ECO4016S: Econometrics Exam Notes — University of Pretoria (UP) South African Focus

Econometrics is where economic theory meets data: it teaches you how to estimate relationships, test hypotheses, and make reliable inferences under uncertainty. For ECO4016S, exam questions typically combine econometric intuition with formal methods—especially around OLS assumptions, inference (t/F tests), heteroskedasticity, autocorrelation, model specification, and the practical use of diagnostics and robust methods. These notes are built to help you answer both calculation-heavy and concept-heavy questions using South African higher education expectations and common exam styles.

Pretoria University Cluster: ECO4016S Econometrics Exam Notes (University of Pretoria)

Course framing for ECO4016S: what the exam tends to test

At South African universities—especially in departments where ECO modules align with core econometrics syllabi—the ECO4016S exam often rewards students who can:

1. Translate a problem statement into a regression model
   • Define dependent and independent variables correctly.
   • Decide whether terms like logs, dummies, interactions, or lags are needed.
2. Apply econometric reasoning
   • State assumptions behind OLS (linearity in parameters, exogeneity, no perfect multicollinearity).
   • Explain when OLS remains valid or fails (e.g., endogeneity, omitted variables).
3. Compute and interpret key statistics
   • Standard errors, t-statistics, F-statistics, confidence intervals.
   • Interpretation of coefficients in levels vs logs, and dummy variable effects.
4. Diagnose and correct common violations
   • Heteroskedasticity (and using robust/GLS approaches).
   • Serial correlation (and using HAC/Newey–West or AR errors).
5. Write exam-ready justification
   • Not just “use robust SE”, but why standard SE are wrong and what assumptions change.

A typical exam flow may look like: a short data generating process description → estimate OLS → test assumptions → adjust standard errors → interpret results and possibly discuss policy implications.

To help you practice, these notes repeatedly use a consistent “exam style” template: (i) model specification, (ii) estimation method, (iii) tests/diagnostics, (iv) inference and interpretation, (v) final conclusion.

1) Regression foundations and OLS assumptions (core theory)

The regression model and notation

You must be comfortable with the general linear regression model (multiple regression):

[
y_i = \beta_0 + \beta_1 x_{1i} + \cdots + \beta_k x_{ki} + u_i
]
or in matrix form:
[
\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{u}
]

OLS chooses (\boldsymbol{\beta}) to minimize:
[
\sum_{i=1}^n (y_i – \hat{y}_i)^2
]

The OLS estimator:
[
\hat{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}
]
requires full column rank (i.e., no perfect multicollinearity).

OLS assumptions examiners commonly use

A minimal “exam package” includes:

1. Linearity in parameters
   • Even if you transform variables (logs, squares), the model must be linear in coefficients.
2. Random sampling / correct model form
   • You’re not missing the systematic part of the relationship (at least not in a way correlated with included regressors).
3. Exogeneity (zero conditional mean)
   [
   \mathbb{E}[u_i \mid \mathbf{X}] = 0
   ]
   This is the key assumption for unbiasedness and valid t-tests.
4. Homoskedasticity and no autocorrelation (for standard formulas)
   • Homoskedastic:
     [
     \mathbb{V}[u_i\mid \mathbf{X}] = \sigma^2
     ]
   • No autocorrelation (in time series settings):
     [
     \text{Cov}(u_i, u_j\mid \mathbf{X}) = 0 \quad \text{for } i\neq j
     ]
5. No perfect multicollinearity
   • At least one regressor must vary independently.

Consequences you must state

Under the full set of classical assumptions:

• (\mathbb{E}[\hat{\boldsymbol{\beta}}]=\boldsymbol{\beta}) (unbiasedness)
• (\text{Var}(\hat{\boldsymbol{\beta}})=\sigma^2(\mathbf{X}'\mathbf{X})^{-1})
• Standard error formula is valid:
  [
  \text{se}(\hat{\beta}j)=\sqrt{\hat{\sigma}^2 c{jj}}
  ]
  where (c_{jj}) is the (j)-th diagonal element of ((\mathbf{X}'\mathbf{X})^{-1}).

When assumptions fail, coefficients may remain unbiased but standard inference may break. Exogeneity failure (endogeneity) is particularly dangerous: it makes coefficient estimates biased/inconsistent.

2) Interpreting coefficients: levels, logs, and dummy variables

Marginal effects in a linear model

In a levels-levels regression:
[
y_i = \beta_0 + \beta_1 x_i + u_i
]
(\beta_1) is the expected change in (y) for a 1-unit change in (x).

Semi-log models (log of dependent or independent variable)

1. Log of dependent variable:
   [
   \ln(y_i)=\beta_0+\beta_1 x_i+u_i
   ]
   Then approximately:
   [
   % \Delta y \approx 100 \cdot \beta_1 \cdot \Delta x
   ]
   So if (x) increases by 1 unit, (\beta_1) is approximately the percentage change in (y) divided by 100.

2. Log of independent variable:
   [
   y_i=\beta_0+\beta_1 \ln(x_i)+u_i
   ]
   A 1% increase in (x) increases (y) by approximately:
   [
   \Delta y \approx \beta_1 \cdot 0.01
   ]

Double-log (elasticity)

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

• A 1% increase in (x) changes (y) by approximately (\beta_1%).

Dummy variables: careful interpretation

If you include a dummy (D_i\in{0,1}):

• In a levels model (y_i = \dots + \delta D_i + u_i), the coefficient (\delta) is the difference in average (y) between groups.
• In a log-dependent model, the dummy coefficient requires an exponential interpretation:
  [
  \ln(y_i)=\dots+\delta D_i+u_i
  ]
  If (D) switches from 0 to 1, predicted (y) multiplies by (e^\delta), so the percentage change is:
  [
  (e^\delta – 1)\times 100%
  ]
  In exams, they may test whether you can distinguish between “approx” and “exact exponential” interpretations. The clean rule: log-dependent + dummy → exponentiate.

3) Hypothesis testing and inference: t-tests and F-tests

t-test logic

Given:
[
H_0:\beta_j=\beta_{j0}
]
The test statistic:
[
t = \frac{\hat{\beta}j-\beta{j0}}{\text{se}(\hat{\beta}_j)}
]
Decision rule uses critical values from the t distribution (or asymptotically normal depending on assumptions, sample size, and course style).

Interpretation matters:

• A statistically significant coefficient (e.g., 5% level) means you reject (H_0) and conclude evidence suggests (\beta_j\neq \beta_{j0}).
• But statistical significance does not automatically imply economic significance.

F-test logic (joint restrictions)

For joint hypotheses:
[
H_0: \mathbf{R}\boldsymbol{\beta}=\mathbf{r}
]

• Compare unrestricted model with restricted model.
• Compute:
  • Either the classical F statistic using SSR (sum of squared residuals),
  • Or using (R^2) and degrees of freedom.

A common exam form:
[
F = \frac{(SSR_r – SSR_{ur})/q}{SSR_{ur}/(n-k-1)}
]
where:

• (q) is number of restrictions,
• (SSR_r) is restricted sum of squared residuals,
• (SSR_{ur}) is unrestricted SSR.

You should always state degrees of freedom correctly; examiners often deduct marks for mismatched df.

Example-style question pattern

A typical short question might ask:

“Test whether (x_1) and (x_2) jointly have no effect on (y) at the 5% level.”

You respond with:

1. Estimate unrestricted model.
2. Estimate restricted model with (x_1) and (x_2) excluded.
3. Compute F statistic.
4. Compare with critical value or compute p-value.
5. Conclude: reject or fail to reject.

If robust SE are used for heteroskedasticity, the F-test logic may require robust versions (e.g., Wald test). Exams often check whether you understand that standard F based on classical homoskedasticity may not apply if errors are heteroskedastic.

4) Error terms: heteroskedasticity and autocorrelation diagnostics

Why heteroskedasticity matters

If:
[
\mathbb{V}[u_i\mid \mathbf{X}] = \sigma_i^2
]
then OLS remains unbiased if exogeneity holds, but standard errors computed under homoskedasticity are wrong. This affects t-tests and confidence intervals.

Symptoms:

• Residual variance increases with fitted values or a regressor.
• Known theoretical reasons: measurement error, variance proportionality, different groups, micro data with different volatility.

Breusch–Pagan / White tests (common exam content)

Even if the exam does not require you to implement the test formula from scratch, you should know:

• The idea: regress squared residuals on variables (or functions) to detect systematic variance patterns.
• Under (H_0) (homoskedasticity), residual variance is constant.
• The test statistic is compared to a chi-square distribution (common in large-sample approximations).

A typical question might provide:

• A test statistic value and p-value.
• Or a regression output for auxiliary regression of (e_i^2) on regressors.

What you should do:

1. State the null hypothesis of constant variance.
2. Interpret p-value relative to significance level.
3. Conclude if heteroskedasticity is present.
4. Mention solution: robust standard errors (e.g., HC variants) or GLS if structure is known.

Autocorrelation in time series (common in applied economics)

If errors are correlated over time:
[
\text{Cov}(u_t, u_{t-s}) \neq 0
]
then:

• OLS coefficients remain unbiased under exogeneity,
• but usual variance formulas are wrong because covariance structure differs.

Classic diagnostic: Durbin–Watson (often taught) for AR(1) patterns, but more robust methods include Breusch–Godfrey and HAC/Newey–West.

If exam uses AR(1) error:
[
u_t = \rho u_{t-1} + \varepsilon_t
]
then standard errors need adjustment, or you may perform feasible GLS.

HAC / Newey–West (what to say)

A strong exam answer states:

• Newey–West provides consistent standard errors under general forms of heteroskedasticity and autocorrelation.
• You choose a lag length (bandwidth) based on sample size and expected correlation length.
• The regression coefficients remain the same, but inference changes.

If a question provides a “lag truncation” value and asks for interpretation, you can say: larger lag truncation protects against longer-run autocorrelation but may increase estimator variability.

5) Model specification: omitted variables, functional form, and endogeneity

Omitted variable bias: a must-know derivation

Suppose the true model is:
[
y_i = \beta_1 x_i + \gamma z_i + u_i
]
but you estimate:
[
y_i = \alpha_0 + \alpha_1 x_i + v_i
]
where (z_i) is omitted. Then:
[
\text{plim}(\hat{\alpha}_1) = \beta_1 + \gamma \frac{\text{Cov}(x_i,z_i)}{\text{Var}(x_i)}
]

• If (z_i) affects (y) ((\gamma\neq 0)) and (z_i) correlates with (x_i), your estimate is biased.
• Direction depends on the signs of (\gamma) and (\text{Cov}(x,z)).

Exams love conceptual questions like:

• “Assume omitted variable is positively correlated with (x). Does coefficient overestimate or underestimate?”
  You answer:
• If omitted variable increases (y) and is positively correlated with (x), the coefficient on (x) tends to be biased upward.

Endogeneity sources you can list

Examiners like when you can name endogeneity sources:

• Omitted variables
• Simultaneity / reverse causality
• Measurement error in regressors
• Selection bias
• Dynamic panel bias if using pooled OLS when lagged dependent variable correlates with fixed effects (if panels were covered)

Even in a cross-sectional econometrics exam, they may introduce endogeneity through story problems:

• Wages and education,
• Interest rates and investment,
• Prices and demand simultaneously.

Functional form errors

Functional form matters: using linear vs log vs polynomial terms incorrectly can bias coefficients if it induces correlation between regressor and error. A test strategy may include:

• Ramsey RESET test (if covered): check for neglected nonlinearities by adding fitted-value powers to auxiliary regression.
• Residual plots and theoretical restrictions.

A strong exam answer:

• States why functional form misspecification leads to misinterpretation and potential bias.
• Suggests remedial steps: transformation, adding interaction terms, using theory-based specification.

6) Multicollinearity, degrees of freedom, and practical inference

What multicollinearity does (and does not do)

If regressors are highly correlated, OLS remains unbiased under exogeneity, but:

• Standard errors inflate because ((\mathbf{X}'\mathbf{X})^{-1}) becomes ill-conditioned.
• Individual t-tests become unreliable.
• Overall fit (F-test for joint significance) may still be significant.

You should be able to:

• Distinguish multicollinearity from heteroskedasticity.
• Mention remedy: drop redundant variables, combine them, collect more data, or use theory-guided regularization (though econometrics courses typically focus on inference rather than machine learning).

Variance Inflation Factor (VIF) concept (if mentioned)

If the exam uses VIF:
[
VIF_j = \frac{1}{1-R_j^2}
]
Large VIF suggests multicollinearity. Provide interpretation rather than recalculating unless asked.

7) Practical exam workflow: from question to final statement

This section translates theory into a reliable “exam method”.

Step-by-step strategy for typical regression questions

1. Read the data-generating description
   • Identify dependent variable (y).
   • Identify regressors (x).
   • Check if any transformation is specified (logs).
2. Write the econometric model explicitly
   • Include intercept.
   • Include dummy variables if group differences exist.
   • Include lags if dynamic behavior is mentioned.
3. State estimation method
   • If classical OLS assumptions seem plausible: OLS.
   • If heteroskedasticity/autocorrelation suspected: robust SE / HAC.
   • If endogeneity suspected: discuss IV or fail to justify (depending on course scope).
4. Compute required statistics
   • Coefficients, standard errors.
   • t-tests for individual coefficients.
   • F-test or joint test for groups.
5. Check diagnostics
   • Heteroskedasticity tests (Breusch–Pagan/White).
   • Autocorrelation tests (Durbin–Watson/Breusch–Godfrey; or Newey–West).
   • Specification tests (RESET) if included.
6. Interpret economically
   • Translate sign and magnitude into plain language.
   • Emphasize what assumptions are needed for validity.
7. Conclude with a correct hypothesis statement
   • Reject/fail to reject with the correct level.

A worked mini-example (with consistent numbers)

Imagine the exam provides output for:
[
\ln(\text{wage}_i)=\beta_0+\beta_1 \text{educ}_i + \beta_2 \text{exper}_i + u_i
]
Suppose the estimated coefficients are:

• (\hat{\beta}_1 = 0.045) with robust SE (=0.015),
• (\hat{\beta}_2 = 0.010) with robust SE (=0.008).

Now interpret (\beta_1):

• A 1-unit increase in educ increases wage by approximately:
  [
  100 \times 0.045 = 4.5%
  ]
  Use t-statistics:
• For education:
  [
  t = 0.045/0.015 = 3.0
  ]
  At 5% level, typically reject (H_0:\beta_1=0) (since |t| large).
• For experience:
  [
  t = 0.010/0.008 = 1.25
  ]
  May fail to reject at 5%.

This is exactly the kind of “interpretation + inference” combination exam questions expect.

8) Heteroskedasticity-robust inference: what changes and why

The “same coefficients, different uncertainty” principle

OLS coefficients:
[
\hat{\boldsymbol{\beta}} = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{y}
]
do not depend on the error variance assumption—only on exogeneity and correct regression form.

But:

• Standard errors assume a variance structure.
• Under heteroskedasticity, standard errors must be recalculated using a robust estimator of:
  [
  \text{Var}(\hat{\boldsymbol{\beta}}) = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\Omega \mathbf{X}(\mathbf{X}'\mathbf{X})^{-1}
  ]
  where (\Omega) is diagonal with (\sigma_i^2) (for heteroskedasticity).

What you can claim in an exam answer

A good answer should include:

• Under heteroskedasticity, OLS remains unbiased if (\mathbb{E}[u\mid X]=0).
• But classical SE (homoskedastic) are inconsistent.
• Therefore, you must use robust SE for correct inference.
• If the exam asks you “what happens to t-stats?”:
  • t-statistics change because SE change.
• If coefficients are significant under classical SE but not under robust SE:
  • conclude the earlier inference was unreliable.

9) Autocorrelation correction: intuition and exam language

Serial correlation and why it breaks standard SE

In time series:

• Even if (\mathbb{E}[u_t\mid X]=0),
• the variance of (\hat{\beta}) depends on covariances (\text{Cov}(u_t,u_s)).

If you wrongly assume independence, you under- or over-estimate variance, thus miscompute t- and F-statistics.

Newey–West/HAC: what you should mention

Newey–West essentially:

• estimates the long-run variance of residuals,
• accounts for autocorrelation up to a specified lag.

If asked to choose lag length:

• you explain the trade-off:
  • too small lag length → miss relevant correlation,
  • too large → noisier SE.

10) Endogeneity and instrumental variables (IV) — when classical OLS is not enough

Even if ECO4016S emphasizes classical regression, endogeneity typically appears as a conceptual hurdle.

The problem

If:
[
\mathbb{E}[u_i\mid X]\neq 0
]
then OLS is biased/inconsistent.

A common scenario:

• education and ability: ability affects wages and is in the error term.
• education also correlates with ability.

IV logic (2SLS) in exam-friendly terms

You need an instrument (z) such that:

1. Relevance:
   [
   \text{Cov}(z,x)\neq 0
   ]
2. Exogeneity:
   [
   \mathbb{E}[z u]=0
   ]

Then use two-stage least squares:

1. First stage:
   [
   x_i = \pi_0 + \pi_1 z_i + \text{other terms} + v_i
   ]

2. Second stage:
   [
   y_i = \beta_0 + \beta_1 \hat{x}_i + \text{other terms} + \varepsilon_i
   ]

In an exam, you might be asked to:

• state why IV corrects endogeneity,
• comment on validity of instrument,
• mention weak instruments and its consequences (large standard errors, bias in finite samples).

Instrument validity tests (conceptual)

If output includes first-stage F-stat or similar:

• a low value suggests weak instrument concerns.
• exogeneity is often tested via overidentification tests if multiple instruments exist (Hansen/Sargan), depending on syllabus.

11) Specification tests: RESET and the idea of checking neglected structure

Ramsey RESET: core intuition

RESET checks whether adding nonlinear combinations of fitted values improves the model.

Procedure:

1. Estimate baseline model.
2. Add powers of fitted values (e.g., (\hat{y}^2), (\hat{y}^3)).
3. Test joint significance of these added terms.

Interpretation:

• If they are significant, it suggests functional form misspecification or omitted nonlinearities.

A good exam answer does not claim that RESET “fixes” the model; it motivates re-specification.

12) Putting it together: exam-style case studies tied to South African policy contexts

These mini case studies help you practice translating economics into regression and inference language. They are written with consistent assumptions and avoid requiring actual SA-specific datasets (since exams often provide output tables).

Case Study A: Education and wages with heteroskedastic errors

Problem: Estimate the effect of education on log wages, controlling for experience. Assume residual variance differs by education group.

Model:
[
\ln(wage)=\beta_0+\beta_1 educ+\beta_2 exper+u
]

Exam tasks:

• Interpret (\beta_1).
• Test whether education matters.
• Decide whether to use robust SE.

Typical reasoning:

• Heteroskedasticity test yields p-value below 0.05.
• Conclusion: reject constant variance.
• Use robust SE.
• Re-evaluate t-tests.

Interpretation:
If (\hat{\beta}_1=0.045), interpret as:

• +1 education unit → wage increases by about 4.5%.
  But final significance depends on robust SE.

Exam pitfall:
Students may use classical SE after rejecting homoskedasticity; that yields invalid inference.

Case Study B: Demand and price with potential simultaneity

Problem: Model consumption demand as a function of price and income. Price may be endogenous because unobserved demand shocks affect pricing.

OLS model:
[
\ln(consumption)=\beta_0+\beta_1 \ln(price)+\beta_2 \ln(income)+u
]

Issue:
(\ln(price)) correlates with (u) (simultaneity/reverse causality).

Exam tasks:

• Explain why OLS is inconsistent.
• Propose IV approach with a plausible instrument.

Instrument example:
A policy-driven or tax-based variation affecting price but not directly consumption shocks.

• Must be relevance and exogeneity.

What you write:

• OLS coefficient on price cannot be interpreted causally.
• IV/2SLS can recover causal effect if instrument validity holds.

Pitfall:
Stating “use IV” without discussing instrument conditions.

13) How to score marks in computations: common formula checks

Even if the exam provides regression output, you may need to compute:

1. t-statistics from coefficients and SE
2. F-tests from SSR values or degrees of freedom
3. Approximate percentage changes from log models
4. Elasticities from double-log coefficients

Quick log interpretation calculator skills

• If coefficient is (0.2) on (\ln(y)) with dummy (D):
  • percentage change:
    [
    (e^{0.2}-1)\times 100% \approx 22.1%
    ]
• If coefficient is (0.03) in (\ln(y)=…+\beta x) and (\Delta x=10):
  • approximate percent change:
    [
    100\times 0.03\times 10=30%
    ]

These are the types of calculations that appear in exams to test understanding rather than rote memorization.

14) Final revision checklist for ECO4016S (UP cluster)

Use this list as your last-day checklist:

Model specification

• Correct dependent variable and regressors
• Correct transformations (logs/dummies)
• Intercept included unless the question says otherwise

OLS assumptions and implications

• Exogeneity stated: (\mathbb{E}[u\mid X]=0)
• Explain consequences if assumptions fail
• Multicollinearity: affects variance, not unbiasedness (under exogeneity)

Inference

• t-tests computed correctly
• F-tests: correct df and restriction count
• Interpretation aligns with units/logs/dummies

Diagnostics

• If heteroskedasticity detected → robust SE / correct inference
• If autocorrelation detected → HAC/Newey–West or appropriate correction
• If specification concerns exist → RESET-type reasoning or re-specify with theory

Endogeneity/IV (if mentioned)

• Explain OLS inconsistency due to (\text{Cov}(z,u)\neq 0)
• State instrument relevance and exogeneity
• Mention weak instruments if asked

15) Example “full marks” answer template (how to structure written responses)

When writing exam answers, aim for structure:

1. Write the econometric model clearly.
2. State what you estimate and under what assumptions.
3. State the test/hypothesis and the decision rule.
4. Perform the calculation (if required) and show key intermediate steps.
5. Interpret the result economically.
6. Conclude with explicit rejection/fail-to-reject wording and a brief justification of what changed (e.g., robust SE).

A full-mark response is not only correct; it is complete and coherent.

If you want, I can also generate a set of ECO4016S-style practice questions (with worked solutions) that match the exact patterns above—short regression interpretation, heteroskedasticity-robust inference, t/F testing, RESET conceptual questions, and IV logic—tailored to a South African econometrics exam marking rubric.