The Data-Generating Process in an Empirical Study

Introduction

An empirical data set is the end of a process, not the starting point. Units have characteristics; some explanatory variables may be assigned, while others arise through decisions or circumstances; outcomes occur; some units enter the data; and measurements are recorded. Making these stages explicit helps clarify what the data describe and where a statistical analysis begins.

A data-generating process (DGP) is a representation of how the relevant quantities and observations arise. It may be expressed in words, equations, or a diagram. It does not have to be a complete account of reality. Its purpose is to state which parts of the process matter for the empirical question and how they are connected. A diagram can organize that account, but it cannot by itself show that a particular causal conclusion is warranted.

Definition: A data-generating process is a description of the mechanisms that produce the variables of interest and the records available for analysis. In a typical empirical study, it distinguishes units and their characteristics, explanatory variables and their assignment or realization, outcomes, selection into the observed data, and the measurement process that creates recorded values.

From Units to Recorded Data

Let $i$ index a unit, such as a person, firm, school, or region. Let $W_i$ denote relevant characteristics or circumstances, $X_i$ other explanatory variables, $A_i$ a focal exposure or assigned variable, and $Y_i$ an outcome. Let $S_i$ indicate whether the unit is included in the data used for analysis: $S_i=1$ means included and $S_i=0$ means not included. Finally, let $Z_i$ denote the recorded values. These may be direct measurements of $A_i$, $X_i$, or $Y_i$, or records constructed from them.

The symbols help separate concepts that may otherwise be blurred. An outcome $Y_i$ is the quantity of interest as it exists in the process being studied; a recorded value $Z_i$ is what the data file contains. The distinction matters even when the record is an accurate measurement, because accuracy is a feature of the measurement process, not something guaranteed by the notation. Likewise, $A_i$ may be assigned by a program, chosen by a unit, or determined by an institution. The word “assignment” names the process that determines its value; it does not imply random assignment.

Units and circumstances.
Specify the units under study and the relevant characteristics or context $W_i$.

Explanatory variables and assignment.
Describe how $X_i$ and the focal variable $A_i$ arise or are assigned.

Outcomes.
Describe how the outcome $Y_i$ is produced in relation to the explanatory variables and other circumstances.

Sampling and measurement.
State which units enter the data and how their underlying values become recorded values $Z_i$.

A compact schematic for one possible ordering is:

W_i \longrightarrow (X_i,A_i) \longrightarrow Y_i \longrightarrow S_i \longrightarrow Z_i

This is a prompt for describing a study, not a universal timeline. In some studies, units are sampled before an assignment is made; in others, measurement and selection may be intertwined. Arrows should reflect the process being described, not merely the order in which variables appear in an equation. A diagram can also include shared causes, such as $W_i$ pointing to both $A_i$ and $Y_i$, if that is part of the proposed account.

A Joint-Distribution Description

The same process can be described probabilistically. For example, a joint distribution for characteristics, explanatory variables, assignment, outcome, inclusion, and recorded values can be factored in a chosen order:

p(W,X,A,Y,S,Z) = p(W,X)\, p(A\mid W,X)\, p(Y\mid A,W,X)\, p(S\mid Y,A,W,X)\, p(Z\mid S,Y,A,W,X).

Here $p(\cdot)$ denotes a probability mass function or density, as appropriate. The expression is a chain-rule factorization: each factor describes a conditional distribution given the quantities that precede it in the chosen order. It does not, by itself, assert that the variables are independent, nor does it establish a causal interpretation of any factor.

To see why, the definition of a conditional distribution gives $p(A\mid W,X)=p(W,X,A)/p(W,X)$, wherever the denominator is positive. Multiplying by $p(W,X)$ recovers $p(W,X,A)$. Applying the same identity successively to $Y$, $S$, and $Z$ yields the displayed product. Thus the factorization is a way to organize a joint distribution. A causal diagram makes additional claims about mechanisms; those claims require justification beyond the algebra of a factorization.

Key distinction: A description of how data are generated is not the same as an estimand, an estimate, or a conclusion. An estimand specifies a target quantity; an estimate is calculated from data; a DGP organizes the process that produces those data. A diagram can expose what an analysis needs to address, but drawing an arrow is not evidence that its interpretation is correct.

Worked Examples

Worked Example: Assignment and an Earnings Outcome

Hypothetical illustration. Suppose a small firm tracks four workers. Let $E_i$ be years of prior experience, $A_i=1$ if the worker is assigned a training course and $A_i=0$ otherwise, and $U_i$ a worker-specific factor not otherwise listed. For illustration, suppose annual earnings $Y_i$, measured in thousands of dollars, are generated by the stipulated equation

$$ Y_i=20+3A_i+2E_i+U_i. $$

The firm assigns training according to its scheduling decisions; the example does not assume that assignment is random. The hypothetical inputs and outcomes are:

Worker	$E_i$ (years)	$A_i$	$U_i$	$Y_i$ (thousands of dollars)
1	0	0	1	21
2	1	1	-1	24
3	2	0	0	24
4	1	1	2	27

For worker 2, substitution gives $Y_2=20+3(1)+2(1)-1=24$, or \$24,000. For worker 4, $Y_4=20+3(1)+2(1)+2=27$, or \$27,000. If the firm records all four workers’ earnings without further transformation, the recorded earnings match these outcomes in this illustration.

The diagram for this stipulated process includes experience and the worker-specific factor as contributors to earnings, while the firm’s scheduling process determines $A_i$. The equation is a constructed description for this example, not an estimated relationship or a causal finding. In particular, computing an association between training and earnings from these four records would not establish what would have happened to the same workers under a different assignment.

Worked Example: Sampling Changes Which Outcomes Are Observed

Hypothetical illustration. A researcher is interested in weekly sales, in dollars, for five shops in a small region. Suppose the underlying outcomes are \$40, \$50, \$60, \$70, and \$80 for shops 1 through 5. A simple draw selects shops 2 and 5 for recording; assume their sales are recorded exactly.

The full-region mean is $(40+50+60+70+80)/5=300/5=\$60$ per shop per week. The observed sample contains \$50 and \$80, so its mean is $(50+80)/2=130/2=\$65$ per shop per week. The sample mean differs from the mean across these five shops by \$5.

A process diagram separates the stages: shops have underlying weekly sales; the draw determines $S_i$; and exact recording turns the selected sales into observed values. The \$65 figure describes the two observed shops. The arithmetic alone does not explain why this particular pair was selected or determine how well it represents the region. Those questions concern the sampling process, which must be described rather than inferred from the outcome values alone.

Worked Example: Measurement Transforms an Outcome Into a Record

Hypothetical illustration. Three sampled commuters have travel times of 12.4, 17.8, and 20.2 minutes. Suppose the recording procedure rounds each time to the nearest five minutes. The recorded values are 10, 20, and 20 minutes, respectively.

The mean of the underlying times is

\frac{12.4+17.8+20.2}{3} = \frac{50.4}{3} = 16.8 \text{ minutes}.

The mean of the recorded values is

\frac{10+20+20}{3} = \frac{50}{3} \approx 16.67 \text{ minutes}.

Both calculations are correct, but they summarize different values: one uses the stipulated travel times and the other uses the records produced by the rounding procedure. The example makes measurement a visible stage in the DGP; it does not require that every study’s measurements be rounded or that rounding always changes a summary in the same direction.

Reading and Using a Process Diagram

A useful diagram is specific enough to clarify how the analyzed records came to exist. It should name the units, identify the main outcome and explanatory variables, and show the assignment, inclusion, and measurement stages that matter for the question. A diagram that includes every conceivable influence can become unreadable; one that omits a consequential stage can mislead. The goal is a compact account of the process relevant to the analysis.

When reviewing a diagram, ask what each arrow means. Does it represent a proposed mechanism, a timing relationship, or simply a connection to be investigated? These are not interchangeable. A process description should also distinguish the full set of relevant units from the units that appear in the observed data. Without that distinction, it is easy to treat a recorded data set as if it directly represented every unit or underlying quantity of interest.

The diagram can guide an empirical analysis by making questions visible: how explanatory variables arise, which outcomes are observed, and how the records were formed. It can also expose where more information or assumptions would be needed before drawing a conclusion. The diagram is a map of the proposed process, not a substitute for evidence or a guarantee that the map is correct.

Common Mistakes

Equating a record with the underlying quantity. A recorded value may be transformed or imperfectly measured. State what the data file contains and how it relates to the quantity of interest.
Treating every explanatory variable as assigned. Some variables are selected or determined by circumstances rather than assigned by a researcher. Describe the relevant process instead of assuming random assignment.
Reading an arrow as proof of causation. An arrow in a proposed diagram expresses a claim about the process; it does not verify that claim or identify a causal effect.
Ignoring selection into the observed data. An outcome can exist for a unit that does not appear in the analysis data. Show how inclusion is determined, even if the process is not yet fully understood.
Confusing the joint-distribution factorization with independence. The chain rule conditions each factor on earlier variables; it does not say those variables are independent.

Key takeaway: Trace the path from units and circumstances through explanatory variables, outcomes, inclusion, and measurement to the records actually analyzed. Keeping these stages distinct makes an empirical question clearer, while reminding us that a process diagram alone does not establish a causal conclusion.

Check Your Understanding

Use the definitions and distinctions in this tutorial to answer the following.

In a study of job training and earnings, what is the difference between an earnings outcome and the earnings value stored in a data file?
Why does calling a variable $A_i$ an assignment variable not imply that it was randomly assigned?
In the chain-rule factorization, what does the factor $p(Y\mid A,W,X)$ describe, and why does its presence not by itself establish a causal interpretation?
A regional survey records only selected households and rounds monthly spending to the nearest ten dollars. Name the two process stages that determine which spending values appear in the data.
In the shop example, the observed sample mean is \$65 and the five-shop mean is \$60. What does each number summarize?

Worker	\(E_i\) (years)	\(A_i\)	\(U_i\)	\(Y_i\) (thousands of dollars)
1	0	0	1	21
2	1	1	-1	24
3	2	0	0	24
4	1	1	2	27