Econometrics Economics

Economic models and statistical evidence

Econometrics connects economic reasoning to empirical measurement. An economic model proposes mechanisms: demand may respond to prices and income, productivity may respond to schooling, and investments may react to interest rates. An econometric model translates these relationships into observables, unknown parameters and a stochastic disturbance representing influences not explicitly incorporated. The purpose of estimation is not merely to draw a line through data. It is to learn which quantitative statements the available variation can support, with explicit uncertainty and limits of interpretation.

Consider a policy question about school class size. Let Y_i denote the average test score of district i, D_i its average class size, and W_i observed characteristics such as household income. A useful starting specification is

The outcome Y_i and explanatory variable D_i are measured, β₁ is the parameter of primary interest, W_i′γ summarises measured controls and u_i contains omitted determinants and idiosyncratic variation. The slope β₁ is a conditional association under the model, unless its identifying assumptions justify interpreting it as a causal effect. If more advantaged districts choose smaller classes, a simple regression may confuse household advantage with school resources. A large sample reduces sampling error, not systematic confounding.

The same distinction applies when an analyst asks whether training raises wages, whether higher cigarette taxes decrease smoking, whether institutions determine income, or whether a yield-curve spread forecasts GDP growth. Forecasting requires useful information about future outcomes; identifying a causal parameter requires credible comparisons between counterfactual outcomes. A variable can be an excellent predictor without producing the outcome it predicts.

Data structures and research designs

Cross-sectional data observe many units at one time, such as household income and education across individuals. The subscript i indexes individuals, firms, districts or countries. A random cross section makes sample averages and regression coefficients natural estimates of population characteristics, although nonresponse, clustering, selection and missingness can invalidate a naive random-sampling model.

Time-series data observe a variable, or several variables, over time. The chronological order matters because Y_t can depend on Y_t₋₁. Trends, changing volatility, seasonal patterns and dynamic responses force us to reconsider assumptions appropriate for independent observations. Time-series data also support genuine forecasting, which must be assessed using observations not used to fit the forecasting model.

Panel or longitudinal data observe the same economic units in several periods and write their outcomes as Y_it. A panel can separate persistent differences between individuals from changes occurring within individuals. For instance, differences in managerial ability across firms may contaminate a cross-sectional correlation between capital and profits, whereas repeated observations may make it possible to control for time-invariant ability.

Repeated or pooled cross sections differ from true panel data: samples from different periods need not contain the same people. Each design implies a different sampling model and affects the credible range of empirical questions.

Potential outcomes and the selection problem

To formalise a causal question, imagine that each unit has two potential outcomes: Y_i(1) if it receives a treatment and Y_i(0) if it does not. Let D_i ∈ {0,1} indicate the treatment actually received. Only one potential outcome is observed:

The individual causal effect is Y_i(1) − Y_i(0), but we cannot directly observe both outcomes for a given unit. We often target the average treatment effect ATE = E[Y(1) − Y(0)] or the average treatment effect on the treated ATT = E[Y(1) − Y(0) | D=1]. These are different quantities when treatment effects are heterogeneous and selection depends on the gains from treatment.

An observed difference in group means can be decomposed exactly:

The expression in braces is selection bias: the difference that would exist between treated and untreated groups even if nobody were treated. If intrinsically sicker patients are more likely to receive hospital care, a simple comparison of health outcomes can suggest that hospitals worsen health even when treatment is beneficial. This illustrates why large and statistically precise associations can be causally misleading.

In a well-run randomised controlled trial, treatment assignment is independent of both potential outcomes, and treatment and control groups have identical expected counterfactual outcomes. Consequently their difference in expected observed outcomes identifies the ATE. Random assignment does not guarantee that finite samples balance perfectly, and noncompliance, attrition, interference or measurement problems still require attention.

Estimand, estimator and estimate

The estimand is the precisely defined population quantity of interest, such as the causal effect of reducing class size by one student. An estimator is the random rule applied to data, such as the OLS coefficient calculated from a sample; the estimate is the numerical value obtained when that rule is applied to the observed data. Confusing these concepts leads to overconfident interpretations: a precisely computed estimate need not target the causal effect described in the research question.

Good empirical work specifies the unit of observation, population, measurement convention, parameter of interest, identifying source of variation, plausible competing mechanisms and inference procedure. A standard error measures sampling variation under a model, not the credibility of the identifying assumptions. Identification is conceptual; inference is statistical; neither can replace the other.

Questions for an empirical project

Before computing any regression, ask what intervention or policy change the research seeks to understand, what comparison approximates the ideal experiment, whether the variation in the explanatory variable is plausibly as good as random conditional on controls, and which population the estimated parameter represents. A second set of questions concerns whether standard errors accommodate clustering or serial dependence, whether there are enough independent units and whether the result is practically meaningful.

From a question to a research specification

Imagine an economist asked to estimate how an increase in university tuition influences enrolment. A careless analysis regresses enrolment on tuition and calls the slope a demand elasticity. A disciplined analysis asks first whether tuition varies across institutions, people or time; whether scholarships change net prices; whether students sort into universities because of quality; and whether universities adjust tuition in response to demand. A valid demand elasticity describes changes in choices when the price changes for reasons not themselves caused by shifts in demand. Prices observed at equilibrium may therefore be endogenous.

The target could be a one-percent increase in tuition, a €1,000 level increase or a discrete policy change from one fee schedule to another. Each intervention has different consequences and may not be identified from the same data. For example, if some institutions admit only wealthy applicants, there may be little overlap in the students who face different prices. Estimands should be specified at the relevant margin and for the population directly exposed to that margin.

Economic interpretation also requires accounting for general equilibrium and spillovers. A programme that raises training among some workers may alter wages or job opportunities for untreated workers; in that case their observed outcomes are not unaffected controls. The stable-unit-treatment-value assumption (SUTVA) combines restrictions on interference and the definition of treatment versions. In many economic applications these restrictions are substantive assumptions, not innocuous statistical conventions.

Representative examples of prediction and causal inference

Suppose an analyst uses prior-year sales, website visits and seasonal indicators to forecast next-month sales. This predictive problem concerns accuracy and stability under future conditions. It need not imply that increasing visits through artificial traffic would raise sales. A second analyst studies whether an advertising subsidy causes sales growth. That causal problem requires treatment assignment independent of potential sales outcomes conditional on an appropriate design. Both analysts can use regression, but they answer different questions and may need different covariates and validation procedures.

Another complication is policy response. If local governments increase spending when unemployment begins rising, a cross-sectional or time-series spending coefficient may be positively related to unemployment even if extra public spending partially offsets job losses. A causal effect can have a different sign from its observational association because policy is targeted where outcomes are expected to deteriorate. State what government would have done in the absence of the policy and how that counterfactual can be approximated.