Simultaneous Equations Models: ILS and 2SLS Explained

What Are Simultaneous Equations Models?

A simultaneous equations model describes several relationships that operate together. An endogenous variable depends on the system itself, while an exogenous variable enters from outside the model. However, a label alone cannot establish the statistical assumptions needed for estimation.

Consider a market for tomatoes. A demand equation relates purchases to price and household income. Meanwhile, a supply equation relates production to price and growing conditions. The market price adjusts until buyers and sellers agree on a quantity.

Consequently, the observed price reflects both sides of the market. A demand shock can increase price and quantity together, even when each household buys less at a higher price, holding everything else constant.

These distinctions follow the standard simultaneous-equations framework in Shalabh’s IIT Kanpur lecture notes. Nevertheless, empirical work must justify exogeneity in its specific setting.

Why a scatterplot cannot automatically identify demand

A collection of equilibrium points can reflect shifts in demand, supply, or both. Therefore, fitting a line through those points does not automatically trace either structural curve. The central question concerns what caused the observed movement.

When only demand shifts and supply stays fixed, equilibrium observations trace the supply curve. Conversely, supply shifts can trace a stable demand curve. An instrument formalizes this idea by isolating a source of movement that satisfies the target equation’s assumptions.

Simultaneity is one source of endogeneity

Endogeneity means that a regressor correlates with the error in the equation of interest. Simultaneous determination can create this problem, but omitted variables and measurement error can also create it. Accordingly, adding a second equation does not necessarily resolve every source of bias.

Likewise, a model can contain several equations without contemporaneous feedback. Recursive systems have a one-way ordering. Under suitable error orthogonality assumptions, equation-by-equation OLS can work for those systems.

Structural Equations and Reduced Forms

Structural equations represent the behavioral relationships you want to understand. Reduced forms instead express endogenous outcomes using the model’s exogenous variables and composite errors. Thus, the two forms answer different questions.

A demand-and-supply system

Assume a₁ < 0, b₁ > 0, and a₂ > 0. In addition, suppose income is orthogonal to both structural errors and has meaningful variation. The exclusion of income from supply provides a potential source of identification for the supply equation.

Equate the right-hand sides and collect the price terms. Provided b₁ − a₁ ≠ 0, the system has a unique algebraic solution for price:

Substitution into supply then gives the quantity reduced form:

For shorter notation, write Q = π_Q0 + π_QYY + v_Q and P = π_P0 + π_PYY + v_P. Notice that the reduced-form errors combine the original shocks. Because income remains orthogonal to those combinations under our assumptions, OLS can consistently estimate the reduced-form coefficients.

Why reduced-form and structural effects differ

The structural coefficient a₂ measures the income effect on demand at a fixed price. By contrast, π_QY measures the equilibrium quantity response after price adjusts. Confusing these effects can lead to a mistaken policy interpretation.

For example, an income increase can raise consumers’ desired purchases while also raising price. The price increase offsets part of the initial demand expansion. Consequently, the final quantity change can be smaller than the fixed-price demand shift.

A compact matrix version

Here, y contains endogenous outcomes and x contains exogenous variables. An invertible B gives a unique reduced form. However, invertibility alone does not identify every structural coefficient: different structural systems can still generate the same reduced-form relationships.

For a complementary teaching treatment of feedback and reduced forms, see Systems of Equations in the SUNY-hosted econometrics text.

Identification: Can the Data Recover the Equation?

Identification concerns whether the model’s restrictions and population relationships uniquely determine the parameters of interest. Estimation comes afterward. Therefore, a large sample cannot rescue an equation that the model leaves unidentified.

The order condition

Let m denote the number of endogenous regressors on the right-hand side of the target equation. Meanwhile, L denotes the number of excluded instruments, after accounting for the included exogenous controls. Under the usual linear exclusion setup, the count provides a first screening rule:

Counting instruments does not establish their usefulness. For instance, two instruments that move two endogenous variables in exactly the same proportion provide only one independent direction of variation.

The rank condition

After removing included exogenous controls, let Z̃ contain the excluded instruments and D̃ contain the endogenous regressors. The relevant population cross-moment matrix must have full column rank:

This expression summarizes the required independent instrument variation in a linear IV equation. In a full simultaneous system, the structural rank condition expresses the same identification problem through restrictions on the other equations. The IIT Kanpur notes develop that system-based treatment.

Relevance and validity are different requirements

A relevant instrument predicts the endogenous regressor after accounting for the controls. In addition, a valid instrument satisfies the target equation’s exclusion and orthogonality assumptions. Accordingly, an excellent predictor can still be an invalid instrument.

Suppose weather affects crop supply. It may help identify demand if it changes market price without independently changing buyers’ demand. However, weather could also change demand for certain foods, shopping behavior, or local income. A useful empirical argument must address those alternative paths.

Similarly, income may help identify supply only if the supply equation can credibly exclude its direct effect. Producers’ financing conditions, local wage costs, or changing product quality could undermine that restriction. Economic reasoning therefore matters as much as algebra.

Lagged variables do not become valid instruments automatically

A lag occurs before the current observation, but timing alone does not prove orthogonality. Serially correlated errors can connect yesterday’s endogenous outcome to today’s disturbance. Consequently, lagged instruments require assumptions about the dynamic process.

A trend also deserves scrutiny. It may capture omitted technological changes, preferences, inflation, or other gradual movements. Excluding a trend from one equation while including it in another imposes an economic restriction, rather than a harmless formatting choice.

How ILS and 2SLS Estimate Simultaneous Equations Models

Indirect least squares

Indirect least squares, or ILS, recovers structural coefficients by transforming estimated reduced-form coefficients. It works naturally when the target equation is exactly identified. For the simple supply equation above, the ratio of income effects recovers its price slope:

Intuitively, income shifts demand and moves the equilibrium along supply. Dividing the quantity response by the price response measures that supply slope. However, a nearly zero denominator makes the ratio unstable.

ILS does not make standard errors impossible. The delta method or a suitable bootstrap can estimate uncertainty for the coefficient transformation. Nevertheless, these approaches require care, particularly when the denominator is weakly estimated.

Two-stage least squares

Two-stage least squares, or 2SLS, uses instrument-driven variation in endogenous regressors. First, project each endogenous regressor onto the complete instrument set. Next, regress the dependent variable on the fitted endogenous regressors and the included exogenous controls.

In this demand example, Y is an included exogenous control and C is an excluded supply shifter. Both belong in the first stage, together with the constant. The exclusion applies to the structural demand equation, not to the first-stage prediction equation.

For actual inference, use an IV estimator. A routine OLS regression on generated fitted values generally reports the wrong standard errors. Stata’s official 2SLS FAQ explains this distinction and the role of the included exogenous variables.

The matrix formula

Here, X contains all structural regressors, including endogenous regressors and exogenous controls. Meanwhile, Z contains the included exogenous regressors and excluded instruments. The inverses require the relevant sample matrices to have full rank.

With exact identification, ILS and IV/2SLS give the same coefficient estimates when they use the same data, restrictions, and specification. In an overidentified equation, 2SLS combines the information from multiple instruments. Separate instrument ratios will generally differ in a finite sample.

OLS, ILS, and 2SLS compared

The official ivregress manual documents 2SLS, LIML, GMM, and supported variance estimators. Estimator choice should follow the design and assumptions, rather than whichever specification gives the preferred sign.

Weak instruments and misleading confidence

An instrument can satisfy the population relevance condition yet provide little information in a finite sample. As a result, conventional IV confidence intervals and tests can behave poorly. A large overall first-stage R² does not solve this problem if included controls explain most of the fit.

Focus on the excluded instruments’ conditional contribution. Report appropriate first-stage diagnostics and, when necessary, weak-instrument-robust inference. The familiar F > 10 rule is a rough screening convention, not a universal guarantee.

Stock and Yogo’s weak-instrument research links critical values to specific bias and test-size criteria. Furthermore, Federal Reserve Bank of New York research addresses robust testing with multiple endogenous regressors. Diagnostic choice must match the covariance assumptions and the number of endogenous variables.

20 Solved Exercises on Simultaneous Equations Models

The following exercises use original constructed numbers unless an exercise explicitly references the supplied historical material. These are teaching calculations, not estimates of a current market. Each answer explains the economic meaning as well as the arithmetic.

Exercise 1: Classify the variables

Question: Which variables are endogenous, and which variable supplies an exclusion?

Solution: The system jointly determines Q and P, so both are endogenous. Income Y is the assumed exogenous variable, and the supply equation excludes it. Therefore, income can potentially identify supply, while this system supplies no excluded exogenous variable for demand.

Exercise 2: Solve the equilibrium at a given income

Question: Set both shocks to zero and let Y = 20. Find equilibrium price and quantity.

Interpretation: At price 22, both sides imply quantity 86. Substituting into demand gives 120 − 44 + 10 = 86, which checks the solution independently.

Exercise 3: Derive both reduced forms

Question: Solve the same model for arbitrary income and shocks.

Solution: The slope difference is 3 − (−2) = 5. Accordingly, the income coefficient in the price equation equals 0.5/5 = 0.1. Substitution into supply produces the remaining coefficients.

Exercise 4: Recover supply through ILS

Question: Use the reduced-form income coefficients and intercepts from Exercise 3 to recover supply.

Answer: Q = 20 + 3P + u_s. Thus, the reduced forms recover the original supply equation exactly in this population example.

Exercise 5: Compare structural and equilibrium income effects

Question: How does a 10-unit income increase affect quantity?

Solution: At a fixed price, demand rises by 0.5 × 10 = 5 units. However, equilibrium price rises by 0.1 × 10 = 1, which reduces desired purchases by 2 units. The final quantity increase equals 5 − 2 = 3 units, matching 0.3 × 10.

Exercise 6: Show why price correlates with the errors

Question: Suppose Y, u_d, and u_s are mutually uncorrelated. What are the price-error covariances?

Solution: Read the shock coefficients from the price reduced form. Unless the corresponding shock has zero variance, price correlates with that equation’s error. Consequently, standard OLS orthogonality fails for both structural equations.

Exercise 7: Calculate the direction of supply OLS bias

Question: Let Var(Y) = 100, Var(u_d) = 25, and Var(u_s) = 25, with zero pairwise covariances. What slope does supply OLS approach?

Interpretation: Even an arbitrarily large sample converges to about 1.33 instead of 3 under these assumptions. Therefore, more observations reduce sampling noise without removing simultaneity bias.

Exercise 8: Apply the order condition

Question: Classify each equation in Exercise 1 using excluded instruments.

Solution: Supply contains one endogenous regressor, P, and excludes one exogenous variable, Y. Its order count permits exact identification, and the nonzero first-stage coefficient establishes rank here. By contrast, demand contains P but excludes no exogenous variable, so it remains underidentified under these restrictions.

Exercise 9: Add a supply shifter to identify demand

Question: If cost C is exogenous and demand excludes it, recover the demand price slope.

Interpretation: Higher cost raises equilibrium price and reduces quantity. Because cost shifts supply alone, the ratio traces demand rather than supply.

Exercise 10: Recover the remaining demand coefficients

Question: Use Exercise 9 to recover the income coefficient and intercept.

Answer: The recovered equation matches the original demand specification. In addition, the calculation separates the direct income effect from the price adjustment.

Exercise 11: Build a 2SLS first stage

Question: Estimate demand from Exercise 9 using the instrument set {1, Y, C}. Which variables belong in the first stage for P?

Solution: Regress P on a constant, Y, and C. Then use fitted P together with Y and a constant for the second-stage coefficient calculation. Although C serves as the excluded instrument, omitting Y from the first-stage projection generally changes the estimator.

For example, first-stage coefficients 20, 0.1, and 0.8 predict P̂ = 26 when Y = 20 and C = 5. Consequently, the demand prediction equals 120 − 2(26) + 0.5(20) = 78.

Exercise 12: Obtain an IV slope from moments

Question: After removing included controls, suppose Cov(Z,Q) = −12 and Cov(Z,P) = 6. Find the just-identified IV slope.

Interpretation: The ratio uses instrument-associated variation. However, the arithmetic alone cannot establish that Z satisfies the exclusion restriction.

Exercise 13: Pass the order condition but fail rank

Question: Two endogenous regressors have two excluded instruments. Their conditional first-stage coefficient matrix equals the following:

Solution: The count gives L = m = 2, but the matrix has rank 1. Both instruments move the endogenous variables in the same proportion. Therefore, the equation remains underidentified despite passing the order count.

Exercise 14: Calculate a conventional first-stage F statistic

Question: A homoskedastic first stage uses n = 100 observations, two excluded instruments, and four coefficients including the intercept. Its restricted and unrestricted residual sums of squares equal 150 and 120.

Interpretation: The excluded instruments jointly improve the first-stage fit. Nevertheless, F = 12 does not certify valid instruments or guarantee reliable conventional inference. Use the appropriate weak-instrument assessment for the model and error structure.

Exercise 15: See why overall R² can mislead

Question: Included controls alone produce R² = 0.900, while adding an instrument raises it to 0.901. What is the instrument’s partial R²?

Solution: The instrument explains only 1% of the variation left after the controls. Thus, an overall first-stage R² above 90% can coexist with limited incremental instrument information. Sample size still matters for statistical strength, so partial R² is not a complete diagnosis by itself.

Exercise 16: Count overidentifying restrictions

Question: An equation contains one endogenous regressor and three excluded instruments. How many overidentifying restrictions does the conventional test have?

Solution: Under full rank and the relevant regularity assumptions, the test has two degrees of freedom. However, a failure to reject does not prove that every instrument is valid. Instruments can share similar violations, and tests can have low power.

Exercise 17: Convert a slope into an elasticity

Question: At P = 22 and Q = 86, convert the demand slope −2 and supply slope 3 into point elasticities.

Interpretation: A one-percent price change corresponds locally to about a 0.512% demand decrease or a 0.767% supply increase, holding the respective shifters constant. Unlike a log-log coefficient, a level-level slope is not itself an elasticity.

Exercise 18: Diagnose a recursive system

Question: Does the second equation automatically require IV?

Solution: No. If X is orthogonal to u₂ and Cov(u₁,u₂) = 0, then y₁ can be orthogonal to u₂. Nevertheless, correlated disturbances would make y₁ endogenous in the second equation even without feedback.

Exercise 19: Recover a macroeconomic coefficient

Question: Suppose the reduced-form effect of investment I on M equals 0.6, while its effect on Y equals 3. Find a₁.

Interpretation: Investment supplies excluded variation for the M equation under the assumed restrictions. By contrast, the Y equation has no excluded exogenous variable in this minimal system, so its structural coefficients cannot all be recovered through the same argument.

Exercise 20: Quantify instrument contamination

Question: After removing controls, suppose the true structural slope equals −2, Cov(Z,P) = 4, and Cov(Z,u) = 0.8. What does the IV ratio approach?

Interpretation: IV remains inconsistent because the instrument violates orthogonality. Moreover, a smaller first-stage covariance would amplify the same contamination. Instrument strength and validity therefore interact, although they remain distinct requirements.

A Historical Crop Example: What the Printed Numbers Reveal

The supplied chapter includes a U.S. crop example covering 1975–1996. Its table describes a crop-output index, a crop-price index, and disposable income per capita in 1996 dollars. The printed source note names the Economic Report of the President, 2000.

These historical numbers illustrate estimation mechanics. They do not describe today’s agricultural market, and this guide does not claim an independent reconstruction of the original official dataset. The discussion below audits the rounded equations visible in the supplied pages.

Recovering the printed supply slope

By comparison, the printed direct OLS supply slope equals 0.5145. Consequently, the ILS slope from the displayed rounded coefficients is about 2.92 times as large. This discrepancy illustrates estimator sensitivity to the identifying assumptions; it does not prove that the IV estimate is correct.

Why four-decimal rounding matters

Small displayed coefficients can hide economically meaningful ratio uncertainty. If ordinary rounding produced 0.0039 and 0.0026, the underlying numbers could lie approximately within the intervals below:

This is a rounding range, not a confidence interval. Nevertheless, it shows why a ratio of printed coefficients need not match a software estimate that retains full precision. Keep the unrounded coefficients until the last reporting step.

A trend-length inconsistency

The photographed demand exercise describes trend values through 30, although the crop table spans 22 years. For a trend indexed to that sample, the natural sequence is T = 1,…,22. This mismatch should trigger a data check before replication.

Changing a trend’s starting point shifts an intercept, while changing its scale changes the associated coefficient. However, neither convention creates eight additional annual observations. Always align the trend with the actual rows used in estimation.

A structural-variable labeling check

The demand model includes income Y, while the supply equation includes trend T. Therefore, recovering demand from the reduced forms should leave Y in the structural demand equation. The printed final demand expression appears to attach its small coefficient to T instead.

Use algebra to resolve the ambiguity. With Q = a₀ + a₁P + a₂Y + u, the excluded trend coefficients identify a₁. Then the difference between the income coefficients identifies a₂.

Using the displayed coefficients gives a price slope near 0.5619 and an income coefficient near 0.0025. Accordingly, that second coefficient belongs to Y under the stated specification. This check prevents a notation error from changing the economic model.

Why the positive demand slope deserves investigation

The chapter’s theoretical demand restriction requires a negative price coefficient, yet its illustrative IV estimate is positive. Calling an estimator consistent under assumptions does not make every sample realization economically convincing. Instead, the sign invites scrutiny of sampling uncertainty, instrument strength, exclusions, trends, and model specification.

The printed first-stage trend t statistic is about 1.95. With one excluded instrument and conventional homoskedastic inference, the corresponding exclusion F statistic is approximately 1.95² = 3.80. Thus, the example also raises a weak-instrument concern.

This calculation does not establish that the instrument is invalid. Rather, it shows that the displayed model supplies limited statistical information for the demand slope. A modern analysis should report uncertainty and consider inference that remains reliable with weak instruments.

A Practical Workflow for Applied Research

Start with the economic mechanism

Write the behavioral equations before searching for instruments. Specify which variables the system determines and which restrictions identify each target equation. Next, explain how the proposed instrument moves the endogenous regressor.

For every excluded instrument, describe a plausible violation. A shipping-cost instrument might also affect product availability or quality. Similarly, a local weather instrument might influence consumer demand directly. Addressing those mechanisms produces a stronger design than simply reporting a significant first stage.

Audit units, samples, and time periods

Check whether each variable uses dollars, thousands of dollars, index points, or logarithms. Confirm that the first stage and structural equation use the same observations. Otherwise, missing values can create an unintended change of sample.

For time-series examples, inspect trends, breaks, serial correlation, and stationarity. Annual macroeconomic series often move together over time. Consequently, a high R² can reflect shared movement without establishing a credible structural relationship.

Use an IV command for final estimation

The following Stata template estimates demand using a cost shifter. Replace the variable names with the corresponding columns in your dataset. Here, quantity is Q, price is endogenous P, income is exogenous Y, and cost is excluded C.

* Demand: quantity depends on price and income
* Cost instruments price; income also enters the instrument set
ivregress 2sls quantity income (price = cost), vce(robust) first

* Inspect first-stage diagnostic output
estat firststage

* Overidentified example with a second credible instrument
ivregress 2sls quantity income (price = cost weather), vce(robust)
estat overid

Only include weather if the demand exclusion is defensible. Adding instruments for the sake of obtaining an overidentification test can weaken the research design. Moreover, robust standard errors address heteroskedasticity, not invalid instruments.

For dependent observations, choose a variance estimator that reflects the sampling process. Clustered market data may require clustering, while time-series data may require autocorrelation-aware inference. The ivregress manual explains the available options.

Understand what overidentification tests can establish

Overidentification tests examine whether the extra moment restrictions conflict with the fitted model. A rejection signals a problem somewhere in the joint assumptions. It does not uniquely identify which instrument caused the problem.

Conversely, a failure to reject is not a certificate of exogeneity. Limited power and common violations can conceal misspecification. Therefore, combine the statistical evidence with institutional knowledge and sensitivity analysis.

A real research example beyond supply and demand

Instrumental variables also appear in research on schooling and earnings. Angrist and Krueger’s study of compulsory school attendance uses quarter of birth to exploit the interaction of school-entry rules and compulsory attendance requirements. The working paper appeared in 1990, followed by the journal article in 1991.

This application illustrates the need for an institutional mechanism, rather than an arbitrary predictor. However, an instrument that supports a particular education design does not automatically work for another population, time period, or outcome. The identifying argument must travel with the estimate.

Common Mistakes and Better Interpretations

Why an IV R² can be negative

OLS minimizes the sum of squared structural residuals over the regressors in its equation. IV instead uses instrument moment conditions, so it need not minimize that same residual sum. Consequently, the familiar centered R² can fall below zero.

A negative IV R² is not, by itself, proof that the computation failed. Likewise, a large IV R² cannot establish correct identification. Stata’s explanation of negative and missing IV R² gives the underlying algebra.

Why standard errors need structural residuals

After computing the IV coefficients, form residuals using the actual structural regressors: û = y − Xβ̂. Do not replace X with the fitted first-stage matrix when constructing those structural residuals. The difference matters for variance estimation.

Meanwhile, the fitted-value regression produces the same 2SLS coefficient vector under the standard setup. However, its default OLS variance calculation answers a different statistical question. This distinction explains why manually reproducing the point estimates does not complete an IV analysis.

Frequently Asked Questions

What is the difference between an endogenous variable and an endogenous regressor?

An endogenous variable is an outcome determined within the system. By contrast, an endogenous regressor appears on the right-hand side of a particular equation and correlates with that equation’s disturbance. Thus, classification within the system and the equation-specific estimation problem are closely related but deserve separate attention.

Can OLS estimate a reduced form?

Yes, provided the included exogenous variables satisfy the relevant orthogonality and regularity assumptions. Reduced-form errors combine structural shocks, so those assumptions must hold for the combination as well. Correcting the reduced form for heteroskedasticity or dependence concerns inference rather than identification.

Does 2SLS estimate the whole system at once?

Standard 2SLS estimates one structural equation using an instrument set. You can apply it separately to several identified equations. System estimators may exploit additional cross-equation information, but they also rely on additional assumptions.

Is 2SLS always better than OLS?

No. If OLS is consistent and efficient under the relevant assumptions, IV can sacrifice precision unnecessarily. Conversely, valid IV can address endogeneity that makes OLS inconsistent. Weak or invalid instruments can make an IV analysis unreliable.

Can an exactly identified equation use an overidentification test?

No conventional overidentification test is available when there are no excess instruments. In that case, the number of overidentifying restrictions is zero. Nevertheless, you can still examine first-stage information, the economic exclusion argument, and sensitivity to the specification.

Does a statistically insignificant instrument prove no identification?

No. Population identification and sample precision are different concepts. However, a weak sample relationship can warn that conventional IV inference is unreliable, even when the population coefficient is nonzero.

How should weak-instrument uncertainty be reported?

Consider confidence sets and tests designed to remain valid under weak identification, alongside the point estimate and first-stage diagnostics. Anderson–Rubin and conditional likelihood-ratio approaches are important examples under their respective assumptions. See Stata’s weak-instrument inference documentation for an implementation overview.

Are the historical coefficients in this guide current estimates?

No. The crop values come from the historical classroom example in the supplied pages. Meanwhile, the other numerical exercises use explicitly constructed models. Neither set should serve as a current agricultural forecast or policy estimate.

Conclusion: Identify the Variation Before Estimating the Effect

Simultaneous equations models turn an important economic insight into an estimation problem: outcomes can influence one another. Consequently, the observed association between price and quantity may combine several behavioral relationships.

Start by distinguishing structural effects from reduced-form responses. Next, justify the exclusions, check the order and rank conditions, and assess the strength of the instruments. Finally, estimate the equation with appropriate IV inference and interpret the result in its economic context.

The most useful habit is to ask what source of variation identifies the coefficient. A correct formula, a large R², or a significant estimate cannot replace that answer.

Sources and Methodology

This guide provides original explanations, derivations, and constructed exercises based on the topics in the supplied chapter pages 228–241. Historical coefficients appear only in the labeled audit section. The following primary or academic teaching sources support the methodological references throughout the article.

1. Shalabh, IIT Kanpur. Simultaneous Equations Models. Academic lecture notes on structural systems and identification.
2. SUNY-hosted Econometrics. Chapter 15: Systems of Equations. Teaching discussion of feedback and reduced forms.
3. Stata. ivregress reference manual. Official estimator and syntax documentation.
4. Stata. Two-stage least-squares regression FAQ. First-stage specification and standard errors.
5. Stock, J. H., and Yogo, M. Testing for Weak Instruments in Linear IV Regression. NBER Technical Working Paper 284, 2002; subsequent chapter publication in 2005.
6. Federal Reserve Bank of New York. A Robust Test for Weak Instruments with Multiple Endogenous Regressors. Staff Report 1020.
7. Stata. Inference robust to weak instruments. Official implementation overview.
8. Stata. Negative and missing R-squared for 2SLS/IV. Interpretation of IV goodness of fit.
9. Angrist, J. D., and Krueger, A. B. Does Compulsory School Attendance Affect Schooling and Earnings? NBER Working Paper 3572, 1990.
10. User-supplied photographs. Chapter 10, Simultaneous-Equations Methods, pages 228–241. Source for the historical classroom audit; bibliographic edition details were not supplied.

Calculation note: the 20 exercises use algebra and explicitly stated assumptions. Historical slope ratios, percentage comparisons, rounding bounds, and the F ≈ t² check are calculations from the displayed values, not newly retrieved official statistics. The document uses native HTML formulas and embedded vector charts, with no external JavaScript or MathJax dependency.