Gerhard Tintner, On Tests of Significance in Time Series (1939)

Tintner’s mathematical note develops significance tests for two time-series problems: comparing the variances of successive finite differences and determining the distribution of a selected serial covariance. Its unifying move is to select observations whose independence permits tractable inference, accepting a loss of information in exchange. Responding to the gap he identifies between theoretical time-series analysis and usable significance tests, Tintner adapts methods associated with Fisher and Snedecor to statistics whose construction otherwise introduces dependence.

The first section begins with the Variate Difference Method. An observed series is decomposed into a smooth mathematical expectation and normally, independently distributed random errors with constant variance. Successive differencing is intended to remove the smooth component:

But the smooth component or mathematical expectation can be eliminated to any desired degree by successive differencing.

This claim depends on the character of the component being removed: Tintner explicitly excludes a “zig-zag” component or a periodic function with a short period. Polynomial trends supply the model case, since sufficiently high differences vanish. Once the smooth component has been eliminated, the appropriately normalized difference variances should agree. Comparing successive orders therefore provides a way to locate the order at which trend removal becomes adequate. Earlier standard-error formulas, he observes, require large samples and knowledge of the true variance.

The obstacle is that independent original errors do not produce independent differences:

But the process of forming finite differences has introduced correlations, even if the original random elements $x_i$ are independently distributed.

Tintner traces these correlations both within each difference series and between successive orders. His solution changes the observations entering the estimates rather than treating the resulting dependence as negligible:

We can make a very simple valid comparison in spite of these correlations if we sacrifice some of the available information.

For first and second differences, he selects every fifth item, with the two selections offset so that they involve disjoint original observations. The resulting variance estimates can be compared using Fisher’s z test or Snedecor’s F table. A significant difference leads to comparison of the next pair of orders. Tintner then generalizes the construction: comparing orders k and k + 1 requires selections spaced by 2k + 3. Different starting positions yield alternative selections, but he stresses that these alternatives are not independent of one another. The section closes with general variance formulas and a large-sample normal approximation for the log-variance comparison.

The second section transfers the selection principle to serial covariance. Assuming independent normal observations with mean zero and unit variance, Tintner retains spaced products of observations separated by lag L. He states that the ensuing formulas are exact only when N is a multiple of L + 1 and otherwise are approximations. The analytical route now changes:

We shall use the method of characteristic functions, [10], in order to establish the distribution of w.

An orthogonal transformation reduces the calculation to a determinant; inversion of the resulting characteristic function gives a density expressed through a Bessel function. Tintner concludes that, for large N, the selected covariance is approximately normal with mean zero and variance (L + 1)/N.

The note’s contribution is thus a concrete strategy for making significance testing possible under explicit normality and independence assumptions. Its two sections show how deliberate selection can recover independence after differencing or lagged multiplication has complicated the sampling structure. The gain is a usable reference distribution; the acknowledged cost is discarded information. That trade-off, together with the restrictions on the trend and error process, defines both the practical relevance and the scope of Tintner’s proposal.