Time-Series Models for Financial Data › Summary

This chapter developed the theory and practice of time-series modeling for financial data. We established stationarity as the foundational property enabling meaningful statistical analysis and showed that while asset prices are non-stationary, returns typically achieve approximate stationarity.

Autoregressive models express current values as linear functions of past values, with the AR coefficient controlling the persistence of shocks. The stationarity condition $|\phi| < 1$ ensures the series remains bounded, while the PACF's cutoff property identifies AR order. Moving average models instead express observations as functions of past innovations, with the ACF cutting off after lag $q$. ARMA combines both structures for parsimonious modeling of complex autocorrelation patterns.

For non-stationary series, ARIMA extends the framework through differencing. The integrated component handles trends and unit roots common in economic and price data. The Box-Jenkins methodology provides a systematic approach: assess stationarity, examine ACF/PACF patterns, estimate candidate models, compare using information criteria, and verify through residual diagnostics.

Forecasting from ARIMA models produces point predictions and uncertainty intervals that widen with horizon. This quantification of forecast uncertainty is essential for risk management applications. The tools developed here, particularly ACF analysis and model diagnostics, will prove valuable throughout the remainder of this book as we explore volatility modeling, regression methods, and machine learning approaches to financial prediction.