¹ Chow, G. C., “Multiplier, Accelerator, and Liquidity Preference in the Determination of the National Income of the United States,” The Review of Economics and Statistics, Vol. XLIX, No. 1 (February, 1967), pp. 1–15.CrossRefGoogle Scholar

² Ibid., p. 9.

³ Ibid., p. 9.

⁴ G. C. Chow, op. cit., pp. 14–15.

⁵ Adelman, I. and Adelman, F. L., “The Dynamic Properties of the Klein-Goldberger Model,” Econometrica. Vol. XXVII, No. 4 (October, 1959), pp. 596–625.CrossRefGoogle Scholar

⁶ A stochastic run being a sample from a universe of behaviors, we may ask if 99 periods allow valid inferences. We would like simulation runs long enough as to generate random vectors distributed according to the original distribution, in spite of sampling fluctuations. For 99 periods they, in fact, reproduce the original distribution very closely. As an additional precaution, we tested the hypothesis that the resulting covariance matrices equal the original ones. To test this hypothesis used the following result: Given y₁,…, yⁿ as observation vectors of components from N(υ, Ψ) the likelihood ratio criterion for testing the hypothesis that H₁: υ=υ_O, where Ψ_O is specified, is , where Under H,¹, −2log λ is asymptotically distributed as χ²with (l/2)p(p+l) degrees of freedom. (Cf. Anderson, T. W., An Introduction to Multivariate Statistical Analysis (New York: Wiley, 1958), pp. 265–267Google Scholar.) For the random vectors applied on the endogenous variables we got −2log lambda; = 14.141, with 10 d. f.; for the vectors applied on the exogenous variables we got −2log λ = 9.793, with 6 d. f.

⁷ I. Adelman and F. L. Adelman, op. cit., pp. 601–602.

⁸ Pseudorandom numbers are numbers generated by some specified deterministic rule that behave in such a way that no reasonable statistical test will detect significant departures from randomness. Cf. Hammersley, J. M. and Handscomb, D. C., Monte Carlo Methods (London: Methuen & Co., 1964), pp. 26–27.CrossRefGoogle Scholar

⁹ The generating recursive relation is χ_i = αχ_i−1 (mod. m), that is, χ_i is the number generated as the remainder of has to be large enough to avoid repetition of the sequence, and it is convenient that a differs by 3 from the nearest multiple of 8, and that χ_o by odd. X_i/m are numbers uniformly distributed on the interval (0, 1). The same sequence can be generated by recalling the same χ_o cf. J. M. Hammersley and D. C. Handscomb, op. cit. pp. 28–31; and M.I.T., Notes on Operations Research—1959 (Cambridge, Mass.: The Technology Press, 1959), pp. 233–234.Google Scholar

¹⁰ Box, G. and Muller, M., “A Note on the Generation of Random Deviates,” The Annals of Mathematical Statistics (Vol. 29, 1958), pp. 610–611Google Scholar. The deviates generated by this method are reliable at the tails of the distribution and the recovery feature mentioned in fn. 9 is preserved. On its computational characteristics, see Muller, M., “A Comparison of Methods of Generating Normal Deviates in Digital Computers,” Journal of the Association of Computing Machinery (6, 1959), pp. 376–383.Google Scholar

¹¹ If r is a random vector distributed according to N(0, I), then v = QLr is a random vector distributed according to N(0,Σ), where L is the diagonal matrix formed with the square root of the characteristic roots and Q is the matrix whose columns are the orthonormal set of characteristic vectors obtained by diagonalizing Σ. Cf. Anderson, T. W., An Introduction to Multivariate Statistical Analysis (New York: Wiley, 1958Google Scholar), Chapter II and Appendix 1.

¹² Wave-like patterns can be the result of chance fluctuations involving cumulative effects. On purely theoretical grounds—in view of the “first arc sine law” for random walks, which applies in the case of our experiment—we ought to expect stochastic simulation to produce wave patterns. Cf. Feller, W., An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. (New York: Wiley, 1957), pp. 66–85.Google Scholar

¹³ Adelman, I., “Business Cycles—Endogenous or Stochastic?”, Economic Journal, Vol. LXX (December, 1960), pp. 783–796.CrossRefGoogle Scholar

¹⁴ I. Adelman and F. L. Adelman, op. cit., pp. 609–611.

¹⁵ A rigorous analysis should be based on the computation of power spectra for the models and the actual series under comparison.

¹⁶ Burns, A. F. and Mitchell, W. C., Measuring Business Cycles (New York: National Bureau of Economic Research, 1947)Google Scholar; Mitchell, W. C., What Happens During Business Cycles (New York: N.B.E.R., 1951).Google Scholar

¹⁷ Cycles in the sense of Burns and Mitchell. Cf. Burns and Mitchell, op. cit., p. 3.

¹⁸ For this and related concepts see W. C. Mitchell, op. cit., pp. 9–12.

¹⁹ Cf. I. Adelman and F. L. Adelman, op. cit., p. 614.

²⁰ Ibid., pp. 614–616; and W. C. Mitchell, op. cit., pp. 154–155.

²¹ We have not attempted to study if the shocked Chow model would produce long cycles because that matter can be handled more effectively by the use of spectral analysis than by the methods used in this paper. Cf. Adelman, I., “Long Cycles—Fact or Artifact?,” American Economic Review. Vol. LV, No. 3 (June, 1965), pp. 444–463.Google Scholar