I want to investigate forecasting before, during and after the financial crisis using daily stock prices, but I need a serious scientific method to identify the beginning of the fall in the stock index and its end. Any ideas?
In terms of method, maybe you can test for multiple structural breaks (see the bfast package (https://cran.r-project.org/web/packages/bfast/bfast.pdf) or strucchange package (https://cran.r-project.org/web/packages/strucchange/strucchange.pdf) for R). The strucchange package has a confidence interval option. Also, given the variety of crises that have been identified (currency, sovereign default, banking crisis, etc.) as well as market crashes this adds an additional layer of complexity to the analysis. Among recent crises, the Tequila crisis, started around December 20th 1994 and may have ended around March 10, 1995 (the GAO report might be the best for details http://www.gao.gov/products/GGD-96-56 ). For the Asian Crisis, Kaminsky and Schmukler's "What triggers market jitters? A chronicle of the Asian crisis." JIMF 1999, suggest the crisis had four phases: July 2, 1997 in Thailand as the beginning, and early February 1999 as the end in Latin America (Argentina did have a follow up crisis see Yeyati, Schmukler and Van Horen's Ec Letters paper http://siteresources.worldbank.org/DEC/Resources/LevyYeyati-Schmukler-VanHoren-ADRsandCrises.pdf ). For the Dot Com crash in the U.S., see Jeremy Siegel's "What is a Bubble?" article in the European Journal of Finance DOI: 10.1111/1468-036X.00206 . For the Global Financial Crisis, there are probably lots of time lines out there, but try the St. Louis Fed for a start https://www.stlouisfed.org/financial-crisis/full-timeline
Trying to predict a financial crisis is very challenging. Some would say it is almost impossible. However, most research suggests looking for a 'bubble' in the stock market prices and inflated stock values. Here's the tricky part: you have to determine if the bubble and inflated stock prices are the result of economic growth, company profits, or risky speculative actions by investors?
I suggest you read "The End of Finance" by Massimo Amato and Luca Fantacci (both on ResearchGate). Amato and Fantacci argue convincingly that Western finance (Capitalism) is inherently and permanently in crisis (alternating booms and busts), due to the fact that such a commodified system of finance is structurally incapable "of performing the exquisitely financial function of settling accounts" between creditors and debtors.
As a result, irredeemable debt builds up within the financial system "consisting [at the national level] of the banknotes of the central banks" and "at the international level of dollars stockpiled in the currency reserves of the Middle and Far East.".
According to Amato and Fantacci, "the present [episode of financial markets] crisis marks the end of a conception and practice of finance grounded in the systematic suppression of the end" (i.e. the settling of accounts between debtors and creditors).
From my personal perspective, this end to the viability of the incumbent system of commodified finance corresponds to the incapacity of the real economy to convert a sufficient quantity of Nature into monetised credit to maintain the liquidity of the system (i.e. paying down past and creating new irredeemable debt), despite technological progress, because planetary limits to such exponential "economistic" growth required for financial market parasitism was breached in the 1970s (when Bretton Woods collapsed).
Since that juncture, continued operation of the financial system could only be maintained through a cannibalisation of itself, presented as new financial "innovations" - including: creative accounting for the virtualisation of profits of corporations linked to financial markets; financial market credit-forgery for asset-inflation followed by capital gains extraction; and the recycling of past interest-bearing debt, within the financial system itself, through derivatives etc.
However, as Amato and Fantacci, advocate, this terminal phase of the inherent crisis of financial market capitalism is also an opportunity to re-search and re-think financial practice and theory. As they say: "What comes to an end in this crisis is the idea of finance grounded in the representation of money and credit as commodities", which means we need to establish "a radically different institutional and theoretical perspective (...) capable of supporting real economic activity", where "money and credit are not traded", and "relations between debtors and creditors are constructed so as to come to an end" (accounts between them are settled).
I suggest that revising your research in this direction could be more fruitful. I also opinion that if you read the book of Amato and Fantacci, you will probably be persuaded that there is no way/method, scientific or otherwise, "to investigate forecasting before, during and after the financial crisis".
If the purpose is to test for multiple structural beaks, of the many tools available (several have been already mentioned), another choice is this one at https://github.com/zhaokg/Rbeast. It is a Bayesian changepoint algorithm. Maybe it is not that useful, but just in case.