Non-linear relationship

[Dominik Miksch]

Hello,

I have a quick question: In order to justify propensity score matching, I need to assume a non linear relationship between outcome and covariates.
Would you see this as a non linear relationship?

Kind regards
Dominik

[Phil Bromiley]

You'll increase your chances of a useful answer by following the FAQ for asking questions - provide Stata code in code delimiters, readable Stata output, and sample data using dataex.

This sure looks linear to me.

[Bruce Weaver]

Hello Dominik. Your plot looks linear to me too.

Why do you say you need to assume a nonlinear relationship between outcome and covariates to justify propensity score matching (PSM)? Can you provide references?

[Carlo Lazzaro]

Dominic:
I share previous comments: your scatter plot looks linear, perhaps with a bit of heteroskeadsticity.
As Bruce highlighted, I do not remember (but that may well be my fault) any recommendation about non-linearity when it comes to apply propensity score matching.

[Dominik Miksch]

Thank you very much for your answers.
Carlo Lazzaro Bruce Weaver That was my interpretation because of

"There are several technical advantages of propensity score analysis over multiple linear regression. In particular, when covariate balance is achieved and no further regression adjustment is necessary, propensity score analysis does not rely on the correct specification of the functional form of the relationship (e.g., linearity or log linearity) between the outcome and the covariates. " from: http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf

Ist this incorrect? If yes, the only motivation of propensity score matching over regression is the selection effect/ selection bias since OLS does not account for endogeneity?
I also argue that in case of a very close match of observables, the set of control variables is better shaped compared to an OLS regression where we look at averages and given the large number of potential predictors, we would suffer from the so called curse of dimensionality.

Or is there any better justification why PSM is 'better' than OLS, or under which circumstances PSM is more appropriate?

Kind regards
Dominik

[Bruce Weaver]

Thanks for posting that, Dominik. Not requiring linearity is not the same thing as requiring non-linearity. The former does not exclude linearity, whereas the latter does. HTH.

[Dominik Miksch]

Bruce Weaver So is it correct if I say I need to assume linearity for OLS/ linear regression but as this is a problem I use PSM what does not require linearity?
Than I show that non linearity in the following way as I say ‘we compare the kernel density estimate of Log sales for ICT and non ICT firms - apart from the difference in mean, meaning the location of the distribution, there is also a clear difference in the shape of the distribution - for non-ICT firms the distribution has a sharper and more pronounced peak, meaning higher kurtosis than for ICT firms. This is seen incidentally in looking at the sample kurtosis in each case - 3.25 vs 2.63. This provides evidence for a non linear relationship between the outcome and the covariates and PSM does not rely on the functional form in the way linear regression does'. But I guess a better answer would talk about why these differences in the shape of the distribution imply a linear data generating process is unlikely which I don't find literature for/ I can't explain it in meaningful words. Can you maybe help me with that please?





Kind regards
Dominik

[Carlo Lazzaro]

Dominik:
it seems also important to stress that, in OLS, it's required that the regression function is linear in the coefficients, not the independent variables: it's perfectly legal to have both a linear and a squared terms for (say) age in an OLS equation.

[Sven-Kristjan Bormann]

As a side note: Can you cite sources or further reason why a difference in sample kurtosis implies a non-linear relationship between the outcome and the covariates? I am just curious because this connection is new for me.

[Dominik Miksch]

Carlo Lazzaro Thank you for that hint, I will indeed stress that.
Regarding the linkage between Kurtosis and non linearity, Sven-Kristjan Bormann mentioned, I read that paper: https://pdfs.semanticscholar.org/c1d...9095e63397.pdf

"An attractive approach to assessing the nonlinearity associated with a parameter of interest is that based on a comparison of the closeness or otherwiseof the Wald and the profile likelihood intervals for that parameter. This idea ispresented in the paper of Cook and Tsai (1990), following the earlier and moregeneral results of Jennings (1986) and Hodges (1987), and complements methodsbased on measures of curvature. It is interesting therefore to relate the measures of skewness and kurtosis developed in Sections 2 and 3 to the discrepancy between the Wald and the profile likelihood intervals of an individual parameterand hence, albeit tentatively, to the nonlinearity associated with that parameter."



I am not sure if this gives enough proof to assume a non linear relation between the outcome and the covariates to say a linear data generating process would not fully explain the observed data

Kind regards
Dominik

[Jeff Wooldridge]

Here are my thoughts.

1. PS Matching does not, theoretically, solve the selection problem any better than regression adjustment. They both rely on ignorability of assignment.
2. There are various forms of nonlinear regression adjustment. For example, if you use Sales, rather than LogSales, as your dependent variable, you can use an exponential mean function and Poisson quasi-MLE. This is available in Stata's teffects.
3. As others have pointed out, even linear RA can be quite flexible by putting in squares and interactions, and maybe more.
4. Sven-Kristjan raised an important point: those univariate plots of the PDFs of LogSales have nothing to do with whether E(LogSales|X) is linear or nonlinear. The paper you cited has nothing to do with the functional form of the mean function. It is about approximating the distribution of the nonlinear least squares estimator. It says nothing about the form of E(Y|X).
5. One question, and this should've come first: Did you really plot the OLS residuals against Y, and not Yhat? I'm not sure what that former tells you. You are looking for neglected nonlinearities in E(Y|X), and that's based on finding correlation between Uhat and functions of X, such as the fitted values.
6. Generally, once you have achieved balance in covariates, the particular method used is less important. You should try IPWRA, too, as this is a "doubly robust" estimator. What is your treatment variable?

JW

[Bruce Weaver]

5. One question, and this should've come first: Did you really plot the OLS residuals against Y, and not Yhat? I'm not sure what that former tells you. You are looking for neglected nonlinearities in E(Y|X), and that's based on finding correlation between Uhat and functions of X, such as the fitted values.
JW

Well spotted Jeff! Not sure how I missed that.

[Dominik Miksch]

Jeff Wooldridge Thank you very much for your answer. First, I think it is worth mention that I'm doing research on a Master's level. I was also thinking about what you mentioned at 1) 2) and 3): I might cannot claim that PSM is any better but I already used it and I should at least explain why I did what I did. But if I understand you correct, your 5) and 6) points towards this?

Yes, I did plot the OLS against Y. What comes out is:

But does it show evidence for a potentially nonlinear relationship? Because the values are not on a straight line?
I also used rvfplot right after:

My treatment variable is ICT adoption.
ICT adoption among Sub Sahara African firms.

I also appreciate your hint regarding IPWRA. I will try this as well.

Kind regards
Dominik

[Jeff Wooldridge]

Dominik: That cloud of dots in the first graph is consisent with a linear conditional mean function. You can't expect to get a straight line! That would be a perfect fit. The second plot is the one usually used as an (informal) check for nonlinearity. That cloud seems consistent with the residuals NOT being a nonlinear function of the fitted values. But the reason we have statistical tests is so we don't have to try to figure things out by eyeballing a large cloud of points.

If you're really interested in seeing whether nonlinearity is present, use a heteroskedasticity-robust RESET, or something like it. You can do it "by hand":

reg y x1 x2 ... xK
predict yhat
gen yhatsq = yhat^2
gen yhatcu = yhatsq*yhat
reg y x1 x2 ... xK yhatsq yhatcu, vce(robust)
test yhatsq yhatcu

If the (robust) F test does not reject, it's unlikely you have missed important nonlinearities. But, again, for applying PS methods this is neither here nor there. PS can be justified whether or not the mean functions are linear. But it's not necessarily "better" than regression adjustment. They both require the same ignorability assumption, so hopefully you have good control variables.

[Dominik Miksch]

Jeff Wooldridge Thank you very much again for your support.
I used the code you provided and stata gives me the following:

test yhatsq yhatcu

( 1) yhatsq = 0
( 2) yhatcu = 0

F( 2, 7619) = 5.58
Prob > F = 0.0038

.
end of do-file

Therefore I reject H0 what indicates nonlinearity is present?
Yes, I got the point that PS methods are not 'better' but if my previous sentence is correct that would help me a lot.
I also found out that I have heteroscedasticity in my model:

estat hettest

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
         Ho: Constant variance
         Variables: fitted values of LogSales

         chi2(1)      =    36.71
         Prob > chi2  =   0.0000

Thank you very much again!

Kind regards
Dominik