rho: Objective function of Generalized Empirical Likelihood (GEL)

Description

It computes the objective function of GEL, its first and second analytical derivatives. It is called by gel.

Usage

rho(x, lamb, derive = 0, type = c("EL", "ET", "CUE"), drop = TRUE, k = 1)

Arguments

A $n\times q$ 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 $\lambda$ and 2 for the second derivative with respect to $\lambda$.

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

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).

Value

'rho' returns a scalar if "derive=0", a $q\time 1$ vector if "derive" = 1 and a $q\times q$ matrix if derive = 2.

Details

The objective function is $\frac{1}{n}\sum_{t=1}^n \rho()$, where $\rho(v)=\log{(1-v)}$ for empirical likelihood, $-e^v$ for exponential tilting and $(-v-0.5v^2)$ for continuous updated estimator.

References

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.