[Bruce Weaver]

This thread only caught my eye today, and I confess that I have not read all of the posts yet. But reading #1 made me wonder if piecewise regression would address the question of interest.

• https://stats.oarc.ucla.edu/stata/fa...sion-in-stata/

Apologies if this has already been suggested, or if something posted above rules it out.

[George Ford]

I think PW regression was the plan, but the proposal is to mark the kink on Y values and not X values.

[Bruce Weaver]

I think PW regression was the plan, but the proposal is to mark the kink on Y values and not X values.

Ah, right. My mistake. Thanks George Ford.

[Jia Gao]

As best I can tesll, Wooldridge said "no".

I think you need to stick with a threshold for X. If Y = bX + e and at some point X gets big enough to turn Y positive and then has no effect, then you can condition on X.

A spline function on X, cycling through cutoffs? Semipar might give you some insights.

Hi Mr.Ford, thanks for your insights. As my reply to wooldridge, in the case of Y = bX + e, there is no theoretical argument about why X should predict the direction of Y and my empirical results show X indeed does not predict the direction of Y. I think Dr. Wooldridge's concern is that it is meaningless to condition the relatinoship between X and Y on the positive/negative of Y if X itself could predict the direction of Y.
So the question is: is it econometrically sound to estimate Y = b1X + b2Z(Y>0) + b3X*Z(Y>0) + u if X has no statistically significant predictive power for Z?

Per semipar, I'm not familiar with it and still learning, But we did include descriptive evidence (i.e., check the correlation between X and Y conditional on the diretion of Y). Is this similar to semipar?

[Jia Gao]

What you're doing is the same as estimating two different linear models -- assuming you're including an overall intercept. One model is for Y > 0 and the other for Y < 0. It's unusual to do that because what sense does it make to hold Z fixed and then change X when changing X might make Y go from negative to positive? So I think you need to rethink what you want to learn.

Although it is theoretically clear that X should not predict the direction of Y and the empirical evidence support this claim, we removed observations with Y close to 0 to further alleviate such concern you mentioned. So far, we have not had a satisfying model to investigate the research question. So any suggestions are welcome.

[Jia Gao]

I think this problem generalize to any model specification in the form

Y = b1X + b2Z + b3X*Z

in which

Z is an indicator variable takes the value of 0 and 1

. Regardless of the relationship between Z and Y, as long as X could predict Z in a statistically significant sense, that is problematic, per our discussion above. So two questions arise:
1, what to do if X could indeed predict Z?
2, Is this sepcification econometrically sound if X could not predict Z?

[George Ford]

I recommend you construct a simple Monte Carlo of what you are thinking about. You'll figure out a lot when you do. The trick is that in the MC you create X's and then the Y. Thus, you can't specify the relationship between Y and X by using Y, so then you'll turn to the residual, which introduces bias. You've got to make it happen with the Xs.

[Jia Gao]

I recommend you construct a simple Monte Carlo of what you are thinking about. You'll figure out a lot when you do. The trick is that in the MC you create X's and then the Y. Thus, you can't specify the relationship between Y and X by using Y, so then you'll turn to the residual, which introduces bias. You've got to make it happen with the Xs.

Thanks. You are saying that even though X do not predict the direction of Y, investigate the relationship between Y and X conditions on the direction of Y using Y = X + Z (Y>0)+ Z*X is still problematic?

I have not done semipar or MC before, I'll look for some help and may take some time to get back to you on them.

[George Ford]

Here's a simple MC.

clear

set obs 1000

g x1 = rgamma(5,1)

g y = -2.5 + 0.5*x1 + rnormal()

reg y x1

Now try to make x1 irrelevant for y > 0. How? That's the problem.

You probably need some type of endogenous switching model on the Xs, not the Y + e.

For semipar, as an investigative tool.

clear

set obs 1000

g x1 = rgamma(5,1)

g x2 = runiform()

gen y = cond(x1<5, -2.5 + 0.5*x1 - 2*x2, -2.5 - 2*x2) + rnormal(0,0.5)

reg y x1 x2

g ydum = y>0

reg y x1 c.ydum#c.x1 x2
** biased

g xdum = x1>5

reg y x1 c.xdum#c.x1 x2
## not biased
lincom x1 + c.xdum#c.x1

semipar y x2 , nonpar(x1)

[George Ford]

It might be helpful if you provided more details about the data. What is Y? What is X? Why would X not affect Y if Y > 0?

I'm having trouble coming up with an example. It's easy to think of cases where X stops at some threshold Y, but simply where dY/dX stops at some threshold Y.

If you have time series data, you might look at the SETAR (Self-Exciting Threshold AutoRegressive) model. There, l.Y is used.

[Jia Gao]

It might be helpful if you provided more details about the data. What is Y? What is X? Why would X not affect Y if Y > 0?

I'm having trouble coming up with an example. It's easy to think of cases where X stops at some threshold Y, but simply where dY/dX stops at some threshold Y.

If you have time series data, you might look at the SETAR (Self-Exciting Threshold AutoRegressive) model. There, l.Y is used.


[ATTACH=CONFIG]n1782760[/ATTACH]

I know this is late as I have been occupied by other papers. The detailed context of this problem is the following:
We are interested in how firm a treat its employee historically influence market reaction to investor's reaction to the firm's layoff announcement. They are theories in the related filed (strategic management, accounting) that predict null relationship of these two but positive (negative) relationship when stakeholders perceive the layoff is for survival (profit maximization).

The literature offered suggestions of factors that predict the direction of market reaction to layoff announcements, but we can not replicate them. For instance, earlier study using around 200 layoff announcement show that proactive or reactive layoff can predict the direction of market reaction to layoff announcement. But in our sample, we have nearly 4,000 layoff annnouncements over 20 years. The preliminary analysis show that proactive or reactive layoff announcement, labelled by LLM models and our research assistant, can not predict the direction of market reaction to layoff announcement.

The attached figure is a visulization of the problem I described. The blue line represent the overall relationship between historical employee treatment and market reaction to layoff announcement. The red line is also this relationship conditional on the market reaction is positive while the green line is the relationship conditional on the market reaction is negative.

We have strong theoretical case in our field to argue for the results, but the econometrics issue is really holding us back.

[Jia Gao]

I recommend you construct a simple Monte Carlo of what you are thinking about. You'll figure out a lot when you do. The trick is that in the MC you create X's and then the Y. Thus, you can't specify the relationship between Y and X by using Y, so then you'll turn to the residual, which introduces bias. You've got to make it happen with the Xs.

I'm not familiar with Monte Carlo. But I'll try to seek help from others