Time Series Analysis – John Robin Inston

1 Time Series Analysis

Time series analysis is the study of describing, modeling, and forecasting data recorded sequentially over time, where — unlike in i.i.d. settings — the ordering of observations carries information and successive values are typically dependent. The field rests on a small set of core ideas that recur across every sub-area below: (1) reducing a dependent sequence to a stationary process (by differencing, decomposition, or transformation) so that classical asymptotic theory applies; (2) characterizing dependence through the autocovariance/autocorrelation function in the time domain, or the spectral density in the frequency domain; (3) building parsimonious generative models (AR, MA, ARMA and their many descendants) whose parameters can be estimated and whose structure gives interpretable, testable predictions; and (4) using the fitted model to forecast, filter, and quantify uncertainty out of sample.

Classical (academic/Box-Jenkins) time series analysis focuses on the conditional mean of a process and linear dependence structure. Quantitative finance adds a second axis largely absent from the classical treatment: modeling the conditional variance (volatility clustering, leverage effects) of financial returns, since means are close to unpredictable but variances are highly persistent and directly priced (options, risk management). The notes below are organized as a sequence of textbook-style chapters moving from foundations, through classical linear models, into frequency-domain methods, non-stationarity/multivariate extensions, and finally the volatility- and mixed-frequency-focused topics central to financial econometrics.

2 Topics

2.1 1. Foundations

Time Series — discrete- vs continuous-time series, the basic object of study.
Stationary Processes — strict vs weak (covariance) stationarity, the assumption underlying nearly everything that follows.
White Noise — the elementary building-block “shock” process.
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.

2.2 2. Linear Time Series Models (Box-Jenkins)

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 Criteria (AIC BIC) — AIC/BIC/AICc for choosing model order.
Ljung-Box Test — portmanteau residual white-noise diagnostic.

2.3 3. Frequency-Domain Methods

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.

2.4 4. Non-Stationarity, Long Memory & Multivariate Extensions

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.

3 5. Financial Econometrics & Volatility Modeling (QF)

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.

4 Foundational References

[[introduction-to-time-series-and-forecasting-brockwell-pdf]] (and solutions) — primary textbook reference for Sections 1-2.
Time Series Analysis and its Applications (with R examples) — Shumway & Stoffer, time series analysis and its applications.
[[pstat274-time-series-analysis]] · [[pstat274-time-series-summary-notes]] — course notes underlying much of Sections 1-2.
Time Series Office Hours — worked example problems (ACVF/ACF derivations).