Stage 1: Elasticity calculations and quasi-equilibrium sample counting

• Select sample size and data granularity (monthly, daily, etc.).

• Calibrate (i.e., index) ERP and variance series to the first observation of each sample period.

• Calculate point elasticities of variance with respect to ERP for each sample. These are expected values, E(εerp).

• Calculate the length of the path traversed in each sample for each calibrated series.

• Count the number of up and down in-sample observations for ERP, variance, and transactions volume. This forms the basis of quasi-equilibrium definitions.

• Roll ahead to next sample window.

Stage 2: Exponential curve fitting – finding bubbles (and crashes)

• Reset the sample counter.

• Fit exponential curves that estimate a trajectory of changes in variance elasticity over each sample period's interval (using the elasticities derived in stage 1). The initial value for each estimate is the first point elasticity in that sample's interval.

• In both bubbles and crashes, variance elasticities in the interval will trace an exponential trajectory with statistically significant parameters. If the main parameter is not significantly different from zero, there is no microbubble in that sample (because y = e° = 1). Sign criteria are used for separating variance elasticities of bubbles from those of crashes.