MATH 545 Spring 2025: Introduction to Time Series

Instructor: Yizhe Zhu, yizhezhu@usc.edu

Office Hours: Wednesday 2-3 pm at KAP 464B 

TA: Jiawei Wang, wang535@usc.edu

TA office hours: Monday from 3 to 5 pm (online at Math Center’s Zoom link) and Tuesday from 2 to 3 pm both in KAP263 and online.

Class schedule: MWF 9:00-9:50 am at KAP 158

Prerequisite: Probability theory and linear algebra

Course Description:  This is an introductory course to time series for graduate-level students.

Topics:  Stationary, nonstationary processes; moving average, autoregressive models; spectral analysis; estimation of mean, autocorrelation, spectrum; seasonal time series.

Course syllabus

Textbook: 

•  Brockwell and Davis. Time Series: Theory and Methods, 2nd edition.

•  (Not required) Brockwell and Davis. Introduction to Time Series and Forecasting, 2nd edition.  

Exam dates: 

• Midterm: Wednesday, March 5, 9-10 am, KAP 158

• Final: Friday, May 9, 8:30-10 am, KAP 158

Homework: Homework will be posted on Brightspace

Course Schedule:  Below is a tentative schedule, to be updated as the semester progresses.

Week 1

• Jan 13:  Introduction, examples of time series, stochastic processes, Kolmogorov Existence Theorem

• Jan 15:  Linear algebra review, covariance and correlation matrices

• Jan 17: Characteristic function, Multivariate normal distribution

Week 2

• Jan 20: No class

• Jan 22:  Conditional Gaussian distribution

• Jan 24: Autocovariance, stationarity

Week 3

• Jan 27:  Properties of the autocovariance function, estimating the mean in a stationary process

• Jan 29:  Estimator for autocovariance, inner product space

• Jan 31:  Hilbert space

Week 4

• February 3:  The projection theorem, properties of the projection map, prediction equations

• February 5:  properties of the projection map, linear prediction of a stationary process 

• February 7:  orthonormal sets

Week 5

• February 10:  Projection in R^n, general linear regression 

• February 12:  conditional expectation as a projection, best predictor

• February 14:  Best predictor of Gaussian, white noise, ARMA process definition

Week 6

• February 17: No class

• February 19:  MA process, AR(1) process

• February 21:  Causal ARMA process

Week 7

• February 24:  Necessary and sufficient condition for causal ARMA process

• February 26:  Invertible process

• February 28:  Autocovariance function of ARMA process

Week 8

• March 3:  Autocovariance function of ARMA process

• March 5:  Midterm

• March 7:   Partial autocorrelation function

Week 9

• March 10:  One-step linear predictor

• March 12:  Durbin-Levinson algorithm 

• March 14:   Innovations algorithm

Week 10

• March 17: No class

• March 19: No class

• March 21: No class

Week 11

• March 24:  The Yule-Walker equations

• March 26:   Estimation for AR processes using the Durbin-Levinson Algorithm

• March 28:  Estimation for MA process using the Innovations Algorithm

Week 12

• March 31:  Preliminary estimation for ARMA(p,q) model. Maximal likelihood estimator

• April 2:  Maximal likelihood estimator, least squares estimation for ARMA processes

• April 4:  Asymptotic properties of MLE

Week 13

• April 7:  Order selection, FPE, KL divergence

• April 9:  AIC, AICC

• April 11:  Spectral density of a stationary process

Week 14

• April 14:  Spectral density of ARMA processes

• April 16:  Causality, Invertibility and the Spectral Density

• April 18:   Causality, Invertibility and the Spectral Density

Week 15

• April 21:  The periodogram

• April 23:  The periodogram

• April 25:  Asymptotic properties of the  periodogram

Week 16

• April 28:  State space models, definition and examples

• April 30:  State space models for stationary process

• May 2:  Review