Time Series Analysis

Author

John Robin Inston

Published

September 25, 2026

1 Foundations

Time Series Fundamentals (discrete vs continuous, stationarity, white noise)
Autocovariance and Autocorrelation Function — ACVF/ACF, sample estimation, Bartlett’s formula.
Partial Autocorrelation Function — PACF and its role in identifying AR order.
Time Series Best Linear Predictors — optimal linear forecasting given known second-order structure.
Innovations Algorithm — recursive one-step prediction for general (non-stationary-friendly) processes.

Moving Average Processes — MA(q): always stationary, ACF cuts off at lag \(q\).
Autoregressive Processes — AR(p): causality condition, ACF decays, PACF cuts off at lag \(p\).
ARMA Processes — ARMA(p,q), combining AR and MA structure.
Causality and Invertibility — root conditions unifying the AR/MA stationarity and invertibility results.
Yule-Walker Equations — moment-based estimation of AR coefficients from the ACF.
Durbin-Levinson Algorithm — recursive solution of the Yule-Walker system and sample PACF.
ARIMA Processes — ARIMA(p,d,q): differencing to handle non-stationary (integrated) series.
SARIMA Processes — seasonal ARIMA(p,d,q)(P,D,Q)\(_s\).
Time Series Decomposition — classical/STL trend-seasonal-remainder decomposition, an alternative to differencing out seasonality.
Model Selection — AIC/BIC/AICc for choosing model order.
Ljung-Box Test — portmanteau residual white-noise diagnostic.

Spectral Distributions — spectral density, the spectral representation theorem.
Periodogram and Spectral Estimation — periodogram, smoothing/tapering, Whittle likelihood.
Wavelet Analysis — joint time-frequency (multiresolution) analysis for non-stationary signals.

Unit Root Tests — (Augmented) Dickey-Fuller, KPSS, Phillips-Perron: testing \(I(1)\) vs stationary.
Long Memory Processes (ARFIMA) — fractional differencing, hyperbolically decaying ACF.
Cointegration — shared stochastic trends, Engle-Granger and Johansen procedures, VECM.
Vector Autoregression (VAR) — multivariate AR, impulse responses, variance decomposition.
Granger Causality — testing predictive (lead-lag) information content across series.
State-Space Models and the Kalman Filter — unifying latent-state representation, exact filtering/smoothing/MLE.
Regime-Switching Models — Markov-switching, TAR/SETAR, STAR: nonlinear regime-dependent dynamics.
Space Time Modeling — extending time series methods to spatially indexed data.

Volatility Models — overview: constant, deterministic, local, and stochastic volatility in derivatives pricing.
Time-Dependent Volatility Model · Local Volatility Model — deterministic volatility extensions of Black-Scholes.
ARCH Models — conditional heteroskedasticity, volatility clustering.
GARCH Models — GARCH(p,q) and extensions (IGARCH, EGARCH/GJR, GARCH-M, multivariate GARCH).
Stochastic Volatility Model — volatility as its own latent diffusion, correlated with returns (leverage).
Dealing with Stochastic Volatility in Time Series Using the R package stochvol · Heavy-Tailed Innovations in the R Package stochvol · Bayesian Analysis of a Stochastic Volatility Model — Bayesian/MCMC estimation of SV models (paper notes).
Realized Volatility — nonparametric, high-frequency-based volatility measurement; jumps and bipower variation.
HAR-RV Model — heterogeneous-horizon linear forecasting of realized volatility, mimicking long memory.
MIDAS Models — mixed-frequency regressions (e.g. GARCH-MIDAS, nowcasting) bridging high- and low-frequency data.

Brockwell, P. J. and Davis, R. A. Introduction to Time Series and Forecasting.
Shumway, R. H. and Stoffer, D. S. Time Series Analysis and Its Applications.