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Inspired by the question of identifying the start time $\tau$ of financialbubbles, we address the calibration of time series in which the inception ofthe latest regime of interest is unknown. By taking into account the tendencyof a given model to overfit data, we introduce the Lagrange regularisation ofthe normalised sum of the squared residuals, $\chi^{2}_{np}(\Phi)$, toendogenously detect the optimal fitting window size := $w^* \in[\tau:\bar{t}_2]$ that should be used for calibration purposes for a fixedpseudo present time $\bar{t}_2$. The performance of the Lagrange regularisationof $\chi^{2}_{np}(\Phi)$ defined as $\chi^{2}_{\lambda (\Phi)}$ is exemplifiedon a simple Linear Regression problem with a change point and compared againstthe 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 ofdegrees of freedom. Applied to synthetic models of financial bubbles with awell-defined transition regime and to a number of financial time series (USS\&P500, Brazil IBovespa and China SSEC Indices), the Lagrange regularisationof $\chi^{2}_{\lambda}(\Phi)$ is found to provide well-defined reasonabledeterminations of the starting times for major bubbles such as the bubblesending with the 1987 Black-Monday, the 2008 Sub-prime crisis and minorspeculative bubbles on other Indexes, without any further exogenousinformation. It thus allows one to endogenise the determination of thebeginning time of bubbles, a problem that had not received previously asystematic objective solution.

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