Fibonacci numbers and the golden ratio seem to be shrouded in myths and misconceptions.
The majority of plants have a Fibonacci number of petals due to the optimal (uniform) packing of the seeds which creates a Fibonacci number of spirals in each direction. They exist, it seems amazing, but we understand why.
Fibonacci numbers do not exist in ancient architecture, old paintings, the human body or financial markets. Nor is there anything inherently aesthetically pleasing about the ratio. These are all urban legends.
Fibonacci numbers are of mathematical interest. It is also interesting that the human mind extrapolates something that exists in nature for reasons that are understood to areas where it doesn't exist, and has no reason to exist. I guess we all enjoy a mystery. :-)
"The Kalman filter can help construct better smoothers. Although we do not delve deeply into the matter, we believe that this approach may very well contain the seeds of reasoning for the observation of the so-called Fibonacci retracements, which is well documented in the area of technical analysis."
Vidyamurthy (2004) page 54
Yeung et al. (2002) found that using Fibonnacci Retracrments failed to work.
Zielonka (2000) found that more than 60% of respondents market Fibonacci proporsions as having predictive value.