Lesson 21 of 22

Multivariate Time Series and Cointegration

Advanced Topic – Optional. Model joint dynamics while separating predictive content, shock identification, long-run relations, and causal claims.

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Advanced Topic – Optional

This chapter extends the core pathway. Complete Chapters 1–14 first, then use this material when your question and data justify the added model complexity.

Motivation

Why begin here

Variables interact, but multivariate fit does not by itself identify interventions or structural shocks.

Why this matters

Model joint dynamics while separating predictive content, shock identification, long-run relations, and causal claims.

Packages and data

Reproducible setup

Base-R demonstrations; vars, urca, tsDyn, rmgarch, and MTS are 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

VAR

Every system variable depends on lags of all included variables.

Joint forecasting system

Granger causality

Incremental predictive content conditional on included history.

Not structural causation

Cointegration

A stationary linear combination of nonstationary variables.

Long-run equilibrium relation

Notation and formulas

Key relationship

yt = c + A1 y(t-1) + ... + Ap y(t-p) + error_t

Impulse responses require a specified shock-identification rule because innovations can be contemporaneously correlated.

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

Worked example

Worked analysis

Scenario

Can a stationary spread be recovered from two simulated nonstationary levels?

Variable: A random walk x and y = 4 + 1.25x + stationary spread.
Model choice: Impulse responses require a specified shock-identification rule because innovations can be contemporaneously correlated.
Calculation: The book run estimated slope 1.244, close to the generating 1.25; residual spread returns toward zero.
Interpretation: The controlled simulation illustrates cointegration; observed data need integration and residual-stationarity assessment.
Reasonableness check: A high R-squared between trending levels can be spurious, and VAR shock responses depend on identification assumptions.

R check

set.seed(2121)
x <- cumsum(rnorm(220,sd=.8))
spread <- arima.sim(list(ar=.55),n=220,sd=.5)
y <- 4+1.25*x+spread
fit <- lm(y~x)
print(coef(fit))
print(acf(residuals(fit),plot=FALSE)$acf[2])
plot(residuals(fit),type='l',ylab='Estimated spread')

Hints

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

The controlled simulation illustrates cointegration; observed data need integration and residual-stationarity assessment. Limitation: A high R-squared between trending levels can be spurious, and VAR shock responses depend on identification assumptions.

Visual

Mean-reverting spread

The simulated cointegrating residual fluctuates around zero instead of drifting.

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-CH21: Multivariate Time Series and Cointegration

Run a self-contained example, verify its output, and explain one limitation for multivariate time series and cointegration.

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

Do not call Granger predictability causal impact or interpret trending-level OLS before checking cointegration.

STATLAB Tip

Label shock variable, size, identification rule, response unit, horizon, and uncertainty method on every impulse response.

Guided practice

Scenario

Reverse a Cholesky ordering and explain why the impulse response changes.

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

Compare VAR lag candidates.

Explanation 2

Compare stationarity of levels and residual spread.

Explanation 3

Explain when DCC is preferable to unrestricted covariance.

Selected solutions

Check one defensible route

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

Explanation 1

Compare information criteria and residual serial correlation.

Explanation 2

Use plots plus ADF/KPSS with explicit deterministic terms.

Explanation 3

DCC offers parsimonious changing correlation when dimension makes free covariance dynamics impractical.

Chapter summary

What to carry forward

Model joint dynamics while separating predictive content, shock identification, long-run relations, and causal claims.

Explanation 1

VAR models joint lag dynamics.

Explanation 2

Granger means prediction, not intervention.

Explanation 3

Impulse responses require identification.

Explanation 4

Cointegration creates stationary combinations.

Terminology

Chapter cheat sheet

ccf()

cross-correlation

Use in Multivariate Time Series and Cointegration.

vars::VAR()

vector autoregression

Use in Multivariate Time Series and Cointegration.

vars::causality()

Granger test

Use in Multivariate Time Series and Cointegration.

urca::ca.jo()

cointegration

Use in Multivariate Time Series and Cointegration.

References and provenance

Engle & Granger (1987); Safavi (2026), Chapter 21. 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 multivariate time series and cointegration 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