Continuous Frequency-limited Processes and Their Macroeconomic Applications
نویسنده
چکیده
Econometricians are liable to regard discrete-time autoregressive moving-average (ARMA) models as approximations to continuous-time stochastic differential equations. (For a recent exposition of this point of view, see Sims 2010.) In common with other statisticians, they tend to assume that the differential equations are powered by the increments of a Wiener process. A Wiener process is formed from the accumulation of a continuous stream of infinitesimal impulses. This generates a trajectory that is everywhere continuous but nowhere differentiable. The impulses that underlie a Wiener process correspond to Dirac delta functions. These generalised functions are zero-valued everywhere except at a single point. The Fourier transform of a Dirac function indicates that it comprises sinusoidal elements of uniform but infinitesimal amplitudes that cover an unbounded range of frequencies. A Wiener process is a non-anticipatory process that has the martingale property, whereby its future evolution cannot be predicted—and the best estimate of its future values is its present value. This property is exploited extensively in financial mathematics, which is one of the reasons for the prevalence of Wiener processes. In many cases of financial modelling, the concomitant property of the unbounded frequencies, which must seem unreasonable, can be overlooked. However, in the context of macroeconomics, where the trajectories of the variables are dominated by trends and by motions of low frequency, it seems inappropriate to propose that the forcing functions should be Wiener processes. The notion of unbounded frequency is also at odds with the modern technology of digital signal processing and communications engineering. Here, the assumption
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