Fitting Autoregressive Models via Yule-Walker Equations Allowing Heavy Tail Innovations

Modern treatments of actuarial risk decision problems increasingly involve heavy tailed data and distributions. Here we consider the setting of time series and the problem of fitting an autogressive model with heavy tailed innovations. Assuming only finite first moments, we introduce a linear system of equations similar to the least squares approach but using Gini covariances instead of the usual ones. This leads to convenient, easily interpreted, closed form expressions for the estimated parameters, thereby capturing the advantages of the least squares approach without requiring second order moment assumptions. A “Gini autocovariance function” is introduced, along with certain other novel types. Related results for the multiple regression problem with heavy tailed errors are treated briefly as well. A numerical comparison of the Gini approach and the least squares approach is provided. AMS 2000 Subject Classification: Primary 62M10 Secondary 62G05