The next step after a simple linear regression is multiple linear regression. Now, what does that mean? That means that originally when we were dealing with simple linear regression, all we had was Y equals some foundational coefficient Beta 0, which we spoke about what that really meant intuitively, at least, plus Beta1 times 1X. When we spoke about it, it was how does, for example, how does TV budget affect sales? One variable that's affecting one variable. For multiple linear regression, we have p unique variables. We have p unique variables. This was simple linear regression, now this whole thing, well technically the error term is also in there for simple linear regression. But either way, we change it from one X-variable to p X-variables. Bi will still do the same thing it did before. Before B1 quantified how X1 and Y dealt with each other. Now for a simple linear regression, that was the whole model. That's all we had to worry about. That was our entire goal. In this case, we have p different Betas. Actually p plus one if you include this one, but p different Betas that attach to an X-variable. This quantifies each one of these for all p of them. Beta3, Beta4, maybe p is seven so all the way up to Beta7. This all deal with their respective effect of that X that's attached to it and that X's relationship with Y. For instance, let's maybe downgrade this hypothetical talk into something tangible. We've been dealing with this pretty much forever now, which is how do sales, how does that deal with our three advertising budgets? In simple linear regression, we dealt only with TV, in this case, maybe we want to know, well, how do all three of these budgets deal with sales? We might have something like sales equals some constant. That is telling us, well if all our budgets are nothing, we don't put anything into any budget. This is where we're at, and that's the same with linear regression. This is the same error term that is always going to be stuck on there. The difference between real life and estimates. They're never going to be equal they are always going to have some error and so we account for that in our formula. This part deals with instead of one variable like TV, we now have three, TV, radio, and newspaper budgets. All three of these, Beta1 specifically, will quantify how does TV and sales, how do they relate to each other? Beta2 will specifically quantify how does my radio budget and sales, how do those interact with each other? What's the relationship there? Beta3, again, was the relationship specifically between newspaper budget and sales. Then if I know all of them, I can make decisions. Now, Beta2 might be negative. It might be saying the more money I put into radio, I'm actually losing sales. I'm not saying that's a realistic finding, I'm just saying it could happen. If it is negative and you might tell your client who was wondering how much budget is put into use, say, I have a pretty good multiple linear regression model and my coefficient for radio is negative or maybe it's in terms of profits, so it's negative. It's saying, radio may increase sales but not more than it cost us to do the radio head and so our coefficient is actually negative for dealing with profits and so radio advertising isn't worth it. None of this is real, this is just a made-up situation, a hypothetical situation. But if, when these Beta terms that we deal with, they give us a lot of information about each one of these X-variables and their interaction with our target variable.
