The stylised facts establish that daily log-returns are not iid. To model their dynamics we need the language of time series: processes indexed by time, with a formal notion of stationarity (the behaviour doesn't drift over the sample), autocorrelation functions (ACF) that measure serial dependence, and estimation tools to diagnose whether a fitted model's residuals are serially independent.
ARMA models — autoregressive moving-average processes — are the classical machinery for modelling serial structure in the conditional mean . For risk modelling of daily returns, pure ARMA models are rarely sufficient: the raw return series shows little autocorrelation, so the conditional mean is close to zero and ARMA adds little. But the tools of time-series analysis (ACF, PACF, Ljung–Box, residual analysis) are essential for the next step: diagnosing whether the conditional variance is constant, and fitting GARCH models when it is not.
Learning objectives. By the end of this note you should be able to:
- state the definitions of strict and covariance stationarity, and give an example of a process that is one but not the other;
- define white noise, strict white noise, and martingale difference, and state the relationships among them;
- write down ARMA in lag-polynomial form and state the causality condition;
- compute the ACF of an AR and an MA process by hand;
- apply and interpret the Ljung–Box portmanteau test on raw and absolute return series;
- describe the standard time-series model-fitting programme: plot, ACF/PACF, fit, residuals, model comparison.
In this lesson
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- Stationarity
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- Autocovariance and autocorrelation functions
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- White noise, strict white noise, and martingale differences
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- ARMA processes
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- Model fitting workflow
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- Why ARMA alone is insufficient for financial risk
Before you start
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