Stochastic instability: a dynamic quantile approach

This paper examines the nature of instability in stochastic dynamical systems. Relying on a quantile approach, we propose to measure dynamic instability by the average rate of divergence ( $$AR{D}^{\text{s}}$$ A R D s ) of the state along a finite forward stochastic path. Under stochastic shocks, $$AR{D}^{\text{s}}$$ A R D s is a random variable with a given distribution function that depends on the nature of the underlying dynamic process as well as the nature of the shocks. We show how our approach can be made empirically tractable using a quantile autoregression (QAR) model. In an empirical application to futures price, the QAR estimates provide statistical evidence that futures price instability varies with market conditions: instability increases with the maturity of the futures contract as well as with higher quantiles (representing positive shocks located in the upper tail of the price distribution). We find that neglecting stochastic shocks (e.g., under a deterministic dynamic analysis) tends to overstate the presence of instability. The results stress the importance of evaluating the dynamic impacts of shocks across the whole distribution.