Computing and using residuals in time series models

The most often used approaches to obtaining and using residuals in applied work with time series models, are unified and documented with both partially-known and new features. Specifically, three different types of residuals, namely "conditional residuals", "unconditional residuals" and "innovations", are considered with regard to (i) their precise definitions, (ii) their computation in practice after model estimation, (iii) their approximate distributional properties in finite samples, and (iv) potential applications of their properties in model diagnostic checking. The focus is on both conditional and unconditional residuals, whose properties have received very limited attention in the literature. However, innovations are also briefly considered in order to provide a comprehensive description of the various classes of residuals a time series analyst might find in applied work. Theoretical discussion is accompanied by practical examples, illustrating (a) that routine application of standard model-building procedures may lead to inaccurate models in cases of practical interest which are easy to come across, and (b) that such inaccuracies can be avoided by using some of the new results on conditional and unconditional residuals developed with regard to points (iii) and (iv) above. For ease and clarity of exposition, only stationary univariate autoregressive moving average models are considered in detail, although extensions to the multivariate case are briefly discussed as well.