Early warnings indicators of financial crises via auto regressive moving average models

We address the problem of defining early warning indicators of financial crises. To this purpose, we fit the relevant time series through a class of linear models, known as auto-regressive moving-average (ARMA(p, q)) models. By running such a fit on intervals of the time series that can be considered stationary, we first determine the typical ARMA(p-,q-). Such a model exists over windows of about 60 days and turns out to be an AR(1). For each of them, we estimate the relative parameters, i.e. ϕ<inf>i</inf> and θ<inf>i</inf> on the same running windows. Then, we define a distance ϒ from such typical model in the space of the likelihood functions and compute it on short intervals of stocks indexes. Such a distance is expected to increase when the stock market deviates from its normal state for the modifications of the volatility which happen commonly before a crisis. We observe that ϒ computed for the Dow Jones, Standard and Poor's and EURO STOXX 50 indexes provides an effective early warning indicator which allows for detection of the crisis events that showed precursors.