Early warnings indicators of financial crises via auto regressive moving average models
Résumé
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 i and θ i 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.