Question

What is the difference between the population and sample regression functions? Is this a distinction without difference?

Answer 1

Answer:

See explanation below.

Step-by-step explanation:

When we want to fit a linear model given by:

$y = \beta_0 + \beta_1 x$

Where y is a vector with the observations of the dependent variable, $\beta_0 , \beta_1$ the parameters of the model and x the vector with the observations of the independent variable.

For this case this population regression function represent the conditional mean of the variable Y with values of X constant. And since is a population regression the parameters are not known, for this reason we use the sample data to obtain the sample regression in order to estimate the parameters of interest $\beta_0, \beta_1$

We can use any method in order to estimate the parameters for example least squares minimizing the difference between the fitted and the real observations for the dependenet variable.  After we find the estimators for the regression model then we have a model with the estimated parameters like this one:

$\hat y = \hat b_0 +\hat b_1 x$

With $\hat \beta_0 = b_o , \hat \beta_1 = b_1$

And this model represent the sample regression function, and this equation shows to use the estimated relation between the dependent and the independent variable.