A time series is data with a memory: a sequence of observations indexed by time — a share price tick by tick, a river's height day by day, the CO₂ over Mauna Loa month by month, the electrocardiogram beat by beat. What sets it apart from ordinary statistics is that the observations are not independent. Yesterday leans on the day before; December looks like last December. Time series analysis is the mathematics of that dependence — how to model it, test it, decompose it, and turn it into a forecast with honest error bars.
This master's-level course builds the subject from its probabilistic foundations to the models that run in production every day. We start with stationary stochastic processes and the autocorrelation function, construct the ARMA family and its non-stationary and seasonal extensions (ARIMA, SARIMA), learn to estimate, forecast and diagnose them the Box–Jenkins way, then cross into the frequency domain for spectral analysis. The final third covers the modern toolkit: state-space models and the Kalman filter, GARCH volatility models for finance, and multivariate methods — vector autoregression, cointegration and Granger causality.
It assumes fluency with
The probabilistic bedrock: a time series as one realization of a stochastic process, what it means for that process to be stationary, and the autocorrelation function that fingerprints it.
Before you model, you pull the series apart: trend, season, and the stationary remainder. Smoothing, differencing and the decompositions that make a series ready for ARMA.
The heart of classical time series: autoregressive and moving-average models, the backshift operator that manipulates them, and the ACF/PACF signatures that let you read a model off the data.
Real series trend and cycle. Integrating differencing into the model gives ARIMA; testing for unit roots tells you how much to difference; seasonal terms and the Box–Jenkins loop tie it together.
Fitting a model to data (least squares and maximum likelihood), producing forecasts and prediction intervals, and checking — with residual tests and information criteria — that the model is any good.
The same series seen through Fourier's eyes. The spectral density decomposes variance across frequency; the periodogram estimates it; filters are understood by what they do to each frequency.
The unifying modern framework: write the model as a hidden state evolving in time, and the Kalman filter estimates it recursively. Exponential smoothing and structural models are special cases.
Financial returns are uncorrelated but not independent: their variance clusters. ARCH and GARCH model that changing volatility — the workhorses of quantitative risk.
When several series move together: vector autoregression, the cross-correlation between them, cointegration of trending series, and Granger's operational test for "does X help predict Y?".