Although it is acknowledged that countries with monarchies are generally more stable than republics, I don't think there is any evidence to support the suggestion that there is a direct causal link between monarchy and increased stability.
Most of the countries that still retain monarchies are relatively wealthy (e.g. Bahrain, Belgium, Denmark, Japan, Portugal, Jordan, Kuwait, Luxembourg, Monaca, Netherlands, Norway, Oman, Qatar, Saudi Arabia, Spain, Sweden, UK, etc.), and so are likely to be more stable anyway. Even in "developed" countries - monarchies or republics - it is fairly evident that political stability is threatened when economic difficulties arise.
Some monarchies have been the source of great instability, having failed to adapt to changing opinions and demands, and have collapsed or been overthrown. It is only those countries who have retained their monarchies - by making it legally impossible to question their legitimacy (such as in Saudi Arabia or Thailand) or by successfully managing any rumblings of discontent from their own populations - where monarchies survive.
I would also suggest that several monarchical countries, such as the UK, Spain, the Netherlands, Portugal, etc, have in the past colonised and exploited the poorer parts of the world and continue to exert their power over smaller nations. I believe it is not the lack of a monarch that has led to their instability but the lack of control over their own resources and the exploitation of those resources by more powerful countries. Those countries with monarchies that have not colonised or exploited other parts of the world but that have instead relied on exploiting their own people, often rule by fear.
