Lagrange regularisation approach to compare nested data sets and determine objectively financial bubbles' inceptions

TL;DR

This paper introduces a Lagrange regularisation method to objectively identify the start times of financial bubbles by optimizing the fitting window in time series analysis, demonstrated on synthetic and real market data.

Contribution

The paper presents a novel Lagrange regularisation approach for endogenously detecting the inception of financial bubbles without external input.

Findings

Effectively identifies bubble start times in historical market data.

Provides a systematic, objective method for regime change detection.

Outperforms traditional residual-based methods in bubble inception detection.

Abstract

Inspired by the question of identifying the start time $\tau$ of financial bubbles, we address the calibration of time series in which the inception of the latest regime of interest is unknown. By taking into account the tendency of a given model to overfit data, we introduce the Lagrange regularisation of the normalised sum of the squared residuals, $\chi^{2}_{np}(\Phi)$, to endogenously detect the optimal fitting window size := $w^* \in [\tau:\bar{t}_2]$ that should be used for calibration purposes for a fixed pseudo present time $\bar{t}_2$. The performance of the Lagrange regularisation of $\chi^{2}_{np}(\Phi)$ defined as $\chi^{2}_{\lambda (\Phi)}$ is exemplified on a simple Linear Regression problem with a change point and compared against the Residual Sum of Squares (RSS) := $\chi^{2}(\Phi)$ and RSS/(N-p):= $\chi^{2}_{np}(\Phi)$, where $N$ is the sample size and p is the number of degrees of freedom. Applied to synthetic models of financial bubbles with a well-defined transition regime and to a number of financial time series (US S\&P500, Brazil IBovespa and China SSEC Indices), the Lagrange regularisation of $\chi^{2}_{\lambda}(\Phi)$ is found to provide well-defined reasonable determinations of the starting times for major bubbles such as the bubbles ending with the 1987 Black-Monday, the 2008 Sub-prime crisis and minor speculative bubbles on other Indexes, without any further exogenous information. It thus allows one to endogenise the determination of the beginning time of bubbles, a problem that had not received previously a systematic objective solution.