# Population vs. Data-Generating Process

A lot of elementary statistical and econometric literature bases on so called "population models". An example is econometric handbook "Introductory Econometrics: A Modern Approach" by J. M. Wooldridge. or very influential paper of P. W. Holland "Statistics and Causal Inference".

On the other hand, some, especially new papers in Causal Infrence, instead to population refer to Data-Generating Process (DGP). An example could be "The Identification Zoo: Meanings of Identification in Econometrics" by A. Lewbel.

My understanding of that matter is that those concepts are very closely related, close enough to think of them as one thing. Of course they are not exactly the same entity, and population could be understood as an infinite number of units generated by particular DGP. Such understanding is derived from idea to join this two approaches.

As I am not strongly convinced to this, I would like to ask: What are the differences and similarities between Population and DGP as the bases of statistical and econometric modelling? And also, relatedly: What is the relation between these two concepts?

• Unless you have a finite population, I’d consider them synonyms. – Dave Oct 15 '20 at 0:12
• @Dave Would the process of sampling the population be an important consideration? – whuber Oct 15 '20 at 16:27
• @whuber What do you mean, something like a simple random sample vs a stratified sample? – Dave Oct 15 '20 at 16:29
• @Dave Yes, or probability sampling, transect sampling, hierarchical sampling, ranked set sampling, adaptive sampling, etc., etc. – whuber Oct 15 '20 at 16:31

In population approach, the model that you are fitting to a data can potentially be a reduced from of the true DGP. A crude example:

Say $$X_t$$ is a time-series that actually grows with time with white noise ($$e_t$$). Specifically, let DGP is

$$X_t = a_0+a_1t+a_2t^2+e_t$$

$$\implies X_{t-1} = a_0+a_1(t-1)+a_2(t-1)^2 + e_{t-1}$$

$$\implies X_{t-1} = X_t-a_1+a_2-2a_2t + e_{t-1}-e_t$$

$$\implies \Delta X_t = a_1-a_2+2a_2t + \Delta e_t$$

$$\implies \Delta X_t-\Delta X_{t-1}=2a_2 + \Delta e_t - \Delta e_{t-1}$$

Therefore, the DGP can be simplified to the following MA(1) type process ($$u_t \equiv \Delta e_t$$):

$$Z_t \equiv \Delta X_t-\Delta X_{t-1}=\beta + \Delta u_t$$

So the random variable $$Z_t$$ has a particular distribution with mean value $$\beta$$, which will be estimated from given observations. And while that is true, it is not unique to the original DGP, because information about at least $$a_1$$ is permanently lost.

If, on the other hand, you model $$\Delta X_t-\Delta X_{t-1}=\beta + u_t$$ as DGP, you are saying that realized value of $$X_t$$ is, by process design, is a function of last two period's values - which is very different from our earlier case.

So the two approaches, I think, will have different implications on interpretation and causal inference.