I am a beginner in learning Statistics with some knowledge of machine learning, some concepts really confused me recently.

Starting from the linear regression and OLS, I think the former is a model with parameters/coefficient and the latter is the method I used to estimate the parameters for the linear regression model. So the relationship between them is the model and estimating method.

However, I read papers using the Gravity Model to describe some phenomena or test hypotheses, some of them estimate the parameters by OLS, but the others use "models" like Poisson, negative binomial to "estimate" the parameters.

I am confused, how can I use one "model" to estimate the parameters of another?

It must be something wrong with my concepts system, and I really need someone to correct me.

In addition, I want to use econometrics methods/models to perform my research (not in the field of statistics or econometrics), is there any books with practical examples and code I can read? Since textbooks I read pay a lot of attention to prove something and lack a macro view of the whole system, I am still not able to perform actual experiments even I obtain that knowledge. Maybe books with some end-to-end examples can help.

  • $\begingroup$ Hi: A regression model is used when the model is linear and the error term is normally distributed. For other assumed distributions ( poisson, binomial whatever ), there are generalized linear models which can be viewed as generalizations of regression models. I would look at a book or notes on glms for a detailed explanation of them. John Fox has a textbook that does a pretty good job with glm's. There are also specific texts just focus on glm's but, I think, for someone starting out, John's book would be a good starting point. Also, I'm sure there should be notes on the internet. $\endgroup$ – mlofton Mar 17 at 5:21
  • $\begingroup$ @mlofton Thanks! But I still don't know which category should Gravity Model, OLS, Poisson, and binomial fall in. According to your comment, Poisson and binomial are GLMs, what about OLS? How can I use GLMs to estimate the parameter of another model (like Gravity Model)? Could you elaborate? Thanks again! $\endgroup$ – Tom Leung Mar 17 at 5:40
  • $\begingroup$ @mlofton Instead of the regression model, what are the others and where they locate in the system of econometrics? I can't find books on this topic. $\endgroup$ – Tom Leung Mar 17 at 5:46
  • $\begingroup$ You may find Gravity Model of International Trade: A User Guide by Ben Shepherd illuminating. See the section on Alternative Estimators in particular. There are Stata and R versions of the code as well. $\endgroup$ – Dimitriy V. Masterov Mar 18 at 6:16

Hi: It is too broad of a topic to explain here. But, the first thing you should do, in order to keep things clear, is not use the term "normal linear regression model.". Call it a linear model with a normally distributed error term. In this way, every single model ( poisson, binomial, linear model with normally distributed error term, probit ) can be viewed as a generalized linear model. I don't know what a gravity model is but any decent econometrics book will have a chapter on a few generalized linear models. The problem is that the econometrics text will not call them generalized linear models. Still, if you want to look in econometrics texts, look at william greene's, judge and hill or madalla. I suspect that each of those will have a chapter on what they call limited dependent variables which are a smaller specific set of glms and therefore make things more confusing. ( econometrics only considers a few glms ). That's why I said to look at John Fox's text. He will actually call them generalized linear models and he'll cover most if not all of them. The drawback with using his text is that he won't come at them from a econometric standpoint. Still, I think it's better to look at generalized linear models first and then, once you have those straight, then look at limited dependent variable models for specific "econometric" type glm's. This is not an answer but I decided to write it here since there was more space. I hope it helps a little.

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  • $\begingroup$ Your answer really helps a lot! I am currently reading Gelman‘s Data Analysis Using Regression and Multilevel/Hierachical Models. $\endgroup$ – Tom Leung Mar 18 at 2:20
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    $\begingroup$ @Tom Leung: I'm glad it helped. I know that's a famous, good book but it tends to come at things from a bayesian perspective so keep that in mind. and the hierarchical models complicate things even further. When I have time, I'll look around for some good notes on the internet for just plain old GLMs. McCullough is the bible but it's pretty theoretical. If anyone can recommend any applied, clear GLM books, feel free to chime in. I used them a long time ago so I'm not up to speed on what's out there. Fox is decent even though it's general and GLMs are just a chapter or two. $\endgroup$ – mlofton Mar 19 at 2:01
  • $\begingroup$ After I read the chapter on GLMs, I think I understand my original question a little bit more. The gravity model was transformed into a Poisson model by a link function and the authors used an estimate method called Poisson Pseudo-Maximum Likelihood Estimator to get the coefficients. Do I get it right? $\endgroup$ – Tom Leung Mar 19 at 4:58
  • $\begingroup$ That sounds okay but I'm not sure about pseudo part. In GLM's, you are essentially doing MLE ( in an indirect way ) so maybe they are using a slight variant of the standard Poisson GLM for the gravity model ? I'm not familiar with that model. Maybe someone can comment on the term "pseudo" in the context of GLMs. This link looks like like a decent set of notes for GLMs. I would try to understand them in their generality first ( i.e: why can a linear model with normal errors be viewed as a GLM ) and then take a look at the Poisson chapter. data.princeton.edu/wws509/notes/# $\endgroup$ – mlofton Mar 19 at 18:13

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