Mediation using Threshold models

[sedki zn]

Dear all,

i hope you are doing good in this period of Covid

in my work i have studied the following relationships:

1. impact of financial constraints on CEO stock option using a threshold model
2. impact of CEO stock option on Risk taking using quantile regression
3. impact of financial constraints on risk taking using threshold model

Now i would study the mediating role of CEO stock option in the relationship between financial constraints and risk taking using threshold model following The Baron and Kenny (1986) method.

Step 1: The independent variable (financial constraints) is shown to significantly influence the dependent variable (R&D expenditure) as a first regression equation. This step is already verified in the third chapter.

Step 2: Independent variable (financial constraints) is shown to significantly influence the mediator (CEO stock option) in the second regression equation. This step as well has been verified in the first chapter.

Step 3: The Mediator must significantly influence the dependent variable in third equation. Here, the independent variable and the mediator are entered as predictors. This step is the subject of the following regression:

Is this technically doable ?

kind regards

[sedki zn]

Dear Clyde Schechter

i would appreciate your help

kind regards

[Clyde Schechter]

It looks technically doable to me; it's just another fixed effects model, with firm and time fixed effects. Actually, perhaps I'm missing something--why do you think it might not be?

If you have a specific question about how to implement it, set out your question, and, of course, post example data using -dataex-.

[sedki zn]

It looks technically doable to me; it's just another fixed effects model, with firm and time fixed effects. Actually, perhaps I'm missing something--why do you think it might not be?

If you have a specific question about how to implement it, set out your question, and, of course, post example data using -dataex-.

Thank you Dear Professor for your answer,

Concerning your question, i was just wondering if th The Baron and Kenny (1986) ( applied with linear models) could be applied with nonlinear models. so that's why i needed to get sure of it.

kind regards

[Clyde Schechter]

The entire area of mediation is riddled with controversy. I had originally interpreted your question as being about whether or not it is possible to fit the model you described in Stata. I understand now that you are questioning the legitimacy of interpreting it as an assessment of mediation. Well, some say yes, and some say no. The same is true about every approach to mediation that is in use. It is perhaps more concerning in non-linear models than in linear models, but even in linear models there is controversy about how, and even whether, to model and analyze mediation. I do not have a deep understanding of the theoretical and conceptual underpinnings of these issues, so I do not take sides. I can only tell you that when you propose to discuss mediation, pretty much no matter how you choose to do it, you will find supporters and detractors, both with strong opinions.

That said, it isn't apparent to me why you think the model in #1 is non-linear. Unless you have left out something, it looks like a linear model to me. The fact that the variable FC enters the model as a piecewise-linear function does not make the model non-linear. Remember that the term "linear" in linear model refers to the relationship between the outcome variable and the coefficients. It is not uncommon for regression models to contain terms that are non-linear transformations of the variables (quadratics, logs, exponentials, etc.)--but that doesn't make them non-linear models.

y = b0 + b1*X + b2*X2 is a linear model, despite the quadratic X term.

y = b0 + b1*X + log(b1)*Z is a non-linear model, even though the variables X and Z are entered linearly.

[sedki zn]

The entire area of mediation is riddled with controversy. I had originally interpreted your question as being about whether or not it is possible to fit the model you described in Stata. I understand now that you are questioning the legitimacy of interpreting it as an assessment of mediation. Well, some say yes, and some say no. The same is true about every approach to mediation that is in use. It is perhaps more concerning in non-linear models than in linear models, but even in linear models there is controversy about how, and even whether, to model and analyze mediation. I do not have a deep understanding of the theoretical and conceptual underpinnings of these issues, so I do not take sides. I can only tell you that when you propose to discuss mediation, pretty much no matter how you choose to do it, you will find supporters and detractors, both with strong opinions.

That said, it isn't apparent to me why you think the model in #1 is non-linear. Unless you have left out something, it looks like a linear model to me. The fact that the variable FC enters the model as a piecewise-linear function does not make the model non-linear. Remember that the term "linear" in linear model refers to the relationship between the outcome variable and the coefficients. It is not uncommon for regression models to contain terms that are non-linear transformations of the variables (quadratics, logs, exponentials, etc.)--but that doesn't make them non-linear models.

y = b0 + b1*X + b2*X2 is a linear model, despite the quadratic X term.

y = b0 + b1*X + log(b1)*Z is a non-linear model, even though the variables X and Z are entered linearly.

Dear Professor,

The threshold model is linear in the slope coefficients (psi, beta_1, beta_2, phi_1', phi_2') but is nonlinear in gamma. Hence the threshold model is a nonlinear dynamic panel data method. It is discontinuous at q_it = gamma.

so that's why the present model is supposed to be nonlinear

Kind regards
Sedki

[Clyde Schechter]

I'm confused. In the model shown in #1, I do not see any q_it, nor any gamma. For that matter, I don't see any psi or phi_* coefficients either.

[sedki zn]

I'm confused. In the model shown in #1, I do not see any q_it, nor any gamma. For that matter, I don't see any psi or phi_* coefficients either.

Dear Professor,

sorry for my late reply

Actually this is the general model as presented following :

where:
pi_it is the variable that depends on the threshold (regime dependent
variable)

q_it is the threshold variable

gamma is the threshold parameter to be estimated



Kind regards
SEDKI

[Clyde Schechter]

Oh! If the threshold parameter is to be estimated, then, yes, you have a non-linear model here. I had assumed that the thresholds were pre-specified constant cutoffs.

I am not aware of any Stata command that would be able to fit a model with this kind of complexity on top of estimating a cut-off for an explanatory variable. Perhaps that says more about the limits of my knowledge than the limits of Stata commands, but I don't know of any. I could imagine somebody writing a maximum-likelihood program to do that, but that's a pretty advanced task and not one that I would be comfortable undertaking.

I also don't know what to say about whether such a model, assuming you could get it fit, is suitable for mediation analysis. As I indicated earlier, the entire construct of mediation is controversial, and not one I have studied in enough depth to understand the implications of this kind of model for it. In particular, I find it difficult to think about whether the usual approaches to interpreting median hold up in the presence of an estimated explanatory variable threshold, the estimate of which might differ in the different regressions used to assess mediation.

Sorry, but I think your question takes us beyond the limits of my knowledge here. Perhaps somebody with a deeper understanding of the issues will pick up the thread.