Gerhard Tintner, Some Applications of Multivariate Analysis to Economic Data (1946)

Tintner’s journal article introduces economic statisticians to four methods—discriminant analysis, principal components, canonical correlation, and weighted regression—through a common notation and eleven empirical illustrations. Its organizing argument is that multivariate analysis extends familiar regression techniques while distinguishing tasks too often conflated: classifying observations, constructing aggregate indexes, predicting between sets of variables, and estimating economically meaningful relationships. The emphasis is practical and methodological:

The emphasis will be on methods of estimation and not on tests of hypotheses.

Tintner supplies significance tests where available, but avoids extensive derivations and computational instructions. Examples drawn from other researchers accompany his own investigations of commodity prices, production indexes, agricultural markets, and aggregate production. These are demonstrations of possible applications, not settled empirical findings. Economic time series introduce a particularly important qualification: the analysis largely neglects serial correlation, which can reduce estimation efficiency and compromise significance tests. The methods also assume essentially linear population relationships, leaving nonlinear extensions as a future task.

After establishing notation for means, variances, covariances, and standardized observations, Tintner presents discriminant analysis as the construction of a linear index that best separates two previously classified groups. Fisher’s method maximizes squared separation relative to variance; its estimating equations resemble the normal equations of multiple regression. Durand’s distinction between good and bad installment loans provides an established application. Tintner then asks whether cyclical behavior distinguishes consumers’ goods from producers’ goods, using nineteen English wholesale-price series from 1860–1913. Cycle length, duration of rising prices, amplitude, and rate of change produce an index that misclassifies only two commodities. The result passes the five-percent significance threshold, but adding pepper removes significance. Its sensitivity to sample composition, questionable normality, and possible nonlinear relationships restrain the economic interpretation.

Principal components shifts the inquiry from separation to representation. Orthogonal components reproduce observed correlations, with the first accounting for the greatest share of standardized variance. Tintner also presents Girshick’s interpretation of the method as constructing a linear combination least subject to random errors under the stipulated assumptions. Statistical optimality, however, does not automatically supply economic content:

The results of this method need not, however, be meaningful in economic terms.

This distinction frames the article’s discussion of aggregation. General-equilibrium analysis potentially involves immense numbers of prices and quantities, whereas empirical macroeconomic models require manageable indexes. Principal components can help assess how faithfully an index represents the variation within its constituent group. For four American production indexes over 1919–1939, the first component accounts for about 76 percent of total standardized variance. Three wholesale-price indexes yield a first component accounting for more than 95 percent, closely correlated with the published all-commodities index. These findings tentatively support common production and price movements across sectors. Tintner stresses that such unity is an empirical result, not a logical necessity: a less integrated economy might require distinct agricultural and industrial factors. Tests using individual commodities remain necessary.

Canonical correlation addresses relationships between two sets of variables by choosing a linear combination from each set to maximize their correlation. Waugh’s studies of meat prices and consumption, and of wheat and flour characteristics, illustrate the method before Tintner applies it to production and wholesale-price indexes. His canonical correlation of 0.8831 links combinations weighted especially toward contrasts between durable-goods and mineral production, and between farm and other prices. These patterns suggest connections with business-cycle theories, but the optimized indexes remain instruments of mutual prediction rather than established structural equations.

Weighted regression makes that distinction central. Ordinary regression optimizes prediction of one variable conditional on fixed values of others. Economic policy instead requires estimates of relationships that can explain responses to changed conditions. Tintner’s wheat-market illustration shows why an observed price–quantity regression need represent neither supply nor demand. Fixing wheat prices would make demand elasticity, rather than the best historical predictor, essential:

Weighted regression is designed for this particular purpose rather than for the prediction of values of one particular variable like classical multiple regression.

Following Koopmans, Tintner treats every observed variable as potentially disturbed. He distinguishes disturbances in variables from disturbances in equations, then develops the case that neglects the latter. This requires inclusion of most important explanatory variables. Estimated error variances provide inverse-variance weights, and a smallest-latent-root procedure determines the fitted relationship. Further root-based tests investigate whether several independent relationships exist. Identification remains a separate economic requirement: in his agricultural-market example, national income helps identify demand, while prices paid by farmers help identify supply.

The final, extended application estimates a Douglas-type production function for the American economy over 1921–1941. Successive differences furnish error-variance estimates, treated as constants despite acknowledged uncertainty. Weighted regression yields estimated labor and capital elasticities of approximately 2.13 and 0.34, with a time trend. Approximate tests support the labor coefficient and trend but not the capital coefficient; confidence limits emphasize the imprecision of the conclusions. Tintner closes by questioning the adequacy of both the data and the aggregate concept:

The economic meaning of a production function representing all enterprises is also somewhat doubtful.

Industry-specific investigations would therefore be preferable. The article’s lasting conceptual contribution is its disciplined separation of statistical construction, predictive success, and structural economic interpretation. Multivariate methods enlarge the economist’s empirical repertoire, but their relevance depends on identification, assumptions about disturbances, and the economic meaning of the variables being combined.