Write a paper detailing your failure to replicate their results.

If you replicated their methodology and got a different result, then that's something noteworthy in itself. As a result, the most logical thing to do is to write a paper citing their paper, and then explaining how in your attempt at replicating their paper, you have obtained different results. Given that the ongoing replication crisis is apparently affecting economics, producing proof that a paper has failed to replicate should be eminently publishable.

It might be polite to give the authors of the original paper a little bit of warning of what you're planning on doing, though.

I get far more emails than I can reply to, but occasionally I do reply to random emails from people I don't know concerning papers I have written. The main factors that lead to me replying are: (1) I can see that the person is genuinely interested and has tried pretty hard to understand. (2) The person has a very concrete question that I can easily write down the answer to. (3) I am not overly pressed for time from other obligations.

In your case I cannot tell what kind of email you wrote. But if it says roughly "Dear X, I am interested in your paper but I don't understand your data collection methodology. Could you explain it in more detail? Best regards, Y." then I would ignore it for sure. Why? Basically the person has written a minimal-effort email, and so I assume they probably read the paper with minimal effort too. And what they want from me is completely unreasonable: They want me to write an expanded methods section just for them.

On the other hand, if the email would show a deeper understanding of the topic, and they would clearly explain their confusion and ask a specific question that I can easily clarify, then I am happy to do so, even if it takes a couple of paragraphs of explanation. However, such emails are rare.

When you publish a paper, the paper stands on its own, with all its strengths and weaknesses. A paper does not come with any warranty or guarantee that the author will freely provide all interested readers with further personalized instruction on the topic of the paper.

It is the choice of the author to answer emails, or not. Presumably the contents were sufficiently clear to warrant publication, so if the authors choose not to reply there remains the indirect route of extracting details from several papers on the topic, either by the same author or by others cited in the bibliography.

Note that you might have more leverage in getting data from the authors, if this is possible.

Ideally, all researchers respond to reasonable* requests, however, quite often they don't. Here are a few things you can do to increase your chances they even get your email (adding to Fabian Meier's answer on how to increase your chance to get a response):

• make sure you have the correct email address - many people change institutes and are no longer checking old accounts, or have left science altogether, even for papers that were just published

• try contacting co-authors, if any. If possible, aim for those that might know something (in case author contributions are published; generally reserch assistants, PhD students, postdocs; less likely old professors and co-authors from a different institute)

• for recently published papers, check outlets such as twitter for posts by the (co)authors metioning the paper, and reply to those

However, in the end it is the authors choice to answer your question. Unless you are a reviewer, then you have a bit more leverage - but still cannot force them (the editor might, but they could refuse and even retract the paper).

** something not obviously stated in the paper, basic knowledge in the field, or probably suggestions for future studies etc

I would consider it all right if you write one or two reminder emails, maybe after three weeks and six weeks.

A lot of people just forget about emails, or put them aside, especially if they cannot answer the questions immediately.

Furthermore, try to make your email as "easy" as possible, so if your email is a wall of text or more than five bullet points, it is much less likely to receive an answer as if you just ask one short and precise question.