Regression over the whole population

Question

What's the meaning of the standard error of a coefficient in a regression when the whole population is included?

I've been so puzzled by this question. Because it seems to me, standard errors make no sense when the whole population is included -- there is no need for statistical inference since you already have the whole population.

But it is so widely used even by many articles published in top journals. For example, if I am examining the relationship between a country's GDP growth rate and its population density, I run the regression:

$$ GDP_i = \alpha + \beta Pop_i + \gamma \mathbf{X}_i + \epsilon_i$$

with all 195 countries on the earth. In the case, all countries (the population) are included. But all literature still talk about the statistical significance of the coefficients.

Could somebody explain is it a misuse of statistical inference when regressing over the whole population?

Answer 1

I had initially flagged this question for moderators to examine whether it would be better to migrate over to the statistics SE site Cross Validated. But since the OP introduced a very specific econometrics example, I believe the (very deep) concept of "population/sample" can be usefully discussed for the purposes of this example.

A first issue is that discussed in @AdamBailey answer: if one considers "all the countries in the world" for a given year or years, and it labels the data as "population", then the next year should belong to a different population. If it belongs to a different population, then how are we to use results from one population to make inference for another population? So indeed, here our "population" is two-dimensional, country and time period -and in that sense, with the time horizon open-ended, we only have a sample in our hands.

The second issue (partly implied in @luchonacho answer) is the following: our population is not the actually observed realizations of the random variables "$GDP_i, i=1,..n$. This is the data. Our population is the collection of random variables themselves, which are functions, not values.

So our data is just one of the possible combined realizations of these random variables. These realizations came about not only as a result of deterministic/engineering relations/causality (reflected in the coefficients), but also under the effect of inherently random factors. In that sense, the data is not a "pure/typical" image of the "population" -it contains noise, non-structural disturbances, one-off shocks etc.

Then this uncertainty will carry over to the estimation of the coefficients we are trying to estimate, because we assume that these coefficients describe causality or co-movement prior to the random elements affecting the final value of the dependent variable.

Due to both aspects above, talking about "standard error of estimates" is totally valid, in this case too, and then apply statistical tests as usual.

Answer 2

It's important to consider what exactly the population is about which an inference is being drawn. It's easy to overlook the time aspect in this context.

Suppose for example that the aim is to forecast the next two years' GDP for each country in the world. Then the population of interest is a set of pairs of the form "country, year". It isn't simply "all countries", and even if a forecast model has been estimated by regression on current and past years' data for each country, that doesn't mean that the whole population of interest has been included.

If one really does start from a complete dataset for the whole population of interest, then all one can do is calculate summary statistics. That could include standard deviations, but it would be inappropriate to call these standard errors, since that term relates to a sampling distribution whereas the only "sample" in this case is the whole population.