4.5 Summary
Hundreds of models have been proposed over the past decades for financial time series attempting to incorporate temporal structure, both for mean modeling, \(\bm{\mu}_t\), and variance modeling, \(\bm{\Sigma}_t\), with the following takeaways:
Mean models range from simple moving averages to more sophisticated ARMA models (or even VECM). However, it is debatable whether they can outperform the simple i.i.d. model, particularly considering the small autocorrelation exhibited by typical financial time series. Nonetheless, the conclusion may greatly depend on the nature and frequency of the financial data.
Variance (or volatility) models are undoubtedly practical, as financial data clearly displays a significant degree of temporal structure in variance (or volatility). Two main approaches exist: GARCH modeling, which is by far the most popular direction in econometrics, and stochastic volatility modeling, which arguably produces a more desirable volatility envelope. Interestingly, stochastic volatility has not gained the same popularity as GARCH models, perhaps due to its higher computational complexity (although this can be remedied via Kalman filtering).
State space modeling provides a general and convenient framework for financial time series. In fact, it embraces most of the common models for the mean and it approximates reasonably well the variance models, such as stochastic volatility modeling.
The Kalman filter is an efficient algorithm for fitting financial time series that can be represented as a state space model. Moreover, it enables time-varying modeling, which is essential for financial data. However, its usage does not seem to be as widespread as it deserves within the financial community, despite being covered in standard time series textbooks.