stochastic term and the regressor.

2.3 The commodity demand model

Consider the commodity demand function representing demand for commodity $C$:

$$
Q_{ic}=\alpha_0+\alpha_1P_i+\alpha_2P_{ir}+u_i \tag{1.6}
$$

where

• $Q_{ic}$ = quantity of commodity $C$ demanded,
• $P_i$ = price of commodity $C$,
• $P_{ir}$ = price of a related commodity,
• $u_i$ = error term,
• $\alpha_0$, $\alpha_1$ and $\alpha_2$ are the parameters.

The model makes the following assumptions on the error term:

$$
E(u_i)=0,\quad E(u_i^2)=\sigma^2,\quad E(u_iu_j)=0\ (i\neq j),\quad E(u_iP_{ir})=0.
$$

Another important dynamic that governs the system is the price equation

$$
P_i=\pi_0+\pi_1Q_{ic}+\pi_2P_{ir}+v_i, \tag{1.7}
$$

with

$$
E(v_i)=0,\quad E(v_i^2)=\sigma_v^2,\quad E(v_iv_j)=0\ (i\neq j),\quad E(v_iP_{ir})=0,\quad E(u_iv_i)=0.
$$

Here $P_{ir}$ is the exogenous variable, while $Q_{ic}$ and $P_i$ are endogenous.

2.4 The general structural system

A two-equation simultaneous system in general form may be written as

$$
Y_{1t}=\beta_{10}+\beta_{12}Y_{2t}+\gamma_{11}X_{1t}+u_{1t}
$$

$$
Y_{2t}=\beta_{20}+\beta_{21}Y_{1t}+\gamma_{22}X_{2t}+u_{2t}
$$

where the $Y$ variables are endogenous and the $X$ variables are exogenous. The reduced form is obtained by solving these two equations for $Y_{1t}$ and $Y_{2t}$ in terms of $X_{1t}$, $X_{2t}$, $u_{1t}$ and $u_{2t}$. This is the algebraic operation required by learning outcome 2: writing the structural form into reduced form.