Gerhard Tintner and R. Narayanan, A Multi-dimensional Stochastic Process for the Explanation of Economic Development (1967)

Tintner and Narayanan’s journal article develops a multivariate stochastic model of economic development and illustrates it with post-war United States data. Its central move is to replace separate probabilistic descriptions of economic variables with a joint process that explicitly represents their interdependence. The article proceeds from transition assumptions to a probability-generating function, derives moments and a limiting normal distribution, and then estimates a two-variable model for gross national product and private consumer expenditures. An appendix supplies the annual observations for 1946–63.

The authors locate their contribution within stochastic approaches to economic evolution associated with Havelmo, arguing that earlier studies chiefly employed univariate processes or independent combinations of them.

In this paper we propose a multi-dimensional stochastic process taking into account the inter-dependence among the variables.

Interdependence is not merely a descriptive correlation added after fitting separate trends. It enters the specification of possible events: a change may affect one component alone or several components simultaneously. Although the formal exposition uses three variables, the authors regard the method as extensible to any number. National income, capital stock, and employment provide examples of the dimensions through which development might be characterized; the empirical application uses a narrower pair.

The underlying process has nonnegative integer-valued components. Over an infinitesimal interval, each permitted event increases a selected combination of components by one, with a constant intensity attached to that combination. Other transitions have negligible first-order probability. These assumptions produce a system of difference-differential equations, solved through a joint probability-generating function under an initial condition placing all components at zero. Each marginal distribution is Poisson, but shared events make the components dependent. The authors also indicate how time-varying intensities can be accommodated by replacing intensity multiplied by elapsed time with its integral.

To connect this counting process to economic quantities, the authors introduce a separate location and scale transformation for each component:

In order to apply the above process to economic data we transform the state space of each component $X_{it}$ by writing $Y_{it} = a_{it} + u_{it} X_{it}$ where $a_{it}$ and $u_{it}$ are functions of time.

This transformation supplies economic levels and measurement scales while retaining the joint event structure. Differentiating the generating function yields means, variances, covariances, and a third-order joint central moment. A component’s variance depends on every event intensity involving it; covariance between two components depends only on events involving both. The third-order moment isolates the intensity of events shared by all three components. The model therefore connects different orders of dependence to identifiable categories of simultaneous change rather than treating all association as an undifferentiated residual. Its limiting normal distribution subsequently makes joint uncertainty calculations possible.

The empirical section simplifies this framework substantially. It uses gross national product and private consumer expenditures at constant 1954 prices, holding location parameters, scale parameters, and event intensities constant:

Further, for simplicity, we assume that all the quantities $a_{it}, u_{it}, \lambda_\delta(t)$ are constants, independent of time.

Under this restriction, expected levels follow linear trends, while variances and covariance grow proportionally with elapsed time. The authors fit the trends using squared residuals weighted by the inverse of time, consistent with the model’s increasing variance, and estimate the shared-event intensity from weighted cross-products of residuals. The fitted annual trend increments are approximately $11.91 billion for gross national product and $7.47 billion for consumption. Both reported trend correlations are 0.99; these are separate fit statistics, not the correlation between the two economic variables.

The fitted intensities give the model a concrete probabilistic interpretation. Events affecting gross national product alone occur at an estimated rate of 4.1135, consumption alone at 5.5532, and both together at 4.7170. The associated increments are 1.3483 and 0.7276 billion dollars. Thus the estimated covariance has a specific source within the model: simultaneous increments in output and consumption. These rates describe infinitesimal transition probabilities, rather than annual probabilities or evidence that one variable causally determines the other.

The final section turns from estimation to joint prediction:

We may also employ the property of asymptotic normality to derive joint-confidence ellipse for the distribution of $(Y_{1t}, Y_{2t})$.

For 1965, the fitted trends predict gross national product of 501.8695 and consumption of 329.8269 billion dollars. Using their estimated variance-covariance matrix, the authors construct a quadratic form with an approximate chi-square distribution and obtain a 95% joint ellipse. This is the practical payoff of retaining dependence: uncertainty is represented around the two quantities together, rather than through unrelated intervals.

The article’s significance lies in this compact connection between shared stochastic events, observable covariance, and joint forecasting. Its economic explanation remains deliberately stylized: the application models linear development trends through constant-rate upward jumps, without specifying production, consumption behaviour, or causal feedback. Indeed, the appendix includes annual declines in gross national product, whereas the fitted constant-parameter process permits only nonnegative increments. The illustration consequently supports the usefulness of a joint trend-and-uncertainty framework more directly than it establishes an exhaustive account of observed economic dynamics.