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.
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