Gerhard Tintner, A Note on Rank, Multicollinearity and Multiple Regression (1945)

Gerhard Tintner’s mathematical journal article connects two problems: determining how many independent linear relationships hold among the systematic components of observed variables, and estimating those relationships when observations contain disturbances. Its central move is to use the same generalized eigenvalue problem for both rank testing and structural estimation, extending earlier work by Koopmans and Tintner from one relationship to several.

Tintner begins by decomposing each observation into a systematic component and a normally distributed disturbance. The disturbances have zero means and time-invariant variances and covariances. Their covariance matrix must be estimated separately, whether through deviations from trends, Fourier series, variate differences, or prior knowledge about measurement error. The object of inference is therefore not simply the covariance structure of the observations:

We want to determine the rank of the matrix of the variances and covariances of $M_{it}$.

This distinction makes rank a question about underlying relationships rather than observed correlations alone. Tintner compares the observed covariance matrix with the estimated disturbance covariance matrix through the equation $|a_{ij}-\lambda V_{ij}|=0$. If there are $r$ independent linear relationships among the systematic components, then, apart from sampling fluctuations, $r$ roots should equal one, and the systematic covariance matrix has rank $M-r$. Following Hsu and Fisher, he proposes a test statistic equal to $(N-1)$ times the sum of the $r$ smallest roots. He states its large-sample chi-square distribution, with $r(N-M-1+r)$ degrees of freedom, under the assumption that the disturbance-covariance estimate rests on a large number of observations.

The practical motivation joins structural econometrics to the problem of redundant predictors. Drawing on Wald and Haavelmo, Tintner distinguishes estimating population relationships from merely producing predictions. Rank also matters for prediction because adding highly correlated variables can consume information without appreciably improving the fit:

The inclusion of strongly correlated predictors cuts down on the number of degrees of freedom without contributing significantly to the reduction of the variance.

Multicollinearity thus supplies a reason to ask how many independent relationships exist, but the paper’s longer second stage pursues structural estimation. Once $R$ relationships have been accepted, Tintner writes them as linear constraints on the systematic components and seeks their coefficients:

Our purpose here is not prediction but estimation of the structural coefficients $k_{vj}$.

Treating the estimated disturbance covariance matrix as effectively fixed when its estimation sample is large, Tintner converts maximum likelihood into weighted least squares. The quantity minimized measures discrepancies between observed and systematic components using the inverse disturbance covariance matrix. This weighting makes the assumed error structure integral to the fitted relationships.

The derivation then introduces normalization and orthogonality conditions to select a representation of the system of relationships. One set makes the coefficient vectors orthonormal with respect to the disturbance covariance matrix; another makes the fitted residual combinations mutually orthogonal across observations. Through successive Lagrange-multiplier calculations, Tintner reduces the coefficient problem to homogeneous linear equations whose nontrivial solutions lead back to the opening determinantal equation. The estimated relationships are obtained from its $R$ smallest latent roots and corresponding characteristic vectors, while their constants are chosen so that the fitted relations pass through the observed means.

The article’s contribution is this unification of rank assessment and the fitting of multiple error-affected structural relations. Its scope remains conditional on an adequately estimated disturbance covariance matrix and large-sample reasoning. There is also a concrete inconsistency in the exposition: after equation (12), the text calls for maximizing $Q$, although the earlier least-squares formulation requires minimization and the concluding procedure selects the smallest roots. The stated endpoint nevertheless makes the intended connection clear: the directions used to diagnose systematic linear dependence also supply the coefficients used to estimate it.