The identification problem in a supply-demand system arises because:

Observed price-quantity data are equilibrium points from BOTH curves shifting, so a simple regression traces a mixture of the two rather than either curve alone

Supply is always perfectly elastic

Prices are fixed by the government

The order condition for identification requires that the number of exogenous variables excluded from an equation be:

At least as large as the number of included endogenous variables minus one

Equal to the total number of equations in the system

Irrelevant to whether the equation is identified

In a recursive (triangular) simultaneous-equation system with uncorrelated disturbances, consistent estimation can proceed by:

Applying OLS to each equation one at a time

Estimating only the reduced form and never the structural form

The linear probability model for a binary dependent variable has which structural flaw that Probit and Logit correct?

Fitted probabilities are not bounded within [0,1] and errors are heteroskedastic

It cannot be estimated by any method

It requires the outcome to be continuous

Regressing two unrelated but independently trending non-stationary time series on each other typically produces:

An unbiased and efficient estimate of a true causal relationship

A result that requires no further testing

Two individually non-stationary series are said to be cointegrated when:

A specific linear combination of them is itself stationary

Both series are constant over time

They have identical numerical values at every point

An MA(q) time-series model expresses the current value as a linear function of:

The current and q past error (shock) terms

A single deterministic trend with no randomness

Which of the following are consequences of estimating a non-recursive (feedback) simultaneous-equation system by ordinary equation-by-equation OLS?

Inconsistency, even as the sample size grows large

A need for instrumental-variable methods such as 2SLS

Inconsistency, even as the sample size grows large

A need for instrumental-variable methods such as 2SLS

An automatic guarantee of unbiasedness

The Identification Problem, Simultaneous Equation Models, Discrete Choice Models and Time Series Analysis

Unit 3 closes with four topics that arise once a single-equation regression is no longer enough: when observed data are jointly determined by several equations, when the outcome itself is a discrete choice rather than a continuous number, and when data are ordered in time.

1. The Identification Problem

In a market where both demand and supply are shifting over the sample period, observed price-quantity pairs are equilibrium points — the intersections of a moving demand curve with a moving supply curve — and a simple regression of quantity on price recovers neither curve individually; it traces some mixture of the two, which is the identification problem. A structural equation is IDENTIFIED only if enough information (usually, other exogenous variables that shift one curve but not the other) is available to isolate it. The ORDER CONDITION gives a necessary rule of thumb: an equation is identified only if the number of exogenous variables excluded from that equation (but present elsewhere in the system) is at least as large as the number of endogenous variables included in it minus one; the RANK CONDITION is the more demanding necessary-and-sufficient condition, checking that the excluded variables' coefficients in the OTHER equations are not all proportional to each other in a way that would leave the equation under-identified despite satisfying the order condition.

⚠️ Under-identified, exactly identified and over-identified

If the order condition is not met at all, the equation is under-identified and NO consistent estimation method can recover its parameters from the available data. If it is met with equality, the equation is exactly identified (a unique solution exists, estimable by indirect least squares). If excluded exogenous variables exceed the requirement, the equation is over-identified, and two-stage least squares (2SLS) is the standard estimator, since it can combine the extra information rather than being paralysed by it.

2. Simultaneous Equation Models: Recursive and Non-Recursive

In a RECURSIVE (triangular) system, the endogenous variables can be ordered so that each equation's endogenous variable depends only on PRECEDING endogenous variables (and exogenous ones), with no feedback loop back to it from later equations — if, in addition, the disturbance terms across equations are uncorrelated, each equation can be consistently estimated one at a time by ordinary OLS. In a NON-RECURSIVE system, two or more endogenous variables depend on each other simultaneously (feedback), and applying OLS equation-by-equation produces SIMULTANEITY BIAS: the regressor is correlated with the error term (because the error in one equation feeds back, through the system, into the very variable used as a regressor in another), so OLS is biased and inconsistent even in large samples.

Indirect Least Squares (ILS) — for an exactly identified equation, solve the reduced form (each endogenous variable expressed purely in terms of exogenous variables) by OLS, then algebraically recover the structural parameters.
Two-Stage Least Squares (2SLS) — for an over-identified equation, first regress the endogenous regressor on all exogenous variables to obtain its predicted (instrumented) values, then use those predicted values in place of the original regressor in the structural equation, purging the correlation with the error.

3. Discrete Choice Models

When the dependent variable is a binary outcome (e.g. labour-force participation, buy/not-buy), the linear probability model (a plain OLS regression of a 0/1 variable on regressors) has two structural flaws: its fitted 'probabilities' are not bounded within [0, 1] and can fall below 0 or exceed 1, and its errors are mechanically heteroskedastic (their variance depends on X). The Probit model instead assumes an underlying latent, continuous variable and maps it through the CUMULATIVE STANDARD NORMAL distribution function to a probability in (0, 1); the Logit model does the same using the cumulative LOGISTIC distribution, which is analytically more convenient and gives coefficients directly interpretable (after transformation) as changes in the log-odds of the outcome. Both are typically estimated by maximum likelihood rather than OLS, precisely because the model is inherently non-linear in its parameters.

ℹ️ Note

Extensions to more than two unordered choices (multinomial logit/probit — e.g. choosing among several modes of transport) and to ordered outcomes (ordered logit/probit — e.g. a survey response on a Likert scale) are standard generalisations of the same latent-variable idea, adjusted for the specific choice structure.

4. Time Series Analysis

A time series is (weakly/covariance) STATIONARY if its mean, variance and autocovariance structure do not depend on the time period itself, only on the lag between observations; most raw macroeconomic series (GDP, price levels) are NON-stationary, typically trending. Regressing one non-stationary series on an unrelated non-stationary series routinely produces SPURIOUS REGRESSION — a high R² and significant-looking t-statistics despite there being no genuine relationship, because both series are simply trending independently — the Granger-Newbold warning. The Augmented Dickey-Fuller (ADF) test is the standard test for a UNIT ROOT (non-stationarity); if two individually non-stationary series share a common underlying stochastic trend such that a specific linear combination of them IS stationary, the series are COINTEGRATED (Engle-Granger), which justifies estimating an error-correction model capturing both their long-run equilibrium relationship and short-run adjustment toward it.

AR, MA and ARIMA — the building blocks of time-series forecasting
Model	Structure
AR(p) — autoregressive	Current value depends linearly on its own p past values plus a current error term.
MA(q) — moving average	Current value depends linearly on the current and q past error (shock) terms.
ARIMA(p,d,q)	Combines AR and MA terms after differencing the series d times to induce stationarity — the standard Box-Jenkins forecasting framework.

Key takeaways

The identification problem arises because observed price-quantity data are equilibrium points of shifting demand and supply; the order condition (necessary) and rank condition (necessary and sufficient) determine whether a structural equation can be recovered at all.
Recursive systems (no feedback, uncorrelated disturbances) can be estimated equation-by-equation by OLS; non-recursive systems (feedback) suffer simultaneity bias under OLS and need indirect least squares (exactly identified) or 2SLS (over-identified).
The linear probability model's fitted values are not bounded in [0,1] and its errors are heteroskedastic; Probit (cumulative normal) and Logit (cumulative logistic) fix both, estimated by maximum likelihood.
Regressing two independently trending non-stationary series produces spurious regression (Granger-Newbold); the ADF test detects a unit root, and cointegration (Engle-Granger) justifies an error-correction model.
AR models regress a series on its own past values, MA models regress it on past error terms, and ARIMA combines both after differencing to induce stationarity — the Box-Jenkins forecasting framework.

Practice questions (9)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

The identification problem in a supply-demand system arises because:
1. Observed price-quantity data are equilibrium points from BOTH curves shifting, so a simple regression traces a mixture of the two rather than either curve alone
2. Demand curves never shift
3. Supply is always perfectly elastic
4. Prices are fixed by the government
Show answer
Answer: A — Observed price-quantity data are equilibrium points from BOTH curves shifting, so a simple regression traces a mixture of the two rather than either curve alone
When both curves move over the sample period, observed points are their successive intersections, and OLS on this data cannot isolate either structural relationship without additional information (excluded exogenous variables).
The order condition for identification requires that the number of exogenous variables excluded from an equation be:
1. At least as large as the number of included endogenous variables minus one
2. Exactly zero
3. Equal to the total number of equations in the system
4. Irrelevant to whether the equation is identified
Show answer
Answer: A — At least as large as the number of included endogenous variables minus one
The order condition is a necessary counting rule: enough excluded exogenous variables must be available elsewhere in the system to isolate the equation's own structural parameters.
In a recursive (triangular) simultaneous-equation system with uncorrelated disturbances, consistent estimation can proceed by:
1. Applying OLS to each equation one at a time
2. Two-stage least squares only
3. Discarding half the equations
4. Estimating only the reduced form and never the structural form
Show answer
Answer: A — Applying OLS to each equation one at a time
Because each equation in a recursive system depends only on preceding endogenous variables (no feedback) and disturbances are uncorrelated across equations, ordinary OLS applied equation-by-equation remains consistent — no simultaneity bias arises.
The linear probability model for a binary dependent variable has which structural flaw that Probit and Logit correct?
1. Fitted probabilities are not bounded within [0,1] and errors are heteroskedastic
2. It cannot be estimated by any method
3. It has no dependent variable
4. It requires the outcome to be continuous
Show answer
Answer: A — Fitted probabilities are not bounded within [0,1] and errors are heteroskedastic
Because OLS is unconstrained, its fitted values can fall outside [0,1] and, since the error variance is inherently tied to X for a binary outcome, the errors are mechanically heteroskedastic — Probit and Logit fix both by mapping a latent index through a bounded cumulative distribution function.
Regressing two unrelated but independently trending non-stationary time series on each other typically produces:
1. A spurious regression — high R² and significant-looking t-statistics despite no genuine relationship
2. An unbiased and efficient estimate of a true causal relationship
3. An R² of exactly zero
4. A result that requires no further testing
Show answer
Answer: A — A spurious regression — high R² and significant-looking t-statistics despite no genuine relationship
The Granger-Newbold warning is precisely this: two series simply trending independently can produce a misleadingly strong-looking regression, which is why stationarity/unit-root testing (ADF) and cointegration checks precede any causal interpretation of a time-series regression.
Two individually non-stationary series are said to be cointegrated when:
1. A specific linear combination of them is itself stationary
2. Both series are constant over time
3. Neither series has a mean
4. They have identical numerical values at every point
Show answer
Answer: A — A specific linear combination of them is itself stationary
Cointegration (Engle-Granger) means the two series share a common stochastic trend such that some linear combination cancels it out, leaving a stationary combination — this justifies an error-correction model capturing their long-run relationship.
An MA(q) time-series model expresses the current value as a linear function of:
1. The current and q past error (shock) terms
2. Its own q past values only
3. The interest rate only
4. A single deterministic trend with no randomness
Show answer
Answer: A — The current and q past error (shock) terms
A moving-average model of order q expresses the series as a linear combination of the current and q lagged error terms; an autoregressive model of order p, by contrast, expresses it in terms of its OWN p past values.
Which of the following are consequences of estimating a non-recursive (feedback) simultaneous-equation system by ordinary equation-by-equation OLS?
1. Simultaneity bias — the regressor is correlated with the error term
2. Inconsistency, even as the sample size grows large
3. A need for instrumental-variable methods such as 2SLS
4. An automatic guarantee of unbiasedness
Show answer
Answer: A — Simultaneity bias — the regressor is correlated with the error term; B — Inconsistency, even as the sample size grows large; C — A need for instrumental-variable methods such as 2SLS
The first three correctly describe the standard consequence of feedback among endogenous variables; OLS is NOT unbiased in this setting, which is precisely why 2SLS or indirect least squares is needed instead.
Name the standard test used to check whether a time series contains a unit root (i.e. is non-stationary).
Numerical answer — type the value.
Show answer
Answer: Augmented Dickey-Fuller test
The (Augmented) Dickey-Fuller test is the standard unit-root test; failing to reject its null hypothesis of a unit root indicates the series is non-stationary, which then requires differencing or cointegration analysis before any regression is run on it.