1. The problem statement, all variables and given/known data Let yi denote the number of times individual i buys tobacco in a given month. Suppose a random sample of N individuals is available, for which we observe values 0,1,2,... for yi. Let xi be an observed characterisitc of these individuals (for example, gender). If we assume that for a given xi, yi has a Poisson distribution with mean λi = exp (β1 + β2xi). That is, the distribution function is: Pr (yi = y |xi) =[exp(-λi)λi^y]/y! y = 0,1,2,... (a) Obtain Bmle and derive its limiting distribution. (b) Now suppose that yi does not take Poisson distribution. However, the orthogonality condition holds: E (yi - exp (β1 + β2xi) | xi) = 0: Propose a consistent estimator for and derive its limiting distribution. 2. Relevant equations 3. The attempt at a solution I'm not sure how to approach this. Do I just plug exp(β1+β2xi) into λi, and derive the MLE, or am I supposed to do something else? When I tried it that way, I get dL/dβ1=ny-Ʃexp(β1+β2xi)=0 dL/dβ2=yƩxi-Ʃxi[exp(β1+β2xi)]=0 And I'm not sure how to solve for β1 and β2 with this.