Lagrange Regularisation Approach to Compare Nested Data Sets and Determine Objectively Financial Bubbles' Inceptions

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.