ECO4006F Econometrics Exam Preparation

Econometrics exams reward two things in equal measure: (1) the ability to reason correctly about identification, assumptions, and estimators, and (2) the ability to execute procedures under time pressure—deriving formulas, checking conditions, interpreting results, and writing coherent answers. This study guide is designed for ECO4006F Econometrics preparation with an emphasis on approaches commonly assessed in South African universities, colleges, and TVETs, while keeping the focus on clear, exam-ready econometrics workflows.

The guide is structured to mirror how many students experience the exam: starting from core models and assumptions, moving through estimation and inference, then covering diagnostics and specification problems, and finally consolidating with practical “how to answer” strategies. Each section includes worked conceptual steps, typical question patterns, and examples aligned with the realities of empirical research in South Africa (dataset quirks, policy evaluation contexts, and frequent econometric pitfalls).

Section 1: Core Econometric Foundations and Exam-Style Setup (ECO4006F)

What the exam is really testing

Even when the course title is “Econometrics,” the exam usually checks a set of competencies that repeat across universities:

1. Model specification skills

   • Can you write down the correct regression equation (including functional form and error structure)?
   • Can you explain what each variable represents economically?
   • Can you distinguish between levels, logs, growth rates, and dummy variables?

2. Assumption literacy

   • Do you clearly understand exogeneity vs endogeneity?
   • Do you know which assumptions justify OLS unbiasedness, consistency, and valid inference?
   • Can you interpret independence/uncorrelatedness conditions in an errors-in-variables context?

3. Estimator and inference fluency

   • Can you derive or at least correctly state OLS, GLS/FGLS, and IV estimators?
   • Can you interpret standard errors, t-stats, F-tests, and confidence intervals?
   • Can you connect theory to outputs: coefficient meaning, partial effects, and marginal interpretations?

4. Diagnostics and robustness

   • Can you identify likely violations (heteroskedasticity, serial correlation, multicollinearity, model misspecification)?
   • Can you choose correct tests and interpret outcomes?
   • Can you propose remedies (robust SEs, transformations, differencing, HAC, WLS/GLS, re-specification)?

5. Communication

   • Can you structure solutions with definitions → steps → final result?
   • Can you explain “why it matters,” not just compute numbers?

Your answers should sound like an econometrics argument, not only like a calculation.

The standard linear regression model (SLRM) you must master

Most exam questions anchor on the multiple linear regression model:

[
y_i = x_i'\beta + u_i,\quad i=1,\dots,n
]

where:

• (y_i) is the dependent variable (e.g., earnings, expenditure, unemployment duration proxy),
• (x_i) is a vector of regressors,
• (\beta) is the vector of parameters,
• (u_i) is the error term containing unobserved influences.

Key assumptions and what they buy you

Although courses vary in how they formalize assumptions, exams usually rely on a core set. The “money assumptions” for OLS are:

1. Linear-in-parameters: model is correctly written in terms of (\beta).
2. Random sampling / independence across observations (sometimes relaxed depending on serial correlation topics).
3. Exogeneity (or conditional mean zero):
   [
   \mathbb{E}[u_i|x_i]=0
   ]
4. No perfect multicollinearity: regressors are not perfectly linearly dependent.
5. Finite variance (for asymptotic results and for standard errors):
   [
   \mathbb{E}[u_i^2|x_i]<\infty
   ]

Under these, OLS is:

• Unbiased for (\beta),
• Consistent (in large samples),
• Asymptotically normal (enables inference),
• BLUE (best linear unbiased estimator) under homoskedasticity.

Homoskedastic vs heteroskedastic errors

A frequent exam branching is whether the variance of errors is constant:

• Homoskedastic:
  [
  Var(u_i|x_i)=\sigma^2
  ]
• Heteroskedastic:
  [
  Var(u_i|x_i)=\sigma_i^2 \neq \sigma^2
  ]

Even with exogeneity, heteroskedasticity affects inference:

• OLS coefficients remain unbiased/consistent (under exogeneity),
• but the usual OLS standard errors are wrong unless you use robust methods (e.g., HC-type robust SEs).

Log-linear, level-level, and semi-log interpretations (exam traps)

Students often lose marks by interpreting coefficients incorrectly. You should be comfortable with common forms:

1) Level-level model

[
y_i = \beta_0 + \beta_1 x_i + u_i
]
Here, (\beta_1) is a marginal effect:

• a one-unit increase in (x_i) changes (y_i) by (\beta_1) units.

2) Log-level (semi-log) model

[
\ln(y_i) = \beta_0 + \beta_1 x_i + u_i
]
Approximate interpretation:

• a one-unit increase in (x_i) changes (y_i) by about (100\times\beta_1%) (for small (\beta_1)).

Exact percent change:

• (100 \times (e^{\beta_1}-1)%).

3) Level-log model

[
y_i = \beta_0 + \beta_1 \ln(x_i) + u_i
]
Then (\beta_1) is the change in (y_i) for a proportional change in (x_i).

4) Log-log model (elasticities)

[
\ln(y_i)=\beta_0+\beta_1\ln(x_i)+u_i
]
Then:

• (\beta_1) is the elasticity of (y) with respect to (x): a 1% increase in (x) changes (y) by (\beta_1%).

5) Dummy variables in logs

Exams commonly include dummy regressors. For example, in a log-linear model:
[
\ln(y_i)=\beta_0+\beta_1D_i+u_i
]
If (D_i) changes from 0 to 1, the expected (y) changes by approximately:

• (100 \times (e^{\beta_1}-1)%).

Many students incorrectly say “( \beta_1 \times 100%)” without exponential adjustment. Use the exponential form for correctness; if exams allow approximations, state the approximation clearly.

Writing the econometric question answer structure

A typical ECO4006F-style short problem might ask: “Explain the econometric model and interpret coefficient(s).” Use a consistent skeleton:

1. State the model (equation in correct form).
2. Define variables (economic meaning).
3. State assumptions needed for estimation/inference (e.g., exogeneity).
4. Interpret the coefficient (including correct units/logs).
5. Discuss what could go wrong (endogeneity, omitted variables, measurement error).

This yields points even if the numerical part is weak.

South African empirical contexts that fit these foundations

While this guide does not assume one single dataset, the models you use should match real research themes common in South Africa:

• Education and earnings: log wage on years of education, school quality proxies, gender, and experience.
• Labour markets: unemployment duration proxies, informality, and household structure.
• Housing and inequality: rent/expenditure on property characteristics.
• Policy evaluation: impacts of interventions (e.g., training programmes) requiring credible identification.

These settings often include:

• survey measurement issues,
• region fixed effects,
• cluster sampling (affects standard errors),
• endogeneity from selection (education → earnings; migration → labour outcomes).

Your exam answers should show awareness of these.

Section 2: Estimation and Inference — From OLS to Hypothesis Testing

The OLS estimator: derivation you may be asked to show

OLS minimizes the sum of squared residuals:
[
S(\beta)=\sum_{i=1}^n (y_i-x_i'\beta)^2
]

The closed-form solution is:
[
\hat{\beta}_{OLS}=(X'X)^{-1}X'y
]
when (X'X) is invertible.

Residuals and fitted values

• Fitted values: (\hat{y} = X\hat{\beta})
• Residuals: (\hat{u} = y – X\hat{\beta})

Standard properties

Under exogeneity:

• (\mathbb{E}[\hat{\beta}]=\beta) (unbiasedness in finite samples if strict assumptions hold),
• (\hat{\beta}) is consistent in large samples.

Variance of OLS and standard errors

A standard exam statement is:

• If errors are homoskedastic and uncorrelated:
  [
  Var(\hat{\beta}|X)=\sigma^2 (X'X)^{-1}
  ]
• Estimate (\sigma^2) with:
  [
  \hat{\sigma}^2 = \frac{1}{n-k}\sum_{i=1}^n \hat{u}_i^2
  ]
  where (k) is the number of parameters including the intercept (if included).

Then:
[
\widehat{Var}(\hat{\beta})=\hat{\sigma}^2(X'X)^{-1}
]

If heteroskedasticity is present

If:
[
Var(u_i|X)=\sigma_i^2
]
then:
[
Var(\hat{\beta}|X) = (X'X)^{-1}X' \Omega X (X'X)^{-1}
]
where (\Omega) is diagonal with (\sigma_i^2).

Exams often expect you to state the solution at a conceptual level:

• use heteroskedasticity-robust standard errors (e.g., HC0/HC1/HC3 depending on course).

Even if you do not compute them numerically, you should explain why they fix inference.

Hypothesis testing: t-tests and F-tests in practice

t-test for single coefficient

For testing:
[
H_0:\beta_j=\beta_{j0}
]
the statistic is:
[
t=\frac{\hat{\beta}j-\beta{j0}}{SE(\hat{\beta}_j)}
]
Under (H_0), it follows a t-distribution with appropriate degrees of freedom (often (n-k) in small samples under homoskedasticity).

Interpretation:

• reject if (|t|) exceeds critical value (or if p-value < significance level).

F-test for joint restrictions (common exam task)

Testing:
[
H_0: R\beta = r
]
where (R) is an (m \times k) restriction matrix and (r) is an (m \times 1) vector.

Key logic:

• Compare restricted vs unrestricted sum of squared residuals.
• Statistic:
  [
  F=\frac{(SSR_r-SSR_u)/m}{SSR_u/(n-k)}
  ]
  In terms of R-squared:
• Many courses also express via (R^2) but the SSR form is more robust for exams.

Example exam prompt

Suppose you test whether “education variables jointly matter”:

• Model A includes education, experience, and years since education.
• Model B restricts coefficients on education-related variables to zero.

In your answer:

1. State the null: coefficients on those education variables jointly equal zero.
2. Identify restricted/unrestricted models.
3. Compute or state the F-statistic formula as appropriate.
4. Conclude: reject/not reject and interpret economically.

Confidence intervals: more than just “add margin of error”

A confidence interval for (\beta_j) at level (1-\alpha) is:
[
\hat{\beta}j \pm t{\alpha/2,,df} \cdot SE(\hat{\beta}_j)
]
You should interpret intervals in words:

• “We cannot rule out effects of size …” not “It is true that…”

Partial effects, marginal interpretations, and policy meaning

A common exam mark scheme values interpretation tied to economics.

If the dependent variable is income (levels)

A coefficient on log expenditures cannot be treated as a unit change; you must convert.

If the dependent variable is log wage

If (\ln(wage)) is regressed on education years:

• the coefficient indicates the percentage effect of one extra year (approx/exact).

In policy evaluation contexts

If there is a binary treatment dummy (D) in log outcome model:

• interpret (e^{\beta_D}-1) as approximate percent change in expected outcomes comparing treated vs untreated.

Model selection: avoiding “wrong model” arguments

Exams often include:

• Which variables should be included?
• Should you transform variables?
• Should you use interaction terms or polynomial terms?

Answer with econometric reasoning:

• Use theory for inclusion (avoid post hoc variable hunting).
• Use transformations if they address:
  • nonlinearity,
  • variance instability,
  • interpretability.
• Use interactions when theory suggests heterogeneity:
  [
  y_i = \beta_0 + \beta_1 x_i + \beta_2 Z_i + \beta_3 (x_i Z_i)+u_i
  ]
  and interpret (\beta_3) as “how the marginal effect of (x) changes with (Z).”

Section 3: Assumption Violations, Diagnostics, and Robustness

Why diagnostics are exam-friendly

Many econometrics exams contain a diagnostic component because it tests whether you can connect:

• theory → observable symptoms → tests → consequences → remedies.

Even if you cannot compute a test statistic, you should be able to:

• state what test would detect the issue,
• describe what it means,
• propose an adjustment.

Multicollinearity: signs and remedies

Multicollinearity occurs when regressors are linearly related enough that it becomes hard to separately identify effects.

Symptoms:

• large standard errors,
• coefficients with inconsistent signs across specifications,
• weak t-tests despite plausible individual variables.

The exam-grade explanation:

• OLS estimates remain unbiased under exogeneity, but standard errors inflate.
• Predictions may still be accurate; interpretation of individual coefficients becomes less reliable.

Common diagnostic: Variance Inflation Factor (VIF).

• Many courses expect:
  • compute VIF for each regressor or define it.
  • If VIF is high (often >10 in practice), multicollinearity is a concern.

Remedies:

• drop redundant variables (theoretically justified),
• combine correlated variables (index construction),
• center variables when interactions are present,
• increase sample size (if possible).

Heteroskedasticity: testing and what changes

Why heteroskedasticity is common

Real-world survey data frequently produce heteroskedasticity because:

• different groups have different variances,
• measurement error scales with income or education,
• cap/ceiling effects (e.g., minimum wage, data censoring approximations).

Tests

Common tests taught:

• Breusch–Pagan (based on auxiliary regression of squared residuals on regressors).
• White test (based on squared residuals regressed on all regressors and their squares/cross-products).
• Sometimes Goldfeld–Quandt for specific structures.

In an exam answer:

1. State what heteroskedasticity implies:
   • OLS standard errors under homoskedasticity are biased.
2. Mention how it affects inference:
   • t-tests and F-tests can be invalid.
3. Provide remedy:
   • use heteroskedasticity-robust SEs.

Example interpretation strategy

If a coefficient is significant under naive SEs but not with robust SEs, you should conclude:

• “The point estimate may still be consistent, but statistical significance is sensitive to variance assumptions.”

That kind of nuanced interpretation scores well.

Serial correlation / autocorrelation

Serial correlation arises when errors across time are correlated:
[
Cov(u_t, u_{t-1}) \neq 0
]
This is especially plausible with:

• time-series macro variables,
• repeated cross-sections with panel components,
• variables with persistence (inflation, unemployment rates, expenditure smoothing).

Symptoms:

• residual plots show clustering,
• Durbin–Watson statistic deviates from 2,
• inefficient standard errors under naive assumptions.

Remedies:

• Newey–West (HAC) robust standard errors for certain time-series contexts,
• use appropriate dynamic specifications,
• consider differencing or GLS when structure is known (but exams often focus on SE adjustments).

Endogeneity and omitted variable bias

This is one of the most exam-critical topics because it links directly to bias and identification.

Omitted variable bias (OVB)

Suppose the true model is:
[
y = \beta x + \gamma z + u
]
but you regress:
[
y = \delta x + e
]
If (z) is omitted and correlated with (x), then (\delta) is biased:
[
\delta = \beta + \gamma \cdot \frac{Cov(x,z)}{Var(x)}
]
(under simplifying assumptions).

In SA data contexts, omitted variables are common:

• ability affects education and earnings,
• local quality affects schooling and employment,
• household motivation affects both consumption and savings behavior.

Exam question patterns:

• “Explain why OLS estimates are biased.”
• “Describe an IV strategy to address endogeneity.”

Your response should be explicit about correlation paths:

• “Since (z) influences both (x) and (y), (E[u|x]\neq 0).”

Reverse causality

Even if there is no omitted variable bias, endogeneity can arise through reverse causality:

• e.g., higher income could increase education, rather than education increasing income.

Exam solution:

• state simultaneity,
• suggest instruments or natural experiments,
• explain why time ordering or external variation might help.

Functional form and specification error

Functional form matters. If the true relationship is nonlinear but you estimate a linear model, you get:

• biased estimates (in nonlinear settings),
• misleading inference,
• poor residual diagnostics.

Diagnostics/approaches:

• include polynomial terms (x^2),
• include logs,
• consider piecewise linear splines if allowed,
• plot residuals against fitted values or against key regressors.

Exam writing:

• show how you would detect misspecification (e.g., residual patterns),
• propose re-specification and re-estimation,
• interpret changes in sign/magnitude.

Interactions and heterogeneous effects diagnostics

If you suspect treatment effects differ by subgroup (e.g., urban vs rural, male vs female, employed vs unemployed), you may specify:
[
y_i = \beta_0 + \beta_1 D_i + \beta_2 W_i + \beta_3 (D_i W_i) + u_i
]
Then:

• effect of (D) when (W=0): (\beta_1),
• effect of (D) when (W=1): (\beta_1+\beta_3).

Diagnostics:

• test (\beta_3=0) for homogeneity,
• interpret robustly if standard errors are adjusted.

Section 4: Advanced Estimation — GLS/FGLS, IV/2SLS, and Consistency Logic

GLS and the intuition for efficiency

When errors are heteroskedastic and/or correlated, OLS remains unbiased under exogeneity but inefficient. If you know the structure of (\Omega), you can use GLS.

General regression form:
[
y = X\beta + u,\quad Var(u|X)=\Omega
]
Then:
[
\hat{\beta}_{GLS}=(X'\Omega^{-1}X)^{-1}X'\Omega^{-1}y
]

FGLS (Feasible GLS):

• estimate (\Omega) from the data,
• then run GLS using the estimated (\hat{\Omega}).

Exam answers should include:

• what GLS does conceptually (weights observations inversely proportional to variance),
• when it is appropriate (known/estimable error structure),
• how it improves efficiency (smaller variance than OLS).

Practical exam example: groupwise heteroskedasticity

Suppose you have a regression where variance differs by region. If you model:

• (Var(u_i|X)=\sigma_g^2) for group (g),
  you can estimate (\sigma_g^2) from residuals, construct a diagonal (\hat{\Omega}), and apply FGLS.

Then you may discuss:

• OLS coefficient consistency under exogeneity,
• improved standard errors under GLS/FGLS,
• interpretation stability.

Instrumental Variables (IV): the identification core

Endogeneity makes OLS fail. IV/2SLS can restore consistency if instruments satisfy conditions.

Instrument relevance

The instrument (Z) must be correlated with the endogenous regressor (x):
[
Cov(Z,x)\neq 0
]
In exam answers you can phrase as “first-stage strength.”

Instrument exogeneity (exclusion)

The instrument affects (y) only through (x), not through the error term (u):
[
Cov(Z,u)=0
]
This is the exclusion restriction.

The structural vs reduced form

Structural equation:
[
y = X\beta + u
]
If one regressor in (X) (say (x_1)) is endogenous, split:
[
y = \beta_0 + \beta_1 x_{1} + \beta_2 x_2 + u
]
Instrumented by (Z) that relates to (x_1).

First stage:
[
x_{1} = \pi_0 + \pi_1 Z + \pi_2 x_2 + v
]
Second stage:
[
y = \beta_0 + \beta_1 \hat{x}_1 + \beta_2 x_2 + u
]
2SLS uses (\hat{x}_1) from first stage in second stage.

Weak instruments: a common exam theme

Weak instruments lead to:

• biased IV estimates,
• incorrect inference (standard errors unreliable),
• first stage F-statistics that are too low.

Many exam problems expect you to:

• describe what happens when relevance is weak,
• mention robustness checks:
  • weak-instrument robust inference (if taught),
  • alternative instruments,
  • larger samples.

You must be careful in writing: “weak instruments bias estimates toward OLS” is often the exam-level statement.

Local Average Treatment Effect (LATE) language (if covered)

In treatment contexts with noncompliance, IV identifies LATE for “compliers.” If your course covers it, ensure consistency:

• discuss that IV identifies the causal effect for those who change treatment status due to the instrument.

Even if not detailed, exam questions may use LATE vocabulary to test conceptual understanding.

Overidentification tests (Sargan/Hansen)

If you have more instruments than endogenous variables, you can test overidentifying restrictions.

Concept:

• if instruments are valid, residuals should not be correlated with the “extra” instruments after controlling for endogenous regressor(s).

Exam writing:

• “Rejecting the null suggests instruments are not exogenous jointly.”
• “Failing to reject does not prove validity; it only indicates no strong evidence against exogeneity.”

Example endogeneity and IV strategy relevant to South Africa

A classic research narrative:

• studying effect of education on wages,
• education may be endogenous due to ability/quality unobserved factors.

Possible instruments in principle:

• policy-driven variation (e.g., changes in schooling mandates or targeted funding),
• geographic distance to institutions,
• variation in historical access.

Exam answers typically require:

1. identify endogenous regressor,
2. propose instrument(s),
3. justify relevance and exclusion,
4. describe two-stage estimation.

Because real instrument selection is contextual, the marks usually reward clear logic rather than naming a specific historical policy—unless your course provided examples.

Interpreting IV estimates

In IV, coefficient interpretation differs from OLS in presence of heterogeneity. But exam-level interpretation often remains:

• “The estimated effect of (x_1) on (y) among instrument-induced variation.”

If the exam expects formal statements, keep them consistent with the IV assumptions.

Section 5: Putting It All Together — Exam Execution, Common Question Types, and Rapid-Response Checklists (ECO4006F)

How to structure answers under time pressure

ECO4006F exams often include mixed question types:

• derivation,
• conceptual explanation,
• multiple parts,
• short calculations.

Use an answer structure that ensures each part is “complete by itself.”

A high-scoring template for multi-part problems

For each sub-question:

1. Identify what is being asked
   • estimation? testing? interpretation? diagnostics? remedy?
2. Write the key formula(s)
   • estimator, test statistic, or model equation.
3. State assumptions explicitly
   • e.g., “under homoskedasticity,” “under exogeneity,” “given instrument validity.”
4. Compute conceptually or mathematically
   • if numerical values exist, compute cleanly.
5. Conclude in plain language
   • “Reject at 5%,” “evidence suggests heteroskedasticity,” “OLS inconsistent due to endogeneity,” etc.

Even when you struggle with computation, step 2–3–5 usually gains partial marks.

Rapid checklist: OLS inference reliability

Before claiming “significant,” ensure:

• Exogeneity: Is (E[u|X]=0) plausible?
• Multicollinearity: are standard errors inflated?
• Homoskedasticity: do you need robust SEs?
• Independence: any clustering/serial correlation?
• Correct functional form: logs vs levels interpreted properly.

If you cannot confirm all, the exam expects you to acknowledge what could invalidate inference and propose robust SEs or re-specification.

Rapid checklist: diagnosing heteroskedasticity and responding

If asked:

• “Would you use OLS or WLS?”
• “What test would you perform?”
• “How would you correct standard errors?”

Answer structure:

1. State problem: (Var(u_i|X)) not constant.
2. Mention test: Breusch–Pagan/White (depending on what course emphasizes).
3. State implication: usual SEs wrong.
4. Recommend remedy:
   • robust SEs if you don’t want to impose structure,
   • WLS/GLS if you can model variance structure.

Rapid checklist: handling serial correlation

If asked about time series/panel:

1. define serial correlation,
2. identify consequence: inefficient estimates and biased standard errors under naive formulae,
3. mention remedy: Newey–West (HAC) or model adjustment (depending on course),
4. interpret results cautiously.

Rapid checklist: IV/2SLS exam questions

When you see “endogeneity” or “causal effect with OLS bias,” do:

1. Identify endogenous regressor (x_1).
2. Propose instrument(s) (Z).
3. State assumptions:
   • relevance: (Cov(Z,x_1)\neq 0),
   • exogeneity/exclusion: (Cov(Z,u)=0).
4. Write first stage and second stage (2SLS).
5. Mention weak instrument concerns.
6. If overidentified: mention Sargan/Hansen logic.

Typical question types and what markers look for

Type A: derive and interpret coefficients

What markers reward:

• correct algebra (if derivation),
• correct unit interpretation,
• correct log interpretation and dummy-variable interpretation.

Common mistake:

• interpreting (\beta) in a log model as linear units.

Type B: “Explain why OLS is biased” (conceptual)

Markers reward:

• clear causal chain explaining endogeneity:
  • omitted variable affects both (x) and (y),
  • measurement error correlates with regressor,
  • reverse causality.

Type C: “Propose an econometric remedy”

Markers reward:

• correct technique match:
  • heteroskedasticity → robust SEs/WLS,
  • serial correlation → HAC or GLS,
  • endogeneity → IV/2SLS or redesign.

Type D: “Choose and justify a model specification”

Markers reward:

• theory + econometric reasoning:
  • include interactions if heterogeneity expected,
  • include fixed effects if omitted time-invariant heterogeneity is likely (if covered),
  • consider transformations when variance changes with scale.

Type E: “Diagnostic and interpretation”

Markers reward:

• linking test result to consequences:
  • e.g., “Reject homoskedasticity → use robust SEs to restore valid inference.”

Mini case studies: practice scenarios (exam realism)

Because exam questions often mirror empirical work, practice answering in “research narrative” form.

Case Study 1: Wage equation with education and heteroskedasticity concern

Suppose you estimate:
[
\ln(wage_i)=\beta_0+\beta_1 education_i + \beta_2 experience_i + u_i
]
You suspect heteroskedasticity because higher education groups may have different variance in wages.

Exam answer should include:

• why heteroskedasticity affects SEs,
• that OLS coefficients can still be consistent if exogeneity holds,
• solution: robust SEs or GLS if structure known,
• interpret (\beta_1) as percent change in expected wage for one additional year of education:
  • (100\times (e^{\beta_1}-1)%).

If a test rejects homoskedasticity, conclude:

• “Statistical significance may change; robust SEs should be used.”

Case Study 2: Endogeneity in training programmes

You estimate:
[
y_i=\beta_0+\beta_1 training_i+\beta_2 X_i+u_i
]
where (training_i) is endogenous due to selection (motivated individuals seek training).

Exam answer should:

• explain why (E[u|training]\neq 0),
• propose IV:
  • define instrument (Z) affecting training participation but not (u) directly,
• outline first and second stages,
• interpret IV coefficient as effect of training among compliers (if LATE is taught; otherwise, among instrument-induced variation).

Case Study 3: Serial correlation in monthly inflation regression

If a course covers time series, you may be asked:

• inflation depends on lagged inflation and other macro variables,
• residuals may be serially correlated.

Exam answer:

• define serial correlation,
• explain effect on standard errors and tests,
• propose HAC/Newey–West corrections or correct dynamic structure if appropriate.

Worked “answer writing” examples (no numbers needed, logic counts)

Example: interpret a coefficient on a dummy in log model

Question-style: “In a model (\ln(y)=\beta_0+\beta_1 D + u), interpret (\beta_1).”

High-score response:

• “When (D) changes from 0 to 1, expected (y) changes by (100\times(e^{\beta_1}-1)%). This is a multiplicative effect on the outcome.”

Example: state consequences of heteroskedasticity

Question-style: “Errors are heteroskedastic. Are OLS estimates biased?”

High-score response:

• “Under exogeneity (E[u|X]=0), OLS coefficients remain unbiased/consistent. However, the usual homoskedastic standard errors are incorrect, so t- and F-tests can be invalid unless robust SEs (or WLS/GLS) are used.”

Example: explain IV assumptions

Question-style: “Justify why IV gives consistent estimates.”

High-score response:

• “Consistency requires instrument relevance ((Z) correlated with endogenous regressor) and instrument exogeneity/exclusion ((Z) uncorrelated with the structural error). Then variation in (Z) isolates exogenous movement in the endogenous regressor.”

Final preparation strategy for ECO4006F

To succeed, you need both conceptual mastery and procedural repetition. A strong final-week plan:

1. Rehearse derivations once

   • OLS formula,
   • variance formula,
   • t/F test expressions,
   • 2SLS first/second stage logic.

2. Rehearse diagnostic choices

   • heteroskedasticity → which test → what remedy,
   • serial correlation → which correction/remedy,
   • endogeneity → IV logic.

3. Rehearse interpretation

   • logs and dummies,
   • interaction marginal effects.

4. Write answers in complete sentences

   • Even when formulas are right, lacking explanation can lose marks.

5. Do timed past-paper practice

   • Focus on finishing structured answers rather than perfect algebra.

South African institutional study considerations (UCT Economics Study Guides cluster relevance)

Because this document belongs to the broader collection of University of Cape Town (UCT) Economics Study Guides, it aligns with an exam culture that often emphasizes rigorous reasoning, clean mathematical statements, and strong interpretation. In SA settings, students also benefit from:

• treating standard errors and diagnostics as central (not optional),
• tying econometric choices to credible research problems familiar in local labour, housing, education, and policy analysis,
• using robust language when assumptions are contested.

Many students lose marks by claiming certainty when assumptions are questionable. The exam generally rewards cautious, assumption-aware econometrics.

Consolidated “What to Memorize” list (for the last 48 hours)

• OLS estimator: (\hat{\beta}=(X'X)^{-1}X'y)
• Exogeneity: (E[u|X]=0)
• Homoskedastic variance: (Var(u_i|X)=\sigma^2)
• t-test: ((\hat{\beta}j-\beta{0})/SE(\hat{\beta}_j))
• F-test joint restrictions: compare SSR restricted/unrestricted
• Heteroskedasticity: robust SEs fix inference
• GLS: ((X'\Omega^{-1}X)^{-1}X'\Omega^{-1}y)
• 2SLS:
  • first stage: endogenous regressor on instruments and exogenous covariates,
  • second stage: outcome on fitted endogenous regressor
• IV assumptions:
  • relevance and exclusion/exogeneity
• Log interpretations:
  • log-log coefficients are elasticities,
  • dummy in log model: percent change (100(e^{\beta}-1)%)

With these in place, you can convert most ECO4006F exam questions into a methodical solution rather than a panic response.