- 11
- Granger, C. W. J. (1980), Long memory relationships and the aggregation of dynamic models, Journal of Econometrics, 25, 227-238.

Total in-text references: 2- In-text reference with the coordinate start=13131
- Prefix
- of the sample period.3 3 Subsample estimates are not reported here, but are available upon request from the authors. -6Analysis of Components of the Monetary Aggregates Given the presence of a fractional exponent in the differencing process for the monetary aggregates, we now attempt to determine the sources of fractional dynamics. One explanation, attributed to
- Exact
- Granger (1980),
- Suffix
- is that a persistent process can arise from the aggregation of constituent processes each of which has short memory. Granger (1980) showed that if a time series ty is the sum of an infinite number of independent first-order Markov processes which have equal variances and whose autoregressive parameters are drawn independently from a beta distributio

- In-text reference with the coordinate start=13281
- Prefix
- the Monetary Aggregates Given the presence of a fractional exponent in the differencing process for the monetary aggregates, we now attempt to determine the sources of fractional dynamics. One explanation, attributed to Granger (1980), is that a persistent process can arise from the aggregation of constituent processes each of which has short memory.
- Exact
- Granger (1980)
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- showed that if a time series ty is the sum of an infinite number of independent first-order Markov processes which have equal variances and whose autoregressive parameters are drawn independently from a beta distribution with support 0,1(), then the aggregated series is asymptotically fractionally integrated with d<0. 5.

- In-text reference with the coordinate start=13131
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- Haraf, W. S. (1986), Monetary Velocity and Monetary Rules, Cato Journal, 6, 641-62.

Total in-text references: 1- In-text reference with the coordinate start=18960
- Prefix
- The time series properties behavior of the velocity of money in the U.S. has attracted a great deal of attention in the literature given its implications for the monetarist position. Gould and Nelson (1974), Nelson and Plosser (1982), and
- Exact
- Haraf (1986)
- Suffix
- conclude that money velocity contains a unit root. A similar conclusion is reached by Serletis (1995), even after allowing for the possibility of a -9one-time break in the intercept and the slope of the trend function at an unknown point in time.

- In-text reference with the coordinate start=18960
- 15
- Hassler, U. (1993), Regression of spectral estimators with fractionally integrated time series, Journal of Time Series Analysis, 14, 369-380.

Total in-text references: 1- In-text reference with the coordinate start=7091
- Prefix
- Assuming that T→∞ limgT()=∞, T→∞ lim gT() T =0, and T→∞ lim lnT()2 gT() =0, the negative of the OLS estimate of the slope coefficient in (3) provides an estimate of d. Geweke and Porter-Hudak (1983) prove consistency and asymptotic normality for d<0, while
- Exact
- Robinson (1995) and Hassler (1993)
- Suffix
- prove consistency and asymptotic normality for d∈0, 0. 5() in the case of Gaussian ARMA innovations in (1). Other authors have used maximum likelihood methods (i.e., the exact maximum likelihood method proposed by Sowell (1992b)) or the approximate frequency domain maximum likelihood method proposed by Fox and Taqqu (1986)), which simultaneously estima

- In-text reference with the coordinate start=7091
- 17
- Hosking, J. R. M. (1981), Fractional Differencing, Biometrika, 68, 165-176.

Total in-text references: 2- In-text reference with the coordinate start=1747
- Prefix
- , whether an IS-LM based structure or a much more elaborate framework, will contain a number of economic variables which have been empirically identified as possessing fractional dynamics, or elements of strong persistence, in their time series representation. The model of fractionally integrated timeseries developed by Granger and Joyeux (1980) and
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- Hosking (1981)
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- allows for a fractional, as opposed to an integer, exponent in the differencing process of the time series. This avoids the 'knife-edge' unit root distinction while permitting a modelled series to exhibit the persistence, or 'long memory,' which characterizes many macroeconomic timeseries.

- In-text reference with the coordinate start=5279
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- The stochastic process yt is both stationary and invertible if all roots of Φ(L) and Θ(L) lie outside the unit circle and d<0. 5. The process is nonstationary for d≥0. 5, as it possesses infinite variance, i.e. see Granger and Joyeux (1980). Assuming that d∈0, 0. 5() and d≠0,
- Exact
- Hosking (1981)
- Suffix
- showed that the correlation function, ρ(⋅), of an ARFIMA process is proportional to 2d−1k as k→∞. Consequently, the autocorrelations of the ARFIMA process decay hyperbolically to zero as k→∞ which is contrary to the faster, geometric decay of a stationary ARMA process.

- In-text reference with the coordinate start=1747
- 24
- Porter-Hudak, S. (1990), An application of the seasonal fractionally differenced model to the monetary aggregates, Journal of the American Statistical Association, 85, 338-344.

Total in-text references: 3- In-text reference with the coordinate start=2574
- Prefix
- Such persistence is also evident in consumption (Diebold and Rudebusch (1991)), interest rates (Shea (1991)), and inflation rates (Baillie, Chung, and Tieslau (1996), Hassler and Wolters (1995), Baum, Barkoulas, and Caglayan (1997)). The importance of measures of money in a macroeconomic modelling framework led
- Exact
- Porter-Hudak (1990) to
- Suffix
- examine M1, M2 and M3 aggregates for fractional integration. The latter study provides the motivation for this paper, in which we extend Porter-Hudak's study of fractional integration in the monetary aggregates in several important ways in order to provide comprehensive evidence on the nature of fractional dynamic behavior in these series.

- In-text reference with the coordinate start=9936
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- The unit-root hypothesis in the growth rates of the simple-sum monetary indices is decidedly rejected and evidence of fractional dynamics with long memory features is established. If we compare the range of these estimates to those estimated by
- Exact
- Porter-Hudak (1990)
- Suffix
- over the 19591986 period, we find values that are broadly comparable.2 1 We also applied the Phillips-Perron (PP, 1988) and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS, 1992) unit-root tests to the growth rates of the monetary series.

- In-text reference with the coordinate start=21556
- Prefix
- Since long memory represents nonlinear dependence in the first moment of the distribution and hence a potentially predictable component in the series dynamics, the possibility of improved forecasting via the estimation of an ARFIMA model arises.
- Exact
- Porter-Hudak (1990)
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- found superior out-of-sample forecasting performance of an ARFIMA model for the M1 aggregate versus a benchmark ARIMA model. Given the substantial fractional exponent in the differencing process in our series, similar improvements in forecasting accuracy may be expected to result from the estimation of an appropriate ARFIMA model for our data series.

- In-text reference with the coordinate start=2574
- 25
- Robinson, P. (1995), Log-periodogram regression of time series with long range dependence, Annals of Statistics, 23, 1048-1072.

Total in-text references: 1- In-text reference with the coordinate start=7091
- Prefix
- Assuming that T→∞ limgT()=∞, T→∞ lim gT() T =0, and T→∞ lim lnT()2 gT() =0, the negative of the OLS estimate of the slope coefficient in (3) provides an estimate of d. Geweke and Porter-Hudak (1983) prove consistency and asymptotic normality for d<0, while
- Exact
- Robinson (1995) and Hassler (1993)
- Suffix
- prove consistency and asymptotic normality for d∈0, 0. 5() in the case of Gaussian ARMA innovations in (1). Other authors have used maximum likelihood methods (i.e., the exact maximum likelihood method proposed by Sowell (1992b)) or the approximate frequency domain maximum likelihood method proposed by Fox and Taqqu (1986)), which simultaneously estima

- In-text reference with the coordinate start=7091
- 27
- Serletis, A. (1995), Random Walks, Breaking Trend Functions, and the Chaotic Structure of the Velocity of Money, Journal of Business and Economic Statistics, 13(4), 453-58.

Total in-text references: 1- In-text reference with the coordinate start=19073
- Prefix
- The time series properties behavior of the velocity of money in the U.S. has attracted a great deal of attention in the literature given its implications for the monetarist position. Gould and Nelson (1974), Nelson and Plosser (1982), and Haraf (1986) conclude that money velocity contains a unit root. A similar conclusion is reached by
- Exact
- Serletis (1995),
- Suffix
- even after allowing for the possibility of a -9one-time break in the intercept and the slope of the trend function at an unknown point in time. Table 4 reports the fractional-exponent estimates for the growth rates of both simple-sum and Divisia velocities.

- In-text reference with the coordinate start=19073
- 28
- Shea, G. S. (1991), Uncertainty and implied variance bounds in long-memory models of the interest rate term structure, Empirical Economics, 16, 287-312.

Total in-text references: 1- In-text reference with the coordinate start=2334
- Prefix
- For instance, fractionally integrated output series have been identified by Diebold and Rudebusch (1989) and Sowell (1992a). Such persistence is also evident in consumption (Diebold and Rudebusch (1991)), interest rates
- Exact
- (Shea (1991)), and
- Suffix
- inflation rates (Baillie, Chung, and Tieslau (1996), Hassler and Wolters (1995), Baum, Barkoulas, and Caglayan (1997)). The importance of measures of money in a macroeconomic modelling framework led Porter-Hudak (1990) to examine M1, M2 and M3 aggregates for fractional integration.

- In-text reference with the coordinate start=2334
- 29
- Sowell, F. (1992a), Modeling long-run behavior with the fractional ARIMA model, Journal of Monetary Economics, 29, 277-302.

Total in-text references: 1- In-text reference with the coordinate start=2210
- Prefix
- This avoids the 'knife-edge' unit root distinction while permitting a modelled series to exhibit the persistence, or 'long memory,' which characterizes many macroeconomic timeseries. For instance, fractionally integrated output series have been identified by Diebold and Rudebusch (1989) and
- Exact
- Sowell (1992a).
- Suffix
- Such persistence is also evident in consumption (Diebold and Rudebusch (1991)), interest rates (Shea (1991)), and inflation rates (Baillie, Chung, and Tieslau (1996), Hassler and Wolters (1995), Baum, Barkoulas, and Caglayan (1997)).

- In-text reference with the coordinate start=2210
- 30
- Sowell, F. (1992b), Maximum likelihood estimation of stationary univariate fractionally-integrated time-series models, Journal of Econometrics, 53, 165188.

Total in-text references: 1- In-text reference with the coordinate start=7370
- Prefix
- Geweke and Porter-Hudak (1983) prove consistency and asymptotic normality for d<0, while Robinson (1995) and Hassler (1993) prove consistency and asymptotic normality for d∈0, 0. 5() in the case of Gaussian ARMA innovations in (1). Other authors have used maximum likelihood methods (i.e., the exact maximum likelihood method proposed by
- Exact
- Sowell (1992b))
- Suffix
- or the approximate frequency domain maximum likelihood method proposed by Fox and Taqqu (1986)), which simultaneously estimate both the short-memory and long-memory parameters of the model.

- In-text reference with the coordinate start=7370