The Massachusetts Bay Colony can be said to have been partly theocratic based on a couple of reasons. For example, the Puritans believed that they had a covenant with God to build a society that would be a model for humankind. This shows partly theocracy because everyone believed in and worshipped a God to help them build a society. Another example is that the colonies franchise was extended to all adult males who were a part of the Puritans congregation. This caused all non-domination or churchless people unable to vote. This also shows the theocratic idea behind this government and how the church and God influenced the decisions of the colony and provided a background to establish their government on.
The Bay Colony was also based on a partly democratic government. The government was partly democratic because the people of the colony elected a governor and his assistants annually. This wasn't a true democracy because only the men that were true members of the church were eligible, which in turn goes back to the theocratic idea of the government. The government was also partly democratic because it treated everyone the same when dealing with laws and taxes. Basically no matter who you were the laws still applied to you and so did the taxes.
The colony was partly oligarchic in a few ways also. The oligarchic part of the government was from the governor and his assistants. Although they were elected into office, the colony was just a colony and not a state or part of a country yet,