Maximization methods

[imccart]

• two-step cmclogit, setting alpha=0 for markets with alpha<0
• manual logit (address convergence)

[sethrs-jhu]

Program in R: likelihood and gradient, estimation command

[imccart]

R multinomial logit with inequality constraints: multinomineq. Employs a bayesian approach and might be quite slow or impossible with our number of fixed effects.

When I last estimated a hospital choice model in R, I used the mclogit command: logit.reg <- mclogit(cbind(choice,id) ~ vars, data=data), where choice is a dummy variable for the observed choice and ID is a factor variable defining each case (in this setting, it was patient/admission).

[sethrs-jhu]

1992 article in Marketing Science suggests an iterative algorithm https://www.jstor.org/stable/184095

How it would work in our case (and I will email some sample code):

• let lna denote the log of alpha, and let m denote the "index" (the mean belief)
• we want to use exp(lna) * m to constrain alpha to be positive
• the article shows a simple iterative procedure to include a term that is nonlinear in parameters, in a multinomial logit
• it involves a Taylor series expansion, but really only the first derivative term
• there is some algebra that I won't put here, but note that the derivative of exp(lna)m with respect to lna is exp(lna)m
• the iteration works as follows:
• given a value of lna, say lna_1, we update as: lna_2 = lna_1 + (coef - 1)
• where "coef" is the coefficient on a constructed variable = exp(lna_1)*m
• you keep going (updating the value of the constructed variable, and reestimating the logit to get new values of "coef") until the estimates of lna converge
• I've run this on simulated data and it seems to work well
• finally, the article also shows it is easy to get standard errors

[imccart]

Iterative procedure:

// ITERATIVE PROCEDURE FOR NONLINEAR PARAMETER

• Define variable to hold values of exp(lna)*index: gen belief_s = .

• Define a scalar to count the number of iterations until convergence: scalar iter8 = 0

• Initialze values of lna (the scalar(alpha) is the true value used to simulate data): scalar lna_old = 4 scalar lna_new = ln(scalar(alpha))

• Iterate until convergence:

• (note the tolerance and max number of iterations) while abs( exp(scalar(lna_new)) - exp(scalar(lna_old)) ) > 0.005 & iter8 < 20 {

  • advance counter: scalar iter8 = scalar(iter8) + 1

  • update the constructed variable: scalar lna_old = scalar(lna_new) quietly replace belief_s = exp(scalar(lna_old))*m

  • estimate the logit quietly asclogit choice dist log_exp i.doc belief_s, nocons case(pat) alt(doc)

  • update the parameter scalar lna_new = scalar(lna_old) + _b[belief_s] - 1

  • report cycle dis "Iteration " iter8 " done" }

• alpha coefficient is then exp(lna_new), with standard error exp(lna_new) * SE(coef on belief_s)