# Instrumental variable identification for price elasticity

I am working on a price elasticity problem and i am using Log-Log linear model. Below are the set of variables

Dependent variable: logarithm of my quantity sold (LogQ)

Independent variable: logarithm of my price(LogMyP), logarithm of price of my competitor price(for cross-price elasticity)(LogCoP), logarithm of my social media rating(LogMyR), logarithm of my competitors social media rating(LogCoR), logarithm of my unsold stock(LogMyS), logarithm of remaining shelf life of my product(LogMyShl), logarithm of market supply in percentage(LogMktspl), day of the week (6 dummy variables)

I need help in identifying set of endogenous and instrumental variables so that i can specify that in Instrumental Variable regression. I believe only logarithm of my price and logarithm of my competitor price are for the rest i do believe that it impacts my demand. The problem i am facing is how to decide which of the independent variable is exogenous or instrumental variable. Is there any test available or any procedure to identify if a variable is instrumental variable or is this purely a subjective thing? in any case can anyone help me identify the instrumental variables?

I provide an answer to your question here. In addition, initial selection of the instruments should be based on theory or hypothesized relationships from your conceptual model to provide a "good story" why the instrument is valid in your case. After selecting the instruments, you need to formally test whether they satisfy the criteria I listed in the linked response. If your instrument fails to meet those criteria, you probably should come up with a better instrument or not run the IV model at all, because you would run into the problem of weak instruments if you do (see Bound et al. 1993 and Stock & Yogo 2002 for more information). As a final note, the third criterion, the criterion of exogeneity of the instrument, can only be tested in an overidentified model (Wooldridge, 2009), so it's a good idea to come up with more than 1 instrument (although in practice, it is difficult to come up with even 1 instrument).

References

1. Bound, J., Jaeger, D. a, & Baker, R. (1993). The cure can be worse than the disease: A cautionary tale regarding instrumental variables. NBER Technical Paper Series, 1–23. Retrieved from http://www.nber.org/papers/t0137.pdf
2. Stock, J. H., & Yogo, M. (2002). Testing for Weak Instruments in Linear IV Regression. The National Bureau of Economic Research, (Technical working paper No. 284), 1–73. Retrieved from http://www.nber.org/papers/t0284.pdf
3. Wooldridge, J. M. (2009). Introductory Econometrics: A Modern Approach (4th ed.). Mason, OH, USA: South-Western, Cengage Learning.

I found the answer myself. IV is uncorrelated with error, however correlated with endogenous variable. So all the variables that fulfill this criteria will be instrumental variable.

• Excellent link. Another consideration is that the Xi's need to be independent of price for the obvious reason that you want a clean estimate of price elasticity. Any collinearity will weaken that statement. – Mike Hunter Mar 23 '16 at 12:16
• Sorry didn't understand what is Xi? – Enthusiast Mar 23 '16 at 12:28
• The independent variables. – Mike Hunter Mar 23 '16 at 12:30
• Yes, independent variables should be exogenous. – Enthusiast Mar 23 '16 at 12:43