**Modified Profile Likelihood Inference And Interval Forecast Of The Burst Of Financial Bubbles**

Swiss Federal Institute of Technology Zurich (ETH Zurich)

ETH Zurich

Swiss Finance Institute; ETH Zürich – Department of Management, Technology, and Economics (D-MTEC)

February 26, 2016

*Swiss Finance Institute Research Paper No. 16-11*

**Abstract: **

We present a detailed methodological study of the application of the modified profile likelihood method for the calibration of nonlinear financial models characterised by a large number of parameters. We apply the general approach to the Log-Periodic Power Law Singularity (LPPLS) model of financial bubbles. This model is particularly relevant because one of its parameters, the critical time *t _{c}* signalling the burst of the bubble, is arguably the target of choice for dynamical risk management. However, previous calibrations of the LPPLS model have shown that the estimation of

*t*is in general quite unstable. Here, we provide a rigorous likelihood inference approach to determine

_{c}*t*, which takes into account the impact of the other nonlinear (so-called “nuisance”) parameters for the correct adjustment of the uncertainty on

_{c}*t*. This provides a rigorous interval estimation for the critical time, rather than a point estimation in previous approaches. As a bonus, the interval estimations can also be obtained for the nuisance parameters (

_{c}*m,w,*damping), which can be used to improve filtering of the calibration results. We show that the use of the modified profile likelihood method dramatically reduces the number of local extrema by constructing much simpler smoother log-likelihood landscapes. The remaining distinct solutions can be interpreted as genuine scenarios that unfold as the time of the analysis flows, which can be compared directly via their likelihood ratio. Finally, we develop a multi-scale profile likelihood analysis to visualize the structure of the financial data at different scales (typically from 100 to 750 days). We test the methodology successfully on synthetic price time series and on three well-known historical financial bubbles.

### Modified Profile Likelihood Inference And Interval Forecast Of The Burst Of Financial Bubbles – Introduction

Financial bubbles and their subsequent crashes provide arguably the most visible departures from well-functional efficient markets. There is an extensive literature (see e.g. the reviews of Kaizoji and Sornette (2010)1, Jiang et al. (2010); Brunnermeier and Oehmke (2012); Xiong (2013)) on the causes of bubbles as well as the reasons for bubbles to be sustained over surprising long period of times. One of these views emphasises the role of herding behaviour on bubble inflation (Johansen and Sornette, 1999b). When imitation is sufficiently strong, a high demand for the asset pushes the price upwards, which itself, and somewhat paradoxically, increases the demand, propelling further the price upward, and so on, in self-fulfilling positive feedback loops. In such regimes, the market is mainly driven by sentiment and becomes detached from any underlying economic value. This process is intrinsically unsustainable and the mispricing ends at a critical time, either smoothly (with a correction phase) or abruptly (via a crash). The formulation of this hypothesis of collective herding behavior within rational expectations theory resulted in the so-called Log-Periodic Power-Law Singularity (LPPLS) model, which has been used for many successful ex-post and ex-ante predictions of bubble bursts (see e.g. a partial list in (Sornette et al., 2013) and a recent implementation for the Chinese bubble and its burst in 2015 (Sornette et al., 2015)).

Notwithstanding a number of improvements concerning the calibration of the LPPLS model, including meta-search heuristics (Sornette and Zhou, 2006) and reformulation of the equations to reduce the number of nonlinear parameters (Filimonov and Sornette, 2013), the calibration of the LPPLS model remains a bottleneck towards achieving robust forecasts and a matter of contention (Bree et al., 2013; Sornette et al., 2013). In this context, the aim of the present paper is to present a fundamental revision of the calibration procedure of the LPPLS model. Specifically, we deviate from the traditional ordinary least squares (OLS) calibration that provides point estimates of parameters, which has been used since the introduction of the model in 1999 (Johansen et al., 1999, 2000). Instead, we employ a rigorous likelihood approach and, for the first time to the best of our knowledge, we provide interval estimates of the parameters, including the most important critical times of market regime changes.

We deliberately avoid dwelling on the derivation of the model and its foundations, and take it as given. We do not discuss supporting evidence and critiques of the model, nor address how to apply the LPPLS model to construct robust signals for extensive backtests or real-time ex-ante predictions. These questions require extensive analyses and are beyond the scope of the present manuscript. See (Johansen and Sornette, 2010; Jiang et al., 2010; Sornette et al., 2013, 2015; Zhang et al., 2015) for investigations in these directions.

The purpose of the present paper is methodological, and the main focus is on the statistical aspects of the theory and the corresponding mathematical derivations. One of the major advances of this paper is to formulate the calibration procedure so that the critical time tc is the major parameter of interest in the likelihood inference, while other model parameters are treated as so-called nuisance parameters. Of course, these other parameters are also intrinsic to the model but their existence contributes to the variance of the parameter of key interest. Such reformulation of the calibration procedure has its roots in an original idea proposed by Filimonov and Sornette (2013), which was however developed in a crude way and without the proper statistical methodology.

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