Sub-topic 2: Time Series Analysis for CFA Candidates
Time series analysis is a crucial tool in finance for understanding past patterns, forecasting future trends, and making informed investment decisions. This module will cover the fundamental concepts and techniques of time series analysis as relevant to the CFA curriculum.
Understanding Time Series Data
A time series is a sequence of data points collected over time, typically at regular intervals. Examples include daily stock prices, monthly inflation rates, or quarterly GDP figures. Analyzing these series helps us identify patterns, seasonality, and trends.
A sequence of data points collected over time at regular intervals.
Components of a Time Series
Time series data can often be decomposed into four main components:
- Trend: The long-term direction of the series (upward, downward, or stable).
- Seasonality: Predictable, repeating patterns within a fixed period (e.g., daily, weekly, yearly).
- Cyclical: Fluctuations that are not of a fixed period, often related to business cycles.
- Irregular (or Random) Component: Unpredictable variations that remain after accounting for the other components.
Stationarity
A stationary time series is one whose statistical properties (like mean, variance, and autocorrelation) do not change over time. Many time series models require the data to be stationary. Non-stationary data can often be made stationary through transformations like differencing.
Stationarity is a key assumption for many time series models. A stationary series has a constant mean, constant variance, and its autocorrelation structure is independent of time. For example, a random walk is non-stationary because its mean and variance change over time. Differencing a non-stationary series (e.g., taking the difference between consecutive observations) can often make it stationary.
Text-based content
Library pages focus on text content
Autocorrelation and Partial Autocorrelation
Autocorrelation (ACF) measures the correlation of a time series with its own lagged values. Partial autocorrelation (PACF) measures the correlation between a time series and its lagged values after removing the effect of intermediate lags. These are crucial for identifying the order of ARIMA models.
ACF measures the total correlation between a series and its lag, while PACF measures the direct correlation after accounting for intermediate lags.
Time Series Models
Several models are used for time series analysis and forecasting:
- Moving Averages (MA): Models that use past forecast errors to predict future values.
- Autoregressive (AR): Models that use past values of the series to predict future values.
- Autoregressive Moving Average (ARMA): Combines AR and MA components.
- Autoregressive Integrated Moving Average (ARIMA): Extends ARMA by including differencing to handle non-stationary data. An ARIMA(p, d, q) model has 'p' autoregressive terms, 'd' differencing orders, and 'q' moving average terms.
- Seasonal ARIMA (SARIMA): Extends ARIMA to handle seasonal patterns.
- GARCH (Generalized Autoregressive Conditional Heteroskedasticity): Models that capture volatility clustering, where periods of high volatility are followed by high volatility, and vice versa.
| Model | Key Components | Primary Use Case |
|---|---|---|
| AR | Past values of the series | Forecasting based on historical levels |
| MA | Past forecast errors | Modeling random shocks and their persistence |
| ARMA | Past values and past errors | Modeling stationary series with both autoregressive and moving average properties |
| ARIMA | Differencing, past values, past errors | Modeling non-stationary series |
| GARCH | Past squared errors and past conditional variances | Modeling time-varying volatility |
Model Selection and Evaluation
Selecting the appropriate time series model involves examining ACF and PACF plots to determine the 'p' and 'q' orders for ARIMA models. Model evaluation typically uses metrics like Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Mean Absolute Error (MAE) on a hold-out sample. Information criteria like AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) can also guide model selection by balancing model fit with complexity.
Remember that simpler models are often preferred if they provide comparable predictive accuracy to more complex ones (parsimony principle).
Forecasting with Time Series Models
Once a model is selected and estimated, it can be used to forecast future values. The forecast horizon and the inherent uncertainty of the data will influence the confidence intervals around these forecasts. For financial applications, understanding the limitations and assumptions of these models is paramount.
Learning Resources
Official curriculum material from the CFA Institute covering time series analysis, essential for exam preparation.
A foundational video explaining the basics of time series analysis and its components.
A comprehensive blog post detailing various time series concepts, models, and practical applications.
Detailed documentation on ARIMA and other time series models, including implementation in Python.
Explains the concept of stationarity and its importance in time series analysis with clear examples.
A series of tutorials covering the fundamentals of time series analysis, including decomposition and forecasting.
An explanation of GARCH models, commonly used in finance for modeling and forecasting volatility.
A broad overview of time series analysis, its history, methods, and applications.
A widely respected online book covering time series forecasting principles and practical applications with R examples.
A clear explanation of Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots and their role in model identification.