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

Abstract

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, χ 2 np (Φ), to endogenously detect the optimal fitting window size := w * ∈ [τ :t2] that should be used for calibration purposes for a fixed pseudo present timē t2. The performance of the Lagrange regularisation of χ 2 np (Φ) defined as χ 2 λ (Φ) is exemplified on a simple Linear Regression problem with a change point and compared against the Residual Sum of Squares (RSS) := χ 2 (Φ) and RSS/(N-p):= χ 2 np (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 χ 2 λ (Φ) is found to provide welldefined 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.

Dissection of Bitcoin's Multiscale Bubble History

“…Therefore, it is important to determine the start of a bubble at first, and then apply the LPPLS for time windows with start points later than the found bubble start date. A solution for the bubble start time identification problem which itself is based on application of the LPPLS model has been introduced recently by Demos and Sornette [32], who propose the Lagrange Regularisation Approach.…”

“…However, this procedure is incorrect as it does not account for the fact that smaller windows will be favoured, due to their smaller number of degrees of freedom. Demos and Sornette [32] observed that applying a simple correction to account for this bias, which is linear in the window size (t 2 − t 1 ), behaves quite well and was efficient in a number of tests and real-life case studies. This Lagrange Regularisation Approach is recalled in Appendix D. We apply the method to the set of LPPLS fits calculated at each of the obtained short and long bubble peaks on the btc/usd rate in the same timeframe as before.…”

“…To determine the beginning of a bubble, we implement the Lagrange regularisation approach introduced by [32]. For a fixed t 2 , we scan t 1 from t 2 − 29 to t 2 − 719, corresponding to 691 windows ranging from 30 to 720 days.…”