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

Inspired by the question of identifying the start time τ 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, χ²_np(Φ), to endogenously detect the optimal fitting window size := w∗ ∈ [τ : t̄₂] that should be used for calibration purposes for a fixed pseudo present time t̄₂. The performance of the Lagrange regularisation of χnp(Φ) defined as χ2λ(Φ) is exemplified on a simple Linear Regression problem with a change point and compared against the Residual Sum of Squares (RSS) := χ2 (Φ) and RSS/(N-p):= χ2np (Φ), 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 χ2λ(Φ) 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