Econometrics questions : Functional form choice

[Aimee Kingsada]

Hi statalist,

I try to estimate the effect of a monetary bonus on physicians’ activity.
I would like to estimate this regression by first-difference between 2008 and 2011 (I only have two points for each individual):

Log(Nb of consultation_it) = B Log(Bonus_it) + H (X_it) + Eit

where Bonus is a equal to 0 for all in 2008 and positive for some in 2011.

I would like to estimate elasticities and I also suspect a nonlinear relationship between bonus and number of consultations. Since the calculation of Log(0) is not possible, I tried to input 0 instead, but it seems to be wrong because the variation of bonus estimated is not really the variation of measured bonus.

Questions :

1. Do I have the same issue of estimating an artificial variation if I only take the variable Bonus and not transform it in log?
2. I was thinking of adding a dummy « Bonus equals 0 » and a variable « Log(bonus) » conditionally of bonus being positive to take into account the nonlinear effect of bonus. But bonus is endogenous so the two variable would be too. The problem is that I have only one unique instrument that works : any ideas of tackling this issue?
3. I also tried to use quadratic functional form for bonus. The results with instrumental variables show that coefficient of bonus^2 is not significant : does it mean that my intuition for a nonlinear effect of the bonus is not right ?

Many thanks for those who read and for your answers.
Aimée Kingsada

[John Mullahy]

Aimee: My instinct would be simply to enter Bonus untransformed (your point #1).

You said you expect a nonlinear relationship and your log transformation of your LHS variable Nb of consultation_it will result in one form of nonlinearity between Bonus and Nb of consultation_it (whether that is the form of nonlinearity that suits your analysis is a separate question.

Since you are conducting a first-difference analysis using a log-linear model presumably simplifies matters for you.

One other idea to consider is also estimating the model in levels-levels form

Nb of consultation_it = B (Bonus_it) + H (X_it) + Eit

and seeing if your elasticity results are meaningfully different than those obtained from estimation of the log-linear form

Log(Nb of consultation_it) = B (Bonus_it) + H (X_it) + Eit

[Aimee Kingsada]

John Mullahy Thank you very much for your answer. I am very grateful !

If I may ask one last question : my variable Bonus shows a "jump" for individuals who earned it (for example, Bonus(2008) = 0 and Bonus(2011)=3000 for individuals who received the bonus ; the distribution of bonus in 2011 is between 200 and 8000 euros when bonus is positive). Is it a problem for the estimation ?

Thank you