rho {gmm} | R Documentation |
It computes the objective function of GEL, its first and second analytical derivatives. It is called by gel.
rho(x, lamb, derive = 0, type = c("EL", "ET", "CUE"), drop = TRUE, k = 1)
x |
A ntimes q matrix with typical element (t,i), g_i(theta,x_t) |
lamb |
A q times 1 vector of lagrange multipliers |
derive |
0 for the objective function, 1 for the first derivative with respect to λ and 2 for the second derivative with respect to λ. |
type |
"EL" for empirical likelihood, "ET" for exponential tilting and "CUE" for continuous updated estimator. |
drop |
Because the solution may not be in the domain of rho(v) forall t in small sample, we can drop those observations to avoid the return of NaN |
k |
It represents the ratio k1/k2, where k1=int_{-infty}^{infty} k(s)ds and k2=int_{-infty}^{infty} k(s)^2 ds. See Smith(2004). |
The objective function is frac{1}{n}sum_{t=1}^n rho(<g(theta,x_t),λ>), where rho(v)=log{(1-v)} for empirical likelihood, -e^v for exponential tilting and (-v-0.5v^2) for continuous updated estimator.
'rho' returns a scalar if "derive=0", a qtime 1 vector if "derive" = 1 and a qtimes q matrix if derive = 2.
Newey, W.K. and Smith, R.J. (2004), Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators. Econometrica, 72, 219-255.
Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative GMM Estimators. Journal of Business and Economic Statistics, 14 262-280.
Smith, R.J. (2004), GEL Criteria for Moment Condition Models. Working paper, CEMMAP.