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).
We add two aspects to this literature: a general characterization of time aggregation in the frequency domain, and an investigation how assumptions for semiparametric inference of a fractionally integrated model are affected under temporal aggregation. In greater detail our contributions are the following. First, we study the effect of temporal aggregation (cumulating flow variables or systematic skip sampling stock variables) of an arbitrary stationary process in the frequency domain. Several results that are implied or at least suggested in the literature are here explicitly collected under general conditions (Proposition 1). Second, the aggregation result is applied to fractionally integrated processes. In particular, we investigate whether typical assumptions on fractionally integrated processes, which are made in the literature to obtain consistency or limiting normality of semiparametric estimators, are closed with respect to aggregation. In other words: if {yt} satisfies a set of assumptions A (which are sufficient to prove properties of some estimator or test), does the temporal aggregate fulfill A, too? If not, then we should be worried, because in most cases there is no “true” or “natural” frequency of the data generating process (DGP), i.e. our observed data must be considered as aggregates. If they do not satisfy A upon aggregation, then we lose grounds for reliable inference. Third, it turns out that typical spectral assumptions made in the semiparametric long memory literature are closed with respect to cumulating and averaging the data (Proposition 2 and Corollary 1). Fourth, it is established that certain spectral assumptions are not closed with respect to skip sampling fractional integration (Proposition 3). Fifth, we discuss repair proposals to this shortcoming.
The rest of this paper is organized as follows. Section 2 treats aggregation in terms of spectral densities. In Section 3, the aggregation result is applied to fractional integration, and the effects of temporal aggregation on integration are studied in some detail. The last section contains a non-technical summary. Proofs are relegated to the Appendix.
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