In the book of Sanford Weisberg "Applied linear regression" there is an introduction of the concept of "(true) mean function", as well as the "variance function". They are only briefly presented. In figure 2.2. a true mean function is shown with beta0=0.7, and beta1=0.8. However, the author does not explain how it was computed. He clearly states what we all know, that such a function is inferior to the OLS function, but I would still like to understand how such a mean function is computed (for the sake of understanding). Could somebody help me?
Dear Robert Ancuceanu
I expecte that you mean expected value of (y/x) which is define as regression equation so that is mean the the expected mean value of Y given that X =certain value x) so that x (small letter ) in statistics mean that (certain value of the variable X), where the variables in statistics allwayes is defined by capital letter such(Y, X).
And the mean equation is estimated by OLS method by which we get the estimated value of B^0 and B^1(Beta not hat and Beta 1 hat).
The estimated equation is
Y^=B^0+B^x wher x here is small letter that is mean certain value (X=x) for example (X=110).
With best regards
The question is how the computation is done and how are the coefficients computed (differently from the ordinary least squares method). In every statistics handbook I’ve read up to date only the OLS method was described. S. Weisberg describes it shortly and shows its graph. But I would like to see the formulas for estimating beta zero and beta one.
I expecte you need the dravative of ols and the formla of betas.
Also you can see
Ben Vogelavang ,2005"Econometrics Theory and Application with EVIEWS" Printce Hall, pp55-61.
Or by the following link
Www.pearsoned.co.uk/vogelvang
Dear Abbas Lafta Kneehr, thank you for your attempt to be helpful. As I stated explicitly when formulating the question, I am not interested in OLS, which I've been familiar with for almost three decades. It would help if you could look at Fig. 2.2 of the S. Weisberg book.
Sorry I don have S. Weisberg, if you can sent to me the picture of figur 2.2 and picture of text that expliend this figur
Robert -
I have that book, but I don't see what you describe. Mine is the 1980 edition. I do not think there is another edition. Is there? I checked Figures 2.2, and also 1.2 and 3.2 to be sure. I looked for "mean function," "true mean function," and "variance function" in the Index, and found nothing. Mean function sounds like you just have an intercept term and an estimated residual, but that is not what you describe.
Figure 2.2, page 37 in my copy of that book is a scatterplot showing how one of the regressors/predictors/independent variables, the "proportion of population with driver's licenses," from among several, would do on its own, with an intercept term, in predicting for "motor fuel consumption [in] gallons per person." This is OLS. I used a couple of the multiple regression versions there in a paper showing what to do to go to WLS (weighted least squares) regression.
For Figure 2.2, beta0 = 22.73, and beta1 = 1409.8.
Sorry. I didn't see what you asked about.
Cheers - Jim
Robert -
The figure you sent me was not Figure 2.2 in my copy of that book, and on page 25, instead of page 37. Don't know how that happened, except I have heard of a book being published in Europe and in the US, and they were different versions. But this seems very different, and with the page numbers showing chapter lengths off that much seems odd.
Anybody else looking for that book may see what I described before, rather than what I'll comment on now.
I think the point made in Figure 2.2 in your copy, not mine, is that if you had populations where the data for y and for x had a relationship such that b1=0.7 and b2=0.8, given by the dashed line, then that would be the "true" relationship, but the solid line would be the OLS regression, which is an estimate. There seems to be an error on that page, as traditionally we would use epsilon, not e, in the "true," but unknown model.
This highlights the fact that people routinely talk about "residuals," when they really mean "estimated residuals," a point Ken Brewer used to emphasize to me. In OLS, or generally better, WLS of which OLS is a special case, the estimated residuals are designated with an "e."
Glancing over that page, I caught an interesting comment that indicated the estimated model overfit in a restricted range. I think that is often true.
Unfortunate that the point of Figure 2.2 in your copy seems to be to point out the difference between e and epsilon, and then he uses e in the wrong place.
Cheers - Jim
PS - Oh. I see from your note above that there have been three more editions. Hadn't heard of that before. Well, that explains the page numbering and figure differences.
Dear James,
thank you very much, indeed, what you say here make good sense, and this is probably how one should interpret it. Yes, you are right that residuals are always estimated, with shorten it usually just because we (people) are lazy. Many thanks again, Robert.
Ah! Now I think I see why you asked about "computing" the true, unknown model. The book (4th ed) notes "computing" and then shows the regression equation he gives with the e in it, but the "true" regression coefficients. He is talking about a simulation, I think, where the true relationship is as stated. You need to run a simulation with those populations as defined to make that relationship true. The sample used for estimating the model gives you the solid line, not the dashed one.
I think the simulation only simulates errors, always assuming the given model, though it seems like sample size matters there also. Well, somewhat, because as long as sum of epsilon comes out zero, you are OK. If he knows what "true" populations he is generating, the difference is starting from a known relationship, rather than taking a sample and trying to estimate an unknown relationship. If he knows the relationship, "computing" seems a little out of place. That is confusing to me. I often get confused on details anyway.
:-)
But I think the point is just that what we often call "residuals," are really "estimated residuals" for an estimated, not "true" model.
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