Comparing nested data sets and objectively determining financial bubbles’ inceptions

Motivated by the question of identifying the start time τ of financial bubbles, we propose an improved calibration approach for 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, χnp2(Φ), to endogenously detect the optimal fitting window size ≔w∗∈[τ:t̄2] that should be used for calibration, assuming a fixed pseudo present time t̄2. The Lagrange regularisation of χnp2(Φ) defines the Lagrange regularised sum of the squared residuals, χλ2(Φ). Its performance is exemplified on a simple Linear Regression problem with a change point and compared against the performances of the Residual Sum of Squares (RSS) ≔χ2(Φ) and RSS/(N-p) ≔χnp2(Φ), where N is the sample size, p is the number of degrees of freedom and Φ is the parameter vector. 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), χλ2(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. The application of the method thus allows one to endogenise the determination of the starting time of bubbles, a problem that has yet not received a systematic objective solution. Moreover, the technique appears as a practical solution for comparing goodness-of-fit across unbalanced sample sizes.