Lesson 13 of 22

Seasonality and Dynamic Regression

Combine calendar structure and predictors with time-series errors while respecting future availability.

Motivation

Why begin here

Ordinary regression standard errors fail when errors remain autocorrelated; dynamic regression models the mean and serial dependence together.

Why this matters

Combine calendar structure and predictors with time-series errors while respecting future availability.

Packages and data

Reproducible setup

Base stats; forecast/fable optional.

Explanation 1

The live starter code is self-contained and uses only runner-compatible functions.

Explanation 2

Book-only outputs are labelled when the separate source dataset or advanced package was not supplied.

Explanation 3

R code, paths, function names, and formulas remain left-to-right in every locale.

Terminology

Core ideas and notation

Seasonal difference

Current value minus the same seasonal position one cycle earlier.

yt - y(t-12)

Fourier terms

Smooth sine/cosine seasonal basis.

Parsimonious calendar pattern

Intervention

Pulse, step, or ramp encoding a known event.

Not automatically causal

Notation and formulas

Key relationship

yt = beta0 + beta' xt + nt, where nt follows ARIMA

Predictor coefficients describe the conditional mean while the error model handles remaining dependence.

Step 1Define every variable and unit.
Step 2Check assumptions and information timing.
Step 3Calculate, verify, and interpret in context.

Worked example

Worked analysis

Scenario

How can monthly seasonality be encoded without twelve separate parameters?

Variable: Five years of monthly indices.
Model choice: Predictor coefficients describe the conditional mean while the error model handles remaining dependence.
Calculation: Four Fourier columns encode two harmonics; fitted residual checks show what calendar structure remains.
Interpretation: Calendar terms are known in advance, so they can support forecasts without requiring an external predictor scenario.
Reasonableness check: Policy and exchange-rate futures are not known; using realized future values would be leakage.

R check

t <- 1:60; s <- 12
fourier <- cbind(sin1=sin(2*pi*t/s), cos1=cos(2*pi*t/s), sin2=sin(4*pi*t/s), cos2=cos(4*pi*t/s))
y <- 2 + .03*t + 1.2*fourier[,1] + .6*fourier[,2]
fit <- lm(y ~ t + fourier)
print(coef(fit))
print(Box.test(residuals(fit),lag=12,type='Ljung-Box'))

Hints

Start by naming the question and unit.
Inspect intermediate values before the final result.
Explain what the calculation cannot establish.

Calendar terms are known in advance, so they can support forecasts without requiring an external predictor scenario. Limitation: Policy and exchange-rate futures are not known; using realized future values would be leakage.

Visual

Seasonal cycle

Illustrative monthly index repeats a smooth annual pattern.

R connection

Live, editable R practice

Run the self-contained starter code in the protected STATLAB R runner. The code prints an auditable result and avoids network or unrestricted file access.

Live R Lab

INTRO-R-CH13: Seasonality and Dynamic Regression

Run a self-contained example, verify its output, and explain one limitation for seasonality and dynamic regression.

Editable R code

Results

Ready to run

Ready to run. Results, errors, tables, plots, and fallback messages will appear here.

State the question, data, and units.

Run and inspect the printed output.

Interpret the result and one limitation.

Common mistake

A lag establishes timing, not exogeneity or causation.

STATLAB Tip

Draw a timeline of when every predictor becomes available relative to the forecast origin.

Guided practice

Scenario

Fit monthly indicator effects and compare residual ACF with the Fourier model.

Hints

Reuse a small tested object.
Change one element at a time.
Finish with an interpretation and limitation.

Compare your result with the definition, units, and model assumptions—not only with a target number.

Exercises

Independent practice

Complete these without looking at the selected solutions. More than one defensible program may exist.

Explanation 1

Create December pulse and post-2020 step variables.

Explanation 2

Generate two harmonic pairs.

Explanation 3

Define two future predictor scenarios.

Selected solutions

Check one defensible route

These are compact solution routes. Confirm dimensions, units, and any changed modelling choices.

Explanation 1

december <- as.integer(month==12); post <- as.integer(date>=as.Date('2020-03-01'))

Explanation 2

Use sin(2*pi*k*t/12) and cos(...) for k=1,2.

Explanation 3

State both the assumed path and how forecast uncertainty depends on it.

Chapter summary

What to carry forward

Combine calendar structure and predictors with time-series errors while respecting future availability.

Explanation 1

Seasonality can be deterministic or stochastic.

Explanation 2

Indicators and Fourier terms impose different structures.

Explanation 3

ARIMA errors repair serial-dependence inference.

Explanation 4

Future predictor availability governs forecasting.

Terminology

Chapter cheat sheet

diff(x,lag=12)

seasonal change

Use in Seasonality and Dynamic Regression.

factor(month)

indicators

Use in Seasonality and Dynamic Regression.

sin()/cos()

Fourier terms

Use in Seasonality and Dynamic Regression.

lag()

distributed lag

Use in Seasonality and Dynamic Regression.

References and provenance

Safavi (2026), Chapter 13. Student notes: Mohammad Safavi, Ph.D., STATLAB Academy, Version 1.0.

Resource

Download the complete student notes

The authoritative 125-page English PDF accompanies this native lesson.

Download

Reflection

Pause and write

What assumption in seasonality and dynamic regression most affects the conclusion?

Name the assumption, evidence you would seek, and how the recommendation might change.

Exit check

Chapter knowledge check

Reach 70% to complete the chapter. Explanations appear after submission.

Checkpoint