📚 Introductory Econometrics with MORISTAT

A Practical Guide to Empirical Analysis

Naguib Lallmahomed · naglal@linux-mauritius.com · 2026-06-14

Chapter 9: Time Series Analysis and Forecasting

🎯 Learning Objectives

Upon completing this chapter, you will be able to:

Understand the characteristics of time series data
Identify trends, seasonality, and cycles
Test for stationarity (unit roots)
Understand ARIMA models (AR, MA, ARIMA)
Generate and evaluate forecasts
Understand Vector Autoregression (VAR)

9.1 What is Time Series Data?

Time series data consists of observations of a variable (or variables) recorded over time at regular intervals:

Daily: Stock prices, weather data
Monthly: Unemployment, inflation, sales
Quarterly: GDP, corporate earnings
Yearly: National accounts, population

Example: Quarterly GDP data for South Africa (2010–2023).

Observations are ordered in time
Current values often depend on past values
Classical regression assumptions may be violated

9.1.1 Components of Time Series

Trend: Long-term movement (upward or downward)
Seasonality: Regular patterns within a year
Cyclical: Business cycle fluctuations (longer than seasonal)
Irregular: Random (unpredictable) fluctuations

9.2 Stationarity and Unit Roots

9.2.1 What is Stationarity?

A time series is stationary if its statistical properties (mean, variance, autocorrelation) do not change over time.

Weak (covariance) stationarity:
- Constant mean: E(Y_t) = μ
- Constant variance: Var(Y_t) = σ²
- Covariance depends only on lag: Cov(Y_t, Y_t-k) = γ_k

9.2.2 Why Stationarity Matters

Non-stationary data can lead to spurious regression (high R² but meaningless relationships)
Many time series models assume stationarity
If non-stationary, we often difference the data

9.2.3 Unit Root Tests

Augmented Dickey-Fuller (ADF) test: Tests if a unit root is present
H₀: Unit root exists (non-stationary)
H₁: No unit root (stationary)
If p-value < 0.05 → reject H₀ → series is stationary

9.3 ARIMA Models

9.3.1 Autoregressive (AR) Models

Y_t = δ + φ₁Y_t-1 + φ₂Y_t-2 + ⋯ + φ_pY_t-p + ε_t

AR(p): p lags of Y
Current value depends on past values
Requires stationarity (|φ| < 1 for AR(1))

9.3.2 Moving Average (MA) Models

Y_t = μ + ε_t + θ₁ε_t-1 + θ₂ε_t-2 + ⋯ + θ_qε_t-q

MA(q): q lags of the error term
Current value depends on past shocks
Always stationary

9.3.3 ARIMA (p, d, q) Models

ARIMA = AR + I (Integrated) + MA
p: AR order
d: Degree of differencing
q: MA order

Example: ARIMA(1,1,1) model:

ΔY_t = δ + φ₁ΔY_t-1 + ε_t + θ₁ε_t-1

where ΔY_t = Y_t − Y_t-1 (first difference).

9.4 Model Selection and Forecasting

9.4.1 Information Criteria

AIC (Akaike Information Criterion): AIC = 2k − 2ln(L)
BIC (Bayesian Information Criterion): BIC = kln(n) − 2ln(L)
Lower values indicate better model fit (penalises complexity)
BIC penalises more heavily than AIC

9.4.2 Forecast Evaluation

RMSE (Root Mean Squared Error): √(Σ(Y_t − Ŷ_t)² / n)
MAE (Mean Absolute Error): Σ|Y_t − Ŷ_t| / n
MAPE (Mean Absolute Percentage Error): Σ|(Y_t − Ŷ_t)/Y_t| × 100%
Lower values indicate better forecasting accuracy

9.5 Vector Autoregression (VAR)

VAR models capture the interdependencies between multiple time series variables:

Y_t = α₁ + β₁₁Y_t-1 + β₁₂X_t-1 + ε_1t

X_t = α₂ + β₂₁Y_t-1 + β₂₂X_t-1 + ε_2t

Each variable is a function of its own lags and lags of other variables
Allows for Granger causality testing
Used for impulse response analysis and forecasting

9.6 MORISTAT Examples

9.6.1 Time Series Data Format

Year,GDP,Inflation,Unemployment
2010,100,5.0,25.0
2011,105,4.5,24.0
2012,110,4.0,23.0
2013,115,3.5,22.0
2014,120,3.0,21.0
2015,125,2.5,20.0

9.6.2 Running a Time Series Regression

MORISTAT> LOAD ts_data.csv
MORISTAT> SUMMARY
MORISTAT> REGRESS GDP ~ Inflation Unemployment

9.7 Practical Exercises

Exercise 9.1: Identifying Time Series Components

You have the following time series data:

Retail sales (monthly) – increasing trend with Christmas peaks
GDP (quarterly) – upward trend with business cycle fluctuations
Stock prices (daily) – random walk with no clear trend

Identify the components present in each series.
Which series is likely non-stationary?
Which series might require seasonal adjustment?

Solution:

Retail sales: Trend + Seasonality + Irregular → non-stationary
GDP: Trend + Cyclical + Irregular → non-stationary
Stock prices: Random walk → non-stationary
All three series are likely non-stationary
Retail sales require seasonal adjustment

Exercise 9.2: ARIMA Model Identification

You observe the following autocorrelation (ACF) and partial autocorrelation (PACF) patterns:

ACF: decays slowly
PACF: cuts off after lag 1

What type of model is suggested?
What if the ACF decays slowly and PACF decays slowly?
What if the ACF cuts off after lag 1 and PACF decays slowly?

Solution:

ACF decays slowly, PACF cuts off at lag 1 → AR(1)
ACF decays slowly, PACF decays slowly → ARIMA(p, d, q) or AR(p) with p > 1
ACF cuts off at lag 1, PACF decays slowly → MA(1)

Exercise 9.3: Forecasting Evaluation

You have two forecasting models with the following errors:

Model A: Errors: 2, -3, 1, -2, 4
Model B: Errors: 1, -1, 0, -1, 1

Calculate the RMSE for both models.
Calculate the MAE for both models.
Which model is better?

Solution:

Model A RMSE: √((4+9+1+4+16)/5) = √(34/5) = √6.8 = 2.607
Model B RMSE: √((1+1+0+1+1)/5) = √(4/5) = √0.8 = 0.894
Model A MAE: (2+3+1+2+4)/5 = 12/5 = 2.4
Model B MAE: (1+1+0+1+1)/5 = 4/5 = 0.8
Model B is better (lower RMSE and MAE)

9.8 Key Terms

Time Series Trend Seasonality Stationarity Unit Root Augmented Dickey-Fuller (ADF) Test ARIMA Autoregressive (AR) Moving Average (MA) Differencing AIC (Akaike Information Criterion) BIC (Bayesian Information Criterion) RMSE MAE MAPE Vector Autoregression (VAR) Granger Causality Impulse Response

9.9 Further Reading

Wooldridge, J.M. (2020). Introductory Econometrics: A Modern Approach, Chapter 10 (Basic Regression with Time Series Data), Chapter 11 (Further Issues in Time Series), and Chapter 18 (Advanced Time Series Topics).
Stock, J.H. & Watson, M.W. (2019). Introduction to Econometrics, Chapter 14 (Introduction to Time Series Regression and Forecasting) and Chapter 16 (More on Time Series).
Hyndman, R.J. & Athanasopoulos, G. (2018). Forecasting: Principles and Practice (free online).

9.10 What's Next?

In Chapter 10, we will cover Limited Dependent Variable Models:

Binary choice models (Logit and Probit)
Multinomial models
Ordered models
MARGINAL EFFECTS

→ Proceed to Chapter 10 (Coming Soon)

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