Can you elaborate how exactly your response outcome is truncated? That is, in truncation, you don’t have data on independent variables as well for the cases where the outcome is truncated. If this is the case, the analysis of truncated outcome in an instrumental variable setup becomes messy and has several caveats. In this case, I would personally consider an approach that is less sensitive to truncation limits such as median or quantile regression. (Of course in a 2SLS framework.)

In case your dependent variable is exhibiting corner-solution (or even true censoring), you can use Stata’s ivtobit for a maximum likelihood instrumental variable (IV) or two-step IV approach.

On a side note, the fact that your endogenous variable is truncated shouldn’t be a major issue. That is, broadly speaking, for a RHS endogenous regressor, standard IV is okay whether your endogenous regressor is binary, truncated, or whatever. Ofcourse, using IV methods that take into account the form of endogenous regressor will likely lead to efficiency gains, but plain IV is still consistent. In other words, if it’s correctly (or well) specified when it’s exogenous (i.e., stage 2 regression), it’s also well specified when it’s endogenous (stage 1 regression) and estimated via standard IV with standard conditions on instrument. It’s the usual efficiency-robustness tradeoff - see standard textbooks such as by Wooldridge for details.

Dear Behram Wali :

Thank you very much for responding to my query. My dependent variable is technical efficiency score (estimated using Data Envelopment Analysis) which takes values greater than 0 through 1 for any given case (DMU). So, I would like to fit a truncated regression model whereby those cases with y=1 are dropped. Though I set the lower limit at 0, I do not have any DMUs with TE score = 0.

I have a covariate representing marketing intensity which I believe is endogenous and happens to be bounded at both ends like the TE score. As to this, I appreciate your thought that the bounded nature of the variable would not impede the application of the standard IV approach. I feel reassured.

I will attempt to proceed with my analysis as you have suggested and see how things work out, and let you know of the outcome.

Thank you very much again for helping me out with your expert thoughts. I am absolutely grateful to you.

Kind regards,

Namgay Dorji.

Hi Namgay: What is the true plausible range of efficiency score? Is it bounded between 0 and 1? If so, I suspect that your dependent variable is not truncated per se, but rather a corner-solution. Your dependent variable would have been truncated if the efficiency scores could range b/w 0 and 5 (as an example) but that you decided nonetheless to focus on those between 0 and 1 as per your research objective. In other words, truncation (or censoring for that matter) is a data observability issue. If my understanding about your DV is correct, your DV (efficiency scores) are perfectly observed and the lower (or upper limit) is a true and intuitive outcome (rather than a censored one), however, the fact that there is a spike at 0 or 1 warrants a methodological framework that treats the spike (arising due to bounded space) differently than the rest of the distribution. For DVs characterized by corner-solution (spikes at 0 or 1 in your case), it follows that the application of “censored” regression model, such as a Tobit model, 'mechanically' fits well into corner solution problems. Thus, you can use Stata's ivtobit for your problem.

For more context about my thoughts about truncation, corner-solution, & censoring above, you can read our recent accepted article. "(2019) Examining Correlations Between Motorcyclist’s Conspicuity, Apparel Related Factors and Injury Severity Score: Evidence from New Motorcycle Crash Causation Study. Accepted for Publication in: Accident Analysis and Prevention." (Not online yet, but I can send it out.) Feel free to reach out if you have any further questions. -wali

I apologize for the typos. Sent from my phone.

Hi Behram Wali, Thank you very much for elaborating on your thoughts. Yes, my DV is bounded between 0 and 1 by definition: Loosely speaking, for any given case, it is the ratio of the "observed value" to the "theoretical maximum value" where the latter is computed by a linear combination (weighted summing with the weights summing to 1) of the observed values of the "best performing" cases. As such, it cannot be less than 0 or greater than 1. Your reasoning for why you think ivtobit would solve my problem is very clear and concise, and I greatly appreciate it. I will consider your expert suggestion in relationship to other important things concerning my research, and work out an appropriate solution for the issue I am faced with. Please allow me to congratulate you on your recent paper "Examining Correlations Between Motorcyclist’s Conspicuity, Apparel Related Factors and Injury Severity Score: Evidence from New Motorcycle Crash Causation Study". It would be a privilege for me to read it if you would be so kind as to send me a electronic copy. I will leave my email address in your private message box.

Again, thank you very much for helping me to get a firm grip on a very important issue concerning my research project. I am absolutely grateful to you. I will certainly contact you if I am in need of additional assistance.

With kind regards,

Namgay Dorji.

Dans le cadre de ma thèse, j'utilise la variable instrumentale à traitement multiple mais j'ai je n'arrive pas à avoir une commande appropriée pour faire mes estimations.

Merci de réagir