I am performing a numerical optimization where I try to find the parameters of a statistical model that best match certain moments of the data. I have 6 parameters in total I need to find. I have written a matlab function which takes the parameters as input and gives the sum of squared deviations from the empirical moments as output. I use the fminsearch function to find the parameters and it gives me a solution.
However, I am unsure if this is really a global minimum. What type of checks I could do to ensure the numerical solution is correct? Plotting the function is challenging due to high dimensionality. Any general advice in solving this type of problem is also appreciated.
You are describing the difficulties of a global optimization problem.
As mentioned in one of the comments, fminsearch() and related function fminunc() will return a local minimum. It provides no guarantee that you will get a global minimum.
A simple way to check if the answer you get really is a global minimum, would be to run the function multiple times from various starting points. If the answer all converges to the same value, it might be a global minimum. If you find an answer with lower error values, then the last answer was not the global minimum.
The only way to be perfectly sure that you have the global minima, is to know whether or not your function is convex (i.e. your function has only a single minima.) This will have to be done analytically.
If it is not possible to be done analytically, there are many global optimization methods you may want to consider, including some available as this MATLAB toolbox.