1
Introduction

1.1 Empirical Examples

Figure 1.1 displays 663 annual observations of minimal water levels of the Nile river. This historical data is from Beran (1994, Sect. 12.2) and ranges from the year 622 until 1284. The second panel contains the sample autocorrelations at lag . The maximum value, , is not particularly large, but the autocorrelogram dies out only very slowly with still being significantly positive. Such a slowly declining autocorrelogram is characteristic of what we will define as long memory or strong persistence. It reflects that the series exhibits a very persistent behavior in that we observe very long cyclical movements or (reversing) trends. Note, e.g. that from the year 737 until 805, there are only three data points above the sample average (=11.48), i.e. there are seven decades of data below the average. Then the series moves above the average for a couple of years, only to swing down below the sample mean for another 20 years from the year 826 on. Similarly, there is a long upward trend from 1060 on until about 1125, followed again by a long-lasting decline. Such irregular cycles or trends due to long-range dependence, or persistence, have first been discovered and discussed by Hurst, a British engineer who worked as hydrologist on the Nile river; see in particular Hurst (1951). Mandelbrot and Wallis (1968) coined the term Joseph effect for such a feature; see also Mandelbrot (1969). This alludes to the biblical seven years of great abundance followed by seven years of famine, only that cycles in Figure 1.1 do not have a period of seven years, not even a constant period.

Figure 1.1 Annual minimal water levels of the Nile river.

Long memory in the sense of strong temporal dependence as it is obvious in Figure 1.1 has been reported in many fields of science. Hipel and McLeod (1994, Section 11.5 ) detected long memory in hydrological or meteorological series like annual average rainfall, temperature, and again river flow data; see also Montanari (2003) for a survey. A further technical area beyond geophysics with long memory time series is the field of data network traffic in computing; see Willinger et al. (2003).

The second data set that we look into is from political science. Let denote the poll data on partisanship, i.e. the voting intention measured by monthly opinion polls in England. More precisely, is the portion of people supporting the Labor Party. The sample ranges from September 1960 until October 1996 and has been analyzed by Byers et al. (1997).1 Figure 1.2 contains the logit transformation of this poll data,

such that for ; here stands for the natural logarithm of . We observe long-lasting upswings followed by downswings amounting to a pseudocyclical pattern or reversing trends. This is well reflected and quantified by the sample autocorrelations in the lower panel, decreasing from quite slowly to . Independently of Byers et al. (1997), Box-Steffensmeier and Smith (1996) detected long memory in US opinion poll data on partisanship. Long memory in political popularity has been confirmed in a sequence of papers; see Byers et al. (2000, 2007), and Dolado et al. (2003); see also Byers et al. (2002) for theoretical underpinning of long memory in political popularity. Further evidence on long memory in political science has been presented by Box-Steffensmeier and Tomlinson (2000); see also the special issue of Electoral Studies edited by Lebo and Clarke (2000).

Figure 1.2 Monthly opinion poll in England, 1960-1996.

Since Granger and Joyeux (1980), the fractionally integrated autoregressive moving average () model gained increasing popularity in economics. The empirical example in Granger and Joyeux (1980) was the monthly US index of consumer food prices. Granger (1980) had shown theoretically how the aggregation of a large number of individual series may result in an index that is fractionally integrated, which provided theoretical grounds for long memory as modeled by fractional integration in price indices. A more systematic analysis by Geweke and Porter-Hudak (1983) revealed long memory in different US price indices. These early papers triggered empirical research in long memory in inflation rates in independent work by Delgado and Robinson (1994) for Spain and by Hassler and Wolters (1995) and Baillie et al. (1996) for international evidence. Since then, there has been offered abundant evidence in favor of long memory in inflation rates; see, e.g. Franses and Ooms (1997), Baum et al. (1999), Franses et al. (1999), Hsu (2005), Kumar and Okimoto (2007), Martins and Rodrigues (2014), and Hassler and Meller (2014), where the more recent research focused on breaks in persistence, i.e. in the order of fractional integration. For an early survey article on further applications in economics, see Baillie (1996).

Figure 1.3 gives a flavor of the memory in US inflation. The seasonally adjusted and demeaned data from January 1966 until June 2008 has been analyzed by Hassler and Meller (2014). The autocorrelations fall from to a minimum of , Again, this slowly declining autocorrelogram mirrors the reversing trends in inflation, although Hassler and Meller (2014) suggested that the persistence may be superimposed by additional features like time-varying variance.

Figure 1.3 Monthly US inflation, 1966-2008.

The fourth empirical example is from the field of finance. Figure 1.4 displays daily observations from January 4, 1993, until May 31, 2007. This sample of 3630 days consists of the logarithm of realized volatility of International Business Machines Corporation () returns computed from underlying five-minutes data; see Hassler et al. (2016) for details. Although the dynamics of the series is partly masked by extreme observations, one clearly may distinguish periods of weeks where the data tends to increase, followed by long time spans of decrease. The high degree of persistence becomes more obvious when looking at the sample autocorrelogram. Starting off with , the decline is extremely slow with still being well above 0.2. Long memory in realized volatility is sometimes considered to be a stylized fact since the papers by Andersen et al. (2001, 2003). Such a view is supported by the special issue in Econometric Reviews edited by Maasoumi and McAleer (2008).

Figure 1.4 Daily realized volatility, 1993-2007.

Finally, with the last example we return to economics. Figure 1.5 shows 435 monthly observations from 1972 until 2008. The series is the logarithm of seasonally adjusted US unemployment rates (number of unemployed persons as a percentage of the civilian labor force); see Hassler and Wolters (2009) for details. The sample average of log-unemployment is 1.7926; compare the straight line in the upper panel of Figure 1.5. Here, the trending behavior is so strong that the sample average is crossed only eight times over the period of 35 years. The deviations from the average are very pronounced and very long relative to the sample size. In that sense the series from Figure 1.5 seems to be most persistent of all the five examples considered in this introduction. This is also expressed by the sample autocorrelogram virtually beginning at one and for What is more, the autocorrelations decline almost linearly in , which is indicative of an process or an process with even ; see Hassler (1997, Corollary 3) and Section 7.5. Hence, the log-unemployment data seems to be most persistent, or most strongly trending, among our empirical examples.

Figure 1.5 Monthly unemployment rate, 1972-2008.

1.2 Overview

There are two natural approaches to long memory modeling by fractional integration. The first one takes the nonstationary model as starting point, i.e. processes integrated of order 1. Such processes are often labeled as unit root processes in econometrics, where they play a major role within the cointegration framework; see, for instance, Johansen (1995), Lütkepohl (2005), or Pesaran (2015). The extension from the model to the more general model might be considered as a nearby approach from an econometric point of view. The second approach starts off with the classical stationary time series model, where the moving average coefficients from the Wold decomposition are assumed to be absolutely summable and to sum to a value different from 0. For this model, which may be called integrated of order 0, (see Chapter 6 ), it holds true that the scaled sample average converges with the square root of the sample size to a nondegenerate normal distribution. This model underlying the major body of time series books from Anderson (1971) over Brockwell and Davis (1991) and Hamilton (1994) to Fuller (1996) may be generalized to the stationary process for . The latter can be further extended to the region of nonstationarity ( ). Here, we follow this second route starting with the case. More precisely, the outline of the book is as...