uncertainty from their estimation. It makes us complacent about the choice of models and functional forms because it credits hypothesis tests with undue discriminatory power. And it leads us again and again into crisis situations because it attributes too little probability to extreme events.
We cannot dismiss the use of ergodic stationarity as a mere simplifying assumption, of the sort regularly and sensibly made in order to arrive at an elegant and acceptable approximation to a more complex phenomenon. A model of a stationary time series approximates an object that can never be observed: a time series of infinite length. This says nothing about the model's ability to approximate a time series of any finite length, such as the lifetime of a trading strategy, a career, or a firm. When events deemed to occur 0.01 percent of the time by a risk model happen twice in a year, there may be no opportunity for another hundred years to prove out the assumed stationarity of the risk model.
Modern financial crises are intimately connected with risk modeling built on the assumption of stationarity. For large actors like international banks, brokerage houses, and institutional investors, risk models matter a lot for the formation of expectations. When those models depend on the assumption of stationarity, they lose the ability to adapt to data that are inconsistent with the assumed data-generation process, because any other data-generation process is ruled out by fiat.
Consider what happens when an institution simply recalibrates the same models, without reexamining the specification of the model, over a period when economic expansion is slowing and beginning to turn toward recession. As the rate of economic growth slows the assumption of ergodicity dissolves new data signaling recession into a long-run average indicating growth. Firms and individuals making decisions based on models are therefore unable to observe the signal being sent by the data that a transition in the reality of the market is under way, even as they recalibrate their models. As a result, actors continue to behave as if growth conditions prevail, even as the market is entering a process of retrenchment.
Thinking about a series of forecasts made during this period of transition, one would likely see forecast errors consistently missing in the same direction, though no information about the forecast error would be fed back into the model. When models encompass a large set of variables, small changes in the environment can lead to sharp changes in model parameters, creating significant hedging errors when those parameters inform hedge ratios. Activity is more at odds with reality as the reversal of conditions continues, until the preponderance of new data can no longer be ignored; through successive recalibrations the weight of the new data balances and overtakes the old data. Suddenly actors are confronted by a vastly different reality as their models catch up to the new data. The result is a perception of discontinuity. The available analytics no longer support the viability of the financial institution's chosen risk profile. Management reacts to the apparent discontinuity, past decisions are abruptly reversed, and consequently market prices show extreme movements that were not previously believed to be within the realm of possibility.
Models staked on stationarity thus sow the seeds of their own destruction by encouraging poor decision making, the outcomes of which later register as a realization of the nearly-impossible. Crises are therefore less about tail events “occurring” than about model-based expectations failing to adapt. As a result, perennial efforts to capture extreme risks in stationary models as if they were simply given are, in large part, misguided. They are as much effect as they are cause. Financial firms would do much better to confront the operational task of revising risk measurements continuously, and using the outputs of that continuous learning process to control their business decisions. Relaxing the assumption of stationarity within one's risk models has the goal of enabling revisions of expectations to take place smoothly, to the extent that our expectations of financial markets are formed with the aid of models, in a way that successive recalibrations cannot.
Bayesian Probability as a Means of Handling Discontinuity
The purpose of this book is to set out a particular view of probability and a set of statistical methods that untether risk management calculations from the foundational assumption of time-invariance. Such methods necessarily move away from the classical analysis of time series, and lay bare the uncertainties in statistical and financial models that are typically papered over by the assumption of ergodic stationarity. Thus, our methods will allow us to entertain the possibilities that we know the parameters of a model only within a nontrivial range of values, multiple models may be adequate to the data, and different models may become the best representation of the data as market conditions change. It is the author's conjecture (and hope) that introducing flexibility in modeling procedures along these multiple dimensions will reduce or even eliminate the extreme discontinuities associated with risk models in crisis periods.
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