The log-periodic-AR(1)-GARCH(1,1) model for financial crashes

Abstract

This paper intends to meet recent claims for the attainment of more rigorous statistical methodology within the econophysics literature. To this end, we consider an econometric approach to investigate the outcomes of the log-periodic model of price movements, which has been largely used to forecast financial crashes. In order to accomplish reliable statistical inference for unknown parameters, we incorporate an autoregressive dynamic and a conditional heteroskedasticity structure in the error term of the original model, yielding the log-periodic-AR(1)-GARCH(1,1) model. Both the original and the extended models are fitted to financial indices of U. S. market, namely S&P500 and NASDAQ. Our analysis reveal two main points: (i) the log-periodic-AR(1)-GARCH(1,1) model has residuals with better statistical properties and (ii) the estimation of the parameter concerning the time of the financial crash has been improved.

Clarifications to questions and criticisms on the Johansen–Ledoit–Sornette financial bubble model

“…The choice of an AR(1) process for the noise is supported by the evidence provided in Refs. [53,88] that the residuals of the calibration of the JLS model to a bubble price time series can be reasonably described by an AR(1) process.…”

Are Bitcoin Bubbles Predictable? Combining a Generalized Metcalfe's Law and the LPPLS Model

“…As is typical in time series regression [32], the errors ǫ i are correlated and may have changing variance (hetero-skedasticity), which if ignored leads to sub-optimal estimates, and confidence intervals that are too small (over-optimistic). In this case, generalized least squares (GLS) provides a conventional solution, which has been used with LPPLS [33,34,35] and, if well-specified, has optimal properties. Here, we opt for a simple specification of the error model, being auto-regressive of order 1 19 ,…”

Prediction accuracy and sloppiness of log-periodic functions

“…The associated eigenvectors confirm the wisdom that the stiffer a direction, the more likely that it is close to an axis, and yWhile the beginning of a bubble is supposed to be given by the lowest value since the previous crash, it may be moved from this minimum to some later date; in Johansen and Sornette (2001) the beginnings of half of the eight bubbles on the Hang Seng up to 1998 were thus moved (Bre´e and Joseph 2010). zLomb periodograms ( van Bothmer 2003) or the residuals (Gazola et al 2008, Lin et al 2009) have been proposed, with mixed results. reversely for sloppy eigenvalues (Gutenkunst et al 2007).…”

“…Of the recent progress, the residuals have been shown to be AR(1) (Gazola et al 2008, Lin et al 2009). It makes sense, therefore, to create artificial data with AR(1) noise.…”

“…the predictive power of log-periodic functions in this context. The question is still unsettled as of yet (Laloux et al 1999, Feigenbaum 2001, van Bothmer 2003, Chang and Feigenbaum 2006, Gazola et al 2008. Problems are indeed numerous: what definition of a bubble and a crash to adopt (Jacobsen 2009, Lin et al 2009),?…”