Early Detection of Epidemic Outbreaks and Financial Bubbles Using Autoregressive Models with Structural Changes

This paper is a further development of our previous work presented at SUGI 2006, NESUG 2007 and SGF 2008. Our basic model is AR(1) (first-order autoregression). This model emerges as a natural approximation of a classic susceptible-infectious-recovered (SIR) model. It inherits first principles of SIR models and can be used in both epidemiological and financial applications for early detection of epidemics and financial bubbles. We consider epidemic outbreaks or financial bubbles as structural changes in the autoregressive coefficient in AR(1) models. We propose two methods of estimation of the autoregressive parameter: least-squares and median-ratio based methods, discuss the questions of bias correction and confidence intervals construction. The value of our first-order autoregressive coefficient less than one corresponds to a stationary, noepidemic/no-bubble regime. If the parameter is greater than one, we have an explosive case (outbreak of epidemic or bubble). When the coefficient is equal to one, we have a unit root case. Under some conditions, least-squares estimates and confidence intervals, based on the observed data in a chosen time window, allow us to decide which case is more appropriate. We propose two alternative strategies for early detection of structural changes. In both strategies we use some generalizations of the Fisher’s F-test: so-called supremum F-tests (essentially equivalent to supremum Likelihood-Ratio tests), and end-of-sample breakpoint S-tests. These tests can be used in rather general situations and usually have a better power performance. Also we provide simulation results.