I do not think there is a solution within this non-linear setting without specifing a fully parametric 'first stage'. This is pretty unattractive because (as opposed to the linear model) this first stage has most likely literally to be true to avoid inconsistencies. An alternative is to treat durations as outcome variable and set-up a non-parametric LATE analysis along the lines of Imbens & Angrist. If covariates are needed, the Froelich (2007, Journal of Emetrics) paper would be the appropriate reference for this.

Try endogenous switching regression model (ESRM)

what do you mean by ESRM please

Thank you for the suggestion Benard! I did not know this model, it seems interesting... I have been reading about it, but if I understood it well, this ESRM is useful to deal with sample selection, right? It envolves the estimation of a selection equation, somehow similar to Heckman procedures... Am I right? However, this is not the problem I am dealing with, but that of endogeneous regressors - when the dependent variable in my model may be jointly determined with (at least) one of the explanatory variables. This could be more easily dealt with if my model was linear. For non-linear models, it seems to be still "under development". For duration models in particular, specially for discrete duration models, I have not found yet no recognized model to deal with endogeneous regressors and no command seems to be available for STATA...

Dear Vera,

can you please further specify to what kind of model you would like to apply an instrumental variable approach? Existing approaches depend on the setup you choose. Here some suggestions:

Abbring, J. H. and van den Berg, G. J. (2005). Social experiments and instrumental variables with duration outcomes. Working Paper.

Van den Berg, G. J., Bonev, P. and Mammen, E. (2013), Nonparametric instrumental variable methods for treatment evaluation among survivors (new version coming soon, an old version is available here: http://www.iza.org/conference_files/SPEAC2011/viewProgram?conf_id=2011)

Bijwaard, G. and Ridder, G. (2005). Correcting for selective compliance in a re-employment bonus experiment. Journal of Econometrics.

Chesher, A. (2003). Semiparametric identification in duration models. Working paper, UCL, London.

Robins, J. M. and Tsiatis, A. A. (1991). Correcting for selective compliance in a re-employment bonus experiment. Communications in Statistics - Theory and Methods.

Best wishes

Petyo

Thank you very much Petyo for the references! I am working with a piece-wise constant hazard model with discrete duration data - a complementary log-logistic model (also known as cloglog model).

Dear Rocha,

of what type is the endogeneous variable in your model? Is this a binary treatment variable? Maybe your problem would be easier to solve if you deal with the problem in a regression context (tobit).

Best wishes

Petyo

Petyo, the potentially endogeneous variable(s) are continuous, not discrete or binary. Duration models are really useful and suitable for the question I am dealing with (firm survival). I have been trying to estimate separately the 1st stage regression (for each endogeneous regressor), introducing as instruments the lagged values of the endogeneous variable - I saw similar procedures in the literature. Then, what I have been doing next, is to introduce the fitted residual of these 1st stage regressions on my structural (survival) equation, and test for the significance of these residuals, as a test for the endogeneity of the associated variables. Some studies have been adopted very similar strategies, but frequently using other types of models in the structural equation (not necessarily duration models). I would like to be safer of these procedure using discrete time hazard models...

Vera, this is an interesting procedure, as far as I remember this is something usually done in panel data models. I am not aware of a theoretical test which justifies this procedure for durations, which doesn't mean it is not correct. Sorry for not being really of help. It pushes me though to start thinking myself about an IV approach...

Vera, I have also experienced your problem and have not found a convincing solution (there is obviously no standard procedure). However, instrumenting in a highly non-linear modell is probabably heroic anyway. What about estimating a simple linear probability model where instrumenting is unproblematic?

Claus, thank you for your feedback! Yes, a simple linear probability model could be a solution, but not a desirable one, due to all the problems associated with LPM... LPM could be an option just for comparison of the results, the literature on the questions I am dealing with does not use it as a standard procedure... I will keep testing and searching and thinking about it!...

I do not think there is a solution within this non-linear setting without specifing a fully parametric 'first stage'. This is pretty unattractive because (as opposed to the linear model) this first stage has most likely literally to be true to avoid inconsistencies. An alternative is to treat durations as outcome variable and set-up a non-parametric LATE analysis along the lines of Imbens & Angrist. If covariates are needed, the Froelich (2007, Journal of Emetrics) paper would be the appropriate reference for this.

Hi every body

dear vera

would you please give me some information about simultaneous equations in panel data

The book by Baltagi has a whole chapter on simultaneous panel data models.

Dear Vera, I have worked with Panel Data Models and Hazard function. I did not have the same problem as you. The model that I was working has binary dependent variables and also a sector variables. I think I must not have endogenous variables because of this sector variables that I include in the regression. You should also take a look at Greene´s book, Baum´s book and the following book:

http://www.amazon.com/Analysis-Survival-Monographs-Statistics-Probability/dp/041224490X

This is the best book that I read about Survival Analyses.

Good Luck!

Dear Vera,

If you think you have a valid instrument, why don't you estimate the reduced form equation, that is, an equation in which you just replace the endogenous variable with the instrument? Doing it that way you should be able to identify an intention to treat (ITT) estimator, and in a second step I guess you could use the first stage regression to go from ITT to treatment effects, which should just be a matter of scaling up the ITT estimate.

Best wishes,

Ola

Heckman and Vytlacil (Econometrica 2005) discuss identification of treatment effects for endogenous binary treatment in a non-parametric framework.

Their paper focus on the Generalized Roy model.

The identification relies on a separability property of the choice variable and the instrument, but the method is non-parametric and can be applied to duration models. For papers that use duration analysis and endogenous choices, see, for example, Aakvik, Arild & Holmås, Tor Helge & Kjerstad, Egil, 2012. "Hospital Capacity, Waiting Times and Sick Leave Duration - an Empirical Analysis of a Norwegian Health Policy Reform," Working Papers in Economics 10/12, University of Bergen, Department of Economics or the thesis of Andrea Asoni (University of Chicago).

Thank you very much for all your suggestions! Tânia, in my model I also have some sector-level variables, but the endogeneity problems arise mainly among firm-level variables, so endogeneity is still there and must be corrected. I don't know if you have the same problem... Rodrigo, thank you very much for these references, I will check them. The endogeneous regressors I am dealing with are continuous (not binary like in Heckman and Vytlacil), so maybe I cannot use treatment effects for now. Even so, this literature is important for my research and it will be highly useful for sure! Thanks a lot!

Hi Vera - I'm working on a similar problem. I've come across the articles by Bijwaard 2008 and Abbring/Van den Berg 2004. AVB show that you can just plug in Kaplan-Meier probabilities in the numerator of a Wald estimator; they also show that a causal hazard ratio is identified at t=0 (but not later)... Bijwaard's method depends on perfect compliance in the control group, which I do not have. I'm still looking for elegant, easy to implement solution. Anyway, wanted to alert you to these refs. Have you made any progress?

Dear all,

just a few comments about the papers of Bijwaard 2008 and Abbring&VdB 2004.

The Bijwaard paper has a few "problems". One of them is that the model is too broad and observationally equivalent to a whole range of models. Another one is, if I properly understood it, that it misses proofs for some of the identification statements. II also had an implementation problem - I simulated the model and the estimator produced nonsense. The author promised me help, which eventually never came. The paper of AVB 2004, as Jacob very precisely pointed out, precludes the problem of dynamic selection by looking only at t=0 (at least in one of the cases).

For other possibilities, see my suggestions above, and in particular the paper of Chesher 2003. It suggests a method for identification in a multiplicative context. If you consider the MPH model (and this is actually the time continuous analogon of the cloglog but WITH unobserved heterogeneity), then this might get you the ratio of the coefficients beta1/beta 2, where beta1 is the coeffiecient of the endogeneous variable and beta2 the coefficient of an exogeneous one. If you normalize beta2, you will get beta1.

I am working (and will be working in future) on related problems.

Best wishes

Petyo

I developed an Instrumental variable estimation procedure for (M)PH models, which is availble on the web

http://www.nidi.nl/Pages/NID/24/764.bGFuZz1FTkc.html

The paper appeared in a Springer book:

http://www.springer.com/social+sciences/population+studies/book/978-1-4020-9966-3

but is also available as an IZA working paper

http://www.iza.org/en/webcontent/publications/papers/viewAbstract?dp_id=2896

A related paper also using a rank estimation procedure appeared recently in Journal of econometric methods:

http://www.degruyter.com/view/j/jem.2013.2.issue-1/jem-2012-0005/jem-2012-0005.xml?format=INT

Dear Vera,

I'm also experiencing a similar problem. Applying a survival model in one of my projects and I suspect that one of my continuous explanatory variable is endogenous. I'm not sure if you have found a solution but I was doing some search and saw that these two papers were recommended:

Lillard, Lee A. 1993. “Simultaneous Equations for Hazards: Marriage Duration and Fertility Timing.”Journal of Econometrics 56:189-217.

Lillard, Lee A., and Constantijn W.A. Panis. 2003. aML Multilevel Multiprocess

Statistical Software, Version 2.0. EconWare, Los Angeles, California.

I just found these references so didn't have chance to look at them. I will share my comments if I make some progress.

Vera, thank you for asking this question, as I am experiencing the same problem.

For now, some brief information for the interested ones, I may comment more on this later.

1) If assignment to treatment and control (also in the case of multiple treatments cq. ordinal variables) is fully complied with, that is r = 1, you can directly use the running variable instead of the treatment variable.

2) This, however, is very unlikely to be the case. There is almost certainly non-compliance, at least among the control group. Abbring & Van den Berg (2005): Social Experiments and Instrumental Variables with Duration Outcomes. is your paper. Yet, without programming knowledge, I do know see how to bring this in practice.

3) You can specify a parametric model, for which I believe IV-solutions are available. Yet the same problem applies, I am not aware of translations into R or Stata code.

4) You can specify a parametric model as Poisson regression or as negative binomial regression. This is intuitively appealing, because you simply count deaths over the risk-set, which is equivalent to the hazard. The drawback is, as mentioned, that it estimated the baseline, which has to be parametrically specified: The Poisson model assumes not only are the hazards proportional, but are also constant (https://stats.stackexchange.com/questions/15160/poisson-regression-in-a-survival-setting-on-a-simulated-set). The advantage is that an at-hand IV-Poisson command is available in Stata, called -ivpoisson-. I also believe that the user-written -qvf- offers IV-Poisson and IV-Negative binomial.

5) Your duration variable is (near-)continuous. You could apply IV-OLS to estimate duration, but I discommend this. This is actually what is done in one of the papers previously mentioned, Aakvik, Holmas and Kjerstad. Hospital capacity, waiting times and sick leave duration - an empirical analysis of a Norwegian health policy reform. The problem of linear regression is that you assume the residuals (i.e. duration conditioned on X) to be distributed normally, which is often unreasonable.

Please let me know if you advance to any at-hand solutions, I am very interested.

----

Update: as shown in the literature, poisson count models are equivalent to piecewist constant exponential hazard models. For proof in Stata, see http://www.stata.com/statalist/archive/2013-11/msg00841.html.

To show that Stata's -ivpoisson- estimates the same coefficients. (1) Repetition of the last comment. (2) Again, but display the non-exponentiated coefficients (this does not affect the estimation, you can confirm by typing display exp(_b[1.tx])). (3) Re-estimate using -ivpoisson- without specifying instruments. (4) Now you can introduce the instruments.

(1) -poisson event ibn.tx, nocons exp(ptime) irr-

(2) -poisson event ibn.tx, nocons exp(ptime)-

(3) -ivpoisson gmm event ibn.tx, nocons exp(ptime) add-

(4) -ivpoisson gmm event ibn.tx (x1=z1), nocons exp(ptime) add-

I am also now digging into this with a similar problem on a R&R. Any progress on suggestions (using Stata)?

Have you checked Terza, Basu & Rathouz (2008)? They argue that 2SRI performs well for non-linear models, including duration models.Article Two-Stage Residual Inclusion Estimation: Addressing Endogene...

Am also having the problem of modelling the relationship between poverty rates(scalar variable) and a yield(a continous variable regressor) where i suspect that yield may be endogenous ,why do i handle this? thank you