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Today I faced a term "finite samples" in reading Borusyak (2021)'s work:

We conclude the section by providing simulation evidence that the efficiency gains from using our estimator are sizable, that its sensitivity to some parallel trend violations is no larger than that of the alternatives, and that our inference tools perform well in finite samples

I did a search about the "finite sample" and "infinite sample"

Sampling from an infinite population is handled by regarding the population as represented by a distribution. ... A random sample from an infinite population is therefore considered as a random sample from a distribution

It seems to me that regarding "finite sample", we randomly choose a sample in a population and analyze this sample and draw a conclusion for a population, and in an "infinite sample", we test for the whole population data to draw a conclusion for the population, is it correct?

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The phrase "finite sample" is somewhat of a pleonams as every sample is (by definition) finite.

What they probably refer to with the phrase "finite sample" is a sample that is small or moderate in size.

A large part of statistical inference is based on large sample approximations. For example, if the sample size grows very big then the sample average converges (in probability) to the expectation and the difference between the sample average and the true mean (times $\sqrt{n}$) converges (in distribution) to a Gaussian (normal) distribution.

In reality, however, samples are finite, so these approximations might not perform very well if the sample size is small. In order to show that the approximations are also performing well even for samples that are not too big, one usually performs a Monte-Carlo (i.e. simulation) exercise. I think this is what the paper is referring to.

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As mentioned in tdm’s +1 answer in real life all samples are finite. However, when you theoretically derive some econometric model you can analytically always examine what would happen if the sample you have grows to infinity $(n \rightarrow \infty)$.

This allows you to examine asymptotic properties of your estimator, and in real life these asymptotic properties will hold approximately for sufficiently large enough samples (for example for OLS about 30 observations per independent regressor is sufficient - see Verbeek A guide to modern econometrics).

However, that is not the same as assuming you have access to the whole population because population can also be infinite, and infinite sample from infinite population does not necessarily mean you sampled everyone (infinity does not work like regular number).

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