MATH7970 Applied Time Series
The Complete Exam Bible · Semester 1, 2026-27MATH7970 Overview
- Hong Kong Baptist University
- Semester One 2026/2027
- Postgraduate course
- Mathematics
- Forecasting and model diagnostics
What this course asks you to do
Applied Time Series develops a complete route from an observed sequence to a defensible forecast. The published objectives emphasise explaining time-series models, selecting a suitable model for a real problem, applying the techniques to data and interpreting findings concisely.
- Assessment Two assignments, one project and a final examination.
- Central method Transform, identify, estimate, diagnose and compare forecasts.
- Hardest judgement Choosing a parsimonious model without hiding residual structure.
- Examination details Follow the current HKBU instructions for format and permitted aids.
How MATH7970 is assessed
| Component | Weight | Format |
|---|---|---|
| Assignment 1 | 10% | Individual assignment, tentative in week 4 |
| Assignment 2 | 15% | Individual assignment, tentative in week 9 |
| Project | 15% | Real-life time-series analysis, submission in week 11 |
| Examination | 60% | Final examination; follow current HKBU instructions for format |
The Semester One 2026/2027 teaching plan publishes these four components and their weights total 100%. It does not publish a hurdle rule, examination duration, sections, question types or permitted materials in the captured files. Use the current HKBU examination instructions for those details.
Assessment structure
Segment widths reproduce the published weights and total 100%.
Current MATH7970 dates
| Date | Item | Control |
|---|---|---|
| 27 September 2026 | Assignment 1 | Published due date. |
| 1 November 2026 | Assignment 2 | Published due date. |
| 15 November 2026 | Project | Published due date. |
Dates are as published in the course site and teaching plan for Semester One 2026/2027. Confirm exact deadlines and submission settings in the live LMS.
What MATH7970 covers
Ten chapters follow the published sequence from forecast baselines and smoothing through stationarity, ARIMA modelling, diagnostics, forecasting and conditional volatility.
Forecast Baselines and Error Measures
naive, mean and drift forecasts; signed errors; MAE and MSEBenchmarks before complexity
Naive, mean and drift forecasts encode different beliefs about level and movement.
- Forecast origin
- Naive forecast
- Mean and drift methods
- Signed forecast error
- MAE and MSE
- Rolling-origin comparison
Exponential Smoothing and Holt Forecasts
recursive level updates, exponential weights and two-state trend forecastsRecursive level updating
Simple exponential smoothing blends the newest observation with the previous level.
- Smoothing recursion
- Alpha and memory
- Initial level
- Holt level update
- Holt trend update
- Held-out tuning
Trend, Seasonality and Stationarity
deterministic and stochastic trend, seasonal differencing and stable momentsDiagnose before transforming
Trend changes long-run level, seasonality repeats on a known calendar and stochastic trend accumulates shocks.
- deterministic trend
- stochastic trend
- seasonal period
- weak stationarity
- ordinary differencing
- seasonal differencing
Autocorrelation, PACF and White Noise
lag dependence, cutoff and tail-off patterns, significance and residual noiseDependence by lag
The ACF measures standardised association across lags, while the PACF removes intermediate linear routes.
- autocovariance
- sample ACF
- partial autocorrelation
- cutoff and tail-off
- significance bands
- white-noise residuals
Autoregressive Models and Stability
long-run means, shock persistence, characteristic roots and stable recursionFeedback from observed history
An AR model links the current value to lagged observations and a new innovation.
- AR recursion
- Long-run mean
- Shock decay
- Negative persistence
- Characteristic roots
- Multi-step forecasting
Moving Average and ARMA Models
finite innovation memory, invertibility and mixed dependenceFinite innovation memory
An MA process combines current and recent innovations, so one shock has a direct effect for a fixed number of observations.
- MA innovation filter
- Finite memory
- Invertibility
- ACF cutoff
- ARMA mixture
- Parameter cancellation
ARIMA and Seasonal Models
integration, random walks, seasonal clocks and Box-Jenkins identificationDifference before identifying dependence
ARIMA separates autoregressive order, differencing order and moving-average order.
- Integrated process
- Random walk
- Ordinary difference
- Seasonal difference
- Seasonal ARIMA
- Box-Jenkins identification
Estimation and Model Diagnostics
moments, likelihood, residual analysis and information criteriaNumbers for a candidate
Moments and likelihood estimate a specified model; they do not validate its assumptions.
- method of moments
- maximum likelihood
- numerical optimisation
- residual ACF
- information criteria
- squared residuals
Forecasting and Model Comparison
minimum-MSE forecasts, updating, rolling origins and common loss functionsConditional means and widening uncertainty
Point forecasts replace future innovations by their conditional mean, while intervals include their possible effects.
- minimum-MSE forecast
- multi-step recursion
- forecast updating
- forecast intervals
- rolling origins
- out-of-sample loss
ARCH and GARCH Volatility Models
volatility clustering, conditional variance, persistence and squared-residual checksPredictable variance
Returns can be uncorrelated while their magnitudes cluster.
- Volatility clustering
- ARCH news term
- GARCH persistence
- Long-run variance
- Conditional standard deviation
- Squared-residual ACF
Those goals require more than recognising notation. You must decide what structure the plot supports, keep transformations explicit, calculate with consistent time indices and let residual evidence challenge the fitted specification.
Forecast baselines and smoothing
The course begins with simple methods because every later model needs a benchmark.
Naive, mean and drift forecasts encode different assumptions about level and change. Forecast errors retain a direction as well as a magnitude, and MAE or MSE should be chosen for the decision rather than for appearance.
Exponential smoothing then turns the complete past into a recursive update, while Holt's method gives trend its own state.
Stationarity and lag evidence
Trend, seasonality and changing variance are properties to diagnose before estimating a stochastic model. Weak stationarity stabilises the mean, variance and lag-covariance relation.
The ACF shows how dependence persists by lag; the PACF asks which lag retains a direct linear link after intermediate lags are removed. Their shapes propose AR and MA orders, while residual versions test whether the fitted model removed the structure it claimed to explain.
AR, MA and ARIMA models
Autoregression creates persistent feedback from observed history.
Moving-average terms give innovations a finite direct memory. ARMA combines both, and ARIMA adds differencing when levels contain accumulated shocks. Seasonal models introduce another clock.
The practical discipline is to state whether a coefficient belongs to levels, logs, ordinary differences or seasonal differences before interpreting it.
Estimation, diagnosis and forecasting
Method of moments and maximum likelihood estimate a candidate model; neither validates it. Residual time plots, ACFs, squared-residual checks and information criteria expose what remains.
Forecast comparison then uses common origins, horizons and loss functions so that models face the same future. Point forecasts should be accompanied by uncertainty that expands as unseen innovations accumulate.
Conditional volatility
ARCH and GARCH models address changing conditional variance after the mean equation has been cleaned.
Squared innovations represent news and lagged conditional variance represents memory. Their coefficients describe reaction and persistence, while standardised squared residuals show whether volatility dependence remains.
The model forecasts magnitude, not the sign of the next return.
How assessment shapes revision
The official teaching plan assigns 10% to Assignment 1, 15% to Assignment 2, 15% to the project and 60% to the examination. The assignment papers allocate marks by complete questions, and the answer booklets display calculation sequences and interpretations.
The project asks for a real-life series, plots, plausible models, hypothesis testing, parameter interpretation and forecasting. The captured material does not publish the examination duration, sections, question types or permitted aids, so current HKBU instructions control those details.
Worked example · free
Update a forecast and preserve the error sign
- 2Compute the signed forecast error.
- 3Apply the smoothing update.
- 1Interpret the direction of the change.
Key terms
- Forecast Origin
- The final time point whose information is available when a forecast is made.
- Forecast Error
- The realised value minus its forecast under the sign convention used in this guide.
- Exponential Smoothing
- A recursive forecast method assigning geometrically declining weights to older observations.
- Stationarity
- A stable probabilistic structure with constant mean and variance and lag-based covariance.
- Autocorrelation
- Standardised linear dependence between observations separated by a specified lag.
- Partial Autocorrelation
- The direct linear association at a lag after intermediate lags are removed.
- White Noise
- A zero-mean, constant-variance process with no serial correlation.
- Autoregression
- A model relating the current observation to its own lagged observations and a new innovation.
- Invertibility
- A condition selecting a moving-average representation whose innovations can be recovered from observed history.
- Seasonal Difference
- The current observation minus the observation one complete season earlier.
- Residual Diagnostic
- A check for structure left in fitted errors after a model has been estimated.
- Volatility Clustering
- The tendency for large movements to occur near large movements and quiet periods near quiet periods.
MATH7970 FAQ
How is this course assessed?
The published split is Assignment 1 at 10%, Assignment 2 at 15%, the project at 15% and the examination at 60%. These numeric rows total 100%. The captured teaching plan does not state a hurdle rule, so do not infer one from the examination weight.
What is the final examination format?
The captured teaching plan confirms a final examination and its 60% weight, but it does not publish the duration, section structure, question types, calculator rule or permitted materials. Use the current HKBU examination instructions and course announcements for those operational details.
Which forecast error sign should I use?
This guide uses actual minus forecast. A positive error means the method forecast too low, and a negative error means it forecast too high. Write the convention before calculating because some texts reverse it, and interpretation changes with the sign.
How do I decide whether a series is stationary?
Use the time plot, transformations and correlation evidence together. Look for a stable mean and variance and covariance that depends on lag rather than calendar time. Slow ACF decay, persistent trend or growing seasonal amplitude indicates that transformation may still be required.
How do ACF and PACF identify an ARIMA model?
They propose parsimonious candidate orders after suitable differencing. An AR process often has a tailing ACF and a PACF cutoff, while an MA process often reverses that pattern. Mixed models can tail in both, so estimation and residual diagnostics must complete the decision.
What makes a residual diagnostic acceptable?
Residuals should have no material remaining serial pattern at the inspected lags, and squared residuals should not retain conditional variance structure when a variance model is claimed adequate. This does not prove the model true; it says the selected diagnostics did not expose an unresolved pattern.
What does the project require?
The published project asks for a real-life series, descriptive statistics and plots, plausible model proposals, hypothesis tests supporting the choice, interpretation of estimated parameters and an attempted forecast. It is organised into introduction, analysis and forecasting, and conclusion, with a ten-page limit.
How should competing forecast models be compared?
Give every model the same training information, forecast origins, horizons and realised targets. Choose a loss measure that matches the decision, and compare against an appropriate simple baseline. A lower in-sample residual measure alone is not evidence of better future forecasting.
Questions students ask in each MATH7970 chapter
- Why use a naive forecast?
- It supplies a demanding simple benchmark when changes are hard to predict. A complex model that cannot beat it out of sample has not demonstrated useful additional structure. It also exposes whether the evaluation design rewards persistence rather than genuine predictive improvement. Chapter 01 · Forecast Baselines and Error Measures
- What happens when alpha is large?
- The forecast reacts quickly to recent observations and retains less influence from the distant past, which can reduce lag but transmit more noise. The validation record must decide whether that exchange improves the intended forecast horizon. Chapter 02 · Exponential Smoothing and Holt Forecasts
- Does a stationary series have to look flat?
- No. It can fluctuate sharply while retaining stable moments and lag dependence. Stationarity is probabilistic structure, not visual calm. The time plot and lag dependence must both support the claimed stable representation. Chapter 03 · Trend, Seasonality and Stationarity
- Why use both ACF and PACF?
- They separate persistence visible through all routes from the direct relation remaining after intermediate lags are removed. Their contrasting shapes narrow the set of parsimonious starting models. Chapter 04 · Autocorrelation, PACF and White Noise
- Why is the intercept not the mean?
- Persistence carries part of the expected level forward, so the mean solves a fixed-point equation and equals c divided by one minus phi in a stable AR(1). Substituting the mean into the expectation equation shows why this fixed point is required. Chapter 05 · Autoregressive Models and Stability
- How long does an MA shock last?
- An MA(q) innovation affects its arrival observation and the next q observations, then leaves the direct representation. Adjacent observations can still correlate while their innovation sets overlap. Chapter 06 · Moving Average and ARMA Models
- What does ARIMA(1,1,0) model?
- It models first differences with first-order autoregressive dependence, then accumulates forecasts back to the level scale. Interpretation must therefore state that coefficients describe changes rather than raw levels. Chapter 07 · ARIMA and Seasonal Models
- Does the largest likelihood always win?
- Only comparable models fitted to the same data can be contrasted, and complexity, diagnostics and forecasting still matter. An optimiser can also settle at a poor local solution or boundary. Chapter 08 · Estimation and Model Diagnostics
- Why do intervals widen with horizon?
- Each additional future date introduces another unknown innovation, and persistent dynamics carry earlier uncertainty forward. The expanding interval is therefore a model implication rather than a decorative addition. Chapter 09 · Forecasting and Model Comparison
- Can returns be uncorrelated but volatile in clusters?
- Yes. Linear mean dependence can be absent while squared or absolute returns retain strong serial structure. That distinction is why squared residuals are inspected after the conditional mean is fitted. Chapter 10 · ARCH and GARCH Volatility Models
How to study for the exam
Build one running model-selection sheet. For each series, record the forecast origin, visible structure, transformation, ACF and PACF pattern, candidate order, estimated coefficients, residual evidence and held-out forecast score. Recalculate at least one example in every chapter without a calculator so that signs and indices become automatic.
Then change one condition: raise alpha, reverse an AR sign, add a seasonal difference or increase volatility persistence, and predict the qualitative effect before computing it. For assessment practice, show the formula, substituted values, result, unit and interpretation on separate lines.
Finish each study session by explaining why the preferred model beats a naive or seasonal-naive benchmark and naming one diagnostic that could still overturn the choice. This routine rehearses the course objectives of selecting suitable models, applying the techniques and interpreting findings concisely. It also prevents a familiar software command from standing in for statistical reasoning.
Keep exam logistics separate: the source confirms the weight but not the format, so verify duration, sections, calculator rules and permitted aids in current HKBU instructions.
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