How to do maximum likelihood estimation when numerical derivatives cannot be calculated

Question

I am trying to estimate a model from Camerer, Ho and Chong 2002 p151.

$i$ is individual, $t$ is time

$j\in m$ which is action set

$s, a$ are two types of individuals

$s_i(t)$ is chosen action by subject $i$ in period $t$.

$P$ is probability.

$$ A^j_i(a,t)=\frac{\phi N(t-1)A^j_i(t-1)}{N(t)} +\frac{[\delta+(1-\delta)I(s^j_i,s_i(t))]\pi(s^j_i,s_{-i}(t))}{N(t)} $$ $$ N(t)=(1-\kappa)\phi N(t-1)+1, \quad t\geq 1 $$ $$ P^j_i(a,t+1)=\frac{exp(\lambda A^j_i(a,t))}{\sum^{m_i}_{k=1}exp(\lambda A^k_i(a,t))} $$ $$A^j_i(s,t)=\sum^{m_{-i}}_{k=1}[\alpha'P^k_{-i}(s,t+1)+(1-\alpha')P^k_{-i}(a,t+1)]\pi_i(s_i^j,s^k_{-i}) $$ $$ P^j_i(s,t+1)=\frac{exp(\lambda A^j_i(s,t))}{\sum^{m_i}_{k=1}exp(\lambda A^k_i(s,t))} $$ $$ L_i=\alpha[\prod^T_{t=1}P^{s_i(t)}_i(s,t)]+(1-\alpha)[\prod^T_{t=1}P^{s_i(t)}_i(a,t)] $$ $$Log L=\sum_i^n log L_i$$ Parameters needed to be estimated are $\phi$, $\delta$, $\kappa$, $\lambda$, $\alpha$, $\alpha'$.

My question is how to estimate this model using maximum likelihood model in Stata. I got error ''could not calculate numerical derivatives''. I think this is because $A^j_i(s,t)$ and $P^j_i(s,t+1)$ are dependent, but I don't know how to solve this problem.

I could also try R or Python or Matlab.

Answer 1

There are optimisation algorithms that don't require derivatives. You can divide them into

• algorithms that assume derivatives exist but don't require them
• algorithms that don't assume smoothness

A good example of the first is Powell's quadratic approximation algorithms newuoa and bobyqa (available in the R minqa package and used by the lme4 package). In relatively low-dimensional problems these are often faster than using a gradient-based algorithm with numeric derivatives.

Algorithms that don't assume smoothness include the famous Nelder-Mead simplex algorithm and simulated annealing algorithms.

In your case I think the objective function should actually be smooth, but it's a bit hard to tell since so many things aren't defined.

M. J. D. Powell (2007) "Developments of NEWUOA for unconstrained minimization without derivatives", Cambridge University, Department of Applied Mathematics and Theoretical Physics, Numerical Analysis Group, Report NA2007/05, http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2007_05.pdf.

M. J. D. Powell (2009), "The BOBYQA algorithm for bound constrained optimization without derivatives", Report No. DAMTP 2009/NA06, Centre for Mathematical Sciences, University of Cambridge, UK. http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf.