Linear or Nonlinear LS estimation?

[Anton Ivanov]

Hello everybody

Please help me to figure out the following issue.

One of the main assumptions in OLS is linearity. Specifically, the linearity of the parameters, and not necessarily linearity of predictors (Williams et al. 2013).

So I estimates the multiple regression model and ran the testparm command. Null (that parameters equal to ) was rejected. Thus, I assume that the assumption mentioned above does not hold.

Are there any ways to fix this?

OR

Does it mean I have to use nonlinear LS regression then?

Thank you in advance.

[Carlo Lazzaro]

Kiton (real surname and full reference are much appreciated on the list):
- to increase the chance to get helpful replies, please show us exactly what you typed and the results Stata gave you back.

Kind regards,
Carlo

[Anton Ivanov]

Carlo, thank you for response. My name is Anton. The full reference is: Williams, M. N., Grajales, C. A. G., & Kurkiewicz, D. (2013). Assumptions of multiple regression: correcting two misconceptions. Pract. Assess. Res. Eval, 18. Specifically, page 8 for parameter assumptions.

1. reg y x1 x2 x3 x4 x5 x5 (assumptions are met, except (A)heteroskedastisity, working on it, robust SE is one way I believe; and (B) Linearity as far as I can understand).
2. testparm x1 x2 x3 x4 x5 x5 (F(6,518)=78.23, p>F)

[Anton Ivanov]

STATA outputs

[Maarten Buis]

testparm did not test the linearity of the effects of the different variables, but whether the effects of all these variables are simultaneously equal to 0. So that command will not help.

The standard "test" for linearity is the "interocular trauma test", in other words just plot various types of residuals and stare at the plots till you know the answer.

Anton: the use of full names (first and last) has a long tradition on this list. We believe that this has helped maintain a friendly and professional atmosphere on this list. This is the reason that the FAQ asks everybody to sign on using their full name. You can ask to change your loginname using the "contact us" button at the bottom right.

[Anton Ivanov]

testparm did not test the linearity of the effects of the different variables, but whether the effects of all these variables are simultaneously equal to 0. So that command will not help.

The standard "test" for linearity is the "interocular trauma test", in other words just plot various types of residuals and stare at the plots till you know the answer.

Anton: the use of full names (first and last) has a long tradition on this list. We believe that this has helped maintain a friendly and professional atmosphere on this list. This is the reason that the FAQ asks everybody to sign on using their full name. You can ask to change your loginname using the "contact us" button at the bottom right.

Thank you very much for reply. As for the name, I apologize for that and will fix it I was simply not aware of this issue.

[Carlo Lazzaro]

Anton wrote:

A)heteroskedastisity, working on it, robust SE is one way...

SE Robust is actually one solution. I would also give a try with a log-linear model (in which only the dependent variable is logged) with default SE and see whether heteroskedasticity is still an issue.
Set aside heteroskedasticity, if not done yet, I would also perform -estat ovtest- to look for omitted variable bias (a more serious concern than heteroskedasticity); this test can give you some clues about how good the linear specification of your model actually is.

Kind regards,
Carlo

[Anton Ivanov]

Anton wrote:


SE Robust is actually one solution. I would also give a try with a log-linear model (in which only the dependent variable is logged) with default SE and see whether heteroskedasticity is still an issue.
Set aside heteroskedasticity, if not done yet, I would also perform -estat ovtest- to look for omitted variable bias (a more serious concern than heteroskedasticity); this test can give you some clues about how good the linear specification of your model actually is.

Kind regards,
Carlo

Carlos, thank you for the suggestions
.
1. I did fit a model with ln_DV, estat hettest - null rejected, so the issue is still there.
2. I also performed estat ovtest, as you suggested. The result is F(3,515)=7.25, p=.001.Null (there are no omitted variables, I believe) is rejected.

So, how shall I address this problem further? Please suggest.

Anton

[Carlo Lazzaro]

Anton:
-heteroskedasticity might be a clue for heavily influential observations (take a look at cooksd option in -predict-). If these were not the culprits, use robust SE.
- omitted variable bias (a more serious concern than heteroskedasticity). There's piece of evidence of this problem with your regression model. Hence, either you have inadvertently omitted some relevant variables or your regression model is linear in coefficients but not in variables. in other words, you may need to square one or more independent variables to get your model better specified. You may also want to take a look at what others have already done in your research field (in general, this is a good habit, especially if you are intended to submit a paper to disseminate the methods and the results of your research).

I would also consider to seek for the help of a statistical experienced colleague, if feasible, and, even better, invest some time in getting familiar with linear regression topics (which include theory, graphical methods for detecting problems like yours and how to become applied with this tools) and how to deal with them in Stata (we all paid this price at some point in time, but the pay-off is, in my opinion, valuable).

I do hope that this helps.

Kind regards,
Carlo

[Anton Ivanov]

Carlo,
I have been working on this research for quite a long time. The study deals with reputation (multi-deimensional approach) and it is common for researchers in this area to face model fit problems. So, I do have a habit to do thorough literature review Importantly, I was surprised to see that very few comprehensively address the issues of whether the classical assumptions (e.g. OLS in this case) have been met or not. Thus, I decided to highlight these in my methodology section. I have taken many stats, econ, and methodology classes, but some issues are new to me, because I have not come across them in my practice before. That is why I address them on this forum and keep on learning.
Moreover, my data has been collected from 11 different sources (secondary data) and encompasses significant outliers (I am aware of them and the ways to analyze them). That is why I am not surprised to face all this issues and try to address them properly with the help of knowledgeable forum members, like yourself.

Thank you very much for the invaluable comments and advise on including squared IVs. I will explore this topic in more detail and consider in my analysis.

Anton

P.S. BTW you pointed out that my model could be "linear in coefficients, but not in variables". That is also something I tried to address. Specifically, in Williams et al. (2013) (Williams, M. N., Grajales, C. A. G., & Kurkiewicz, D. (2013). Assumptions of multiple regression: correcting two misconceptions. Pract. Assess. Res. Eval, 18.) they highlight:
"response variable is assumed to be a linear function of the parameters (b1, b2, b3... bp), but not necessarily a linear function of the predictor variables X1, X2, X3... Xp.
To clarify this I decided to post my question here, since recently I have also started learning STATA.

[Carlo Lazzaro]

Anton:
no doubt that you're skilled in what you do. My recommendations, as usual, was based on the handful of details I read in your posts.
I agree that there should be more attention in addressing whether OLS (or other statistical procedures) requirements are met or not.
About continuos learning (or "leaving one's comfort zone"), I remember one of Nick Cox's reply of some years ago that ended up like this: "We are all beginners; some of us are simply more experienced".

All the best for your research project.

Kind regards,
Carlo