That is, we don’t need to worry or wonder about which assumptions apply in our case. Why do we do this? By using a random process that we have control over, we can choose the truth. Then, we look at the results over all of our repetitions. Then, we do that over and over and over again. In the context of this chapter, simulation refers to the process of using a random process that we have control over (usually, a data generating process) to produce data that we can evaluate with a given method. But is it necessary to know the proof behind a method in order to use that method well as a tool? I’m skeptical.īut we still must know how our methods work! Enter simulation. I think doing proofs is highly illuminating, and getting good at doing econometric proofs has made me understand how data works much better. 385 385 Whether this is a problem is a matter of opinion. Many (most?) of the people doing active statistical research have forgotten all but the most basic of statistical proofs, if they ever knew them in the first place. Proof-writing is really a task for the people developing methods, not the consumers of those methods. There’s a good chance that you’re not going to go out and learn them, or at least you’re unlikely to go get good at them, even if you do econometric research.
I haven’t taught you how to do econometric proofs. The proofs underlying ordinary least squares tell us how its sampling variation works, which assumptions must be true for our estimate to identify the causal effect, what will happen if those assumptions aren’t correct, and so on. Why go through the trouble of doing mathematical proofs about our statistical methods? Because they let us know how our estimates work. but they’re lurking between the lines, and in the citations I can only imagine you’re breezing past. Declining marginal value, comparative advantage, and all that. Those proof-based books are valuable, but I’m not sure how badly we need another one. The most famous of which at the undergraduate level is by Wooldridge ( 2016), which is definitely worth a look. In my defense, there are already plenty of proof-based econometrics books out there for you to peruse. I have chosen to omit the actual proofs themselves, 384 384 Whether this is a blessing or a deprivation to the reader is up to you (or perhaps your professor). Pretty much all of the methods and conclusions I discuss in this book are supported by mathematical proofs like this. then we can prove mathematically that the ordinary least squares estimate of \(\hat_1\) will be the true \(\beta_1\) on average. If we assume A, B, and C, then based on the laws of statistics we can prove that D is true.įor example, if we assume that the true model is \(Y = \beta_0 + \beta_1X + \varepsilon\), and that \(X\) is unrelated to \(\varepsilon\), 383 383 And some other less well-discussed assumptions. Statistical methods are generally developed by mathematical proof. Unlike statistical research, which is completely made up of things that are at least slightly false, statistics itself is almost entirely true.