Description

1️⃣ Time series analysis studies data evolving through time — focusing here on stationary processes with stable mean and variance.
2️⃣ Linear processes express Xt=μ+∑iaiεt−iXt​=μ+∑i​ai​εt−i​ with white-noise shocks and absolutely summable filters.
3️⃣ Integration order $I(d)$: differencing $d$ times renders a non-stationary process stationary (e.g. $I(0)$, $I(1)$).
4️⃣ Moving-average models $MA(q)$ combine current and past shocks; ACF cuts off after lag q, PACF decays.
5️⃣ Autoregressive models $AR(p)$ use past values; stationary when characteristic roots lie outside the unit circle.
6️⃣ ARMA(p,q) merges both structures; invertibility ensures a unique, well-behaved representation.
7️⃣ Box–Jenkins method: identification → estimation → diagnostic checking for ARIMA modelling.
8️⃣ ACF / PACF diagnostics distinguish AR, MA, ARMA structures; Yule-Walker equations link autocorrelations to parameters.
9️⃣ Stationarity check: explosive AR roots ($|z|<1$) or slowly decaying ACF → non-stationary behavior.
🔟 Applications: simulation, spectral analysis, and QuantLet exercises illustrate model behaviour and stability.