C.W.J. Granger. Long memory relationships and the aggregation of dynamic models. J. Econometrics, 14:227-238, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....kuehng mathematik.tu muenchen.de, http: www m4.ma.tum.de m4 researchers, cf. e.g. Hu and ksendal [14] and the references therein. For an introduction to FBM see Samorodnitsky and Taqqu [22] Certain nancial time series show long memory properties as observed since the 1980s; see Granger [11], resp. Granger and Joyeux [12] and Mandelbrot [18] Such observation has led to an ongoing debate among econometricians and statisticians. It is obvious that any deterministic component like a small trend or business cycle can cause a ctitious long memory e ect in a time series and it has been ....

C.W.J. Granger. Long memory relationships and the aggregation of dynamic models. J. Econometrics, 14:227-238, 1980.

....there are not many 38 economic arguments available to support these statistical findings. Bollerslev and Mikkelson (1994) suggest that long memory in the volatilities of stock market indexes is a consequence of aggregation, because individual returns appear to have less persistent volatility. Granger (1980) shows that the sum of AR(1) processes with coefficients drawn randomly from a suitable distribution approaches a long memory process, as the number of terms in the sum increases. The same result can be derived in the context of short memory stochastic volatility models, with aggregation ....

Granger, C. (1980),"Long memory relationships and the aggregation of dynamic models," Journal of Econometrics 14, 227-238.

....series since [30] The relative success of the LRD concept in economics may also be attributed to the development of a rationale for its presence in macro level economic and financial systems based on the aggregation of micro units. This was originally proposed in [95] and further developed in [49], both in the context of contemporaneous aggregation of heterogeneous AR(1) micro units. Estimation of fully parametric long memory models such as ARFIMA appeared cumbersome, especially in the time domain, and, moreover, practical estimation AMS Subject classification. Primary 62M10# secondary ....

.... of short run dynamics, but also against departures from Gaussianityin the t s (see [104] and [19] x In most existing literature on interest rates, the short rate process has the property Marc Henry,Paolo Zaffaroni These justifications are based on an aggregation result in [95] and [49]with the more general implication that LRD may be found in the aggregate produced from a large number of heterogeneous autoregressive processes describing the microeconomic dynamics of each unit, e.g. the behaviour of eachagent when heterogeneity among agents is allowed. In the generalized ....

Granger, C. W. J. (1980): "Long memory relationships and the aggregation of dynamic models," Journal of Econometrics, 14, 227--238.

....to a short memory context. Three explanatory sections precede the illustrative option pricing results in Section 5, so that this paper provides a self contained description of how to price options with a long memory assumption. A general introduction to the relevant literature is provided by Granger (1980) and Baillie (1996) on long memory, Andersen, Bollerslev, Diebold and Labys (2000) on evidence for long memory in volatility, Duan (1995) on option pricing for ARCH processes, and Bollerslev and Mikkelsen (1999) on applying these pricing methods with long memory specifications. 3 Section 2 ....

....that are proportional to d 2 w and variance ratios whose logarithms are very close to linear functions of the aggregation period n. It is also seen from these papers that estimates of d are between 0.3 and 0.5, with most estimates close to 0.4. 2.4. Explanations of long memory in volatility Granger (1980) shows that long memory can be a consequence of aggregating short memory processes; specifically if AR(1) components are aggregated and if the AR(1) parameters are drawn from a Beta distribution then the aggregated process converges to a long memory process as the number of components increases. ....

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Granger, C.W.J., 1980, Long memory relationships and the aggregation of dynamic models, Journal of Econometrics 14, 227-238.

....# Technical Report TR COSC 03 98 1 Several methods for generating pseudo random self similar sequences have been proposed. They include methods based on fast fractional Gaussian noise [15] fractional ARIMA processes [9] the M G # queue model [11] 13] autoregressive processes [3] [8], spatial renewal processes [26] etc. Some of them generate asymptotically self similar sequences and require large amounts of CPU time. For example, Hosking s method [9] based on the F ARIMA(0,d,0) process, needs many hours to produce a self similar sequence with 131,072 (2 17 ) numbers on a ....

Granger, C. Long Memory Relationships and the Aggregation of Dynamic Models. Journal of Econometrics 14 (1980), 227--238.

....control mechanisms. Several methods for generating pseudo random self similar sequences have been proposed. They include methods based on fast fractional Gaussian noise [14] fractional ARIMA processes [9] the # Technical Report 1 M G # queue model [10] 12] autoregressive processes [3] [8], spatial renewal processes [23] etc. Some of them generate asymptotically self similar sequences and require large amounts of CPU time. For example, Hosking s method [9] based on the F ARIMA(0,d,0) process, needs many hours to produce a self similar sequence with 131,072 (2 17 ) numbers on a ....

Granger, C. Long Memory Relationships and the Aggregation of Dynamic Models. Journal of Econometrics 14 (1980), 227--238.

....example, in the case of the squares of the daily returns of the S P 500 it could happen that squares for the individual stocks do not exhibit long memory and the apparent long memory of the index is just due to aggregation. A motivation for this can be found, for instance, in Robinson (1978) or Granger (1980). We address the nonstationarity problem by splitting up the daily data into arguably stationary periods and the aggregation problem by using daily data on the individual stocks in the Dow Jones Industrial Average. Our conclusions confirm the results of Ding et al. 1993) In particular, for ....

....squared returns are derived from the squared returns of the individual stocks. It may well happen that the specific stocks do not exhibit strong dependence and the apparent long memory of the index is just due to aggregation. A motivation of this can be found, for instance, in Robinson (1978) or Granger (1980). In these papers it is shown that starting with individual independent AR(1) series with random autoregressive coefficients, the aggregate series can exhibit long memory for certain specifications of the distribution function from which these coefficients are drawn. This result can be generalized ....

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Granger, C.W.J. (1980), "Long Memory Relationships and the Aggregation of Dynamic Models", Journal of Econometrics,14,227-238.

.... ffl aggregation methods based on aggregating a large number of single source models such as models based on chaotic maps [19] ON OFF model with innite variance sojourn times [22] and AR(1) processes with the AR(1) parameters chosen from a beta distribution on [0,1] with shape parameters p and q [6]. 3 ffl bisection methods based on generating points of the process path itopdownj by properly interpolating from the existing points. The last group includes, as the simplest case, random midpoint displacement (RMD) which has traditionally been identied as a method appropriate for fast but ....

C.W.J. Granger. Long memory relationships and the aggregation of dynamic models. J. Econometr., 14:227238, 1980.

....) 1 ( t L x L L t t d e q f (2) where f and q are polynomials of orders p and q respectively, with all zeroes of f(L) outside the unit circle, and all zeros of q(L) outside or on the unit circle, and e t is white noise. These kind of models were introduced by Granger and Joyeux (1980) Granger (1980, 1981) and Hosking (1981) although earlier work by Adenstedt (1974) and Taqqu (1975) shows an awareness of the representation) and were justified theoretically in terms of aggregation by Robinson (1978) and Granger (1980) In view of the preceding remarks there is some interest in estimating ....

....noise. These kind of models were introduced by Granger and Joyeux (1980) Granger (1980, 1981) and Hosking (1981) although earlier work by Adenstedt (1974) and Taqqu (1975) shows an awareness of the representation) and were justified theoretically in terms of aggregation by Robinson (1978) and Granger (1980). In view of the preceding remarks there is some interest in estimating the fractional differencing parameter d, along with the other parameters related to the ARMA representation. Sowell (1992) analyzed the exact maximum likelihood estimates of the parameters of a fractional 3 ARIMA model (2) ....

Granger, C. W. J., 1980, Long memory relationships and the aggregation of dynamic models, Journal of Econometrics 14, 227-238.

....various mechanisms that could generate long memory. Most econometric attention has focused on the role of aggregation. Here we briefly review two such aggregation based perspectives on long memory, in order to contrast them to our subsequent perspective, which is quite different. First, following Granger (1980), consider the aggregation of i = 1, N cross sectional 4x it i x i,t 1 g it , f x (T ) N 2B E [var(g it ) m 1 1 e iT 2 dF( dF( 2 B(p,q) 2p 1 (1 2 ) q 1 d , 0# #1, x (J) 2 B(p,q) m 1 0 2p J 1 (1 2 ) q 2 d AJ ....

....is white noise, and for all i, j, t. As , the spectrum of the aggregate g it g it zg jt i zg jt N64 can be approximated as x t j N i 1 x it where F is the c.d.f. governing the s. If F is a beta distribution, then the J th autocovariance of is x t Thus . x t I 1 q 2 Granger s (1980) elegant bridge from cross sectional aggregation to long memory has since been refined by a number of authors. Lippi and Zaffaroni (1999) for example, generalize Granger s result by replacing Granger s assumed beta distribution with weaker semiparametric 5W ( M (Tt) m Tt 0 ( j M m 1 W ....

Granger, C.W.J. (1980), "Long Memory Relationships and the Aggregation of Dynamic Models," Journal of Econometrics, 14, 227-238.

....process, so that although volatility shocks are highly persistent, they eventually dissipate at a slow hyperbolic rate. 25 Hence, we now turn to an investigation of fractional integration in the daily realized volatility. Fractionally integrated long memory processes were introduced by Granger (1980, 1981) and Granger and Joyeux (1980) for a recent survey of their applications in economics see Baillie (1996) The slow hyperbolic decay of the long lag sample autocorrelations and the log linear explosion of the low frequency spectrum are distinguishing features of a covariance stationary ....

Granger, C.W.J. (1980), "Long Memory Relationships and the Aggregation of Dynamic Models," Journal of Econometrics, 14, 227-238.

....and over long time periods, this procedure introduces significant temporal dependence. These attributes are shared by fractional Brownian motion. A similar construction of asymptotic self similarity, employing certain AR(1) processes in lieu of the renewal reward process W , may be found in [38]. 2.6.7 Parsimonious Modeling Since empirical data sets are necessarily finite, it is not possible to determine with certainty whether or not the asymptotic relations (9) 13) 14) etc. hold for data records. However, 20 if sufficient data (empirical or simulated) are available and the ....

Granger, C.W.J., "Long Memory Relationships and the Aggregation of Dynamic Models ", J. Econometrics 14 (1980), 227--238.

....that each individual reacts to publicly available news in their own way, so that, for example, interest rate rises are good news for lenders and bad news for borrowers. This aspect of the model is not essential to the present development, and readers are referred to BDP for the details. 4 due to Granger (1980), is that under a beta(u,v) distribution for the a , the time series representation of t y approximates (large N) to a process of the form = o k k t k t y e a (3) where ) v k k O = a , and t e is a shock process depending on news. Simply stated, this says that averaging a mixture of ....

Granger, C. W. J. (1980) `Long memory relationships and the aggregation of dynamic models', Journal of Econometrics 14, 227-238 10 Granger, C. W. J. and R. Joyeux, (1980) 'An Introduction to Long Memory Time Series Models And Fractional Differencing' Journal Of Time Series Analysis 1 (1) 1529.

....1 Z 0 (ae) ae n Gamma1 dae h 0 = 1 (2) On a ainsi agr eg e continument des suites g eom etriques ae n Gamma1 au moyen d un poids, ou densit e d une mesure (ae) sin(ff ) 1 ae Gamma 1) Gammaff . Une agr egation analogue avait d ej a et e propos ee dans un cadre al eatoire [4]. 2.3 Lemme de Watson discret La formulation int egrale de la r eponse impulsionnelle permet une etude plus raffin ee du comportement asymptotique de la r eponse impulsionnelle. Une formulation semblable avait d ej a permis des r esultats sur la stabilit e de filtres a m emoire longue [6] Le ....

C.W.J. Granger. Long Memory Relationships and the Aggregation of Dynamic Models. J. of Econometrics, 14:227--238, 1980.

....seems implausible that variations in the wealth distribution could occur with sufficient frequency to drive the ARCH effects observed in stock prices. The literature on theoretical explanations of the observed persistence in economics timeseries is even sparser. The earliest result is the paper by Granger [1980] showing how aggregation of independent AR(1) processes can generate a process which is fractionally integrated. Since many financial and economic decision problems in stochastic environments lead to decision rules and pricing relations which are autoregressive, this would seem a plausible ....

Granger, C. W. J., Long Memory Relationships and the Aggregation of Dynamic Models. Journal of Econometrics 14 (1980) 227-238.

.... features are considered in [28, 156, 160, 274, 275, 369, 371] based on shot noise processes) 373] linear models with long range dependence) 30,261, 262, 276, 284, 411, 412] renewal reward processes and their superposition) 116, 283,401] renewal processes or zero rate processes) [165] (aggregation of simple short range dependent models) and [135, 294, 414, 416] wavelet analysis) Further models are considered in [16,19,176,313,379,386,403,417] A radically different approach to modeling self similar phenomena relies on ideas from the theories of chaos and fractals ....

....algorithm [37, 394] are discussed in [24, 146, 196, 198, 394] They are generally impractical for long time series. Approximate methods are described in [30, 74, 120, 215, 245, 252, 258, 285, 290, 331, 334, 337, 356, 374, 376, 404,412] some of these methods rely on earlier results reported in [77, 165,391], derived for a different purpose and re interpreted here in the context of synthetic traffic generation. These methods are generally very fast and feasible for even A Bibliographical Guide 7 very long time series. However, the statistical quality of the generated sequences is, in general, not ....

C. W. J. Granger. Long memory relationships and aggregation of dynamic models. Journal of Econometrics, 14:227--238, 1980.

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C.W.J. Granger. Long memory relationships and the aggregation of dynamic models. J. Econometrics, 14:227-238, 1980.

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C.W.J. Granger, Long memory relationships and the aggregation of dynamic models, J. Econometrics 14 (1980) 227--238.

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Granger, C.W.J. (1980) "Long Memory Relationships and the Aggregation of Dynamic Models ", Journal of Econometrics , 14, 227-238.

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GRANGER, C.W.J. (1980): \Long-Memory Relationships and the Aggregation of Dynamic Models", Journal of Econometrics, 14, 227-238.

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GRANGER, C.W.J. (1980): \Long-Memory Relationships and the Aggregation of Dynamic Models", Journal of Econometrics, ##, 227-238.

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C. W. J. Granger, "Long Memory Relationships and the Aggregation of Dynamic Models", J. Econometr. 14, 227-238, 1980.

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