This course introduces basic time series analysis and forecasting methods. Topics include stationary processes, ARMA models, modeling and forecasting using ARMA models, nonstationary and seasonal time series models, state-space models, and forecasting techniques.
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By the end of this course, students will be able to:
- Describe important time series models and their applications in various fields.
- Formulate real life problems using time series models.
- Use statistical software to estimate models from real data and draw conclusions and develop solutions from the estimated models.
- Use visual and numerical diagnostics to assess the soundness of their models.
- Communicate the statistical analyses of substantial data sets through explanatory text, tables, and graphs.
- Combine and adapt different statistical models to analyze larger and more complex data.
Syllabus
Module 1: Course Introduction and Intuition for Stationarity
Welcome to Introduction to Time Series! This module introduces students to the foundational concepts and tools for time series analysis, equipping them with the necessary skills to understand, model, and analyze data that change over time. Through a blend of theoretical lessons and practical exercises, students will explore the nature of time series data, the principles of stationarity, and begin their journey into time series modeling.
Module 2: Basic Analysis of Stationary Processes
This module offers a comprehensive exploration of ARMA (Auto Regressive Moving Average) processes, equipping students with the ability to dissect and comprehend the mechanics of ARMA models, including terminology and mathematical foundations like the backward shift operator. Through hands-on experience, students will learn to classify ARMA processes based on causality and invertibility, estimate key statistical properties of stationary time series such as the sample mean, autocovariance, and autocorrelation using R, and employ visual tools to determine the order of MA processes. This module also provides an introduction to forecasting, where students apply prediction operators to achieve optimal forecasts for stationary processes, rounding out their foundational understanding of time series analysis.
Module 3: ARMA processes and their Autocorrelation Functions
This module delves into advanced concepts surrounding ARMA(p,q) processes, focusing on the intricacies of causality and invertibility. Students will gain the analytical skills necessary to categorize ARMA processes, enhancing their understanding through the computation and interpretation of the Partial Autocorrelation Function (PACF) for both theoretical processes and real time series data using R. This module emphasizes the practical application of ACF and PACF plots in determining the order of ARMA processes and it introduces advanced forecasting techniques, providing students with the tools to implement precise forecasting methods for a variety of ARMA models, preparing them for complex challenges in time series analysis.
Module 4: More About the ACF; Best Linear Predictors, Autocorrelation, and Partial Autocorrelation
In this module, students will continue to delve into the intricate world of time series analysis, exploring techniques for parameter estimation and model selection. Students will master the Yule-Walker equations, a fundamental tool for preliminary parameter estimation, and gain proficiency in maximum likelihood estimation for ARMA processes. Additionally, students will learn to determine the optimal model order through various statistical criteria, ensuring parsimonious yet accurate representations of time series data.
Module 5: Fitting Data to ARMA models
This module equips students with advanced techniques for modeling and forecasting time series data. Students will explore the ARIMA framework, learning how to handle trends and seasonality through differencing and seasonal components. They will master the interpretation of ACF and PACF plots, enabling them to determine appropriate model orders. Additionally, students will explore the intricacies of SARIMA processes, incorporating both seasonal and non-seasonal components. This module culminates with forecasting techniques tailored for ARIMA and SARIMA models, empowering students to make accurate predictions for complex time series scenarios.
Module 6: Diagnostics and Order Selection
This module focuses on refining ARMA model selection and diagnostics, teaching students to critically evaluate model fit using standardized residuals and various diagnostic plots in R, including the normal Q-Q plot and the Ljung-Box test. Additionally, the module covers the principles of model order selection, emphasizing the avoidance of overfitting and the application of the Akaike Information Criterion (AIC) and its correction (AICC) in choosing the optimal ARMA model for specific time series data.
Module 7: Nonstationary processes: ARIMA and SARIMA Models
This module introduces students to ARIMA and SARIMA modeling techniques, essential for analyzing non-stationary and seasonal time series data. In the first lesson, students will learn to define ARIMA processes, use the Dickey-Fuller test to determine the need for differencing, and fit ARIMA models using R, incorporating the concept of exponential smoothing. The second lesson extends these skills to SARIMA models, focusing on identifying seasonality and fitting these models to capture seasonal patterns in data, providing a comprehensive toolkit for sophisticated time series analysis.
Module 8: More on Forecasting
This module equips students with advanced forecasting techniques beyond one-step-ahead predictions, focusing on ARMA, ARIMA, and SARIMA processes using R. In the initial lesson, learners will explore methodologies for generating multi-step forecasts with these models. The subsequent lesson dives into exponential smoothing, teaching students to handle models with additive errors, trends, and seasonality, and how to effectively use R's HoltWinters and forecast functions to fit and interpret time series forecasts, thus providing a thorough grounding in dynamic forecasting methods.
Summative Course Assessment
This module contains the summative course assessment that has been designed to evaluate your understanding of the course material and assess your ability to apply the knowledge you have acquired throughout the course.
