micEcon (version 0.2-4)

maxNR: Newton-Raphson maximisation

Description

unconstrained maximisation algorithm based on Newton-Raphson method.

Usage

maxNR(fn, grad = NULL, hess = NULL, theta, print.level = 0,
      tol = 1e-06, gradtol = 1e-06, steptol = 1e-06, lambdatol = 1e-06,
      qrtol = 1e-10,
      iterlim = 15,
      constPar = NULL, activePar=rep(TRUE, NParam), ...)

Arguments

fn

function to be maximised. In order to use numeric gradient and BHHH method, fn must return vector of observation-specific likelihood values. Those are summed by maxNR if necessary. If the parameters are out of range, fn

grad

gradient of the function. If NULL, numeric gradient is used. It must return a gradient vector, or matrix where columns correspond to individual parameters. Note that this corresponds to t(numericGradient(fn))

hess

hessian matrix of the function. If missing, numeric hessian, based on gradient, is used.

theta

initial value for the parameter vector

print.level

this argument determines the level of printing which is done during the minimization process. The default value of 0 means that no printing occurs, a value of 1 means that initial and final details are printed and a value of 2 means that f

tol

stopping condition. Stop if the absolute difference between successive iterations less than tol, return code=2.

gradtol

stopping condition. Stop if the norm of the gradient less than gradtol, return code=1.

steptol

stopping/error condition. If not find acceptable/higher value of the function with step=1, step is divided by 2 and tried again. This is repeated until step < steptol, then code=3 is returned.

lambdatol

control whether the hessian is treated as negative definite. If the largest of the eigenvalues of the hessian is larger than -lambdatol, a suitable diagonal matrix is subtracted from the hessian (quadratic hill-climbing).

qrtol

?

iterlim

stopping condition. Stop if more than iterlim iterations, return code=4.

constPar

index vector. Which of the parameters must be treated as constants.

activePar

logical vector, which parameters are treated as free (resp constant)

...

further argument to fn, grad and hess.

Value

• list of class "maximisation" which following components:
• maximum
  fn value at maximum (the last calculated value if not converged).
• estimate
  estimated parameter value.
• gradient
  last gradient value which was calculated. Should be close to 0 if normal convergence.
• hessian
  hessian at the maximum (the last calculated value if not converged). May be used as basis for variance-covariance matrix.
• code
  return code: 1{gradient close to zero (normal convergence).} 2{successive function values within tolerance limit (normal convergence).} 3{last step could not find higher value (probably not converged).} 4{iteration limit exceeded.} 100{Initial value out of range.}
• message
  a short message, describing code.
• last.step
  list describing the last unsuccessful step if code=3 with following components: theta0{previous parameter value} f0{fn value at theta0} theta1{new parameter value. Probably close to the previous one.}
• activePar
  logical vector, which parameters are not constants.
• iterations
  number of iterations.
• type
  character string, type of maximisation.

Details

The algorithm can treat constant parameters and related changes of parameter values. Constant parameters are useful if a parameter value is converging toward the edge of support or for testing.

One way is to put constPar to non-NULL. Second possibility is to signal by fn which parameters are constant and change the values of the parameter vector. The value of fn may have following attributes: constPar{index vector. Which parameters are redefined to constant} constVal{numeric vector. Values of the constant parameters.} newVal{a list with following components: index{which parameters will have a new value} val{the new value of parameters} } The difference between constVal and newVal is that the latter parameters are not set to constants. If the attribute newVal is present, the new function value is allowed to be below the previous one.

References

W. Greene: "Advanced Econometrics"; S.M. Goldfeld, R.E. Quandt: "Nonlinear Methods in Econometrics". Amsterdam, North-Holland 1972.

See Also

nlm for Newton-Raphson optimisation, optim for different gradient-based optimisation methods.

Examples

Run this code

## ML estimation of exponential duration model:
t <- rexp(100, 2)
loglik <- function(theta) sum(log(theta) - theta*t)
## Note the log-likelihood and gradient are summed over observations
gradlik <- function(theta) sum(1/theta - t)
hesslik <- function(theta) -100/theta^2
## Estimate with numeric gradient and hessian
a <- maxNR(loglik, theta=1, print.level=2)
summary(a)
## You would probably prefer 1/mean(t) instead ;-)
## Estimate with analytic gradient and hessian
a <- maxNR(loglik, gradlik, hesslik, theta=1)
summary(a)

Run the code above in your browser using DataLab