Time-series analysis - AR MA ARMA ARIMA and stationarity

Learning Outcomes

This article explains how to classify and interpret AR, MA, ARMA, and ARIMA time-series models in the context of CFA Level II quantitative methods. It clarifies the structure and notation of each model class, including the roles of the order parameters p, d, and q, and links these to observable features in financial data such as serial correlation and shock persistence. It examines the requirement for covariance stationarity, shows how to recognise nonstationary behaviour using plots and formal unit-root tests, and highlights why ignoring nonstationarity leads to biased parameter estimates, unreliable forecasts, and invalid inference. It details how to transform nonstationary series through differencing, and how that transformation changes the interpretation of model parameters and forecasts. It discusses the practical selection of AR versus ARMA versus ARIMA specifications for asset returns, macroeconomic series, and risk variables, with an explicit focus on how such decisions are tested in the CFA exam. It also reviews common exam traps, including misreading random walks, confusing trend and unit-root nonstationarity, and mis-identifying appropriate models.

CFA Level 2 Syllabus

For the CFA Level 2 exam, you are expected to understand the use and limitations of time-series analysis in financial modelling, with a focus on the following syllabus points:

• Identify characteristics of AR, MA, ARMA, and ARIMA models
• Explain the requirement for covariance stationarity and implications of nonstationarity
• Test time series for unit roots and explain the use of differencing to achieve stationarity
• Interpret model results, select between trend and autoregressive models, and justify model selection when analysing investment data

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. What is the key requirement for using an ARMA model on a financial time series?
2. Which model is appropriate for a time series displaying both autoregressive and moving average patterns?
3. Describe in one sentence the primary purpose of differencing in time-series analysis.
4. How does the presence of a unit root impact stationarity and model suitability?

Introduction

Time-series analysis is essential in financial modelling and forecasting. Many financial variables, such as interest rates and asset prices, exist as time-dependent sequences, requiring special consideration compared to cross-sectional data. This article will introduce the main time-series frameworks—AR, MA, ARMA, and ARIMA models—clarify the principle of stationarity, and highlight their importance and proper use in the CFA exam context.

Key Term: time series
A sequence of data points collected or recorded at time intervals, typically evenly spaced.

AR, MA, ARMA, AND ARIMA MODELS

Autoregressive (AR) Models

An AR model expresses the current value of a series as a linear function of its own previous values plus a random error. The most common form is the AR(1):

$x*t = b_0 + b_1 x*{t-1} + \epsilon_t$

Where $x_t$ is the value at time $t$, $b_0$ is a constant, $b_1$ is the lag coefficient, and $\epsilon_t$ is a white noise error term.

Key Term: autoregressive (AR) model
A model in which current values of a variable depend linearly on previous values (lags) of itself and a random error.

Moving Average (MA) Models

MA models define each value as a function of past error terms (shocks), not past values:

$x*t = \mu + \epsilon_t + \theta_1 \epsilon*{t-1} + \theta*2 \epsilon*{t-2} + ... + \theta*q \epsilon*{t-q}$

Where $\mu$ is the mean, $\epsilon_t$ is the current shock, and $\theta_j$ are coefficients for $q$ previous shocks.

Key Term: moving average (MA) model
A model in which each observation is represented as a mean plus a linear combination of past random error terms.

ARMA Models

Combining both AR and MA elements, ARMA models are more flexible for series with autocorrelation and moving average effects:

$x*t = b_0 + b_1 x*{t-1} + ... + b*p x*{t-p} + \epsilon*t + \theta_1 \epsilon*{t-1} + ... + \theta*q \epsilon*{t-q}$

$p$ is the number of AR lags, $q$ is the number of MA lags.

Key Term: ARMA model
A time-series model containing both autoregressive (AR) and moving average (MA) terms.

ARIMA Models

ARIMA (Autoregressive Integrated Moving Average) models extend ARMA to series that are not stationary, by including differencing ($d$):

• If first differences are used, $d=1$.
• General ARIMA($p$, $d$, $q$): $d$ indicates order of differencing.

First differencing: $y_t = x_t - x_{t-1}$.

Key Term: ARIMA model
An ARMA model applied to differenced data in order to handle nonstationary time series.

STATIONARITY IN TIME-SERIES ANALYSIS

What Is Stationarity

A time series is stationary if its mean, variance, and covariance are constant over time. Stationarity allows valid estimation of model parameters. Most classical time-series models require covariance stationarity.

Key Term: covariance stationary
A time series with constant mean and variance over time, and autocovariances that depend only on lag, not on time.

Recognising and Testing Stationarity

• Mean: Should not change over time.
• Variance: Should remain constant.
• Covariances: Should depend only on lag, not time.
• Series with clear upward or downward trends or changing variance are nonstationary.

Testing tools:

• Visual inspection (plots for trend/variance).
• Formal tests (e.g., Dickey-Fuller test for unit roots).

Key Term: unit root
A characteristic of a time series with a lag coefficient of 1, indicating nonstationarity and potentially a random walk pattern.

WHAT TO DO WITH NONSTATIONARY SERIES

When a series is nonstationary (e.g., contains a unit root), AR, MA, or ARMA models are not appropriate without transformation. Differencing the data can render it stationary.

Key Term: first differencing
Transforming a time series by subtracting the previous value from each observation to eliminate trends and induce stationarity.

If a single differencing is not sufficient, further differencing may be required.

Worked Example 1.1

Suppose you analyse a series of quarterly GDP figures exhibiting a persistent upward trend. How can you make this series suitable for an ARMA model?

Answer:
The persistent trend indicates nonstationarity, likely due to a unit root. Apply first differencing to the GDP series, i.e., use $y_t = x_t - x_{t-1}$. If the differenced series appears stationary (constant mean/variance), you may now fit an ARMA model to the differenced data, which is technically an ARIMA( $p$, 1, $q$ ) model.

Worked Example 1.2

You observe a time series of monthly exchange rates. The Dickey-Fuller test fails to reject the null hypothesis of a unit root. What does this mean and what step should you take?

Answer:
Failing to reject the null hypothesis means the series is nonstationary (contains a unit root). Take first differences of the series. If the differenced series passes the stationarity test, model using ARMA on the differenced data.

Revision Tip

Focus on identifying whether a series is stationary before fitting AR, MA, or ARMA models. Unit root tests and plotting the series are quick diagnostic steps.

Exam Warning

Not all trending series can be made stationary by differencing—the presence of seasonality or multiple unit roots may require more advanced techniques.

Summary

AR, MA, ARMA, and ARIMA models provide a flexible set of tools for modelling time series in finance. However, the key requirement for using these models is that the series be covariance stationary, except for ARIMA models which handle nonstationarity via differencing. Always diagnose stationarity before applying AR, MA, or ARMA models, and use first differencing to transform unit root series.

Key Point Checklist

This article has covered the following key knowledge points:

• Define and distinguish AR, MA, ARMA, and ARIMA time-series models
• Explain and test for stationarity and unit roots in time series
• Understand implications of nonstationarity and the role of differencing
• Select and justify the appropriate time-series model based on stationarity
• Recognise typical CFA exam pitfalls with unit roots and nonstationarity

Key Terms and Concepts

• time series
• autoregressive (AR) model
• moving average (MA) model
• ARMA model
• ARIMA model
• covariance stationary
• unit root
• first differencing