Say if we wanted to conduct MLE to obtain the parameter estimate for a simple linear regression model
wage = β1educ
I understand that we first find the joint probability of y taking on certain values (given certain x values).
E.g. if we had a sample of data with the following results: wage=10 and educ=2, wage=15 and educ=4, wage=20 and educ=6, etc, etc. The first step of MLE would be to compute the joint probability of all these sample values occurring.
If we have assumed that the errors are normally distributed then I have been taught that we do:
Notably, the equation inside the product operator is the formula for the PDF.
Finally, my question is: If we can't use a PDF to say that the probability of y is exactly equal to something, then why are we doing that here?
In my example to compute the MLE we would be essentially saying that the P(y=10|x=2) is equal to some exact value given by the PDF, P(y=15|x=4) is equal to some exact value given by the PDF, etc, etc... and then obviously we multiply these probabilities together to get the likelihood. So my confusion is around the fact that we are saying that an exact probability of something occurring is given by a value on a PDF when I thought that was impossible.