Gerhard Tintner, “A Study in the Analysis of Stationary Time Series. Herman Wold” (1940)

Gerhard Tintner’s review assesses Herman Wold’s 1938 book as an important mathematical advance whose usefulness for empirical economics remains dependent on the development of statistical tests. Its argument moves from Wold’s theoretical framework and classification of time-series models to their economic applications, then to the unresolved question of inference. Tintner’s praise is substantial, but he distinguishes a powerful method of representing fluctuations from a method capable of establishing the validity of empirical findings.

Wold’s subject is stationary time series: series without a trend, exhibiting periodic or “pseudoperiodic” fluctuations. Tintner locates the book’s foundations in the modern probability theory developed by Khintchine and Kolmogoroff. He presents its achievement as the systematic development and amplification of ideas associated with G. U. Yule, Sir Gilbert Walker, and E. Slutsky. The review thus places Wold within an existing mathematical research tradition while crediting him with a significant advance in its organization and extension.

The theoretical survey distinguishes functional schemes, understood as strictly periodic functions treated through harmonic analysis; hidden periodicities, investigated through periodogram analysis; and linear-regression schemes, subdivided into autoregression and moving averages. These distinctions matter because recurring fluctuations need not all arise from the same kind of process. Tintner’s account emphasizes Wold’s comparison of alternative schemes rather than presenting periodicity as a sufficient explanation in itself.

The empirical discussion centers on Beveridge’s wheat-price index for 1770–1869 and Myrdal’s Swedish cost-of-living index for 1840–1913. Wold approaches these through correlograms, graphs of serial correlation coefficients. The wheat-price series appears to correspond approximately to a moving-average scheme, while the cost-of-living series appears to fit linear autoregression. Tintner immediately qualifies the interpretive force of these results:

It is, however, somewhat difficult to see the economic meaning of these results.

The qualification separates statistical correspondence from economic explanation. Tintner suggests that moving averages may have some connection with ideas related to Pigou’s business-cycle theory, whereas autoregression seems compatible with mathematical theories associated with Frisch, Tinbergen, and Kalecki. These are tentative connections, not established explanations of the observed series. His wording preserves the uncertainty surrounding the passage from a mathematical scheme to an economic mechanism.

The review’s decisive criticism concerns statistical inference:

One great shortcoming of the book, which is also realized by the author, is that it fails to treat the subject from the point of view of modern statistical inference as developed by R. A. Fisher, E. Pearson, and J. Neyman.

Tintner says that this omission prevents a judgment about the validity of the results. The issue is therefore more fundamental than choosing among plausible models: the analyst needs a basis for assessing the evidential strength of an apparent correspondence. He identifies the development of significance tests as the condition for giving Wold’s theoretical work practical value in empirical analysis, while acknowledging the serious mathematical difficulties involved. His footnote to his own work on significance tests in time series situates this criticism within an active methodological problem.

The concluding assessment retains both the enthusiasm and the reservation:

This book seems to be a great step forward in the right direction.

Tintner praises Wold’s exposition and mastery, recommending the book to statisticians and economists with substantial mathematical training and knowledge of probability theory grounded in point-set considerations. Its prospective importance lies in opening a route toward solving what he regards as the foremost problem in applying statistics to economic data: the analysis of economic time series. The review’s central distinction is between theoretical progress and empirical warrant. Wold advances the former; the latter still requires methods of inference that can turn suggestive model fits into assessable results.