ECON4001: Econometrics for Honours Students Course Notes

Econometrics in an Honours program is where theory, computation, and empirical judgement converge. ECON4001 builds the skills needed to design identification strategies, estimate models credibly, diagnose econometric problems, and communicate results with statistical and economic interpretation. These course notes synthesise core econometric methods with Honours-level expectations for reasoning, robustness, and applied work—using an emphasis on research contexts relevant to South African universities and research practice, including typical data environments such as labour, education, household expenditure, inflation, crime, health access, firm performance, and policy evaluation.

Section 1: Econometrics Foundations for Honours-Level Research

Honours econometrics assumes you already understand basic statistics and regression. ECON4001 raises the bar by requiring you to (i) define models precisely, (ii) justify assumptions, (iii) translate assumptions into testable implications where possible, (iv) understand what can and cannot be inferred from data, and (v) produce defensible empirical conclusions. Foundations here focus on statistical models, identification, and what “estimation” really means.

1.1 The econometric workflow: from research question to evidence

A strong econometric project begins with a clear research question and a mapping from theory to an estimable model. The key step is translating a substantive claim (e.g., “higher minimum wages reduce youth employment”) into:

1. A conceptual model (what economic mechanism links the variables?).
2. A statistical model (what variables enter? in what functional form?).
3. An identification strategy (why is the variation in the key regressor plausibly exogenous, or how will you address endogeneity?).
4. An estimation plan (which estimator? what assumptions?).
5. A validation plan (diagnostics, robustness checks, alternative specifications, placebo tests).
6. A reporting plan (effect sizes, uncertainty, limitations, and how the results relate to theory).

In South African research environments, data constraints are common: incomplete administrative data, measurement error in surveys, uneven sampling, missing covariates, and policy changes that might be confounded with other shocks. Econometrics helps you deal with these constraints systematically rather than ad hoc.

1.2 Model notation and the econometric object

Consider a generic linear regression model:
[
y_i = x_i'\beta + u_i
]
where (y_i) is the outcome (e.g., log wages), (x_i) is a vector of regressors (e.g., education, age, industry, household characteristics), (\beta) are parameters, and (u_i) is the error term capturing unobserved influences.

In Honours econometrics, the error term is not merely “noise.” It encodes everything unmodelled: omitted variables, measurement error, unobserved heterogeneity, and structural disturbances. Different econometric problems correspond to different properties of (u_i), such as:

• Conditional mean zero: (E[u_i \mid x_i]=0) (needed for unbiasedness and consistency under exogeneity).
• Homoskedasticity vs heteroskedasticity: (\mathrm{Var}(u_i \mid x_i)=\sigma^2) vs (\neq \sigma^2).
• Independence vs correlation: i.i.d. assumptions vs clustering/serial correlation.
• Endogeneity: correlation between regressors and the error due to omitted variables or reverse causality.

1.3 Identification vs estimation

A central conceptual distinction is:

• Identification: whether the data-generating process uniquely determines the parameter(s).
• Estimation: the procedure used to approximate identified parameters using sample data.

For example, in a naive wage regression ( \log(wage)=\beta , schooling + controls + u ), schooling is often endogenous due to ability bias and parental background. Even if you compute OLS estimates precisely, they may not correspond to the causal effect of schooling unless identification assumptions hold. Honours work expects you to discuss:

• Why schooling is endogenous in your context.
• What assumptions would make it exogenous (or what instrument/approach you use).
• Whether those assumptions are plausible and testable indirectly.

1.4 Assumptions as research statements

Econometric assumptions should be written in plain language and connected to the setting. For instance, the assumption (E[u_i|x_i]=0) can be framed as: “After conditioning on observed covariates (x_i), unobserved factors affecting (y_i) do not systematically correlate with the regressors.” In practice:

• If (x_i) includes policy exposure (e.g., access to a grant), then (E[u_i|x_i]=0) requires that unobserved determinants of outcomes are balanced across exposure groups once controls are added.
• If (x_i) includes education, it implies ability and motivation are fully captured by controls, which is usually implausible without stronger strategies (e.g., fixed effects with panel data, or instruments).

1.5 Conditional expectation, projection, and interpretation

Regression estimation can be interpreted as a projection in probability theory. The population linear projection minimises mean squared error:
[
\beta^{} = \arg\min_{\beta} E\big[(y_i-x_i'\beta)^2\big]
]
Under exogeneity, (\beta^) coincides with the causal parameter. Under endogeneity, (\beta^*) becomes a “best linear approximation” rather than a causal effect.

This matters for interpretation in Honours-level writing. If identification is weak, you must communicate whether coefficients are structural, causal, or associational. For example:

• “A 1-year increase in schooling is associated with a 6% higher wage” may be appropriate for descriptive/associational results.
• “Schooling increases wages by X%” requires a causal identification argument.

1.6 Worked conceptual example: labour outcomes and omitted variable bias

Imagine studying whether unemployment insurance (UI) increases job search intensity among working-age adults. Let:

• (y_i): job search intensity (weekly applications).
• (x_i): UI eligibility or receipt indicator.
• (u_i): unobserved job-seeker motivation.

If individuals with high motivation are more likely to obtain UI (or are more likely to report UI receipt), then (UI_i) correlates with (u_i). The OLS estimand becomes biased. Honours-level thinking asks:

• Are eligibility rules based on administrative records (reducing bias) or self-report (increasing bias)?
• Is there a discontinuity or policy threshold that can serve as an instrument?
• Are there panel data where fixed effects capture time-invariant motivation?

This conceptual step sets up later econometric methods—IV, DiD, fixed effects, and discontinuity designs.

Section 2: Classical Regression, Inference, and Diagnostics (with South African Data Context)

This section builds a thorough understanding of the linear regression model, estimation, inference, and the diagnostics you need to trust empirical results. It includes explicit discussion of common econometric failures and how to respond in a defensible Honours workflow.

2.1 OLS estimation and the meaning of (\hat{\beta})

OLS estimates (\beta) by solving:
[
\hat{\beta}_{OLS} = (X'X)^{-1}X'y
]
in the standard full-rank case. The key properties depend on assumptions:

• Unbiasedness: if (E[u|X]=0).
• Consistency: if (E[u|X]=0) and regularity conditions hold.
• Efficiency: if errors are homoskedastic and uncorrelated (Gauss–Markov).

In applied South African work, you often face heteroskedasticity (e.g., variance differs across income levels, provinces, or education groups). Even if OLS coefficients remain unbiased/consistent under exogeneity, the standard errors can be wrong if you assume homoskedasticity when the variance is not constant. That affects hypothesis tests and confidence intervals.

2.2 Inference: standard errors, degrees of freedom, and robust variance

The standard t-test relies on:
[
t = \frac{\hat{\beta}j – \beta{0j}}{se(\hat{\beta}_j)}
]
where (se(\hat{\beta}_j)) is derived from (\mathrm{Var}(\hat{\beta})). When heteroskedasticity or correlation exists, you use robust variance estimators. In practice, Honours-level reporting includes:

• Whether you used heteroskedasticity-robust standard errors.
• Whether you used cluster-robust errors (e.g., clustering at school, district, or firm level).
• Whether you adjusted for serial correlation in time series or panel data.

A typical applied caution: clustering reduces the effective number of independent clusters. With few clusters, asymptotic approximations may be weak. Econometrics Honours notes should therefore emphasise:

• Checking number of clusters.
• Considering alternative robust corrections (e.g., small-sample adjustments) and reporting them.

2.3 Hypothesis testing and model comparison

Common hypothesis forms include:

• Single coefficient tests: (H_0:\beta_j=0).
• Joint tests: (H_0:R\beta=r), using an F-test.
• Nested model comparisons: e.g., compare a baseline model with controls to an expanded model with additional policy variables.

In Honours work, tests should be interpreted with care:

• Statistical significance depends on sample size and variance.
• Economic significance is often more meaningful for policy-relevant decisions.

A robust reporting style includes both:

• p-values / confidence intervals.
• Effect sizes (e.g., in percentage terms if outcomes are logs).
• Discussion of whether the magnitude aligns with economic mechanisms.

2.4 Functional form: logs, levels, and interpretation

You often switch to log transformations to interpret coefficients as elasticities or semi-elasticities.

• If (y) is in logs: (y=\ln(wage)).
• If (x) is in logs: then coefficient on (x) is an elasticity.
• If (x) is a dummy and (y) is log: the coefficient approximates a percentage difference.

A concrete example: if
[
\ln(wage_i) = \alpha + \beta D_i + controls + u_i
]
where (D_i=1) indicates completed vocational training, then (\beta) is approximately the log difference in wages. For more accurate interpretation, one can convert:
[
% \Delta wage \approx (e^\beta -1)\times 100
]
This matters for interpretation and must be applied consistently wherever percentage effects are discussed.

2.5 Multicollinearity: diagnosis and consequences

Multicollinearity occurs when regressors are highly correlated, inflating standard errors. It does not bias OLS coefficients under exogeneity, but it reduces precision and can make individual coefficients statistically insignificant.

Honours-level diagnostics include:

• Variance Inflation Factors (VIFs).
• Correlation matrix inspection.
• Conceptual reasoning: variables may represent overlapping constructs (e.g., multiple measures of socio-economic status).

If multicollinearity is present, Honours work may proceed with:

• Dropping redundant variables only when justified.
• Combining variables into indices (with transparent methodology).
• Interpreting coefficients cautiously and relying more on joint tests or predicted effects.

2.6 Heteroskedasticity and its impacts

Heteroskedasticity means:
[
\mathrm{Var}(u_i|X) = \sigma_i^2
]
OLS remains consistent under exogeneity but standard errors become incorrect if computed under constant variance assumptions.

Diagnosing heteroskedasticity can involve:

• Visual residual plots.
• Formal tests (though not always reliable).
• Comparing conventional vs robust standard errors.

A South African example context: wage data often show larger variance for high-earning occupational groups. Similarly, household expenditure may exhibit different variability across provinces.

The applied response: use heteroskedasticity-robust standard errors unless you have a better modelling strategy (e.g., weighted least squares when justified).

2.7 Autocorrelation and clustering

If observations are correlated over time (serial correlation), tests that assume independence are invalid. Similarly, if observations share unobserved shocks within groups (e.g., students within schools), you need cluster-robust standard errors.

• Serial correlation: relevant in time series or when panel errors are correlated within individual units over time.
• Cluster correlation: relevant in hierarchical data (schools, municipalities, firms).

A careful approach for cluster-robust SEs:

1. Identify the level at which shocks are likely shared.
2. Cluster at that level.
3. Check the number of clusters; very small cluster counts raise concerns.
4. Consider two-way clustering when multiple grouping dimensions matter (e.g., school and cohort).

2.8 Endogeneity: the core threat to causal interpretation

Endogeneity can arise from:

• Omitted variables: unobserved determinants correlated with regressors.
• Measurement error: errors in regressors lead to attenuation bias.
• Reverse causality: outcome influences regressors.
• Simultaneity: simultaneous determination of variables.

To address endogeneity, regression diagnostics are often insufficient—you need identification strategies (IV, DiD, FE, RD, etc.). Honours students are expected to:

• Provide a credible endogeneity argument for the main regressor(s).
• Choose an approach aligned with the data and institutional context.

2.9 A full example regression narrative (labour and education)

Suppose a researcher examines the relationship between years of education and monthly earnings for South African adults. The baseline model:

[
\ln(earnings_i)=\alpha + \beta yearsEdu_i + \gamma age_i + \delta age_i^2 + \theta province_i + \epsilon_i
]

Potential issues:

• Endogeneity: years of education correlates with ability and family background not fully captured by controls.
• Heteroskedasticity: earnings variance differs across education levels.
• Clustering: respondents from the same province or sampling cluster may share shocks.

Honours-level steps:

1. Estimate OLS with robust SEs (at least heteroskedasticity-robust).
2. Add richer controls to improve conditional exogeneity plausibility (e.g., socio-economic status indicators).
3. Consider fixed effects if panel data exists (e.g., individuals observed over time).
4. If endogeneity remains, consider IV using an instrument (e.g., exposure to schooling policy changes, historical school access) or an RD design with a threshold.

This example illustrates that diagnostics and robust inference are necessary but do not solve endogeneity alone.

Section 3: Advanced Econometric Methods for Honours Coursework (Endogeneity, Causality, and Policy Evaluation)

This section develops the principal methods typically tested and required in an Honours econometrics course: instrumental variables, panel data and fixed effects, difference-in-differences, and regression discontinuity. It emphasises identifying assumptions, threats, and practical implementation considerations relevant to policy and labour/education/health settings commonly studied in South Africa.

3.1 Instrumental Variables (IV): theory and conditions

When a regressor (x_i) is endogenous, you require an instrument (z_i) satisfying:

1. Relevance: (Cov(z_i, x_i)\neq 0) (instrument affects the endogenous regressor).
2. Exogeneity (exclusion + independence): (z_i) affects the outcome only through (x_i), i.e., (Cov(z_i, u_i)=0).

In practice, exogeneity is usually the hardest to justify. Honours students must link the instrument to institutional or policy mechanisms:

• A natural experiment that shifts (x_i) but plausibly does not directly affect (y_i) except through (x_i).
• A threshold or administrative rule generating quasi-random variation.

An IV workflow typically includes:

1. First-stage estimation of (x_i) on (z_i) and controls.
2. Second-stage estimation of (y_i) on predicted (x_i) and controls.
3. Reporting diagnostics such as first-stage F-statistics (with caution about specific thresholds and assumptions).
4. Testing overidentifying restrictions if multiple instruments exist.

3.2 Local Average Treatment Effect and interpretation

Under standard IV assumptions, IV estimates the Local Average Treatment Effect (LATE): the causal effect for “compliers,” i.e., those whose treatment status is influenced by the instrument.

This has interpretational consequences. For example, an instrument for education access may shift educational attainment only for some groups. The IV estimate is therefore not necessarily the average treatment effect for all individuals.

A South African policy context where LATE reasoning arises:

• Instruments based on eligibility rules may induce compliance for people near thresholds (or those who respond to policy changes).
• People far from eligibility may not change behaviour, meaning the instrument identifies effects for a subset of the population.

3.3 Weak instruments and how they distort results

A weak instrument leads to:

• Biased IV estimates (finite-sample bias).
• Large standard errors.
• Potential misleading inference.

Honours students should avoid blindly applying IV without assessing instrument strength. If relevance is weak, alternative designs (e.g., DiD or RD) might be better.

3.4 Example: IV for education and earnings using policy exposure

Consider a setting where access to a schooling support program increases years of education for eligible learners. Suppose:

• (y_i = \ln(earnings_i))
• (x_i = yearsEdu_i) (endogenous)
• (z_i = eligibility_i) derived from a documented policy rule

The core argument:

• Relevance: eligibility increases attendance/completion for eligible learners.
• Exclusion: eligibility affects earnings only through education, not through separate channels.

Honours-level critique:

• Eligibility might also correlate with socio-economic status if policy uptake differs by household characteristics.
• The instrument could affect health, job networks, or labour market connections directly.

Potential mitigation:

• Include rich baseline covariates and fixed effects.
• Conduct placebo tests: eligibility should not predict outcomes measured before education differences emerge.
• If possible, use a stronger instrument (e.g., distance to an administrative office that handles eligibility, under plausible assumptions).

3.5 Panel data and fixed effects (FE): controlling for unobservables

In panel data, you observe units (i) over time (t):
[
y_{it} = x_{it}'\beta + \alpha_i + u_{it}
]
where (\alpha_i) captures time-invariant unobserved heterogeneity. Fixed effects estimators remove (\alpha_i) by demeaning or using within transformations:
[
y_{it}-\bar{y}i = (x{it}-\bar{x}i)'\beta + (u{it}-\bar{u}_i)
]

This strategy handles omitted variables that are constant over time, which is common in labour and education where unobserved ability is relatively stable. However, FE does not solve time-varying endogeneity, such as:

• Time-varying omitted variables (e.g., changing health shocks affecting both employment and outcomes).
• Simultaneous causality in the same period.
• Measurement changes over time.

Honours expectations include:

• Justifying that key unobservables are time-invariant (or sufficiently captured).
• Considering whether FE assumptions are plausible in the specific South African context.

3.6 Differences-in-differences (DiD): policy evaluation logic

DiD compares changes over time between a treated group and a control group. The canonical model with two periods:
[
y_{it} = \alpha + \beta (Treated_i \times Post_t) + \gamma Treated_i + \delta Post_t + u_{it}
]
where (\beta) is the DiD estimator capturing the average treatment effect on the treated under assumptions.

Key DiD assumption: Parallel trends.

• In the absence of treatment, treated and control groups would have evolved similarly.

Honours-level practice includes:

• Testing pre-trends where data allows.
• Allowing flexible time trends or using event-study specifications.
• Considering spillovers: treatment might affect control units (e.g., regional policy diffusion).

Concrete example: schooling policy and employment outcomes

Suppose a program begins in 2012 that increases training opportunities for youth in certain municipalities. You want to estimate impact on employment in 2012–2014.

• Treated municipalities: those that receive training centres.
• Control municipalities: those without centres.

Honours-grade DiD work would include:

1. Choosing a comparison window (e.g., 2010–2014).
2. Checking pre-2012 trends in employment for treated vs control.
3. Estimating DiD with robust SEs clustered at municipality level.
4. Performing robustness checks:
   • Alternative control groups.
   • Excluding municipalities with other concurrent interventions.
   • Varying the post-treatment start date if implementation is staggered.

3.7 Event-study and staggered adoption

When treatment timing varies by unit, modern DiD requires careful estimation. Event-study approaches model outcomes relative to time since treatment.

Honours students should understand the risks:

• Heterogeneous treatment effects over time complicate interpretation.
• Incorrect weighting can produce misleading average effects.

Even if you do not implement advanced estimators in every assignment, you should conceptually explain:

• What “average treatment effect” means under staggered adoption.
• Why event-study plots are useful diagnostics for pre-trends and dynamic effects.

3.8 Regression discontinuity design (RD): threshold-based identification

RD uses a discontinuity in treatment assignment at a threshold (c) in an assignment variable (s_i). Treatment (D_i) changes sharply at (s_i=c). The key assumption is that potential outcomes would evolve smoothly in (s_i) absent treatment.

Two main forms:

• Sharp RD: treatment is deterministic at the threshold.
• Fuzzy RD: probability of treatment changes but not deterministically (often uses RD as an IV with threshold as instrument).

Honours-grade RD considerations include:

• Choosing bandwidth around the cutoff.
• Ensuring enough data near the cutoff to estimate precisely.
• Testing for discontinuities in density of the running variable (McCrary test).
• Selecting polynomial order or local linear estimation and reporting sensitivity.

Example: education scholarship at a test score cutoff

Suppose scholarships are awarded if a learner’s grade point score exceeds a threshold (say 65). The aim is to estimate causal impact on later university attendance.

Potential threats:

• Manipulation around the threshold: if learners can influence scores, the continuity assumption may fail.
• Sorting: if advantaged groups can better target scholarship eligibility.

Mitigation:

• Density tests to detect manipulation.
• Visual inspection of covariate balance around the threshold.
• Covariate discontinuity checks.

3.9 Method selection: matching the estimator to the data-generating process

A key Honours skill is justifying which method is appropriate. Use the data and institutional setting:

• If you have a credible instrument and relevance/exclusion are plausible → IV.
• If you have repeated observations and time-invariant unobservables are important → FE/Panel methods.
• If there is a policy intervention with a credible comparison group and parallel trends → DiD.
• If treatment assignment depends on an observable threshold → RD.

Method choice is part of causal inference. In the exam context, students are often asked to propose an identification strategy and justify assumptions.

Section 4: Likelihood, Limited Dependent Variables, and Nonlinear Models

Honours econometrics also addresses nonlinear modelling where OLS assumptions fail or outcomes are not continuous. This includes limited dependent variable models such as logit/probit, count models like Poisson, and conceptual foundations for maximum likelihood estimation (MLE). While not every exam focuses equally on all nonlinear models, the underlying inference principles recur.

4.1 Why OLS can be inappropriate for limited dependent variables

When outcomes are binary, counts, or censored, OLS can produce:

• Predicted values outside the logical range (e.g., probabilities less than 0 or greater than 1).
• Heteroskedastic and non-normal errors by construction.
• Incorrect functional assumptions affecting inference.

Examples in South African datasets:

• Binary outcomes: employed/unemployed; pass/fail; insured/uninsured.
• Count outcomes: number of hospital visits; number of crimes reported; number of job applications.
• Censored outcomes: income observed as “at least zero” due to top-coding.

4.2 Maximum likelihood estimation (MLE): basic framework

MLE chooses parameters (\theta) that maximise the likelihood:
[
L(\theta) = \prod_{i=1}^n f(y_i|x_i,\theta)
]
Equivalently maximise log-likelihood:
[
\ell(\theta)=\sum_{i=1}^n \log f(y_i|x_i,\theta)
]

Properties:

• Under correct specification and regularity, MLE is consistent and asymptotically normal.
• In nonlinear models, marginal effects often matter more than raw coefficients.

Honours work requires you to understand:

• How the likelihood is constructed from distributional assumptions.
• How to compute predicted probabilities and marginal effects.
• Robust standard errors and correct interpretation.

4.3 Logit and probit models for binary outcomes

Let (y_i \in {0,1}). In a latent variable approach:
[
y_i^* = x_i'\beta + \varepsilon_i,\quad y_i = 1{y_i^*>0}
]

• Logit: (\varepsilon_i) follows logistic distribution.
• Probit: (\varepsilon_i) follows standard normal distribution.

The probability:

• Logit: (\Pr(y_i=1|x_i) = \Lambda(x_i'\beta)), where (\Lambda) is logistic CDF.
• Probit: (\Pr(y_i=1|x_i) = \Phi(x_i'\beta)), where (\Phi) is normal CDF.

Marginal effects: interpretability

Unlike OLS, coefficients in logit/probit don’t directly represent marginal changes in probability. The marginal effect of a regressor (x_{ik}) is:
[
\frac{\partial \Pr(y_i=1|x_i)}{\partial x_{ik}}
]
which depends on (x_i). Honours exams often expect students to:

• compute or describe marginal effects at means,
• explain heterogeneity of effects across covariate levels.

4.4 Example: probability of passing a matriculation exam

Suppose (y_i=1) if a learner passes matric mathematics and (x_i) includes study hours, school quality index, and prior test score.

• A logit/probit model estimates (\Pr(\text{Pass}=1|x)).
• You can interpret marginal effects: “An additional hour of study increases the probability of passing by X percentage points for learners with covariate profile (x).”

But Honours work also demands attention to identification:

• Study hours may be endogenous (motivation, parental support).
• If you only estimate a logit/probit association, you must state it is not necessarily causal.
• If causal estimation is required, you need IV for nonlinear models or an alternative design (e.g., DiD if policy changes study conditions).

Even when causal methods are not used, good practice includes:

• Checking overfitting risk (if high-dimensional covariates).
• Assessing predictive performance.
• Using robust SEs if heteroskedasticity or clustering exists.

4.5 Count models: Poisson and overdispersion

For count outcomes (y_i \in {0,1,2,\dots}), a Poisson regression models:
[
E[y_i|x_i] = \exp(x_i'\beta)
]
and assumes:
[
\Pr(y_i=k|x_i) = \frac{(\exp(x_i'\beta))^k e^{-\exp(x_i'\beta)}}{k!}
]

However, real count data often displays overdispersion: variance greater than mean. The Poisson assumption then underestimates variance, leading to overconfident inference.

Common remedy: Negative Binomial regression, which adds an overdispersion parameter.

Honours-level practice includes:

• diagnosing overdispersion (mean-variance relationship),
• choosing an appropriate model,
• interpreting rate ratios or incidence rate effects.

Example: number of healthcare visits

Suppose (y_i) is the number of clinic visits in the past month. Predictors include income, distance to clinic, chronic illness status.

Interpretation in log-link models: coefficients translate to multiplicative changes in expected counts. If (x_{ik}) increases by 1 unit, expected count multiplies by (e^{\beta_k}).

4.6 Censoring: Tobit intuition

When outcomes are censored (e.g., income reported only up to a threshold), Tobit models can be used. Tobit handles the fact that we observe “0” for those whose latent variable is below 0, rather than truly having zero.

Honours students should be able to:

• explain the latent variable logic,
• discuss identification assumptions behind censoring models,
• critique when censoring is non-random.

4.7 Nonlinearity and endogeneity: limits of standard models

Nonlinear models with endogenous regressors are complex. If you ignore endogeneity, maximum likelihood estimates can be inconsistent. Strategies include:

• Using instruments (often more involved, e.g., IV-probit).
• Using control functions in some cases.
• Applying designs like DiD or RD that deliver identification even for nonlinear outcomes (with careful modelling).

The exam logic often tests conceptual understanding:

• “State why endogeneity still threatens causal interpretation in logit/probit.”
• “Propose an approach to address it given available data.”

Section 5: Applied Econometrics for Honours Assessment: Design, Robustness, and Communicating Results

The final section focuses on what exam questions and Honours assignments often reward: coherent empirical design, robust methods, careful diagnostics, and clear communication of econometric results. It integrates previous topics into an end-to-end applied workflow that reflects common South African empirical research settings.

5.1 Building a credible empirical strategy

An Honours empirical strategy should state:

• outcome variable definition,
• key regressor(s) and why they matter,
• control variables and what threats they address,
• identification strategy and assumptions,
• estimation method and inference method,
• robustness checks and why they matter.

A well-structured strategy addresses four major threats:

1. Endogeneity: how do you deal with it?
2. Unobserved heterogeneity: can you control for it using FE or design?
3. Incorrect standard errors: how do you cluster/robustify?
4. Model misspecification: functional form and omitted nonlinearities.

5.2 Specification planning: from baseline to enriched models

A typical Honours modelling sequence:

1. Baseline model: minimal covariates motivated by theory.
2. Add controls: include plausible confounders and fixed effects.
3. Check robustness:
   • alternate functional form (levels vs logs),
   • alternate sample restrictions (excluding outliers, focusing on specific groups),
   • alternate estimators if assumptions differ.
4. Placebo tests:
   • test pre-treatment outcomes in DiD,
   • run RD with placebo thresholds if plausible,
   • test instruments/exposure against outcomes they should not affect.
5. Sensitivity analysis:
   • vary bandwidth in RD,
   • vary covariate sets and polynomial order,
   • compare results with/without additional fixed effects.

Honours marking often reflects not only what you do, but whether you justify why each robustness check addresses a specific risk.

5.3 Diagnostics beyond “significance”: residuals and fit

Even with robust SEs, you should diagnose:

• Residual patterns (heteroskedasticity, nonlinearity).
• Influence and leverage points (outliers can dominate OLS).
• Multicollinearity and redundancy in controls.
• In nonlinear models, check predictive performance and calibration.

For exams, you may not compute complex diagnostics, but you must explain what they detect and why they matter.

5.4 Robustness to specification and sample selection

A common applied problem is sample selection bias: outcomes observed only for a subset (e.g., employment measured only for those in labour force surveys; health outcomes observed only for those who access clinics). If selection correlates with unobservables affecting outcomes, estimates become biased.

Possible solutions depend on data and context:

• Use selection models where appropriate (e.g., Heckman selection).
• Use designs where selection is mitigated (e.g., threshold-based assignment).
• Employ bounding methods if selection is difficult.

Even if not all are required, Honours students should recognize selection as a distinct problem from missing covariates.

5.5 Communication: writing econometric results like a researcher

A high-quality results section should include:

• A clear description of dependent variable and units.
• Interpretation of key coefficient(s) in economic terms.
• Reporting of uncertainty (standard errors, confidence intervals).
• A narrative connecting findings to hypotheses and theory.
• Discussion of limitations: endogeneity, external validity, data constraints.

A strong habit in Honours coursework is to include both:

• Statistical inference: e.g., “coefficient is significant at 5%.”
• Economic interpretation: e.g., “effect size is X percentage points,” “policy implication: small vs large.”

5.6 Tables and consistency in reporting (what matters in assessment)

In empirical work, inconsistencies lead to major grade losses. Key practices:

• Ensure units are consistent: if the outcome is log, translate correctly to percentage impacts.
• Keep sample sizes consistent across models unless you explicitly change sample restrictions.
• For categorical variables, define base categories and interpret relative to that.
• If you use clustering, state the cluster level in every relevant results table/paragraph.
• When reporting robustness, label models clearly: “Column (2) includes province fixed effects; Column (3) uses cluster-robust SEs.”

While exam questions may not require full tables, they often ask you to describe what you would report and how to interpret it.

5.7 Case study template: a South African policy evaluation storyline

Consider a coherent narrative relevant to many South African empirical topics: impact of a vocational training intervention on youth employment prospects.

A plausible Honours exam-style design:

• Outcome: (y_i) = employment status or employment probability.
• Treatment: (D_i) = whether a youth accessed the vocational training.
• Endogeneity threat: motivated youths might seek training (reverse causality/selection).
• Identification: choose one:
  • DiD if program rollout differs by municipality and timing permits pre-trends checking; or
  • RD if eligibility depends on a cutoff in an assessment score; or
  • IV if eligibility is generated by an administrative rule, with exclusion plausible.

Then estimation:

1. Estimate baseline model.
2. Add controls and fixed effects.
3. Use robust/clustered SEs.
4. Conduct placebo and pre-trend checks.
5. Interpret results as causal or associational depending on identification validity.

The point is not the particular policy but the logic: a credible design must be matched to a credible identification assumption, and reporting must match the assumptions.

5.8 Common exam question patterns and expected responses

Honours courses typically test both computation and conceptual reasoning. Expected responses include:

• “Explain why OLS is biased when endogeneity exists.”
  • Mention conditional mean violation, omitted variables, reverse causality, and how that changes interpretation.
• “Propose an identification strategy to estimate causal effects.”
  • Present an IV/DiD/RD plan, state assumptions, and connect to institution/policy details.
• “Discuss the key diagnostics for your chosen estimator.”
  • Robust SEs, clustering level, pre-trend checks, bandwidth sensitivity, weak instrument concerns.
• “Interpret coefficients from a log-linear model.”
  • Translate coefficient into percentage change using appropriate transformation (approximation or exact (e^\beta-1)).
• “Compare FE vs pooled OLS.”
  • Discuss removal of time-invariant unobservables and remaining threats from time-varying endogeneity.

The strongest answers are coherent: they explicitly state assumptions and align them with the empirical strategy and interpretation.

5.9 Summary: what mastery looks like in ECON4001

Mastery of ECON4001 at Honours level means you can:

• Translate economic questions into econometric models.
• Distinguish identification from estimation and communicate the difference.
• Implement and interpret OLS correctly with robust inference.
• Diagnose and respond to heteroskedasticity, autocorrelation, clustering, and multicollinearity.
• Address endogeneity using IV, FE, DiD, or RD—chosen for a credible reason.
• Understand nonlinear models via likelihood and interpret marginal effects.
• Execute a robust empirical workflow and write results in a disciplined, transparent way suited to academic and policy audiences in South Africa.

This combination of technical understanding and research judgement is what turns econometrics from “methods” into “evidence.”