Gerhard Tintner, Eine neue Methode für die Schätzung der logistischen Funktion (1958)

Gerhard Tintner’s journal article proposes a computationally simple method for estimating a logistic trend from observations taken at regular intervals. Presented at the Biometric Symposium in Linz in 1956 and published in 1958, it moves from the econometric problem of trend removal through a critique of derivative-based estimation to a reciprocal transformation and an application to Swedish population data. Its central contribution is to replace the nonlinear estimation problem with a linear difference equation whose coefficients yield the logistic parameters.

Tintner motivates the procedure through the mismatch between statistical methods developed for stationary processes and the trending series commonly encountered in economics:

Aber in der Oekonometrie enthalten fast alle empirischen Zeitreihen einen Trend und sind daher nicht stationär. Es ist häufig notwendig, den Trend zu eliminieren und die statistischen Methoden auf die Residuen vom Trend anzuwenden.

English translation: But in econometrics almost all empirical time series contain a trend and are therefore not stationary. It is frequently necessary to eliminate the trend and apply statistical methods to the residuals from the trend.

Trend estimation thus serves a broader analytical purpose: it makes residual variation accessible to statistical investigation. Yet choosing a trend is not merely a matter of computational convenience. Polynomial and exponential functions are comparatively easy to handle, but their unbounded growth makes their extension beyond observed data questionable:

Aber solche Trends haben die unangenehme Eigenschaft, daß sie mit wachsender Zeit unendlich werden. Sie eignen sich daher vielleicht für die Interpolation, aber kaum für die Extrapolation ökonomischer Zeitreihen.

English translation: But such trends have the unpleasant property that they become infinite as time increases. They may therefore be suitable for interpolation, but hardly for extrapolation of economic time series.

The logistic function supplies the contrasting property Tintner wants: an upper asymptote. He writes it as (y_t=k/(1+be^{-at})), with three generally unknown constants. Here (k) supplies the limiting level, while (a) and (b) determine the curve’s development through time. The article treats this bounded form as potentially appropriate for population movements and related economic series, rather than claiming that all economic growth must follow it. Its practical obstacle is estimation: direct least-squares or maximum-likelihood procedures produce equations nonlinear in the unknown parameters.

Tintner first reconstructs Harold Hotelling’s alternative. The logistic differential equation expresses proportional growth as a linear function of the current level: ((dy_t/dt)/y_t=a-(a/k)y_t). Estimating its coefficients gives (a) and (a/k), and hence (k); a formula attributed to E. C. Rhodes then supplies (b) from equally spaced observations. This approach already shifts attention from fitting the original curve to estimating a simpler relation. Its difficulty, however, is that the relevant growth rate is not directly observed in ordinary economic data:

Daher müssen wir die Wachstumsrate durch Differenzen annähern [9]. Das ist eine komplizierte Angelegenheit und die Resultate sind nicht verläßlich, besonders wenn Beobachtungsfehler in der ursprünglichen Reihe unserer Daten vorliegen.

English translation: We must therefore approximate the growth rate by differences [9]. This is a complicated matter, and the results are not reliable, especially when observation errors are present in our original data series.

The new method avoids this approximation. Assuming that population follows the logistic function, Tintner takes its reciprocal, (z_t=1/y_t=(1+be^{-at})/k). This transformed sequence satisfies the exact first-order linear difference equation [ z_{t+1}=\frac{1-e^{-a}}{k}+e^{-a}z_t. ] The decisive conceptual move is from a differential relation requiring estimated derivatives to a discrete relation connecting successive observations. It preserves the logistic model while matching the temporal form in which the data are available.

Nun können wir die Maximum Likelihood Methode oder die Methode der kleinsten Quadrate direkt auf diese Differenzengleichung anwenden.

English translation: We can now apply the maximum likelihood method or the method of least squares directly to this difference equation.

The slope estimates (e^{-a}), and the intercept estimates ((1-e^{-a})/k); together they determine (a) and (k). Rhodes’s formula again supplies the remaining parameter (b). Tintner’s innovation therefore concerns the route to estimation, not a new growth law. He retains the logistic specification and the existing procedure for recovering (b), but replaces numerical differentiation with estimation of a linear recurrence. The brief article does not develop an explicit stochastic error model, compare estimator performance, or establish that the transformed residuals satisfy the assumptions of either estimation method.

The worked example uses eleven Swedish population observations, recorded at ten-year intervals from 1850 through 1950. Regressing reciprocal population on its preceding value produces an intercept of (0.0000000126494) and a slope of (0.8693468). Tintner reports (a=0.14) and an upper asymptote of 10,328,805 inhabitants. Because the time index advances once per observation, the fitted rate applies to a decade, not a year. Extending the index to (t=12), he reports a population forecast of 7,390,625 for 1960.

The numerical presentation requires caution. The prose reports (b=2.1176), whereas the displayed final curve uses (2.176); its numerator also differs by one inhabitant from the previously reported asymptote. These discrepancies should remain visible rather than be silently reconciled. The example illustrates a procedure, but does not provide an assessment of forecast uncertainty or predictive accuracy. The article’s lasting methodological interest lies in its economical reformulation: a bounded nonlinear trend becomes estimable through a linear relation between reciprocal observations, making the distinction between continuous growth models and discrete measurement central to the estimation problem.