Can The Phases of the Moon Affect Precipitation?

Often dismissed as an old superstition, but does the changing phases of the Moon affect when it would rain or snow?
I don’t know how many heard of it, but I first heard this supposedly old superstition back in 1989 that goes: Wet weather follows the new Moon and the full Moon. Dry weather follows the first quarter Moon and the last quarter Moon. Strangely enough, a correlation was indeed found out at that time using U.S. Weather Bureau precipitation records showing that there is indeed a better than average chance of rain or snow in the week after a full Moon and the week after new Moon. While the driest periods tend to occur the week after the first quarter Moon and the week after the last quarter Moon. Unfortunately at the time, no clear-cut explanation was provided behind the phenomena after the study was published showing a correlation between occurrence of precipitation and the phases of the Moon. Will a renewed study ever shed light on the validity of this old superstition?
My working hypothesis on the matter is that probably during the week after the full Moon and the week after the new Moon, the gravitational effects between the Earth and the Moon during these periods probably allowed higher than average amounts of meteoric and cometary dust to fall into our atmosphere. These comet and meteorite sourced material probably acted as nuclei via the Bergeron-Findeisen Theory of Rain / Precipitation thus causing rainfall and snowfall frequency to increase a week after the full Moon and the new Moon. But is this explanation really satisfactory?
Swedish meteorologist Tor Bergeron first proposed the nuclei theory of precipitation around the mid-1930s, which was later elaborated by German physicist Walter Findeisen and is now widely accepted as the Bergeron-Findeisen Theory of Rain. This theory was later applied as the working principle behind cloud seeding. Artificial seeding of rain clouds to induce precipitation during times of drought was developed in 1946 by General Electric’s Vincent J. Schaefer and Irving Langmuir. They used both silver iodide and dry ice as cloud seeding material.
Silver iodide, whose crystalline structure is similar to that of natural ice and therefore provides hospitable nuclei on which ice crystals readily form. Solid carbon dioxide or dry ice – another good cloud seeding agent - is so cold that it causes water vapor to solidify into enormous numbers of tiny ice crystals. In either case, precipitation should follow, according to the Bergeron-Findeisen Theory. Pellets of dry ice are usually sown into a cloud from airplanes while silver iodide is released as smoke, sometimes from an airplane, sometimes from the ground. Meteoric and cometary dust could act as a cloud seeding nuclei, increasing chances of rain or snow – depending on the season – during the week after full Moon and the week after new Moon.
The jury is still out on whether the phases of the Moon affects the frequency of rainfall. But white tailed deer population in the North American continent seem to pattern their migration routes with the phases of the Moon and rainfall since these species of deer like to consume ashes of brush set off by lighting induced brush fires and the water pools that eventually form in the succeeding rainfall.