The 26 reference contexts in paper John Barkoulas, Christopher F. Baum, Mustafa Caglayan (1998) “Fractional Monetary Dynamics” / RePEc:boc:bocoec:321

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    Any dynamic macroeconomic model, 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
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    Granger and Joyeux (1980) and 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.
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    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
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    Diebold and Rudebusch (1989) and Sowell (1992a).
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    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)).
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    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 Sowell (1992a). Such persistence is also evident in consumption
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    (Diebold and Rudebusch (1991)),
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    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 Porter-Hudak (1990) to examine M1, M2 and M3 aggregates for fractional integration.
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    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
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    (Shea (1991)), and
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    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.
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    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 (Shea (1991)), and inflation rates (Baillie, Chung, and Tieslau (1996),
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    Hassler and Wolters (1995),
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    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.
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    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
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    Porter-Hudak (1990) to
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    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.
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    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. More specifically, we -1test for fractional integration, using the spectral regression method developed by
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    Geweke and Porter-Hudak (1983),
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    in both simple-sum and Divisia monetary aggregates, monetary base, money multipliers, and velocity series. Given clear evidence of fractional integration in the aggregates, we subsequently try to identify which components of the monetary aggregates might be responsible for fractional integration and therefore evaluate Granger's (1980) aggregation hypothesi
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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
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    Granger and Joyeux (1980).
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    Assuming that d∈0, 0. 5() and d≠0, Hosking (1981) 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.
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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,
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    Hosking (1981)
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    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.
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    The process exhibits short memory for d=0, corresponding to stationary and invertible ARMA modeling. For d∈0. 5,1[) the process is mean reverting, even though it is not covariance stationary, as there is no long-run impact of an innovation on future values of the process.
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    Geweke and Porter-Hudak (1983)
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    suggest a semiparametric procedure to obtain an estimate of the fractional differencing parameter d based on the slope of the spectral density function around the angular frequency ξ=0.
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    of the sample λξ πλ ==λ −    2 1 1 T ,...,, T is the number of observations, and ν = gT() << T2 T is the number of Fourier frequencies included in the spectral regression. 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.
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    Geweke and Porter-Hudak (1983)
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    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).
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    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
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    Robinson (1995) and Hassler (1993)
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    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
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    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
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    Sowell (1992b))
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    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.
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    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
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    Fox and Taqqu (1986)),
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    which simultaneously estimate both the short-memory and long-memory parameters of the model. These estimation methods are computationally burdensome, rely on the correct specification of the high-frequency (ARMA) structure to obtain consistent parameter estimates, the final ARFIMA specification chosen generally varies across different selection
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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
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    Porter-Hudak (1990)
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    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.
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    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
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    Granger (1980),
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    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
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    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.
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    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.
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    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 distribution with support 0,1(), then the aggregated series is asymptotically fractionally integrated with d<0. 5.
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    Granger and Ding (1996)
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    extended the aggregation argument to mixtures of Ijd() processes for a range of distributions for dj; they also showed that other data generating mechanisms, like time-varying coefficient models and possibly nonlinear models, can have the long-memory property.
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    However, a more disaggregated data set may be needed in order to fully address the aggregation argument. -8Analysis of Divisia Indices and Velocity Series We subsequently test for a fractional integration order in an alternative set of monetary aggregates: the Divisia indices
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    (Thornton and Yue (1992)).
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    The Divisia monetary aggregates were proposed by Barnett et al. (1984) as superior to simple-sum aggregates which “implicitly view distant substitutes for money as perfect substitutes for currency.” (1984, p.1051) Barnett et al. found that the Divisia aggregates performed considerably better in terms of causality tests, tests of the structural stability of money demand f
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    However, a more disaggregated data set may be needed in order to fully address the aggregation argument. -8Analysis of Divisia Indices and Velocity Series We subsequently test for a fractional integration order in an alternative set of monetary aggregates: the Divisia indices (Thornton and Yue (1992)). The Divisia monetary aggregates were proposed by
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    Barnett et al. (1984)
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    as superior to simple-sum aggregates which “implicitly view distant substitutes for money as perfect substitutes for currency.” (1984, p.1051) Barnett et al. found that the Divisia aggregates performed considerably better in terms of causality tests, tests of the structural stability of money demand functions, and forecasting.
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    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.
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    Gould and Nelson (1974), Nelson and Plosser (1982), and Haraf (1986)
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    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.
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    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
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    Serletis (1995),
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    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.
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    Although not every component of the simple-sum aggregates exhibits long memory, the overall evidence is substantial and robust in support of fractional monetary dynamics with long-memory features. Our findings of fractional integration orders between one and two (and statistically distinguishable from one and two) is contrary to the conclusion reached by
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    King et al. (1991) and Friedman and Kuttner (1992)
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    that nominal money balances are I2() processes. A shock to the growth rate of the monetary series displays significant persistence, but it eventually dissipates. The money velocity series are best characterized as unit-root processes.
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    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.
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    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.
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    The nonlinear relations arising in this context deserve further scrutiny. Finally, care must be exercised in interpreting results from regressions involving the growth rates of monetary series.
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    Tsay and Chung (1995)
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    have shown the existence of spurious effects in regressions involving two -11independent long memory fractionally integrated processes whose orders of integration sum up to a value greater than 0.5.
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    monetary series would be likely to trigger these effects. -12Appendix All data series are seasonally adjusted, monthly observations obtained from the Federal Reserve Bank of St Louis' FRED database, which contains series originally published by the Board of Governors of the Federal Reserve System. The Divisia aggregates series were originally published in
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    Thornton and Yue (1992).
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    For the Divisia M1, M2, M3, and L series, the sample period is 1960:1 to 1992:12. The sample period is 1959:1 to 1995:10 for the following series: simple-sum M1, M2, M3, L, currency in circulation, demand deposits, total checkable deposits, small time deposits at commercial banks, small time deposits at thrift institutions, savings deposits at comm
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