Model fitting notes from a non-statistician

The science fiction author Ray Cummings once wrote[109] that “time is what keeps everything from happening at once”. Time, or the lack of it, is also what keeps some things from happening at all. In my case, the lack of time has meant that statistical analysis is the least developed part of my personal intellectual portfolio.

It is also, for reasons of lack of funding, the least developed aspect of Minsky . We hope to be able to extend Minsky at some stage to implement some methods for parameter estimation in complex systems, but for the moment, if you wish to fit a Minsky model to data, you’re going to have to do that by exporting the model to another program. Minsky currently supports model exporting to Matlab (it’s an option in the “Export Canvas” command on the File menu) for that purpose; it should soon support export to Vensim; at some stage we will add model exporting to R; and you can export the data from a Minsky simulation to a CSV file, which can be loaded into any data analysis program.

Fitting models to data is, of course, a large part of conventional economics, with its own subdiscipline of econometrics—and its own intellectual problems. Fitting a complex systems model to data opens yet more cans of worms, fundamentally because a complex systems model necessarily violates the conditions for linear regression (which is the mainstay of econometrics) that elements of a model predominantly interact additively. The inherent nonlinearity of complex systems models, along with the far-from-equilibrium behavior that most models generate, means that the values of parameters also interact with each other in nonlinear ways—normally multiplicatively, as in my model. This creates a “fitness landscape” for those parameters with numerous mountains and valleys (in terms of the model’s deviation from real world data) that can trap a standard least squares parameter fitting process in a local minima which is close to the initial guess values, but far removed from the model’s actual minimum deviation from real-world data.

This weakness of standard techniques cannot be addressed by a “brute force” approach of examining every permutation of the parameters, because the number of permutations is overwhelming. For example, the Keen-Minsky model shown in Figure 228 has nine parameters. If we try just twenty different values for each parameter, then there are 9[20] , or over 12 million trillion, different combinations of parameters to consider. This is simply too large a number of possibilities to test, so mathematicians and computer scientists have developed a range of techniques to sample a subset of possible combinations, and have reasonable confidence that your eventual choice of parameters is a global minimum, rather than a local one.

These techniques include genetic algorithms, simulated annealing, neural networks, plus a range of variations on least squares techniques—such as the Adam Optimization Algorithm[110] (the name is derived from “adaptive moment estimation”)—all of which are designed to overcome the problem of the parameter estimation technique getting locked onto a local minimum (deviation of the model from the data) which is not the global minimum.

My main interest in fitting models is not finding the best parameter values to enable a given model to replicate real world data, but in seeing whether or not the empirical data qualitatively conforms to a given model. If there are qualitative similarities, then the model, while it might not be able to replicate the empirical data precisely—or even closely—can provide insights into the real-world system.

Page 253 Model fitting notes from a non-statistician

This was the point of Lorenz’s model, which was constructed by stripping down an extremely accurate high-dimensional model of fluid turbulence to an extremely simple model with just 3 variables and 3 parameters. Lorenz didn’t construct this model to fit the data on turbulence, but to show the underlying factors causing that turbulence were the interaction of aspects of the weather—wind speed, temperature, humidity, pressure gradients, etc.—in highly nonlinear ways. In doing so, he “discovered” chaos (though it had first been identified logically by Poincare at the end of the 19th century), and a whole new way of modelling the weather was born.

I had a similar ambition for my model of Minsky’s Financial Instability Hypothesis. I wanted to do what Minsky had not managed to do, to produce a mathematical model of his verbal intuitions about the role of private debt and credit in causing both cycles and crises in capitalism.[111] I did that with a model in which I made a range of simplifying assumptions that removed other possible sources of instability apart from the nonlinear interactions of the model’s system states—the rate of employment, the oncldistribution of income, and the private debt to GDP ratio.

Some obvious sources of cyclical behavior in capitalism are variations in the rate of interest, changes in the capital to output ratio, intersectoral production and monetary dynamics, changes in the rate of population growth and technological change… All these were effectively held constant in my basic model. All that was left were the interactions of those three system states, and out of them arose two unexpected properties—in the sense that neither of them were predictions of Minsky’s verbal model.

The Keen-Minsky model shown in Figure 228 has four key qualitative features:

A rising level of private debt to GDP over time;
An eventual debt-deflationary crisis—which in technical terms is convergence of the model onto the “bad equilibrium” of zero employment, zero wages, and an infinite debt to GDP ratio;
Cycles in the rate of economic growth diminishing and then rising before a crisis; and
The shift in the distribution of income over time, with a rising private debt ratio causing a declining workers’ share of income, while the profit share of income is cyclical but unaffected by the rising debt ratio.

Only the first two characteristics that Minsky had described in his hypothesis, and that I had expected to emanate from the model. The latter two were what are known in complex systems as “emergent properties”: behaviors of a model that are not built into it by its designer, but result from the nonlinear interactions of the components.[112] Page 254 Model fitting notes from a non-statistician

The second property in particular was striking. Though Minsky had famously stated that “Stability— or tranquility—in a world with a cyclical past and capitalist financial institutions is destabilizing” (Minsky 1982, p. 101), this was in relation to the process within one cycle, where a period of tranquil growth would lead to rising expectations, turning a period of tranquil growth into a credit-fuelled boom. Minsky also expected that, in the absence of “Big Government”, there debt to GDP ratio would increase over a series of booms and busts, until a level of debt was accumulated that overwhelmed the economy and caused a Depression.

But he did not expect that these booms and busts would get smaller in magnitude for a while, and then get larger—which was the first emergent property of my model. Nor did he think that the rising debt ratio, and rising debt servicing costs, would come at the expense of workers and not capitalists (until the final crisis occurred).[113] These predictions were a direct product of the mathematical model, which I first developed in August of 1992.

behavior which wasn’t pre-programmed by the modeler, but arose from the nonlinear interactions of the model’s system states.

Model fitting notes from a non-statistician

Footnotes

109 https://quotes.yourdictionary.com/author/quote/592234

110 The site https://www.geeksforgeeks.org/intuition-of-adam-optimizer/ gives a reasonably accessible explanation of the algorithm.

111 Minsky did try to do that, in his PhD thesis, and it led to two of the only three papers by Minsky that were published in leading mainstream journals (Minsky 1957, 1959). He failed, because he used as his underlying model the Hicks-Hansen-Samuelson second-order difference equation known as the “Multiplier-accelerator model” (https://en.wikipedia.org/wiki/Multiplier-accelerator_model). As soon as I read the introduction to that paper, I knew that Minsky wouldn’t succeed, because I had already worked out that this model was based on an economic fallacy of equating actual savings to desired investment, both of which were functions of income. It was therefore asking “what level of GDP ensures that actual savings equals desired investment, when both are lagged functions of income”, to which the only answer was “zero GDP”. I explain why in more detail in the paper “Burying Samuelson’s Multiplier-Accelerator and resurrecting Goodwin’s Growth Cycle in Minsky” (Keen 2020a).

112 People often think that emergent properties can only be found in multi-agent models, where the macro behavior can’t be derived from the micro, but this isn’t the case. Lorenz’s butterfly is the perfect instance of

113 I explain the second phenomenon in Manifesto on pages 87-88, and the first on pages 88-93.