ETC3400 Principles of Econometrics
The Complete Exam Bible · S2 2026ETC3400 Overview
- Monash University
- Semester 2, 2026
- Level 3
- Final examination
ETC3400 Principles of Econometrics gives a formal treatment of the statistical foundations behind econometric inference, with maximum likelihood as the organising method.
- Central method Maximum likelihood connects estimation, asymptotics and testing
- Assessment Three assignments total 40%; final exam is 60%
- Exam preparation Rebuild assignment reasoning in unfamiliar settings
- Computing Report reproducible R code and optimisation diagnostics
How ETC3400 is assessed
| Component | Weight | Format |
|---|---|---|
| Assignment 1 | 10% | Individual; detailed algebra plus reproducible R output; due in Week 6 |
| Assignment 2 | 20% | Individual; theory, numerical likelihood and Monte Carlo work; due in Week 9 |
| Assignment 3 | 10% | Individual; due in Week 12; the captured page does not include its attached brief |
| Final Exam | 60% | Individual examination on the e-assessment platform during the examination period |
Current ETC3400 dates
| Date | Item | Control |
|---|---|---|
| 3 Sep 2026 | Assignment 1 | Current Moodle task date |
| 1 Oct 2026 | Assignment 2 | Current Moodle task date |
| 22 Oct 2026 | Assignment 3 | Current Moodle task date |
Dates are as published in the Semester 2, 2026 Moodle assessment pages. Confirm exact deadlines and submission settings in the live LMS.
What ETC3400 covers
Seven connected chapters follow the course from likelihood construction and estimator behaviour to asymptotic theory, numerical optimisation and classical likelihood tests.
Likelihood and Maximum Likelihood Estimation
Build a probability model, likelihood and log likelihood, solve the score equation, then verify curvature and boundaries.Likelihood and Maximum Likelihood Estimation starts with the statistical model: sample space, parameter space, sampling assumptions and a probability law. Once the sample is observed, its joint density is viewed as a function of the parameter.
- Statistical model: possible samples, admissible parameters and a probability law
- Independent sampling and the product form of the joint density
- Likelihood as a function of the parameter with observed data fixed
- Log likelihood equivalence through strict monotonicity of the logarithm
- Score function and first order condition for an interior candidate
- Hessian, concavity and second order verification
Estimators, Sampling Distributions and Bias
Separate a random estimator from its realised estimate, derive sampling distributions, and compare bias, variance and mean squared error.Estimators, Sampling Distributions and Bias separates the rule used before data arrive from the numerical result produced after observation.
- Function notation, domain and codomain for an estimator
- Estimator based on random sample versus estimate based on observed data
- Sampling distribution created by mapping every possible sample
- Multiplicity when several datasets produce the same estimate
- Repeated sample expectation and the definition of bias
- Sampling variance as a measure of dispersion
Information and Minimum Variance Bounds
Use score variation and expected curvature to quantify information, derive Cramer Rao bounds, and test finite sample efficiency.Information and Minimum Variance Bounds connects likelihood curvature with the precision available from a sample. Under regularity conditions, Fisher information can be expressed as score variance or as negative expected log likelihood curvature.
- Score mean and variance under regular likelihood models
- Observed information from sample curvature
- Expected Fisher information and its two equivalent formulas
- Additivity of information for independent observations
- Scalar Cramer Rao lower bound for an unbiased target estimator
- Attainment and finite sample efficiency within a reference class
Convergence, Consistency and Delta Methods
Distinguish convergence modes, prove consistency, expand the likelihood score, and transform limiting distributions.Convergence, Consistency and Delta Methods distinguishes pathwise, probabilistic and distributional limits before using them in likelihood theory.
- Almost sure convergence and sample path interpretation
- Convergence in probability and consistency
- Convergence in distribution through cumulative distribution functions
- Hierarchy among convergence modes and the constant limit special case
- Continuous mapping for probability and distributional limits
- Taylor expansion of the likelihood score around the truth
Vector Likelihood and Normal Linear Models
Move from scalar likelihood to score vectors, information matrices, projection geometry and variance estimator comparisons.Vector Likelihood and Normal Linear Models extends the scalar likelihood argument to parameter vectors, score vectors, Hessian matrices and covariance ordering.
- Score vector, Hessian matrix and multivariate information
- Classical normal linear model Y equals X beta plus u
- Full column rank and invertibility of X prime X
- Maximum likelihood estimator of the coefficient vector
- Projection and residual maker matrices
- Zero covariance, joint normality and independence
Numerical Likelihood Optimisation
Use Newton Raphson and general optimisers, choose stable parameterisations, and report diagnostics that support the numerical result.Numerical Likelihood Optimisation treats software output as a result that must be audited. Newton Raphson uses the score and Hessian to predict the next root, while general routines such as BFGS search using objective and gradient information.
- Newton Raphson as a local linear approximation to the score
- General numerical minimisation of the negative log likelihood
- Starting values and basins of attraction
- Convergence codes, final gradients and objective values
- Positive scale parameters and log transformations
- Bound constrained versus unconstrained algorithms
Likelihood Ratio, Wald and Score Tests
Compare constrained and unrestricted fits, use chi squared reference laws, and explain the curvature geometry behind the three tests.Likelihood Ratio, Wald and Score Tests compares three ways to measure departure from a null hypothesis. Likelihood Ratio uses the objective drop between unrestricted and constrained fits.
- Null and alternative hypotheses as parameter restrictions
- Unrestricted and constrained maximum likelihood estimates
- Likelihood Ratio statistic from the objective difference
- Wald statistic from scaled estimator distance
- Score or LM statistic from null slope and information
- Chi squared reference distribution and restriction count
The course starts by constructing likelihood and log likelihood functions from a probability model, then separates an observed estimate from the random estimator whose repeated sample behaviour can be studied. Bias, variance, Fisher information and minimum variance bounds establish finite sample criteria.
Convergence, consistency, asymptotic normality and the delta method then explain why likelihood procedures support approximate inference in larger samples.
The later teaching blocks extend the scalar argument to vector parameters and the classical normal linear model, where matrix dimensions, projection geometry and covariance ordering matter.
Numerical optimisation is not treated as a black box: students compare starting values, parameterisations, convergence codes, objective values and evaluation counts.
The course closes with Likelihood Ratio, Wald and Score tests, including the geometry that makes them exactly equal in a known variance normal model and asymptotically equivalent more generally.
The official teaching advice says the assignments develop the knowledge, intuition and way of thinking expected in the final examination.
Students should be able to solve assignment style questions from scratch, explain the reasoning behind each line and apply the same ideas in new settings. The final examination is worth 60% and uses the e-assessment platform. The captured source does not state the duration, section structure or permitted materials, so those logistics should be confirmed in the current Moodle exam notice.
Worked example · free
Compare Wald and Likelihood Ratio evidence for a Bernoulli parameter
- +1The unrestricted estimate is 16 divided by 25, which equals 0.64.
- +2The Wald statistic is 25 times (0.64 minus 0.50) squared, divided by 0.64 times 0.36, which is about 2.13.
- +3Evaluate the Bernoulli log likelihood at 0.64 and 0.50. Twice the unrestricted minus constrained value is about 1.98.
- +1Both statistics are below the chi squared one degree of freedom critical value 3.841, so do not reject the null at 5%.
- +1Explain that the finite sample statistics differ because Wald uses a plug in variance at the unrestricted estimate while LR uses the objective drop.
Key terms
- Likelihood function
- The joint probability or density of the observed sample viewed as a function of the parameter. It ranks candidate parameter values but is not a probability distribution over them.
- Maximum likelihood estimator
- A rule that maps each possible sample to a parameter value that maximises the likelihood over the admissible parameter space.
- Sampling distribution
- The probability distribution of an estimator across repeated samples generated by the statistical model.
- Fisher information
- Expected likelihood curvature, equivalently score variance under regularity conditions, that quantifies local statistical precision.
- Consistency
- Convergence in probability of an estimator to the true parameter as sample size grows.
- Delta method
- A first order transformation rule that converts an asymptotic distribution for an estimator into one for a smooth function of that estimator.
- Observed information
- The negative Hessian of the sample log likelihood, evaluated for the data actually observed.
- Profile likelihood
- A likelihood for parameters of interest obtained by maximising over nuisance parameters at each candidate value.
- Score test
- A hypothesis test that scales the log likelihood slope evaluated at the constrained or null estimate by null information.
ETC3400 FAQ
How hard is this course?
The source shows a mathematically progressive course with detailed algebra, matrix work, simulation and numerical optimisation. Difficulty comes from connecting these layers, so weekly problem solving is more useful than memorising isolated formulas.
How should I prepare for the final exam?
Redo the assignments from scratch, explain why every likelihood contribution and theorem applies, and practise transferring the method to a new distribution. The lecturer explicitly describes assignment style reasoning as the best preparation.
What are the assessment weights?
Assignment 1 is 10%, Assignment 2 is 20%, Assignment 3 is 10%, and the Final Exam is 60%. These four official handbook and schedule rows together sum to 100% of the unit assessment structure.
Are past papers available in the captured materials?
The course page includes a past final exam folder and says solutions are provided later in the semester, but this capture contains no complete past paper or past exam solution. Use the original practice here and check Moodle for current availability.
Why is maximum likelihood the central method?
It links model construction, estimation, information, asymptotic distribution theory and hypothesis testing. The weekly topics are designed as one inferential framework rather than unrelated calculation techniques.
Do I need to use R?
Yes. The course materials use R for numerical likelihood and simulation, while permitting another familiar language if students can translate the tutorial instructions. Reproducible, organised and commented code is explicitly expected.
Questions students ask in each ETC3400 chapter
- Why can the log likelihood replace the likelihood?
- The logarithm is strictly increasing for positive inputs, so it preserves the ordering of all likelihood values and therefore the location of the maximum. It also improves numerical stability by replacing products with sums. Chapter 01 · Likelihood and Maximum Likelihood Estimation
- What is the difference between an estimator and an estimate?
- An estimator is a random rule applied to all possible samples, while an estimate is the fixed value obtained from one realised dataset. Expectations, variances and sampling distributions belong to the estimator. Chapter 02 · Estimators, Sampling Distributions and Bias
- Why are there two formulas for Fisher information?
- Regular likelihood identities make expected squared score equal negative expected second derivative. The equality can fail in irregular settings, so its conditions matter whenever support or differentiation behaves unusually. Chapter 03 · Information and Minimum Variance Bounds
- Does consistency mean an estimator is unbiased?
- No. Consistency describes behaviour as sample size grows, while unbiasedness is an exact finite sample expectation statement. A biased estimator can become consistent when its bias vanishes with sample size. Chapter 04 · Convergence, Consistency and Delta Methods
- Why should I write matrix dimensions?
- Dimensions expose an invalid multiplication or transpose immediately and identify the shape of the final statistical object. They are especially useful for score vectors, covariance sandwiches and linear target transformations. Chapter 05 · Vector Likelihood and Normal Linear Models
- Does a successful convergence code prove the estimate is correct?
- No. It reports the algorithm status, not the validity of the model, derivatives or located optimum. Check the final gradient, restrictions, objective value and agreement across independent solution routes. Chapter 06 · Numerical Likelihood Optimisation
- Why is the Likelihood Ratio statistic nonnegative?
- The unrestricted parameter space contains the constrained one, so its maximum log likelihood cannot be smaller. Twice the unrestricted minus constrained value is therefore at least zero. Chapter 07 · Likelihood Ratio, Wald and Score Tests
How to study for the exam
Work in a fixed order each week. First state the model, sample and parameter space. Second derive the likelihood and log likelihood before differentiating. Third separate the realised estimate from the estimator and ask what repeated sampling property is being studied. Fourth attach every large sample conclusion to its theorem and conditions.
Fifth reproduce numerical results with a recorded seed, starting vector, parameterisation, convergence code, objective value and final score check. Attempt tutorial questions before reading solutions, revisit assignment questions from a blank page, and resolve missing steps early because later topics reuse them.
Before the final, practise compact answers that show the likelihood contribution, restriction, matrix dimension or decision threshold rather than reporting an unsupported number.
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