Determining inflation persistence is a prominent issue when it comes to forecasting (Stock and Watson, 2007), or when monetary policy recommendations are at stake, see e.g. Mishkin (2007). Kumar and Okimoto (2007) addressed the possibility of breaks in inflation persistence within a framework of fractional integration, which can be traced back to Hassler and Wolters (1995) or Baillie, Chung and Tieslau (1996). The effect of temporal aggregation on inflation dynamics has recently been studied by Paya, Duarte and Holden (2007). The question how aggregation and persistence interact is of interest beyond inflation, and has troubled applied economists for a long time, see Christiano, Eichenbaum and Marshall (1991) for empirical evidence in the context of the permanent income hypothesis and Rossana and Seater (1995) for a representative set of economic time series. Using fractionally integrated models, Chambers (1998) found with macroeconomic series that the empirical degree of integration may depend on the level of temporal aggregation, see also Diebold and Rudebusch (1989). In empirical finance, too, one of the core issues with respect to realized volatility is optimal sampling, see e.g. Ait-Sahalia, Mykland and Zhang (2005) or Andersen and Bollerslev (1998).
In this paper we understand by temporal aggregation both: systematic sampling (or skip sampling) of stock variables where only every pth data point is observed, and summation of flow variables where neighbouring observations are cumulated to determine the total flow. Econometricians have devoted their attention to both types of temporal aggregation for decades. Early results for autoregressive moving-average (ARMA) models were obtained by Brewer (1973) and Weiss (1984), and by Geweke (1978) for sta-tionary dynamic regression models. A treatment of integrated (of order one) ARIMA models was provided by Wei (1981) and Stram and Wei (1986), for skip sampling and cumulating, respectively. In particular, skip sampling can be embedded in the more general problem of missing observations, see Palm and Nijman (1984) for an investigation of dynamic regression models. In the frequency domain, temporal aggregation will be accompanied by the socalled aliasing effect, which is well known under discrete-time sampling from a continuous-time process, see e.g. Sims (1971) and Hansen and Sargent (1983). In particular, the aspect of temporal aggregation and forecasting has been addressed by Lütkepohl (1987). Moreover, the potential interaction of seasonal integration and unit roots at frequency zero due to temporal aggregation was studied by Granger and Siklos (1995), see also Pons (2006).
