Comment on “Nonlinear Monetary Dynamics” by DeCoster and Mitchell
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The concept of dimension, its uses, and the difficulties in estimation are not universally well understood. A recent article in the Journal of Business & Economic Statistics by DeCoster and Mitchell (1991) (DM) illustrates our claim. Dimension calculations are useful for characterizing the topological properties of the data. The nonexistence of a lowdimensional attracting set, however, implies nothing about the presence of complicated dynamics. For succinct reference to these ideas, see Ramsey, Sayers, and Rothman (1990). The purpose of this comment is twofold. First, we wish to remind the reader of two practical difficulties in estimating the Grassberger and Procaccia (GP) dimension concept. Second, using an alternative procedure that mitigates many of the problems that stem from small sample sizes, we reestimate DM's strongest result and show that the "finding of nonlinearity in Divisia M2" (see p. 460 of DM) cannot be sustained by the dimension calculations cited. The interested reader can find a more complete discussion of these and related issues in the article by Ramsey and Rothman (1992) and the references cited therein.
[1] James B. Ramsey,et al. The statistical properties of dimension calculations using small data sets , 1990 .
[2] Philip Rothman,et al. The Statistical Properties of Dimension Calculations Using Small Data Sets: Some Economic Applications , 1990 .
[3] Douglas W. Mitchell,et al. Nonlinear Monetary Dynamics , 1991 .