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There is a confusing facet of the concept of a specification test.

According to many textbooks, a specification test is roughly a statistical test investigating whether assumptions in a statistical model are true or not.

Here, I think, it is not clear whether specification tests are arguments for the population of interest or the data in hand.

To be specific, suppose that we imposed an assumption (or specified a model) denoted by M, and we have a data set. In this situation, a specification for M is rejected. Then, does that mean M is not appropriate to the given data? or the population?

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A specification test tests whether a model is appropriate for the underlying population, not just the data. The data is a "proxy" for the population. If the dataset is too small to be a good proxy for the population, the specification tests will not reject the tested model. See e.g. the Ramsey RESET test. There, the linear model is rejected only if the nonlinear terms of the nonlinear model are considered significant, which only happens if you have enough data.

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