Oskar Morgenstern’s Review of Contributions to the Study of Oscillatory Time-Series, by M. G. Kendall (1947)

Oskar Morgenstern’s review examines Kendall’s experimental comparison of methods for analyzing oscillatory time-series. Its central distinction is between demonstrating a method’s performance on artificially generated data and establishing its adequacy for economic observations. Morgenstern values Kendall’s computational labor and clear exposition, but questions the inference from experiments designed around an autoregressive mechanism to a general preference for correlogram analysis.

The review first identifies the scope of Kendall’s inquiry: stationary series without a trend, whose mean is taken as zero. Kendall compares four approaches—measuring distances between peaks or troughs, harmonic or periodogram analysis, correlogram analysis, and variate difference analysis. Morgenstern emphasizes that these are not merely alternative calculating procedures. Each corresponds to assumptions about the mechanism generating the observations. Counting peaks requires relatively few assumptions; variate difference analysis specifically supposes that random elements are superimposed on systematic movement and that their magnitude can be estimated.

Kendall favors an autoregressive model in which disturbances become incorporated into the system and affect its subsequent motion. Morgenstern describes its governing mechanism:

We then have a system capable of damped oscillations but subjected to a stream of external shocks which continually regenerate the oscillations.

The distinction matters because apparent periodicity need not imply an independently sustained cycle: repeated shocks can renew oscillations that would otherwise diminish. Kendall’s principal investigation concerns this model, using four experimental series and two empirical series collected by Beveridge. The artificial series embody the assumptions underlying correlogram analysis. Applied to them, competing methods produce results Kendall finds unacceptable, particularly periodogram analysis. Variate difference analysis receives a less emphatic rejection, while peak-counting is compromised by investigators’ discretionary exclusion of supposedly unimportant effects.

Morgenstern’s objection concerns the evidential reach of these comparisons. Failure on an autoregressive series does not establish failure on economic data unless the same generating mechanism has been demonstrated there. Resemblance between periodograms from artificial and empirical series cannot substitute for that demonstration:

It is not convincing when the author merely implicitly postulates the Beveridge series to be autoregressive and dismisses their periodograms because these appear similar to those considered inadequate—obtained for his artificially constructed autoregressive series.

If the Beveridge series are not simply autoregressive, Morgenstern argues, their correlograms might instead warrant rejection and their periodograms might be acceptable. He points to methods developed by Wiener, Kolmogoroff, and others as relevant to investigating the empirical mechanism. He also stresses limitations Kendall acknowledges: short autoregressive series may not damp sufficiently, and correlogram analysis is insensitive to changes in the autoregressive scheme. Such changes are especially plausible across long economic records, even though historically extensive series may remain too short for the method’s requirements.

The review links this methodological caution to changing computational possibilities. Morgenstern recognizes the labor invested in Kendall’s calculations but regards the discussion of computational shortcuts as already becoming obsolete. High-speed devices permit much larger investigations; machines suited to periodograms and autocorrelations had apparently been available during the research. He suggests wartime restrictions may explain Kendall’s lack of information about these developments.

The conclusion consequently recommends broader experimentation rather than a final ranking of methods:

Experimental work with empirically given series on an enormously larger scale than anything done thus far and with a minimum but variety of assumptions about the underlying generating mechanisms seems to be indicated.

Morgenstern’s positive proposal combines expanded computation with diversity of models. Economic time-series analysis is not yet secure enough to select one mechanism or discard powerful alternatives such as Fourier methods. Kendall’s paper remains, in his judgment, well written and stimulating; its significance lies both in its experimental contribution and in the unresolved question it exposes: how to connect the assumptions that make an analytical method successful to the processes actually producing economic observations.