The first one is a really convoluted structure with a both a passive verb and a passive past participle, but the bottom line is that the writing isn't current or ongoing so the Simple Present isn't the best choice. Would you say, "The food is believed to be eaten by an unknown person" if you were referring to an action completed in the past? I would rearrange the whole structure using "it" as an empty subject: It is believed that the novel was written in the 17th century, but the author is unknown.

Both sentences are excellent. The only change I would make would be to say "author" instead of "writer" to avoid the repetition of using the words "written" and "writer" in the same sentence. It is true that the first option uses passive voice twice, and passive voice is often frowned upon, but that is not a concern here. Passive voice is appropriate when you need to make something happen without specifying what or who makes it happen. That is exactly what you need here. In this situation, replete with uncertainty, past participles are ideal. My favorite of the two is the first ("believed to have been written"). I just prefer the way it sounds. Maybe I like it more because it takes the point of view of the book. Rather than thinking of the book as a "thing" it treats the book with more respect, talking about its experience of having been written. It sort of humanizes the book, in some poetic sense, giving it more respect. But no need to take my advice! That's just the way I like to talk. Language is personal.

hii

Both versions of the phrase are grammatically correct, but they convey slightly different meanings. The novel is believed to have been written by an unknown writer of the 17th century. The novel is believed to be written by an unknown writer of the 17th century. The first version (have been written) suggests a completed action in the past. It implies that the writing of the novel is something that happened in the past, and we are acknowledging that belief. The second version (be written) is more of a general statement. It doesn't explicitly specify when the writing occurred. It suggests that the novel is believed to be the work of an unknown writer from the 17th century, without emphasizing the completion of the writing in the past. Both versions are used, and the choice between them depends on the nuance you want to convey. If you want to emphasize the historical completion of the writing in the 17th century, the first version might be more suitable. If you want to present it as a more general belief without emphasizing the timing of the writing, the second version works well.