What is the conditional expectation function or the population regression function? 2.2. What is the difference between the population and sample regression functions? Is this a distinction without difference?
The Correct Answer and Explanation is:
Correct Answer:
The Conditional Expectation Function (CEF) or Population Regression Function (PRF) is defined as:
E[Y | X] — the expected (mean) value of the dependent variable Y, given the value(s) of the independent variable(s) X.
2.2 Explanation
The Conditional Expectation Function (CEF) or Population Regression Function (PRF) represents the true, underlying relationship between the dependent variable (Y) and the independent variable(s) (X) in the population. Mathematically, it is expressed as:
E[Y | X] = f(X)
This function is theoretical—it captures the average value of Y for each possible value of X across the entire population. It is not directly observable in practice because we rarely have data on the entire population. Instead, we estimate this function using sample data.
The Sample Regression Function (SRF) is the estimated version of the PRF, obtained from a finite sample of data using statistical methods such as Ordinary Least Squares (OLS). While the PRF describes a real but unknown function, the SRF is a concrete, data-driven approximation:
Ŷ = a + bX (in simple linear regression)
Here, a and b are sample estimates of the true parameters (often denoted α and β in the PRF).
The distinction between the PRF and SRF is crucial. The PRF is the true model, while the SRF is an estimate based on limited information. The difference matters because:
- The SRF is subject to sampling variability—different samples will yield different regression estimates.
- Understanding the PRF helps define the assumptions necessary for estimation and inference (e.g., linearity, independence, homoskedasticity).
Calling this distinction “without difference” would be incorrect. While they look similar in form, they differ in scope: one is theoretical, the other empirical. Confusing them can lead to errors in interpretation, such as assuming the SRF perfectly reflects the true population relationship, which is rarely the case.
