Time Series • Multivariate Time Series Models and Forecasting
Time Series / Multivariate Time Series Models and Forecasting

Multivariate Time Series Models and Forecasting

Notes 6 Multivariate Time Series Models and Forecasting
Quick concept

A multivariate time series contains observations of multiple variables recorded over the same time points. The variables may be related to one another, and their joint behaviour can be used for analysis and forecasting. That is different from a univariate series, which follows one variable only.

Notes

Multivariate Time Series Models and Forecasting

Definition

A multivariate time series contains observations of multiple variables recorded over the same time points. The variables may be related to one another, and their joint behaviour can be used for analysis and forecasting. That is different from a univariate series, which follows one variable only.

Example

Monthly sales, advertising expenditure and customer visits for the same shop, recorded for the same months. A rise in visits may accompany a rise in sales; advertising may move with both. Those links are information. They are not, by themselves, proof that advertising caused the extra sales.

Univariate vs Multivariate

Univariate time series Multivariate time series
What is recorded One variable over time (for example sales only) Several variables over the same times (sales, ads, visits)
Main information The past of that one series Own pasts plus relationships among the series
Forecast idea Forecast one series, mainly from itself Can forecast several series together (simultaneous forecasting)

Multiple series can help because one variable may lead another (temperature ahead of electricity demand) or move together with it (GDP, inflation and unemployment in the same quarters). Correlation does not automatically imply causation. A joint model can still be useful for description and forecasting even when the economic “why” is not fully settled.

Joint Behaviour and Simultaneous Forecasting

Joint behaviour means the typical co-movement: when one series is high, is another also high, in the same month or after a lag? Simultaneous forecasting means producing future values for more than one series in a way that respects those links, rather than fitting three unrelated univariate models and ignoring the rest.

Multiple Time Series ↓ Examine Relationships ↓ Model Joint Behavior ↓ Forecast Multiple Variables

At this syllabus level, the modelling idea is broad: keep a common time index, plot the series together, look at contemporaneous and lagged association, then choose a joint structure if the relationships look stable enough. Named systems such as vector autoregressions are not required here; the exam point is the multivariate idea, not a matrix derivation.

A Compact Modelling Flow

Multivariate Data ↓ Explore Relationships ↓ Check Stationarity ↓ Select Model ↓ Fit ↓ Forecast ↓ Evaluate

Exam-Oriented Key Points

  1. Multivariate = several series, same time index, possibly related.
  2. Univariate uses one series; multivariate uses joint behaviour as well.
  3. Variables can help predict each other; correlation is not causation.
  4. Simultaneous forecasting means producing several futures that respect the links.

A Small Multivariate Plot Example

What the Student Should Expect

Constructed monthly teaching series (not a downloaded file): sales, advertising and visits that tend to move together. Expected display: three aligned line plots and a correlation table printed by your run. Describe the co-movement from the graph; do not memorise invented printed numbers.

# Import libraries import numpy as np import pandas as pd import matplotlib.pyplot as plt # Load / prepare constructed monthly series idx = pd.date_range("2024-01-01", periods=24, freq="MS") t = np.arange(24) visits = 80 + 0.8 * t + 8 * np.sin(2 * np.pi * t / 12) ads = 20 + 0.3 * t + 0.15 * (visits - visits.mean()) sales = 40 + 0.4 * visits + 0.2 * ads df = pd.DataFrame( {"sales": sales, "advertising": ads, "visits": visits}, index=idx ) # Plot the three series together df.plot(subplots=True, title="Constructed multivariate monthly series") plt.tight_layout() plt.show() # Association among the same-month values (not a causal test) print(df.corr())

Same-month correlation summarises contemporaneous association. Lagged cross-correlation (sales now vs advertising last month) is the next conceptual step; it still does not prove cause.

Exam-Oriented Key Points

  1. Always align multivariate series on a common time index.
  2. Plot first, then summarise association; do not skip the graph.