Econometrics · Lesson 12 of 18

Simultaneous equations and identification

A price is not chosen by the demand curve or by the supply curve: it is where they meet. When variables are determined together, a regression of one on another mixes the equations up. This lesson explains why OLS fails, how to tell whether an equation can be recovered at all (the order and rank conditions, which the exams ask constantly), and which estimator to use once it can: indirect least squares, two-stage least squares, or plain OLS in the special recursive case.

← Econometrics moduleLesson 12 of 18 · about 55 minutes

A question to hold on to

Given the structural model y₁ = 3y₂ − 2x₁ + x₂ + u₁; y₂ = y₃ + x₃ + u₂; y₃ = y₁ − y₂ − 2x₃ + u₃. Based on the order condition, the second equation is: (A) over identified; (B) exactly identified; (C) under identified; (D) information not sufficient.

UGC NET December 2025Solved at the end ↓

New here?What is the Ashanagar survey?The invented household survey behind this page's examples: 600 households, generated from a model whose true coefficients we publish.

The Ashanagar survey is invented. The statistics modules used one district of 60 households. Econometrics needs more households and more variables, so this is a new, larger survey of 600 households in the same imaginary district: income, spending, household size, whether the household is in the town or a village, the head’s schooling and earnings.

Because we wrote the data-generating process, we know the true answers, and every lab can show whether a regression recovers them. Household spending, in thousands of rupees a month, was generated as 1 + 0.45 × income + 0.9 × household size + 1.5 × town + an error with standard deviation 2.5. A real survey never tells you its true coefficients. This one does, so you can see how close an estimate comes.

It also holds a variable, the head’s ability, that a real survey could not observe. It is kept so that the lessons on omitted variables can show what happens when it is left out. Every figure here is illustrative, not official statistics.

Households: 600
Share in the town: 41%
Mean monthly income: ₹15.4 thousand
Mean monthly spending: ₹12.6 thousand
Mean schooling of the head: 8.2 years

How the module uses it →

The model

Equations determined together

A simultaneous equations model has several equations whose variables are determined together. Variables whose values the model determines are endogenous (price and quantity in a market); variables determined outside it are exogenous (income, rainfall), and lagged endogenous variables behave like exogenous ones and are called predetermined. The model as the theory writes it is the structural form:

A market in structural form

demand: q = a_{0} + a_{1} p + a_{2} y + u_{d} supply: q = b_{0} + b_{1} p + b_{2} w + u_{s}

Each equation is a behavioural claim. Price and quantity are endogenous; income y and rainfall w are exogenous. a₁ < 0 is the demand slope and b₁ > 0 the supply slope.

Solve the system for the endogenous variables in terms of the exogenous ones and the shocks and you get the reduced form: p and q each as a function of y, w and the errors. Every right-hand variable is exogenous, so OLS can estimate the reduced form without bias. The difficulty is getting back from the reduced form to the structural equations, which is what we want to know (the demand elasticity, the supply response).

Interactive

Why OLS fails: simultaneity bias

Regress quantity on price in the data, and you do not get the demand slope or the supply slope, because the data are intersections of the two curves. A demand shock raises both price and quantity along the supply curve; a supply shock moves them along the demand curve. Price is therefore correlated with both equations’ errors: it is endogenous in each. Without shifters, the OLS slope converges to a weighted average of the two true slopes, weighted by the variance of the other equation’s shock:

Simultaneity bias with no shifters

plim b_{O L S} = \frac{a _{1} Var ( u _{s} ) + b _{1} Var ( u _{d} )}{Var ( u _{s} ) + Var ( u _{d} )}

Var(u_d) is the demand shock's variance, Var(u_s) the supply shock's. With equal variances and slopes −1 and +1 the OLS slope is 0, a number belonging to neither curve.

In the lab, change the relative size of the two shocks: when demand shocks dominate, the points trace out the supply curve; when supply shocks dominate, they trace the demand curve; in between, OLS gets a muddle. The 2SLS and indirect least squares estimates, which use income and rain as shifters, recover both true slopes in every case.

True model: demand q = 10 − 1 × p + income + u_d; supply q = 2 + 1 × p + rain + u_s. Income and rain are observed shifters. The true demand slope is −1 and the true supply slope is +1. Each dot is one market-period: the price and quantity at which the two curves met.

Size of demand shocks, sd of u_d1Size of supply shocks, sd of u_s1Number of market-periods, n150

OLS slope of q on p0.102

Demand slope by 2SLS (rain as instrument; truth −1)−0.857

Supply slope by 2SLS (income as instrument; truth +1)1.23

Indirect least squares: demand, supply−0.857, 1.23

OLS slope averaged over 200 samples−0.011

2SLS averaged: demand, supply−1.01, 1.025

Horizontal: price. Vertical: quantity. The dashed lines are the true demand and supply curves when income and rain are at their averages; the solid line is the OLS fit.

Observed market-periods and the OLS lineTrue demand (slope −1)True supply (slope +1)

What it shows

Both curves are being shocked by similar amounts, so the points are a cloud and OLS finds a slope of −0.01 on average, a blend of the two true slopes, belonging to neither. That is simultaneity bias: price is determined together with quantity, so it is correlated with both equations’ errors. The two-stage estimates use the shifter that appears in only one equation. Rain moves supply and not demand, so it traces out the demand curve (estimate −0.86); income moves demand and not supply, so it traces out the supply curve (estimate 1.23). With one excluded shifter per equation the equations are just identified, and indirect least squares (−0.86, 1.23) gives exactly the same numbers as 2SLS.

Watch out for

The mistakes examiners are fishing for

Estimating a demand curve by regressing quantity on price. Without a variable that shifts supply and not demand, you cannot say which curve you have traced.
Saying OLS bias is the same as in lesson 7. The mechanism differs: here the bias comes from the equation’s own structure, since price responds to quantity’s shock, and no omitted variable has to exist.

Interactive

Identification: order and rank

An equation is identified if its structural coefficients can be recovered from the reduced form. Any equation that looks like any combination of the others cannot be told apart from it. Identification comes from exclusion restrictions: variables that appear in other equations but are left out of this one. There are three states:

Under-identified: the reduced form holds too little information. The equation cannot be estimated by any method.
Exactly (just) identified: a unique solution. Indirect least squares or 2SLS.
Over-identified: more than one way to solve it from the reduced form. 2SLS combines them.

The order condition (necessary)

K - k \geq m - 1

m = endogenous variables in the equation, including the left-hand one. K = all exogenous variables in the system, k = those in the equation. Equality: exactly identified; more: over-identified; fewer: under-identified.

The order condition is a count: to estimate the coefficients on m − 1 endogenous right-hand variables you need at least m − 1 exogenous variables excluded from the equation to act as instruments. It is necessary, not sufficient. The rank condition is necessary and sufficient: build the matrix of coefficients, in the other equations, of the variables excluded from this one; the equation is identified if and only if that matrix has rank (number of equations − 1). It guards against an excluded variable that appears only with zero coefficients elsewhere. The counter below checks both on three-equation systems: tick the variables in each equation and read the verdict.

Equation 1 has Y1, Y2, Y3, X1, X2; equation 2 has Y2, Y3, Y1, X2; equation 3 has Y3, X1, X2, X3. Y variables are endogenous, X variables exogenous. The left-hand variable of each equation (Y1 in equation 1, and so on) is always included.

Reading it

The order condition needs at least (included endogenous variables − 1) exogenous variables left out of the equation: a count, necessary but not sufficient. The rank condition is the sufficient one: the coefficients, in the other equations, of the variables left out of this one must form a matrix of rank (number of equations − 1). Here at least one equation fails the rank condition, so it cannot be estimated consistently by any method. An exactly identified one can be estimated by indirect least squares or 2SLS, which give the same answer.

Try the second preset (the exam-style system B). Equation 2 passes both conditions with a surplus. Equation 3 passes the order condition, with two exclusions for the two it needs, and fails the rank condition: the two variables it excludes, x₁ and x₂, appear with non-zero coefficients only in equation 1, and equation 2 contains neither, so the matrix of their coefficients in the other equations has a row of zeros and rank 1, not 2. Counting is not enough, which is why the rank condition exists.

Pause and predict

An equation in a system has 3 endogenous variables (counting the left-hand one) and the system has 4 exogenous variables, 1 of which is in the equation. By the order condition the equation is

Estimation

Estimating: ILS, 2SLS and recursive systems

Indirect least squares (ILS) works for exactly identified equations. The steps, in order: identify the structural equations; derive the reduced form; estimate each reduced-form equation by OLS; solve back for the structural coefficients. Because the reduced-form regressors are exogenous, the OLS step is legitimate.

Derive it

Indirect least squares: recovering structural slopes from the reduced form

We want to show

In a market with demand q = a₀ + a₁p + a₂y + u_d and supply q = b₀ + b₁p + b₂w + u_s, the structural slopes are ratios of reduced-form coefficients: a₁ = (effect of w on q) ÷ (effect of w on p), and b₁ = (effect of y on q) ÷ (effect of y on p).

Set demand equal to supply and solve for price.
$p = \frac{b _{0} - a _{0}}{a _{1} - b _{1}} + \frac{- a _{2}}{a _{1} - b _{1}} y + \frac{b _{2}}{a _{1} - b _{1}} w + \frac{u _{s} - u _{d}}{a _{1} - b _{1}}$
This is the reduced form for p: price as a function of the exogenous variables y and w and the shocks only. Every right-hand variable is exogenous, so OLS estimates it without bias.
Substitute back into one equation for q.
Put p into the supply equation q = b₀ + b₁p + b₂w + u_s. The reduced form for q has coefficients on y and w that combine the structural ones. Write the reduced forms as p = π₁y + π₂w + … and q = π₃y + π₄w + …
Because y is in demand only and w in supply only, the ratio of the effects of y on q and on p is the supply slope, and the ratio of the effects of w gives the demand slope:
$b_{1} = \frac{π _{3}}{π _{1}} = \frac{\partial q / \partial y}{\partial p / \partial y}, a_{1} = \frac{π _{4}}{π _{2}} = \frac{\partial q / \partial w}{\partial p / \partial w}$
A shift of y moves only the demand curve, so the change it causes in q and in p traces out the supply curve, and the ratio of the two changes is its slope.
Check on 400 simulated market-periods.
The true model is q = 10 − p + y + u_d and q = 2 + p + w + u_s, so a₁ = −1 and b₁ = +1. The estimated reduced forms are p = … + 0.471 y − 0.516 w and q = … + 0.512 y + 0.493 w. So the supply slope is 0.512 ÷ 0.471 = 1.088 and the demand slope is 0.493 ÷ −0.516 = −0.956. Two-stage least squares gives 1.088 and −0.956: identical.

Step 1 of 3

Two-stage least squares works for exactly and over-identified equations (lesson 11). First regress each endogenous right-hand variable on all the exogenous variables in the system; then regress y on the fitted values and the equation’s own exogenous variables. It is applied to the structural equation, not the reduced form (which OLS already handles). When the equation is exactly identified 2SLS equals ILS.
Recursive systems. If the system is triangular (the first equation has only exogenous variables on the right, the second adds the first endogenous variable, and so on) and the errors are uncorrelated across equations, there is no simultaneity in the sense that matters: each equation’s regressors are uncorrelated with its error, and OLS is consistent equation by equation. This is the one case in which OLS is appropriate for a system. Neither the order nor the rank condition is what licenses it.
Under-identified equations cannot be estimated consistently by any method. The remedy is theory: find another exclusion restriction, or an additional shifter.

What to say in an exam

In a simultaneous system OLS on a structural equation is inconsistent, because the endogenous right-hand variables are correlated with the error. Check the order condition (K − k ≥ m − 1) and the rank condition. Estimate exactly identified equations by indirect least squares or 2SLS, over-identified ones by 2SLS, and recursive systems by OLS. Under-identified equations cannot be estimated.

Solving the opening question

The opening question, solved

UGC NET December 2025 · hard

y₁ = 3y₂ − 2x₁ + x₂ + u₁; y₂ = y₃ + x₃ + u₂; y₃ = y₁ − y₂ − 2x₃ + u₃. By the order condition, is the second equation under-, exactly or over-identified?

Count what is in the system.
Endogenous variables: y₁, y₂, y₃ (three equations). Exogenous: x₁, x₂, x₃, so K = 3.
Count what is in equation 2.
It contains y₂ and y₃ (so m = 2) and x₃ (so k = 1). It leaves out y₁, x₁ and x₂.
Apply K − k ≥ m − 1.
K − k = 3 − 1 = 2 excluded exogenous variables (x₁, x₂). m − 1 = 1. Since 2 > 1, the order condition is satisfied with a surplus.

Answer

(A) Over-identified. The counter above confirms that the rank condition is also met.

Revise

The lesson on one screen

Forms: Structural: the behavioural equations. Reduced form: each endogenous variable on the exogenous ones only. OLS on the reduced form is fine.
Simultaneity bias: An endogenous right-hand variable is correlated with the error, so OLS on a structural equation is biased and inconsistent. Without shifters the OLS slope is a blend of the demand and supply slopes.
Order condition: K − k ≥ m − 1: excluded exogenous variables at least m − 1, where m counts the endogenous variables in the equation. Necessary, not sufficient.
Rank condition: Necessary and sufficient. The matrix of coefficients, in the other equations, of the variables excluded from this one, has rank G − 1.
Status: Under-identified: cannot be estimated. Exactly identified: unique structural coefficients. Over-identified: more than one solution.
ILS: Identify, derive the reduced form, estimate it by OLS, solve for the structural coefficients. Exactly identified equations only.
2SLS: Applied to the structural equation. Exactly or over-identified. Equals ILS when exactly identified.
Recursive model: Triangular with uncorrelated errors: OLS is consistent, equation by equation.

Practice

Check yourself

0 of 13 correct so far

Compute

How many exclusions are needed

An equation contains three endogenous variables (counting the left-hand one). At least how many exogenous variables of the system must be excluded from it for the order condition to hold?

Excluded exogenous variables

The degree of over-identification

A system has 4 exogenous variables. An equation contains 1 of them and 2 endogenous variables (counting the left-hand one). By how many does the number of excluded exogenous variables exceed the number needed?

Surplus

Indirect least squares

Demand includes income y and supply includes weather w. In the reduced forms, a better season (w up) raises q by 0.6 and lowers p by 0.3. Find the demand slope by ILS.

Demand slope

A reduced-form coefficient from the structure

Demand: q = 10 − 2p + 0.5y + u_d. Supply: q = 2 + 2p + u_s. Find the coefficient on y in the reduced form for p.

Coefficient

The OLS slope in a market

Demand slope −2, supply slope +1, Var(u_d) = 4 and Var(u_s) = 1, with no shifters. To what does the OLS slope of q on p converge?

OLS limit

Counting reduced-form coefficients

A system has 2 endogenous variables and 2 exogenous variables. How many reduced-form coefficients on the exogenous variables are there (ignoring the constants)?

Coefficients

Is it just identified?

An equation has 2 endogenous variables (counting the left-hand one) and excludes exactly 1 exogenous variable. By how many does the number of exclusions exceed the number needed?

Surplus

A supply slope by ILS

The effect of income y on q is 0.9 and on p is 0.6 in the reduced forms, with y excluded from supply. Find the supply slope.

Supply slope

Quick questions

1Two-stage least squares is applied to estimate the coefficients of

2In a simultaneous equations model OLS can be applied equation by equation if the model is

3The order condition for identification is

4Which is the correct sequence for indirect least squares?

5An under-identified equation can be estimated consistently by

Real questions

Past papers, by level

Standard

Restricted against unrestricted regressions, reading output, and statement questions, up to GATE and IIT JAM level.

5 questionsPractise →

All 5 past-paper questions for this lesson, easiest first: practise the full set →

These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.

Questions on the Hausman test and the instrument idea are in lesson 11. The match-the-test questions that mention simultaneity are in the test atlas at the end of the module.

← Lesson 11Endogeneity and instrumental variables Lesson 13 →Panel data: fixed and random effects

Simultaneous equations and identification

Equations determined together

Why OLS fails: simultaneity bias

The mistakes examiners are fishing for

Identification: order and rank

Estimating: ILS, 2SLS and recursive systems

Indirect least squares: recovering structural slopes from the reduced form

The opening question, solved

The lesson on one screen

Check yourself

Compute

Quick questions

Past papers, by level

Standard

Read next

Simultaneous equations and identification

Equations determined together

Why OLS fails: simultaneity bias

The mistakes examiners are fishing for

Identification: order and rank

Estimating: ILS, 2SLS and recursive systems

Indirect least squares: recovering structural slopes from the reduced form

The opening question, solved

The lesson on one screen

Check yourself

Compute

Quick questions

Past papers, by level

Standard

Read next

Equation	Excluded X	Needed	Order condition	Rank condition
1, for Y1	1	2	Under-identified	Fails
2, for Y2	2	2	Exactly identified	Satisfied
3, for Y3	0	0	Exactly identified	Satisfied